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Full text of "A treatise of the natural grounds, and principles of harmony / by William Holder. To which is added by way of appendix, Rules for playing a thorow-bass, with variety of proper lessons, fuges, and examples to explain the said rules, also directions for tuning an harpsichord or spinnet / by the late Mr. Godfrey Keller"

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Natural Grounds, and Vrlncipks 

HARMONY. 

By William Holder, D. D. Fellow of the Royal 
Society y and late Sub-Dean of their M A J E ST Y’j Chapel 
Royal. 


To which is Added, hy v/SLy oi AP P END IX : 

RULES for Playing a Tlwrow-Bafs ; with Variety 
of Proper LeJfonSy Fuges^ and Examples to Explain the fa id 
RULES. Alfo Diredions for Tuning an Harpfichord or 
Spinnet, 


By the late Mr. Godfrey Keller. 


With feveral new Examples, which before were wan- 
ting, the better to explain fome Paffages in the for- 
mer Impreflions. 


The whole being Revis’d, and Corre£led from many 
grofs Miftakes committed in the firft Publication of 
thefe Rules. 


LONDON: 

Printed by W. Pearson, over againft WnghRs Coffee- 
Houfe in Alder/gateffreet for J. Wilcox in \ittk- Britain ; 
and T. Osbqrn e in Graf Inn, 1731. 


THE 


PUBLISHER. 


TO THE 


READER. 


T he Intenti07i of TuUiJhing Mu KellerV Rules, 
(hy way of Affendix to Du Holder’s %ook) 
was chiefly to ref cue them from ma7ty Miflakes 
and Errors, which were occafioned hy the Ignorance 
cf the firfi Tuhlijhers of them on Elates, which wotdd 
never have haffe?Pd, if the judicious Author had liv^d 
to have correBed the Plates himfelf : Nor cotdd he have 
JufeAd thofe Examples, which are now Added, to have 
been waTtting, the better to explain fome of the faid 
Rules, which before were only printed in Figures, with^^ 
out a proper lllufiration of the fanie in Mufical Notes ; 
as is evident by many Inflances of the fa?7ie kind through- 
out the faid Work. 

Aiid as this Book iiiay fall into the Hands of fo77ie, who 
have (not only a Taft for 'Dr. Holder’j Treatife, but 
alfo) a Genius for Compofing as well as for Playing a 
Through Bafs ; it is 7iot hnproper to Ohferve, that there 
are inany excelleiit Rules coJitaiiPd in it, which will he 
found of great Adva7itage to young Co77ipofers, as well 
as to thofe who praBife a Through Bajs ; efpecially with 
Regard to the various Ways of taking Th if cords, which 
is one of the mofi difficult Parts of Compofltion. 

And / humbly prefume, that for thefe Reafons, it will 
v€C07ne at leaft as Ufeful and htBruBive, as any thing 
that has hitherto been Puhliflfd of this kind. 


Vale. i|( 
THE 



THE 




INDEX. 


T Chap. Pag. 
H E IntroduBion. 


Of Sound in General. 

I. 

1. 

of Sound Harmontck. 

ir. 

5 - 

Jppendix to Chap, II, concerning 

the Motions and 

Meafures of' a ^Pendulum. 


‘5 

Of Confonancy and Diffonancy* 

III. 

3 * 

of Concords. 

IV. 

38 

Of proportion. 

V. 

67 

Of Difcords and Decrees. 

VI. 

94 

Digreffion, concerning the*^' 


1 00 

Ancient Greek Mufick.y 



Of Difcords. 

VII. 

1 16 

Of Differences. 

VIII. 

140 

Conclufion. 

IX. 

‘ 47 , 


The Index to Mr. Kellers 

Rules. 

O F Concords and Dijcords. Page l 59 

Of Common Chords differently taken. 1 60 

Of Common Chords and Sixes differently taken. 1 60 
Of Cadences. 161 

A 2 Of 


The INDEX. 

of the feVeral Di/cords and Manner of flaying them. 

Page 165 

H-Ow to mo'Ve the Hands when the 'Bafs afcends or de^ 
Jcends, 1 64 

of Dividing upon Notes in Common Time. 1 6 5 

of Dividing in Triple Time. 1 67 

Of Natural Sixesy and Broper Cadences in a jhirpl{ey. 

169 

Of Natural Sixes^ 8 cc. in a fat p{ey. 170 

Buies how Sixes inay he ufed in Compoftng. i/i 
Btdes about Se\miths and Ninths. 173 

Several Ways of Accompanying ivhen the Bafs Afcends^ 
and Defcends by Degrees. \yy 

Of playing all Sorts of Dif cords in a flat E^y. 174 
Of playing Dif cords in a fharp Ney. i 8 o 

Of making Chords eafy to the Memory. 1 8 j 

Of play ing fome Notes the fame way^ ivhich yet haVe a 
different appearance in Writing. ‘ ^ ‘ 

Of Tranfpofition, 

of Difcordsy how prepar'd and refolVd. 

Some Examples for playing a TJmow-Bafs. 

Short Leffons by way of Fugeing. 
tj^des for Tuning an Harpjiclmdy &c. 


183 
i 84 
186 
194 
200 


Rules 



THE 

'Natural Grounds and Vrinctpks 

o F 

HARMONY. 


The INTRODUCTION. 

H armony confifts of Canfes, Na- 
tuuil and oArtiJicial^ as of Matter 
and Form* The Material Part of 
it^ is Sound or Voice. The Formal Fart 
isj The Ti)iff ofition of Sotutd or Voice i?ito 
Harnio?iy ; which requires^ as a frepara^ 
five Caufe^ shilful Compojitio^i ; and^ as an 
iinmediate Efficient ^ Artful Performance* 
The forfner Party viz. The Matter^ lies 
deep in Nature^ a?id requires much Re- 
fearch into Natural Philofophy to unfold it ; 
to find how Sounds are made^ a7td how 
they are firfi fitted by Nature for Harmony ^ 
before they be difpofed by o.4rt : Both toge- 
ther 7nake Harmony cofnpleat* 

B Har* 


1 


The INTRODUCTION. 

Har/no?iy^ then^ refults from fraclick Mu-^ 
fich^ and is made hy the Natural a7td Arti-' 
ficial Agreement of dif'erent So7mds^ (viz. 
Grave and Acute) hy which the Senfe of 
Hearing is delighted. 

This is frofer vi Symphony^ i. e. Con- 
fent of 7nore Voices in dif}'ere?it Tones \ 
is found alfo in folitary Mufich of one Voice^ 
hy the Ohfervatio?i and Expectation of the 
Earj comparing the Habitudes of the fol- 
lowing Notes to thofe which did precede. 

Now the Theory in Natural Philofophy^ 
cf the Grou7tds and Reajons of this oAgree- 
went of Sounds^ and confeque?it T)elight 
and Pleafure of the Ear^ (leaving the Ma- 
nagement of thefe Sounds to the Maflers of 
Harmonick Coinpoficre^ a?id the shlful Ar- 
tifts in Performance) is the Suh]eEt of this 
Difcourfe. Tioe Defign whereof (for the 
Sake and Service of all Lovers of Miifick^ 
and particularly the Gentlemeft of Their 
Majefly’^s Chapel Royal) is^ to lay down thefe 
Trinciples as f)ort^ a?id intelligible^ as the 
Sul]eH Matter will bear. 

Where the firft thing A^ecejJ'ary^ is a Co?i- 
fideratio7i of fomewhat of the Nature of 
Sound in Ge?ieral ; and then^ more particu- 
larly^ of Harmonick So7{?ids^ &c. 


CHAP. 


of Sound in General. 


1 



CHAP. I. 


Of Sound in General. 

I N Genera!/' to pafs by what is not per- 
tinent to this Dehgn) Sence and Ex- 
perience confirm thefe fbllowino- Pro- 
perties of Sound. ° 

I. All Sound is made by Motion, viz. 
by Percullion with Collifion of the Air. 

2. That Sound may be propa'^atcd 
and carried to Dilbnce/it requires a Afe' 
dium by which to pafs. 

3- This Medium (to our Purpofe) is 


js propagated along 
^le Medium ; (o far alfo the Motion pafictli. 
Eor (If we may not fay that tlie Motion 
and Sound are one and the fame thin? 

B 2 


r? ' 



2 


Of Sound in General. 

yet at lea ft) it is neceftarily confequent, 
that if the Motion ceafe, the Sound muft 
alfo ceafe. 

5. Sound, where it meets with no 
Obftacle, palTetli in a Sphere of the Afe- 

greater or lefs, according to the 
Force and Greatnels of the Sound ; of 
which Sphere the foncx’ous Body is as the 
Centre. 

6 . Sound, fo far as it reacheth, paf- 
feth the Meditm^ not in an Inftant, but in 
a certain uniform Degree of Velocity, cal- 
culated by Gn^e?idus^ to be about the rate 
of 276 Paces, in the fpace of a fecond Mi- 
nute of an Hour. And wdiere it meets 
with any Obftacle, it is fubjeft to the Laws 
of Reflexion, which is the Caufe of Ec- 
cho’s Meliorations, and Augmentations of 
Sound. 

7. Sound, /. e. the Motion of Sound, 
or founding Motion, is carried through the 
'Medium or Sphere of Activity, with an 
Impetus or Force, which fliakes the free 
Medium^ and fti’ikes and ihakes every Ob- 
ftacle it meets with, more or lefs, accord- 
ing to the vehemency of the Sound, and 
Nature of the Obftacle, and Nearnefs of 
it to the Centre, or fonorous Body, Thus 


of Sound in General. 5 

tlic Impetuous Motions of tlie Sound of 
Thunder, or of a Canon, fliake all before 
it, even to the breaking oi Glafs Windows, 
i^e. 

S. The Parts of the founding Body 
are moved with a Motion of Trembling, 
or Vibration, as is evident in a Bell or 
, Pipe, and mod manifeil in the firing of a 
Mufical Inflmment. 

9. This Trembling, or Vibration, is 
either equal and uniform, or elfe unequal 
and irregular ; and again, fwiher or flower, 
according to the Conftitution of the fono- 
rous Body, and Qtiality and Manner of 
of Percuflion ; and from hence arile Diffe- 
rences of Sounds. 

10. The Trembling, or Vibration of 
the fonorous Body, by which the particu- 
lar Sound is conffituted and difci'iminated, 
is itnpreffed upon, and carried along the 
Mediujn in the fame Figure and Meafurc, 
otherwife it would not be the fame Sound, 
when it arrives at a more diftant Ear, /. e. 
the Tremblings and Vibrations (which 
may be called Undulations) of the Air or 

are all along of the fame Velo- 
city and Figure, with thofe of the fono^ 
pus Bodv, by which they are caiifed. 

B 5 The 


4 Of Sound Harmonick. 

The Differences of Sounds, as of one 
Voice from another, £jfc. (befides the Dif- 
ference of Tune, which is caufed by the 
Difference of Vibrations) arife from the 
Conftitudcn and Figure, and other Acci- 
dents of the fonorous Body. 

II. If the fonorous Body be requifitely 
conftituted, /. e, of Parts folid, or tenfe, 
and regular, fit, being ftruck, to receive 
•and exprefs the tremulous Motions of 
Sound, equally and fwiftly, then it will ren- 
der a certain and even Harmonical Tone 
or Tune, received with Pleafure, and judg- 
ed and meafured by the Ear : Otherwife 
it will produce an obtufe or uneven Sound, 
not giving any certain or difcernable Tune, 

Now this Tune, or Tuneable Sound, 
i. e. hi) ^.:av rotV/i', 

An agreeable .Cadence of Voice, at one 
Pitch or Tenfion. This Tuneable Sound 
(I (ay) as it is capable of other Tenfions 
towards Acutenefs, or Gravity, /. e. the 
Tenfions greater or lefs, the Tune graver 
or more acute, /. e. lower or higlier, is the 
firft Afattcr or Element of Mu lick. And 
this Harmonick Scuad comes, next to be 
cpnfidcred. 


CHAR 


of Sound Ir-Lirmonkk: 


5 


C H A P. If. 


Of Sound Htinnonlck- 


*^HE firft and great Principle upon 
which the Nature of Harmonical 


Sounds is to be found out and difeovered, is 
this : That the Tune of a Note (to fpeak in 
our vulgar Phrafe) is conhituted by the 
Meafure and Proportion of Vibrations of 
the fonorous Body ; I mean, of the Y clo> 
city of thofe Vibrations in their Recouiies. 

For, the frequenter the Vibrations are, 
the more Acute is the 'Pune •, the flower 
and fewer they are in the fame Space of 
Time, by fo much the more Grave is tlie 
I'une. So that any given Note of a Tunc, 
is made by one certain Meafure of Ve- 
locity of Vibrations ; viz. Such a certain 
Number of Courfes and Recourfes. e. 
ot a Chord or String, in fuch a certain Space 
of Time, doth conifitute fuch a certain de- 
terminate Tune, And all fuch Sounds as 
are Unifons, or of the fame Tune with that 
given Note, though upon whatfoever dif- 
terent Bodies, (as Strings P/ge^ 

Y)nx^ &c.) are made with Vibrations on 



B 


Trem- 


6 Oj Sound Hat vionki{. 

Tremblings of thofe Bodies, all equal each 
to the other: And whatlbever Tuneable 
Sound is more Acute, is made with Vibra- 
tions more fwift ; and whatfoever is more 
grave, is made with more flow Vibrations : 
And this is univerfally agreed upon, as 
moft evident to Experience, and will be 
more manifeft through the whole Theory. 

And, That the Continuance of the 
Sound in the fame Tune, to the laft, (as 
may be perceived in Wire-firings, which 
being once ftruck will hold their Sound 
long) depends upon the Equality of I'ime 
of the Vibrations, from the greatelf Range 
till they come to ceafe : And this perfeftly 
makes out the following Theory of Confo- 
nancy, and Diflbnancyv 

Some of the ancient Greek Authors of 
Mufick, took Notice of Vibrations: And 
that the fwifter Vibrations caufed Acuter, 
and the flower, graver I'ones. And that 
the Mixture, or not Mixture of Motions 
creating feveral Intervals of Tunc, was the 
Reafon of their being Concord or Dif- 
cord. And like wife, they found out the 
(evcral Lengths of a Monochord^ propor- 
tioned to the feveral Intervals ot Harmo- 
nick Sounds : But diey did not make out 
the Equality of Meafure of Time, of the 


of Sound Harmonic 7 

Vibrations laft fpoken of, neither could be 
prepared to anlwer fucli Objeftions, as 
might be made againft the Continuity of 
the lamenefs of Tune, during the Continu- 
ance of the Sound of a Strin^^ or a Bell 
after it is llruck. Neither did any of them 
offer any Reafons for the* Proportions affio-n- 
ed, only it is faid, that Pjtbagor/a found 
them out by Chance. 

But now, Thefe (fince the Acute Ga- 
lileo hath obferved, and difcovered the Na- 
ture oi Fe7idulums) are eafy to be explain- 
ed, which I fhall do, premifing fome Con- 
fideration of the Properties of the Motions 
of a Bendulunu 



^ Plumbet c on a String or 
Wire, hxed at 0. Bear C to A : Then let 
It I ange freely, and it will move toward 

back towards 
1 he Motion from A to B, I call the 

Courfe, 


8 Of Sound Harmonic^. 

Courfe, and back from B to the Re- 
courfe of the "?e7idulum^ making almoft a 
Semi-Circle, of which 0 is the Centre. 
Then fuffering the Pe7tdulum to move of 
itfelf, forwards and backwards, the Range 
of it will attvery Courfe and Recourfe a- 
bate, and diminifli by degrees, till it come 
to reft perpendicular at 0 C. 

Now that which Galileo firft obferved, 
was, that all the Courfes and Recourfes 
of the "Pendulum^ from the greateft Range 
through all Degrees till it come to relt, 
were made in equal Spaces of Time. That 
is, e. g. The Range between A and B, is 
made in the fame Space of Time, with the 
Range between D and E ; the Plumber 
moving fwifter between A and jB, the great- 
er Space ; and flower between D and £, 
the leffer; in fuch Proportions, that the 
Motions between the Terms A B and D 
£, are performed in equal Space of Time. 

And here it is to be Noted, that where 
ever in this Treatife, the fwiftnefs or flow- 
nefs of Vibrations is fpoke of, it muft be 
always underflood of the frequency of their 
Courfes and Recourfes, and not of the Mo- 
tion by which it paffetli from one fide to 
another. For it is true, that the fame PeTi- 
dulum under the fame Velocity of Returns, 

moves 


of Sound Harmonick; p 

moves from one fide to the other, with 
greater or le(s Velocity, according as the 
Range is, greater or leis. 

And hence it is, that the 1/ibratIons of 
a Pendulum are become fo excellent, and 
ufeful a Meafure of Time ; efpccially when 
a fecond Obfervation is added, that, as you 
fhorten the Te^idu/um^^ by bringing C near- 
er to its Centre 0, fo the Cibrations will 
be made proportionably in a fliorter Mca- 
fure of Time, and the contrary if you 
lengthen it. And this is found to hold in 
a Duplicate proportion of length to Velo- 
city. That is, the length quadrupled, will 
fubduple the Velocity of Vibrations : And 
the Length fubquadriipled, will duple the 
Vibrations, for the Proportion holds reci- 
procally. As you add to the length of the 
Pendulum^ fo you diminifh the frequency 
of Vibrations, and incrcale them by Ihoit- 
ning it. 

Now therefore to make the Courfes of 
a 'Pendulum doubly fwift, e. to move 
twice in the fame Space of Time, in which 
it did before move once-, you muft fubqua- 
druple the Length of it, /. e, make the 
Pendulum but a quarter fo long as it was 
before. And to make the Librations dou- 
bly flow, to pals once in tlie 'I'ime they 

did 


to of Sound Harmonkk* 

did pafs twice ; you mull: quadruple the 
Length ; make the "Pendulum four Times 
as long as it was before, and fo on in what 
Proportion ^ou pleafe. 

Now to apply this to Mufick, make 
two Pendulums^ AB and CT), fatten toge- 
ther the Plumbets B and T), and ttretch 
them at length, (fixing the Centers ^ and 
C.) Then, being ttruck, and put into Mo- 
tion ; the Vibrations, which before were 
dittinft, made by A "B, and C D, will now 
be united (as of one entire String) both 
backward and forward, between E and K 
Which Vibrations (retaining the aforefaid 
Analogy to a Tendulumj will be made in 
equal Spaces of Time, from the firtt to the 
latt ; /. e. from the greatett Range to the 
leatt, until they ceafe. Now, this being a 
double Pekduhim^ to fubduple the fwiftnefs 
of the Vibrations, you do but double the 
length from A to C, which will be qua- 
druple to A B. The lower Figure is the 
fame with that above, only the Plummets 
taken off. 


A P 


of Sound Harmonkk. 1 1 


E 



F 


And here you have the Nature of the 
String of a Mufical Inftrument, refembling 
a double Pe7idulum moving upon two Cen- 
ters, the Nut and the Bridge, and Vibra- 
ting with the greateil Range in the mid- 
dle of its Length; and the Vibrations e- 
qual even to the laft, which muft make it 
keep the fame Tune fo long as it Sounds. 
And becaufe it doth manifeftly keep the 
fame Tune to the laft ; it follows that the 
Vibrations are equal, confirming one ano- 
ther by two of our Senfes ; in that we fee 
the Vibrations of a Pendulum move equally, 
and we hear the Tune of a String, when 
it is ftruck, continue the fame. 

The Meafure of fwiftnefs of Vibra- 
tions of the String or Chord, (as hath been 
faid,) confiicutes and determines the Tune, 
as to the Acutenefs and Gravity of the Note 
which it founds ; And the lengthning or 

fliort- 


T 2 Of Sound Harmontck^. 

flioitning of the String, under the fame 
Tenfion, determines the Meafure of the 
Vibrations which it makes. And thus, 
Harmony comes under mathematical Cal- 
culations of Proportions, of the length of 
Chords, of the Meafure of Time in Vibra- 
tions ; of the Intervals of Tuned Sounds^ 
As the length of one Chord to another, 
CiCteris paribus^ I mean, being of the fame 
Matter, thicknefs and tenfion *, fo is the 
Meafure of the Time of their Vibrations. 
As the Time of Vibrations of one String 
to another, fo is the Interval or Space of 
Acutenefs or Gravity of the Tune of that 
one, to the Tune of the other: And con- 
fequently, as the length is (Cateris paribus) 
fo is the determinate Tune. 

And upon thcfe Proportions in the Dif- 
ferences of Lengths of Vibrations, and of 
Acutenefs and Gravity ; I fliall infill: all a- 
long in this Treatife, very largely and parti- 
cularly, for the full Information of all fucli 
in2;enious Lovers of Mufick, as lhall have 
the Curiolitv to inquire into the Natural 
Caufes of Harmony, and of the Thano- 
mena which occurr therein, though other- 
wife, to the more learned in Mulick and 
Mathematical Proportions, all might be 
expreiled very much fliorter, and Itill be 
more Ihortned by the help of Symbols. 

And 


of Sound Harnionick* i 5 

And here we may fix our Foot : Con- 
cluding, that what is evident to Senfe, of 
the Ph^nomena^ in a Chord, is equally 
(tliough not fo difcernably) true of the 
Motions of all other Bodies which render 
a tuneable Sound, as the Trembling of a 
Fjell or Truinfet^ the forming of the Lary?tx 
in our felves, and other Animals, the throat 
of Pipes and of thofe of an Organ, 

All of them in feveral Proportions fenfibly 
trembling and imprefling the like Undula- 
tions of the Medium^ as is done by the fe- 
veral (more manifeft) Vibrations of Strings 
or Chords. 

In thefe other Bodies, laft fpoken of, we 
manifeftly fee the Reafon of the Difference 
of the fwiftnefs of their Vibrations (though 
we cannot fo well meafure them) from 
their Shape, and other Accidents in their 
Conftitution ; and chiefly from the Propor- 
tions of their Magnitudes; the Greater ge- 
nerally Vibrating flower, and the Lefs 
more Iwiftly, which give the Tunes accord- 
ingly. We fee it in the Greatnefs of a 
String ; a greater and thicker Chord will 
give a graver and lower Tone, than one 
that is more flender, of the fame Tenfion 
and Length ; but they may be made Uni- 
foa by altering their Length and Tenfion. 


T £ K- 


14 of Sound Harmonick^, 

Tension is proper to Chords or 
Strings (except you will account a Drrmi 
for a Mufical Inftrument, which hath a 
Tenfion not in Length, but in the whole 
fuidace) as when we wind up, or let down 
the Strings, /. e. give them a greater or lefs 
Tenfion, in tuning a Lute^ or Harf- 
/ichordj and is of great Concern, and may 
be mealured by hanging Weights on the 
String to give it Tenfion but not eafily, 
nor fb ceixainlv-. 

But the lengths of Chords (becaufe of 
their Analogy to a Pendtihmi) is chiefly 
confidered, in the difcovery of the Propor- 
tions which belong to Harmony, it being 
moft eafie to meaiure and defign the Parts 
of a Monochord, in relation to the whole 
String ; and therefore all Intervals in Har- 
mony may firll be defcribed, and under- 
ftood, by the Proportions of the length 
of Strings, and conicquently of their Vi- 
brations. And it is for that Reafon, that 
in this Trcatife of the Grounds of Harmo- 
ny, Chords come fo much to be confi- 
dered, rather than other founding Bodies^ 
and tliofe, chiefly in their Proportions of 
Length. It is true, that in Wind-Inltru- 
ments, there is a Regard to the Length 
of Pipes, but they are not fo well acco- 
modated (as our Chords) to be examined, 


of Sound JHarmonicf. i 5 

neither are their Vibrations, nor the rriea- 
fure of them fo manifeft. 

There are fome Mufical Sounds which 
fcein to be made, not by Vibrations but 
by Pulfes as by whisking fwiftly over fome 
Silk or Camblet-lfufFs, or over the Teeth 
of a Comb, which render a kind of Tune 
more Acute or Grave, according to the 
fwiftnefs of the Motion- Here the Sound 
is made, not by Vibrations of the fame 
Body, but by Percuflion of feveral equal, 
and equidillant Bodies *, as Threads of the 
Stuff, Teeth of the Comb paffing over 
them with the lame Velocity as Vibrations 
are made* It gives the fame Modification 
to the Tune, and to the Undulations of the 
Air, as is done by Vibrations of the fame 
Meafure ; the Multiplicity of Pulfes or 
Percuffions, anfwering the Multiplicity of 
Vibrations. I take this Notice of it, be- 
caufe others have done fo ; but I think it 
to be of no ufe in Mufick. 




c 


AP- 


APPPENDIX. 


Before T co/iclude this Chapter, it ?nay feem 
needful^ letter to co?ifir7n the Foundation 
‘ik'e have laid^ a?id give the Reader fo?ne 
more ample Satisfaction alout the Moti- 
ons and Meafures of a Pendulum, and the 
jlppiication of it to Harnionick Motioru 



• 

I R S T then, it Is manifeft to Sence 
__ and Experience, and out of all dif- 
putc *, that the Courfes and Recourfes re- 
turn fooner or later, /. e. more or lefs fre- 
quently, according as the Pendulum is fliort- 
iVed, or made longer. And that the Pro- 
portion by v/hich the Frequency increaf- 
cth, is (at lead) very near duplicate, viz^ 
of the length of the Pe7idulum^ to the Num- 
ber of Vi&’ations, but is in rcverfe, /. e* as 
the Length encreafeth, fo the Vibrations 
decreal’e ; and contrary, quadruple the 
Length,’ and the Vibrations will be fub- 
dnpled. Subquadruple the Length, and 
the Vibrations will be dupled. And laft-. 
]y, that the Librations, the Courfes and 
Recourfes of the fame Peridulum^ are all 
made in equal Space of Time, or very 
r.ear to it, trem the greateft Range to the 

lead. 


END I X. 17 

leaft Now though the duplicate Propor- 
tion alTigned, and the equality of Time, 
are a little called in queftion, as not pei- 
feftly exaft, though very near it ; yet in a 
Monochord we find them perfeftly agree, 
v/z as to the length, Duple inftead of Du^ 
plicate, becaufe a String faftned at both^ 
ends is as a double PefidtilufYi^ each ot 
which is quadrupled by dupling the whole 
String. . And on this duple Proportion, de- 
pend all the Rations found in Harmony. 
And again, the Vibrations of a String are 
exa£Uy equal, becaufe they continue to 
give the fame Tune. 

Supposing then fome little difference 
may fometime ieem to be found in either ot 
thele Motions of a Pendulum^ yet the near- 
nefs to Truth is enqugh to fupport our 
Foundation, by Ihewing what is intended 
by Nature, though it fometimes meet with 
fecret Obftacles in the Pe?iduhim^ which 
it does not in a well made String. We 
may juftly make fome Allowance for the 
Accidents, and unfeen Caufes, which hap- 
pen to make fome little Variations iii Tri- 
als of Motion upon grofs Matter, and con- 
fequently the like tor nicer Experiments 
made upon the Pe?idtdmn» It is difficult 
to find exactly the determinate Point of 
the Plumbet, which regulates the Motions 

B 2 ‘ of 


» 8 END IX. 

of the ^e7idulu7n^ and configns its juft length- 
Ihen obferve the Varieties which happen 
through various forts of Matter, upon which 
Experiments are made. Merfennus tells us, 
that heavier Weights of the fame length 
move llower, lo that whillf a Lead Plum- 
bet makes 39 Vibrations, Cork or Wood 
will make at lead: /}o. 

Ac AIN, that a ftifl' Ve?iduhmi vibrates 
more irequently, than that which liangs 
upon a Chord. So that a Bar of Iron, or 
Staff of Wood ought to be lialf as long a- 
gain as the other, to make the Vibrations 
equal. Yet in each of thefe refpeftively 
to itfelf, you will find the duplicate Pro- 
portion and Equality of Vibration, or as 
near as may be. And (as to Equality) 
though in the Extreams of the Ranges of 
Librations, viz* the greateil compared to 
the leali, there may (from unfeen Cau- 
fes) appear fome Difference, yet there is 
no difcernable Difference of the Time of 
Vibrations of a Pe7idulu?n in Ranges, that 
are near to one another, whether greater 
or lefs ; which is the Cafe of the llanges 
of the Vibration of a String being made 
in a very fmall Compafs : And tlierefore 
tlie Librations of a Fendtilmu^ limitted to 
a fmall Difierence of Ranges, do well cor- 
refpond with the Vibrations of a String. 

As 

1 


^JTTENVIX, 15 ? 

As to Strings, the Wliole of Hannony 
depends upon this experimented and un- 
qiidlioned Truth, that Diapafon is duple 
to its Unifon, and confequently Diapente 
is Sefquialterum, Diatefferon Selc|uitertium, 
€ifr. Yet if you happen to divide a faul- 
ty String of an Infliument, you will not 
find the Oftave jull in the Middle, nor 
the other Intervals in their due Proportion, 
whieh is no default in Nature, but tha 
Matter we apply. A falfe String is that, 
which is thicker in one Part of its Length, 
than in another. The thicker Part natu- 
rally vibrates flower, and founds graver ; 
the more flender Parc vibrates fwifter, and 
founds more acute. Thus whilft two 
Sounds fo near one another, are at once 
made upon the fame String, they make 
a rough difeording Jarr, being a hoarfe 
Tunc mixed of both, more or lefs, as the 
String is more or lefs unequal : And if the 
thicker Part be next the FretrS, then the 
Fret (for Example D. F. H. 8tc. in a Viol 
or Lute) will render the Tune of the Note 
too fliarp ; and tlie contrary, if the flen- 
der Part of the String be next the Frets ; 
becaufe in the former, tlie thicker Part is^ 
Hopped, and the thinner founds more of 
the acuter Parc of this unhappy Mixture ; 
As in the later, the thicker Part is left to 
found the graver Tune, and thus the Fret 

C i will 


20 


A <? ND IX. 

will give a wrong Tune though the Fault 
be not in the Fret, but in the String ; which 
yet, by an unwary Experimenter, may hap- 
pen to caufe the Se^w Cano?iis to be cal- 
led in queftion, as well as the Meafures of 
a Tendtdmn are difputed. 

But all this does not difprove the Mea- 
fures found out, and afligned to Harmo- 
nick Intervals, which are verified upon a 
true String or Wire as to their Lengths, 
and as to the Equality of Recourfes in' 
their Vibrations, though Fendulums are 
thought to move flower in their leaft Ran- 
ges ; yet, as to Strings, in the very fmall 
Ranges which they make, (which are much 
lefs in other Inftruments, or founding Bo- 
dies) I need add no more than this, that 
the Continuance of the fame Tune to the 
laft, after a Chord is ftruck, and the con- 
tinued Motion in lefs Vibrations of a fym- 
pathizing String, duriitg the Continuance 
of greater Vibrations of the String which 
is Itruck, do either of them fufficiently de- 
monftrate, that thofe greater or lefs Vibra- 
tions, are both made in the fame Meafure 
of Time, according to their Proportions, 
keeping exaQ: Pace with each other. O- 
therwite ; In the former, the Tune would 
fenfibly alter, and in the latter, the fym- 
pathizing String could not be continued 


jTT END IX. 11 

in its Motion. This was not fo well con- 
cluded, till the late Dlfcoverics of the Pc//- 
dulum gave light to it. 


There is one thing more which I 
muft not omit. That the Motions of a 
Vendulmn^ may feem not fo proper to ex- 
plicate the Motions of a String, becaulc 
the faid Motions depend upon differing 
Principles, viz. thofe ot a Venduhnn upon 
Gravity ; thofe of a String upon Elafticity, 
I fhall therefore endeavour to fliew, how 
the Motions oi a Tendulufn^ agree with 
thofe of a Spring, and how properly the 
Explication of the Vibrations of a String, 
is deduced from the Properties of a Pt?i- 
dulmn. 

The Elahick power of a Spring, in a 
Body indued with Elafticity, feems to be 
nothing elfe, but a natural Propenfion and 
Endeavour of that Body, forced out of its 
own Place, or Pollure, to reftore it lelf a- 
gain into its former, more eafie and natu- 
ral Pofturc of Reft. And this is found in 
feveral Sorts of Bodies, and makes diffe- 
rent Cafes, of which I lhall mention fome% 

If the Violence be by Compreflion, for- 
cing a Body into lefs room than it natu- 
rally requires j then the Endeavour of Re- 

C 4 ftitu- 


21 


JfT END IX. 

ttitutioiij is by Dilatation to gain room e- 
nough. Thus Air may be compreffed in- 
to Id's Space, and then will have a great 
Elafticity, and ft niggle to gain its room. 
Thus, if you fqueeze a dry Sponge, it will 
naturally, when you take off the Force, 
Ipread it fdf, and hll its former Place. So, 
if you prefs with your Finger a blown 
Bladder, it will fpring and rife again to its 
Place. And to this may be reduced the 
Springs of a Watch, and of a Spiral Wire, 
&c. 


Again, a ffiff, but pliable Body, faften- 
ed at one End, and drawn afide at the o- 
ther, will fpring back to its foimer Place ; 
this is the Cafe of Steel-fprings of Locks, 
Snap-haunces, dft’. and Branches of Trees, 
wlien fliaken with the Wind, or pulled a- 
fide, return to their former Poilure : As is 
faid of the Palm, DepyeJJa Rep/rco, And 
there are innumerable inllances of this kind, 
where the force is by bending, and the 
Reftitution by unbending or returning. 


This kind is refembled by a Venduhm^ 
or Plummet hanging on a String, whole 
Gravity, like the Spring in thofe other Bo- 
dies, naturally carries it to its place, which 
here is downward ; all heavy Bodies na- 
turally defcending till they meet with fome 

Ob~ 


'AfTENV IX. 25 

Obftacle to reft upon. And the loweft that 
the Plummet can defcend in its Reftraint 
by the String, is, when it hangs perpendi- 
cular, as to ABj where it is neareft to the 
Horizontal Plane G and therefore low- 
eft. Now, if you force the Plummet up- 
ward (held at length upon the String) from 
B to C, and let it go ; it will, by a Spon- 
taneous Motion, endeavour its Reftitution 
to B : But, having nothing to ftop it but 
Air, the Impulfe of its own Velocity will 
carry it beyond 5, towards D ; and fo 
backward and forward, decreafing at every 
Range, till it come to reft at 'E. 


■ A 



Thus the Eenduluni and Spring agree 
in Nature, if you confider the Force a- 
gainft them, and their Endeavour of Re- 
ftitution. 


?4 APP ENT) IX. 

By T fiuther, if you take a thin ftiff 
.mina of Steel, like a Piece of Two-penny 
Riband of foine length, and nail it fall at 
one End, (the remainder of it being free 
in the Air) then force the other End afide 
and let it go ; it will make Vibrations back- 
ward and forward, perfeftly anfwering 
thofe of a Vendidum, And much more, 
if you contrive it with a little Steel But- 
ton at the End of it, both to help the Mo- 
tion when once fet on foot, and to bear 
it better againft the Rcfiftance of the Air. 
There will be no difference between the 
Vibrations of this Spring, and of a Pendu^ 
lunij which in both, will be alike increafed 
or decreafed in Proportion co their Lengths. 
The fame End (viz. Reft) being, in the 
fame manner, obtained by Gravity in one, 
and Elafticity in the other. 

Further yet, if you nail the Spring 
above, and let it hang down perpendicu- 
lar, with a heavier Weight at the lower 
End, and then fet it on moving, the Vi- 
brations will be continued and carried on 
both by Gravity and Elafticity, the Ten- 
duhan and the Spring will be moft friend- 
ly joyned to caufe a fimple equal Motion 
of Librations, I mean, an equal Meafure 
of Time in the Recourfes*, only the Spring 
anfwerable to ks Strength, may caule the 


APPEr^DIX. 25 

Librations to be fome what fwifcer, as an 
Addition of Tenfion does to a String con- 
tinued in the fame length. 

I come now to confider a String of an 
Inftrument, which is a Spring fanned at 
both Ends. It acquired! a double Elafti- 
city. The firft by Tenfion, and the Spring 
is ftronger or weaker, according as the 
Tenfion is greater or lefs. And by how 
much ftronger the Spring is, fo much more 
frequent are the Vibrations, and by this 
Tenfion therefore, the Strings of an Inftru- 
ment keeping the fame length are put in 
Tune, and this Spring draws length-ways, 
cndeavouving a Relaxation of the Tenfion. 

But then, Secondly^ the String being 
under a ftated Tenfion, hath another E- 
laftick Power fide-ways, depending upon 
the former, by which it endeavours, if it 
be drawn afide, to reftore it felf to the 
cafieft Tenfion, in the fhorteft, viz. ftreight- 
eft line. 

In the former Cafe, Tenfion doth the 
fame with abatement of length, and aftedfs 
the String properly as a Spring, in that 
the String being forcibly ftretched, as for- 
cibly draws back to regain the remifs Pof- 
ture in which it was before : And bears 

' little 


/ 


2 ^ APPENDIX. 

little Analogy with the Pendulum^ except 
in general, in their fpontaneous Motions 
in order to their Reftitution, 

But there is great Correfpondence in 
the fecond Cafe, between the Librations 
of a Pendulum and the Vibrations of a 
String (for fo, for diftinftions fake, I will 
now call them) in that they are both pro- 
portioned to their length, as has been 
ihewm ; and between the Elafticity which 
moves the String, and Gravity which moves 
the Pendulum^ both of them having the 
fame Tendency to Reftitution, and by the 
fame Method. As the Declivity of the 
Motion of a Pendulum^ and coniequcntly 
the Impulfe of its Gravity is ftill lelfened 
in the Arch of its Range from a Semi- 
Circle, till it come to relf perpendicular ; 
the Defcent thereof being more downright 
at the hrft and greatelf Ranges, and more 
Horizontal at the laft and fliorteft Ranges, 
as may be feen in the preceding Figure C I 
IE E 'B ; lb the Impulfe of Spring is If ill 
gradually lelfened as the Ranges Ihorten, 
and as it gains of relaxation, till it come 
to be reftored to rdf in its fliorteft Line. 
And this may be the Caufe of the Equality 
of 'Time of the Librations of a Pendulum ^ 
and alfo of the Vibrations of a String. 
Nov, the Proportions Df Length, to th? 

V elo«* 


appendix. ^7 

Velocity of Vibrations in one, and of Li- 
brations in the other, we are fure of, an J 
find by manifeft Experience to be qua- 
druple in one, and duple in the other. 

Now tack two equal FenduJmns toge- 
thei* (as before) being faftned at both Ends, 
take away the Plumbets, and you make it 
a String, retaining till the fime Properties 
of Motion, only what was faid before to 
be caufed by Gravity, muft now be faid 
to be done by Elafticity. You fee what 
an eafie Step here is out of one into the 
other, and what Agreement there is be- 
tween them. The Thanofnena are the 
fame, but difficultly experimented in a 
String, where the Vibrations are too fwift 
to fall under each exaft Meafure \ but moft; 
eafie in a Pendulum^ whofe flow Libra- 
tions may be meafured at pleafure, and 
numbered by diflant Moments of Time. 

T o bring it nearer, make your Tenfion 
of the String by Gravity, inftead of fcrew- 
ing it up with a Pegg or Pin : Hang weight 
upon a Pulley at one End of the String, and 
as you increafe the Weight, fo you do 
increafe the Tenfion, and as you increafe 
the Tenfion, fo you increafe tlie Velocity 
ot Vibrations. So the Vibrations are pro- 
fortionably regulated immediately by Ten- 


2 S ^'appendix. 

fion, and mediately by Gravity. So that 
Gravity may claim a lliare in the Meafures 
of thefe Harmonick Motions. 

B u T to come ftill nearer, and home to 
our purpofe. Faften a Gut or Wire-ftring 
above, and hang a heavy Weight on it 
below, as the Weight is more or lefs, fo 
will be the Tenfion, and confequently the 
Vibrations. But let the fame Weight con- 
tinue, and the String will have a ftated 
fetled Tenhon. Here you have both in 
one, a Fenduhmi^ and the Spring of a String, 
which refembles a double 'Pendulum : Draw 
the Weight afide, and let it range, and 
it is properly a "Pe?iduhm^ librating after 
the Nature of a "Pendulum. Again, when 
the Weight is at reft, ftrike the String with 
a gentle Pleflrum made of a Quill, on the 
upper part, fo as not to make the Weight 
move, and the String will vibrate, and give 
its Tune, like other Strings faftened at 
botli Ends, as this is alfo, in this Cafe. 
So you have here both a Penduhmi and a 
String, or either, which you pleafe. And 
(the 'Fenfion being fuppofed to be fettled 
under the fame Weight) the common Mea- 
fure and Regulator of the Proportions of 
them is the Length, and as you alter the 
Length, fb proportionably you alter at once 
the Velocity in the Recourfes of the Vi- 


APPENDIX. 2^ 

bratlons of the String, and of the Libra- 
tions of the Te?iduluni. And though the 
Vibrations be fo much fwifter, and more 
frequent than the Librations, yet the Ra- 
tions are altered alike. If you fubduple the 
Length of the String, then the Vibrations 
will be dupled. And if you fubquadruple 
it, then the Librations will be alfo dupled, • 
allowing for fo much of the Body of the 
Weight as mull be taken in, to determine 
the Length of the Pe?idt{hnj. 

The Vibrations are altered in duple 
Proportion to the Librations, becaufe (as 
has been fhewn) the String is as a double 
Tendtdum^ either one of which fuppofed 
Pendulums is but half fo long as the String, 
and is quadrupled by dupling the whole 
String. Still therefore the Proportion of 
their Alterations holds fo certainly and 
regularly with the Proportion of every 
Change of their common Length, that, 
if you have the Comparative Ration of 
either of thefe two, Vibrations or Li- 
brations to the Length, you have them 
both: The increafe of the Velocity of Li- 
brations being fubduple to the increafe of 
the Velocity of Vibrations. And thus the 
Motions of a Tenduhm do fully and pro- 
perly difcover to us, the Motions of a 
String, by the manifeft Correfpondence of 

their 


30 appendix. 

their Properties and Nature* The Temu- 
luniks Motion of Gravity, and the Strings 
of Elaflicity bearing fo certain Proportions 
according to Length, that the Principles 
of Harmony, may be very properly made 
out, and moft eafily comprehended, as ex- 
plained by the Pendulum. And we find, 
that in all Ages, this part of Harmony was 
never fo cheerfully underftood, as fince the 
late Difcoveries about the Ve^idtihm. 

And I chufe to make this Illuftration 
by the Pendulum^ becaufe it is fo eafie for 
Experiment, and for our Comprehenfion; 
and the Elaftick Power fo difficult. 

Having feen the Origine of Tuneable 
or Harmonick Sounds, and of their Dif- 
ference in refpeft of Acutenefs and Gra- 
vity : It is next to be confidered, how 
they come to be aftefted with Confonancy 
and Dilfonancy, and what thefe are. 


CHAP. 


CHAP. III. 




Of Confonancy and DifTonaricy. 

r ^<Onfonancy and DifTonancy are the Re- 
_j fult of the Agreement, mixture or u- 
niting (or the contrary) of the undulated 
Motions of the Air or Me^Jium^ caufed by 
the Vibrations by which the Sounds of 
diftinft Tunes are made. And thofe are 
more or lefs capable of fuch Mixture or 
Coincidence according to the Proportion 
of the Meafures of Velocity in which they 
are made, /. e. according as they are more 
or lefs commenfurate. This I might well 
fet down as a Pofhdatum. But I fhall by 
feveral Inftances endeavour to illuftrate the 
undulating Motions or Undulations of the ' 
Air ; and confirm what is faid of their 
Agreements and Difagreements. And firft 
the Undulations, by fomewhat we fee in 
other Liquids. 

L E T a Stone drop into the Middle of 
a fmall Pond of flanding Water when it is 
quiet, you fliall fee a Motion forthwith im- 
prelfcd upon the Water, palling and dila- 
ting from that Center where the Stone 
fell, in circular Waves one within an other, 

D ftill 


3 i Of Confonancy 

ftill propagated from die Center, fpread- 
ing till they reach and dafli againft the 
Banks, and then returning, if the force of 
the Motion be fufficient, and meeting thofe^ 
inner Circles which purfue the fame Courfe/ 
without giving them any Check. 

And if you drop a Stone in another 
place, from that Centre will likewife fpread 
round Waves; which meeting the other, 
will quickly pafs them, each moving for- 
wards in its own proper Figure. 

The like is better experimented in 
Quick-filver, which being a more denfe 
Body, continues its Motions longer, and 
may be placed nearer your Eye. If you 
try it in a pretty large round Velfel, fup- 
pofe of a Foot Diameter, the Waves will 
keep their own Motion forward and back- 
ward, and quietly pafs by one another as 
they meet. Something of this may be feen 
in a long narrow PalTage, where there is 
not room to advance in Circles. 

Make a wooden Trough or long Box, 
fuppofe of two Inches broad, and two 
deep, and twenty long. Fill in three Quar- 
ters or half full of quick-filver, and place 
it Horizontally, when it is at quiet, give 

it with your Finger a little patt at one End, 

and 


and Diironancy. 5 5 

and it will imprefs a Motion of a ridged 
Wave a crofs, which will pafs on to the 
other End, and dafhing againft it, return 
in the fame Manner, and dafh againft the 
hether End, and go back again, and thus 
backward and forward, till the Motion 
ceafe. Now if after you have fet this 
Motion on foot, you caule fuch another, 
you fhall fee each Wave keep its regular 
Courfe ; and when they meet one another, 
pafs on without any Reluclancy. 

I do not fay thefe Experiments are full 
to my purpofe, becaufe thefe being upon 
(ingle Bodies, are not fufficient to exprefs 
the Difagreementsof Difproportionate Mo- 
tions caufed by diiferent Vibrations of 
diffeient founding Bodies * but thefe may 
ferve to illullrate thofe invifible Undula- 
tions of Air. And how a Voice reflefted 
by the Walls of a Room, or by Eccho be- 
ing of adequate Vibrations, returns from 
the Wall, and meets the commenfurate 
Undulations paffing forwards, without hin- 
dering one another. 

But theie ate Inftances which further 
^nfirm the Reafons of Confonancy and 
Uillonancy, by the manifert agreeing or 

difagreemg Meafures oi* Motions already 

ipoken of. ^ 

D 2 jj. 


34 Of Conlonarcy 

It hath been a common PraSice to 
imitate a Tabour and Pipe upon an Or- 
gan. Sound together two difcording Keys 
(the bale Keys will fliew it beft, becaufe 
their Vibrations are flower) let them, for 
Exariiple, be Gamut with Gamut lliarp, 
or F Faut fliarp, or all three together. 
Though thefe ot themfelves fhould be ex- 
ceeding finooth and well voyced Pipes ; 
yet, when ftruck together, there will be 
Inch a Battel in the Air between their 
difproportioned Motions, fuch a Clatter 
and Thumping, that it will be like the 
beating of a Drum, while a Jigg is play- 
ed to it with the other hand. If you 
ceafe this, and found a full Cloie of Con- 
cords, it will appear furprizingly fmooth 
and fweet, which fhews how Difcords 
well placed, fet off Concords in Compo- 
iition. But I bring this Inftance to fhew, 
how ftrong and vehement thefe undula- 
ting Motions are, and how they corref- 
pond with the Vibrations by which they 
are made. 

I T may be worth the while, to relate 
an Experiment upon which I happened. 
Being in an Arched founding Room near 
a flirill Bell of a Houfe Clock, when the 
Alarm ftruck, I whiilled to it, which I 
did with eafc in the fame Tune with the 
Bell, but, endeavouring to whiftle a Note 

higher 


and DiflToiuncy 35 

higher or lower, the Sound of the Bell 
and its crofs Motions were fo predominant, 
that my Breath and Lips were check’d fo, 
that I could not whiffle at all, nor make 
any Sound of it in that difcording Tune. 
After, I founded a fiirill whiftling Pipe, 
which was out of Tune to the Bell, and 
their Motions fo claffled, that they feeincd to 
found like fwitching one another in tlie Air. 

I 

GALILEO, from this DoSrine of 
Te7ich£hms, eafily and naturally explains 
the fo much admired fympathy of Con- 
fonant ftrings ; one (though untouch’d) 
moving when the other ii ftruck. It is 
perceptible in Strings of the fame, or a- 
nother Inftrument, by trembling fo as to 
fhake off a Straw laid upon the other 
String : But in the fame Inftrument, it 
may ‘be made very vifible, as in a Bafs- 
Viol. Strike one of the lower Strings 
with the Bow, hard and ftrong, and if any 
of the other Strings be Unifon or Oftave 
to it, you fhall plainly fee it vibrate, and 
continue to do fo, as long as you continue 
the Stroke of your Bow, and, all the while, 
the other Strings which are diffonant, reft 
quiet. 

The Reafon hereof is this. When you 
ffrike your String, the Progreffive found 

D 3 of 


3 6 Of Confonancy 

of it ftrikes and flarts all the other Strings^ 
and every of them makes a Move- 
" ment in its own proper Vibration. The 
Confonant itring, keeping meafure in its 
Vibrations with tliofe of the founding String 
hath its Motion continued, and propaga- 
ted by continual agreeing Pulfes or Stokes 
of the other. Whereas the Remainder of 
the Diffonant firings having no help, but 
being checked by the Crofs Motions of the 
founding String, are conftrained to remain 
Ifill and quiet. Like as, if you fiand be- 
fore a Penaz/lu?77^ and blow gently upon 
it as it paffeth from you, and fo again in 
its next Courfes keeping exa£t time with 
it, it is moll: eafily continued in its Mo- 
tion. But if you blow irregularly in Mea- 
lures different from the Meafure of the 
Motion of the Pe?iduhm^ and fo moft fre- 
quently blow againft it, the Motion of it 
will be fo checked, that it muff quickly 
teafc. / 

And here we may take Notice, fas has 
been hinted before) that this alfo confirms 
the aforefaid Equality of the Time of Vi- 
brations to the laft, for that the fmall and 
weak Vibrations of the fympathizing String 
are regulated and continued by the Pulfes 
of the greater and ftronger Vibrations of 
the founding String, which proves, that 

not- 


and Diflonancy. ^7 

notwithftanding that Difpatky of Range, 
they are coiTiiTieniurate in the 1 inie o 
their Motion. 

This Experiment is ancient : I find it 
in Qu'pitilidHus a Author, 

who is (uppofed to have been contempo- 
rary with 'Pbdarch, But the Reafon or it 
deduced from the Penduhiftiy is new, and 
firfi: difeovered by Galileo. 

It is an ordinary Trial, to find out 
the Tune of a Beer-glafs without firiking 
it, by holding it near your Mouth, and 
humming loud to it, in feveral fingle Tunes, 
and when you at laft hitt upon the Tune 
of the Glafs, it will tremble and Eccho to 
you. Which fliews the Confent and Uni- 
formity of Vibrations of the fame Tune,^ 
though in feveral Bodies. 

T o clofe this Chapter. I may conclude 
that Confonancy is the Pafiage of feveral 
Tuneable founds through the Medium^ fre- 
quently mixing and uniting in their undu- 
lated Motions, caufed by the well pro- 
portioned commenfurate Vibrations of the 
ibnorous Bodies, and confequently arriving, 
fmooth, and fweet, and pleafant to the Ear. 
On the contrary, Diffonancy is from dif- 
proportionate Motions of Sounds, not mix- 

D 4 ing 


1 8 Of Confonancy 

ing, but jarring and clafhing as they pafs, 
and arriving to the Ear Harlh, and Gra- 
ting, and Offenfive. And this, in the next 
Chapter lhall be more amply explained. 

hJ o w, what Concords and Difcords are 
thus produced, and in uie, in order to Har- 
mony, I lhall next confider. 

V 


CHAP. IV. 

Of Concords, 

C oncords are Harmonick founds, which 
being joined pleafe and delight the 
Ear ; and Difcords the Contrary. So that 
it is indeed the judgment of the Ear that 
determines which are Concords and which 
are Difcords. And to that we muft firll 
refort to find out their Number. And then 
we may after fearch and examine how the 
natural Produclion of thofe Sounds, dif- 
pofeth tl em to be pleafing or unpleafant. 
Like as the Palate is abiblute Judge of 
Tafts, what is fweet, and what is bitter, 
or lowr, i^c- though there may be alfo 
found out fome natural Caufes of thofe 
(^uaiicies. But the Ear being entertained 
with Motions which fall under exa£t De- 
mon- 


of Concords. 39 

monftrations of their Meafures, the Do- 
ftrine hereof is capable of being more ac- 
curately difcovered. 

First then, (fetting afide the Unifon 
Concord, which is no Space nor Interval, but 
an Indentity of Tune) the Ear allows and 
approves thefe following Intervals, and on- 
ly thefe for Concords to any given Note, 
viz. the Odave or Eighth, the Fifth, then 
the Fourth, (though by later Matters of 
Mufick degraded from his Place) then the 
Third Major^ the Third Minor^ the Sixth 
Majorj and the Sixth Minor. And alfo 
fuch, as in the Compafs of any Voice or 
Inttrument beyond the Oftave, may be 
compounded of thefe, for fuch thofe are, 
I mean compounded, and only the for- 
mer feven are fimple Concords ; not but 
that they may feem to be compounded, 
viz. the greater of the lefs within an O- 
ftave, and therefore may be called Syttems, 
but they are Originals. Whereas beyond 
an octave, all is but Repetition of thefe 
in Compound with the Eighth, as a Tentli 
is an Eighth and a Third ; a Twelfth is 
an Eighth and a Fifth ; a Fifteenth is Dil- 
diapafon, /. e. two OCtaves, gjfc. 

But notwithttanding this DittinClion 
of Original and Compound Concords ) and 

tho^ 


40 Of Concords, 

tho’ thefe Compounded Concords are found, 
and difcerned by their Habitude to the O- 
riginal Concords comprehended in the Sy- 
ftem of Diapafon ; (as a Tenth afcending 
is an Oftave above the Third, or a Third 
above the Octave ; a Twelfth is an 0£tave_ 
to the Fifth, or a Fifth to the Eighth, a 
Fifteenth is an Eighth above the Odave, 
i. e. Dildiapafon two Eighths, iffcj yet 
they muft be own’d, and are to be efteem’d 
good and true Concords, and equally ufe- 
tui in Melody, efpecially in that of Con- 
fort. 

T H B Syftem of an Eighth, containing 
feven Intervals, or Spaces, or Degrees, and 
eight Notes reckoned inclufively, as ex- 
preffed by eight Chords, is called Diapa- 
fon, /. e. a Syltem of all intermediate Con-* 
cords, which were anciently reputed to 
be onlv the Fifth and the Fourth, and it 
comprehends them both, as. being com- 
pounded of them both And now, that 
the Thirds and Sixths are admitted for 
Concords, the Eighth contains them alfo : 
Viz. a I'hird Ma]or and Sixth Minor ^ and 
again a 'Ihird Minor and Sixth Major.^ 
The Oftave being but a Replication of 
the Unifon, or given Note below it, and 
the fame, as it were in Miniature, it clo- 
feth and terminates the ftrft perfed Sylfem, 


of Concords." 4 1 

and the next OGave above it afcends by 
the fame Intervals, and is in like manner 
compounded of them, and fo on, as far as 
you can proceed upwards or downwards 
with Voices or Inllruments, as may be 
feen in an Organ, or Harpfichord. It is 
therefore moft juftly judged by the Ear, 
to be the chief of all Concords, and is 
the only Confonant Syftem, which being 
added to it felt, ftill makes Concords. 

And to it all other Concords agree, 
and are Confonant, though they do not all 
agree to each other ; nor any of them make 
a Concord if added to it felf, and the Com- 
plement or Refidue of any Concord to 
Diapafon, is alfo Concord. 

The next in Dignity is the Fifth, then 
the Fourth, Third Major, Third Minor^ 
Sixth Major, and laftly Sixth Minor ; all 
taken by Afcent from the Unifon or given 
Note. 

By Unifon is meant, fometimes the Ha- 
bitude or Ration of Equality of two Notes 
compared together, being of the very fame 
Tune. Sometimes (as here) for the given 
fingle Note to which the Difiance, or the 
Rations of other Intervals are compared. 
As, if we conlider the Relations to Ga^mut, 

to 


I 


4^ Of Concords. 

to which Jre is a Tone or Second, B mi 
a Third, ^ C a Fourth, D a Fifth, We 
call Gamut the Unifon, for want of a more 
proper Word. Thus C faut^ or any other 
Note to which other Intervals are taken, 
may be called the Unifon. 

And the Reader may eafily difcern, in 
which Senfe it is taken' all along by the 
Coherence of the Difcourfe. 

I come now to confider the natural Rea- 
fons, why Concords pleafe the Ear, by exa- 
mining the Motions by which all Con- 
cords are made, which having been gene- 
rally alledged in the beginning of the third 
Chapter, Riall now more particularly be 
difcuffed. 

And here I hope the Reader will par- 
don fome Repetition in a SubjeQ: that (lands 
in need of all Light that may be, i(, for 
his eafe and more Heady Progrefs, before I 
proceed, I call him back to a Review and 
brief Summary of fome of thofc Notions, 
which have been premis’d and confider- 
ed more at large. I haye fliewed, 

I. That Harmonick Sound or Tune 
is made by equal Vibrations or Tremblings 
of a Body fitly conftituted. 

2. That 


of Concords. 4; 

2. That thofe Vibrations make their 
Courfes and Recourfes in the fame Mea- 
fure of Time ; from the greateft Range 
to the leifer, till they come to reft. 

3. That thofe Vibrations are under 
a certain Meafure of Frequency of Courfes 
and Recourfes in a given Space of Time. 

4. That if the Vibrations be more 
frequent, the Tune will be proportipnably 
more Acute ; if lefs frequent, more Grave. 

5. That the Librations of a Tendulum 
become doubly frequent, if the Pendulum be 
made four times fhorter ; and twice flower, 
if the Pendulum be four times longer. 

6. That a Chord, or String of a Mu- 
fical Inftrument, is as a double Pe 7 idulu?n^ 
or two "Pendulums tacked together at length, 
and therefore hath the fame Effefts by 
dupling ; as a Pendulum by quadrupling, 
/. e> by dupling the Length of the Chord, 
the Vibrations will be fubdupled, /. e. 
be half fo many in a given Time. And 
by fubdupling the length of the Chord, 
the Vibrations will be dupled, and propor- 
tionably fo in all other Meafures of Length, 
the Vibrations bearing a Reciprocal pro* 
portion to the Length. 

7. That 


44 Concords. 

7. That thefe Vibrations imprefs a 
Motion of Undulation or Trembling in the 
Medium (as far as the Motion extends) of 
the fame Meafure with the Vibrations- 

8. That if the Motions made by dif- 
ferent Chords be fo commenfurate, that 
they mix and unite ; bear the fame Courfe 
either altogether, or alternately, or fre- 
quently : I'hen the Sounds of thofe diffe- 
rent Chords, thus mixing, will calmly pafs 
the Medium^ and arrive at the Ear as one 
Sound, or near the fame, and fo will fmooth- 
ly and evenly ftrike the Ear with Pleafure, 
and this is Confonancy, and from the want 
of fuch Mixture is DifTonancy. I may 
add, that as the more frequent Mixture 
• or Coinfidence of Vibrations, render the 
Concords generally fo much the more per- 
fect ; So, the lefs there is of Mixture, the 
greater and more harfh will be the Dif- 
cord. 

From the Premifes, it will be eafie 
to comprehend the natural Reafon, why 
the Ear is delighted with thofe forenamed 
Concords ; and that is, becaufe they all 
unite in their Motions often, and at the 
leaft at every fixth Courfe of Vibration, 
which appears from the Rations by which 
they arc conftituted, which are all contain- 


of Concords. 45 

ed within that Number, and all^ Rations 
contained within that Space of Six, make 
Concords, becaufe the Mixture of their 
Motions is anfwerable to the Ration of 
them, and are made at or before every 
Sixth Courfe. This will appear if we exa- 
mine their Motions. Firft, how and why 
the Unifons agree fo perf^ly ; and then 
finding the Reafon of an Oftave, and fixing 
that, all the reft will follow. 

T o this purpofe, ftrike a Chord of a 
founding Inltrument, and at the fame 
Time, another Chord fuppoled to be in 
all refpcfts Equal, /. e» in Length, Matter, 
Thicknefs and Tenfion. Here then, both 
the Strings give their Sound *, each Sound 
is a certain Tune *, each Tune is made by 
a certain Meafure of V ibrations ; the fame 
Vibrations are impreffed upon, and carried 
every way along the Medium^ in Undula- 
tions of the fame Meafure with them, un- 
til the Sounds arrive at the Ear. Now the 
Chords being fuppofed to be equal in all 
refpefts ; it tollows, that their Vibrations 
mult be alfo equal, and confequently move 
in the fame Meafure, joyning and uniting 
in every Courfe and Recourie, and keep- 
ing Itill the fame Equality, and Mixture of 
Motions of the String, and in the Medimn- 
Therefore the Habitude of thefe two 

Strings 


Of Concords. 

Strings is called Unifon, and is fo peifea. 
ly Confonant, that it is an Identity of 
Tune, there being no Interval or Space 
between them. And the Ear can hardly 
judge, whether the Sound be made by two 
Strings, or by one. 

But Confonancy is more properly con- 
fidered, as an Interval, or Space betw^een 
Tones of different Acutenefs or Gravity. 
And amongft them, the moffperfeft is that 
which comes neareft to Unifon, (I do not 
mean betwixt which there is the leaft Dif- 
ference of Interval ; but, in whofe Motions 
there is the greateft Mixture and Agree- 
ment next to Unifon. The Motions of 
two Unifons are in Ration of i to i, or 
of Equality. The next Ration in whole 
Numbers is 2 to i. Duple. Divide a Mo- 
nochcrd in two Equal parts, half the 
Length compared to the whole, being in 
Subduple Ration, will make double Vi- 
brations, making two Recourfes in the 
fame time that the other makes one, and 
fo uniting and mixing alternately, /. e. eve- 
ry other Motion. Then comparing the 
Sounds of thefe two, and the half will be 
found to found an Oftave to the whole 
ChoVd. Now the Oftave (afcending from 
the Unifon) being thus found and fixed 
to be in duple Proportion of Vibrations, 


Of Concords. 47 

and fubduple of Length ; confequently the 
Proportions of all other Intervali are eafiiy 
found out^ 

They are found out by refolving or di- 
viding the Oftave into the Mean Rations 
which are contain’d in it. Euclidi, in his 
SeEiio Canonif^ Theorem 6 ^ gives two De- 
monftrations to prove, that Duple Ration 
contains, and is compos’d of the two next 
Rations, 7;/^. Sefquialtera and Sefquztertiai 
Therefore an Oftave which is in Duple Ra- 
tion 2 to I is divided into, and compos’d of 
a Fifth, whofe Ration is found to be Stf- 
quialtera 3 to i ; and a Fourth, whofe Ra- 
tion is Sefquitertia 4 to 3. In like manner 
Sefquialtera is compos’d of Sefquiquurta and 
Sefqmquinta ; that is^ a Fifth, 3 to 2, may 
be' divided into a Third y to 4 ; and 

a Third Minor ^ ^ to 5? 

T H E R E is an eafie Way to take a view 
of the Mean Rations, which may be con- 
tain’d in any Ration given, by transferring 
the Prime or Radical Numbers of the given 
Ration into greater Numbers of the tame 
Ration, as i to i into 4 to 2, or 6 to 
which have the fame Ration of Duple.' 
Again, 3 to 2 into 6 to 4, which is hill Sef- 
quialtera. Now in 4 to 2 the Medicty is 3. 
bo that 4 to 3, and 3 to 2, are compiehcn- 

£ ded 


4 8 Of Concords, 

ded 

in 4 to 2 ; that is, a Fourth and a Fifth 
'^^’e comprehended in an Eighth. In 6 to 4 
Mediety is 5, fo 6 to 4 contains 6 to 5, 
^nd 5 to 4 ; /. <?. a Fifth contains the two 
I'hirds. Let (J to 3 be the Odave, and it 
contains 6 to 5 Third lefs^ 5 to 4 Third nia- 
and 4 to 3 a Fourth, and hath tw^oMe- 
dieties, 5 and 4. Of this I fliall fay more 
in the next Chapter. 

Th e s e Rations exprefs the Difference 
of Length in feveral Strings which make 
the Concords ; and confequently the Diffe- 
rence of their Vibrations. Take two Strings 
A B, in all other refpeGs equal, and com- 
pare their Lengths, which, if equal, make 
Unifon, or the lame Tune. If A be double 
in Length to B, /. e. 2 to i, the Vibrations 
of B will be duple to thofc of A, and unite 
alternately, vi2, at every Courle, crofling 
at the Recourfe, and give the Sound of an 
Octave to A. 

I F the Length of A be to that of B as 
3 to 2, and confequently the Vibrations as 
2 to 3, their Sounds wdll confort in a Fifdi, 
and their Motions unite after every fecond 
Recourfc, /• at every other or third 
Courfe. 



of Concords. 49 

If A to B be as ^ to they found a 
Fourth, their Motions uniting after every 
third Recourfe, viz. at every fourth Courfe. 

If A to B be as 5 to 4, they found a 
Ditone, or Third Majors and unite after 
every fourth Recourfe, /. e. every fifth 
Courfe. 

I F A to B be as 6 to y, they found a 
Trihemitone, or Third Minor^ uniting af- 
ter every fifth Recourfe, at every lixch 
Courfe* 

Th u s, by the frequency of their being 
mixM and united, the Harmony of joyn’d 
Concords is found fo very fweet and plea- 
fing ; the Remoter being alfo cortibined by 
fiieir relation to other Concords befides the 
Unifon. The greater Sixth, 5 to 3, is 
within the compais of Rations between 
I and 6 ; but, I confefs, the leffer Sixth, 
^ to 5, is beyond it ; but is the Comple- 
ment of 6 to 5 to an Oftave, and makes a 
better Concord by its Combinations with 
the Oftave, and Fourth from the Unifon ; 
having the Relation of a Third Mimr to 
One, and of a Third M^jor to the Oijier, 
and their Motions uniting accordingly^ 
And the Sixth Ma]or hath the fame Ad- 
vantage. Of thefe Combinations I fliall 

B i have 


50 Of Concords. 

have occafioii to fay fouiewhat more, after 
I have made the Subjeft in hand as plain 
as I can. 

I propos’d the collating of two feveral 
Strings, to exprefs the Confort which is 
made by theni ; . but otherwife, thefe Ra- 
tions are more certainly found upon the 
Meafures of a Monochord, taken by being 
apply’d to the Section of a Canon, or a 
Rule of the String’s length divided into 
Parrs, as occafion requires ; becaule tliere 
is no need fo often to repeat Ci^teris fari^ 
as is when feveral Strings are collated. 
And if you take the Rations as Fraftions, 
it will be more eafie to meafure out the 
given Parts of a Monochord, or fingle 
String extended on an Inftrument : Thofe 
Parts of the String divided by a moveable 
Bridge or Fret put under, and made to 
found ; 'Pliat Sound, related to the Sound 
of the Whole, will give the Interval fought 
after. Ex. gr. f of the Chord gives an 
Eighth, y give a Fifth, found a Fourth, 
T lound a Xhh'd Mrjor^ F a Third Minor^ 
\ a Si.\'th Major., a Sixth Mi?wr : Now we 
thus exprefs thefe Concords. 


JJmfo7U 


of Concoriis. 


51 


Vuifon. 3d 3d Maj. 4th $tli 



1 1 

1 1 

t 1 

1 1 
1 1 
( 1 

— - j 





e-a- 

■— 5— 

■ ei 6 -\ 

• fl -4 — 

— e 5 — 1 

[ — G -+—■ 1 

6 i 3 

: — 5 


6th Min. 6th Maj. Sth 3d & 5th. 4th & 6th. 



Authentic. Ptagal, 


I faid, that all Concords are in Rations 
within the Number Six; and I may add, 
that all Rations within the Number Six 
are Concords : Of which take the following 
Scheme. 


•r 


6 to 5 3d Minor f 
to 4 5th 
to 3 8th 
to 2 lith 
to I 1 9th 

4 to 3 4tii 1 

to 2 8tli ' 

to I 15th 

6 to 5 3 d Minor] 

5 to 4 3d Major. 
4 to 3 Fourth 
3 to 2 Fifth 
i to I Eighth 

3 to 2 5th 
to I 12ch 

j to 4 3d Major. 

2 to I Sth 


to 3 6th \dajor. 



to 2 loih Major, 



to 1 ijthAdajor, 




All that are Concords to the Unifon, 
are alio Concords to the Oftaye And all 
that are Difcords to the Unifon, are Dif- 

E 3 co.rdsi 


5^ Of Concords. 

cords to the 0£i;ave. And fome of the In- 
termediate Concords, are Concords one to 
another ; as, the two Thirds to the Fifth, 
and the Fourth to the two Sixths. So that 
the Unifon, Third, Fifth, and Oftave ; or 
the Unifon, Fourth, Sixth, and Oftave, may 
be founded together to make a compleat 
Clofe of Harmony : I do not mean aClofe 
to conclude with, for the Plagal is not 
luch ; but a compleat Clofe, as it includes 
all Concords within the compafs of Diafa- 
fo?i. A Scheme of which I have fet down 
at the end of the ’foregoing Staff of five 
Lines, wdiich containeth the Notes by 
which the aforefaid Concords are cxprefs’d. 
The former two, which afcend from the 
Unifon, Gamuts by Third Major (p\: Minor^ 
and Fifth, up to the Oftave, are ufually 
call’d Authentick, as relating principally to 
the Unifon, and beft fatisfv ing the Ear to 
refi upon : The other two, which afcend by 
the Fourth and Sixtli Minor (or Mrjor^ up 
to the fame Oftave, are call’d Flngal^ as 
more combining with the Oftave, feeming 
to require a more proper Bafs Note, viz* 
an Eighth below the Fourth, and therefore 
not making a good concluding Clofe : And 
on the continual fliifting tliefe, or often 
changing them, depends the Variety of 
Harmony (as^ffar as Confonancy reachcth, 
nvhich is but as the Body ofMufick) in 


of Concords. 55 

aH Contrapunft chiefly, but indeed in all 
kinds of Compofition. I do not exclude a 
Sprinkling of Difcords, nor here meddle 
with Air, Meafure, and Rythmus, which 
are the Soul and Spirit of Mufick, and give 
it fo great a commanding Power. I'he 
Plagal Moods defcend by the fame Inter- 
vals, by which the Authentick afcend ; 
which is by Thirds and Fifdis ; and the 
Authentick defcend the fame by which the 
Plagal afcend, viz. by Fourths and Sixths ; 
one chiefly relating to the Unifon, the other 
to the Oclave. 

But that, for which I defcrib’d thefe 
full Clofes, was chiefly to give ( as I pro- 
mis’d) a larger account of the beforc- 
mention’d Combinations of Concords, which 
encreafe the Confonancies of each Note, 
and^ make a wonderful Variegation and 
Delightfulnefs of the Harmony. 

Cast your Eye upon the firfl: of 
them in the Authentick Scale, you will Re 
that Umi hath three Relations of Confo- 
nancy, viz. to the Unifon, or given Note 
G ; to the Fifth, and to the Odavc : To 
the Unifon as a Third mi^ior ; to the Fifth 
as a Third ; to the OcFavc a Sixth- 
7U{:ijoy * fo that its M^otions )oyn aftci* every 
fifth Recourfe, /. e. at every fixth Courle, 

E 4 V'ith 


5^ 4 Of Concords. 

with the Unifon ; every fifth with the Dla- 
pente or Fifth ; every fixth Courfe with the 
Oftave. Then confider the Diapente, 
D fol re ) as a Fifth to the Unifon, it joyns 
with it every third Courfe ; and as a 
Fourth to the Oftave, they joyn every 
fourth Courfe. Then, the Octave with the 
Unifon, joyns after every fecond Vibration, 
e. at every Courfe. 

Now take a Review of the Variety of 
Confonancies in thefe four Notes. Here 
are mixM together in one Confort the Ra- 
tions of 2 to I, 3 to 2, 4 to 3, 5 to 4, 6 to y, 
5 to 3. And juft fo it is in the other do- 
les, only changing alternately the Sixths. 

You may fee here, within the fpace of 
three Intervals from the Unifon, viz. 3d, 
jth, and 8th, what a Concourfe there is of 
Confonant Rations, to variegate and give 
(as ’twere) a pleafant Purling to the Har- 
mony within that Space : For now, all this 
Variety is fonuM within one Syftem of 
Diat^afo'i^ juitly bearing that Name. But 
then, chink what it will be when the re- 
mote Compounded Concords are ]oyn’d to 
them ; as when we make a full Clofe with 
both Hands upon an Organ, or Harpfichord, 
or when the higher Part of a Confort of 
Mufick is reconcird to the lower, by the 

middle 


of Concords.^ 5 j 

middle Parts, viz^ the Treble to tlie Bafs, 
by the Mean and Tenor: And all chib, re- 
frcfh’d by the Interchaugin^s made be- 
tween the Plagal and Authentick Moods. 
Add to all this the infinite Variety ot Move- 
ment of fome Parts, thro’ all Spaces, v/ii 
fome Parc moves flowly ; a ^d (as in 
one Part chafing and purluing another. 

The whole Reafon of Confonancy be- 
ing founded upon the Mixture and Uni- 
ting of the Vibrating Motions of fcvcral 
Chords, or founding Bodies, ’tis fit it Ihould 
here be better explain’d and confirm’d. 
That their Mixtures accord to their Ra- 
tions, ’tis eufie to be computed *, but it 
may be renrefented to your Eye. 


of Concords.’ 




V 

0 

ABBA 
AB BA AB BA 

A B B AIA B, &Pc 
AB BA AB BA)aB BA. 


1 V 

AB BA 1 AB BA 
ABC CAB 1 BAG CBA 

AB, Qpc, 
ABC. 


Of Concords. 57, 

L E T V V be a Chord, and ftand for 
the Unifon : Let O O be a Chord half fo 
long, which will be an Oftave to the Uni- 
fon,^and the Vibrations double : Then, I 
fay, they will alternately {L e. at every 
other Vibration) unite. Let from A toB 
be the Courfe of the Vibration, and from 
B to A the Recourfe ; obferving by the 
way, that (in relation to the Figures men- 
tion’d in this Paragraph and the next, as 
alfo in the former Diagram of the ‘Pe/^du- 
him, cap. 2, pag. 9O when I fay, [ fromB 
to A] and [overtakes V in A., i do 

there endeavour to exprefs the Matter brief 
and perfpicuous, without perplexing the 
Figures with many Lines ; and avoiding 
the Incumbrance of fo many Cautions, 
whereby to dift raO: the Reader : Yet I mull 
always be underftood to acknowledge the 
continual Decreale of the Range of Vibra- 
tions between A and B, while the Motion 
continues ; and by A and B mean only the 
Extremities of the Range of all thofe Vi- 
brations, both the Firil greatefl, and allb 
the Succcffive leffen’d, and gradually con- 
trafted Extremities of their Range. And 
the following Demonftration proceeds aud 
holds equally in both, being apply’dtothe 
Velocity of Recourfes,and not to theCom- 
pafs of their Range, which is not at all 
here confider’d. Such a kind of Equity, I 

muft 


5 ^ Of Concords* 

muft lometimes, in other parts of this Dif- 
courfe, beg of the candid Reader. To 
proceed therefore, I fay, whilft V being 
flruck, makes his Courfe from A to B ; O 
( Ifruck likewife) will have his Courfe from 
A to B, and Recourfe from B to A. Next, 
whilft V makes Recourfe from B to A, 
O is making its Courfe contrary, from A 
to B, but recouifeth, and overtakes V in A, 
and then they arc united in A, and begin 
their Courfe together. So you fee, that 
the Vibrations of Diafajon unite alternate- 
ly, joyning at every Courfe of the Unifon, 
and crofting at the Recourfe, 

Thus alfo Diafente^ or Fifth, having 
the Ration of 3 to 2, unites in like manner 
at every third Courfe of the Unifon. Let 
the Chord D D be Diapente to the Unifon 

V ; whilft V courfe th from A to B, the 
Chord D courfeth from A to B, and makes 
half liis Recourfe as far as C ; /. e, to 2. 
Whilft Y recourfeth from B to A, D paf- 
fetli from C to A, and back from A to B. 
Whilft V courfeth again from A to B, D 
palicth from B to A, and back to C. Whilft 

V recourfeth from B to A, D palTeth from 
C to B, and back to A ; and then diey 
unite in A, beginning their Courfes toge- 
ther at every third C'ourfe ol V. In like 
pnanner .the reft of the Concords unite, at 


of Concords. 59 

the 4th, 5th, 6th Couife, according to their 
Rations, as might this fame way be fliewn, 
but it would take up too much room, and is 
necdlefs, being made evident enough from 
thefe Examples already given. 

Thus far the Rates and Meafures of 
Con[o7iance lead us on, and give us the true 
and demonftrable Grounds of Harmony : 
But ftill ’tis not compleat without Difcords 
and Degrees (of which 1 fhall treat ia 
another Chapter) intermix’d with the Con- 
cords, to give them a Foyl, and fet them 
off the bettei. For (to ufe a homely Re- 
femblance) that our Food, taken alone, 
tho’ proper, and wholfome, and natural, 
may not cloy the Palate, and abate the 
Appetite, the Cook finds fuch kinds and 
varieties of Sawce, as quicken and pleafe 
the Palace, and fliarpen the Appetite, tho’ 
not feed the Stomach- as Vinegar,Muftardj 
Pepper, gjfc. which nourifli not, nor are 
taken alone, but carry down the Nourilh- 
ment with better Relifh, and affifi: it in 
Digeftion. So the PraQiical Mahers and 
Skilful Compofers m.ake ufe of Difcords, 
judicioufiy taken, to rclifli the Confort, 
and make the Concords arrive much fwee- 
ter at the Ear, in all forts of Defcant, but 
moft frequently in Cadence to a Clofe. In 
all which, the chief Regard is to be had 

to 


6o Of Concords. 

to what the Ear may expeQ: in the Con* 
duft of the Compofition, and muft be per"* 
form’d with Moderation and Judgment ? 
which I now only mention, not intending 
to treat of Compofmg, which is out of my 
Defign and Sphere, and would be too 
large; but my Defign is, to make thefe 
Grounds as plain as I can, thereby to gra- 
tifie thofe whofe Philofophical Learning, 
without previous Skill in Mufick, will ea- 
fily render them capable of this Theory : 
And alfo thofe Mafters in Praftick Mufick, 
and Lovers of it, who, tho’ wanting Phi- 
lofophy, and the Latin and other Foreign 
Tongues, to read better Authors ; yet, by 
the help of their Knowledge in Mufick, 
may attain to underhand the depth of the 
Grounds and Realbns of Harmony, for 
whofe fakes it is done in this Lan- 
guage. 

I fhall conclude this Chapter with fome 
Remarks, concerning the Names given to 
the feverai Concords : We call them 
Fourth^ Fifths SicctJ:., and Eighth. Of thefe 
the Thirds being Two, and Sixths being 
alfo Two, want better diftliiguifliing Names. 
To call them Fiat and Sharp Thirds, and 
Flat and Sharp Sixths, is not enough, and 
lies under a Miiiake ; I mean, it is not a 
fufficient Diftiriftion, to call the greater 

Third 


of Concords* 6 i 

Third and Sixth, Sharp Third and Sharp 
Sixth, and the leffer, Flat. They are fo 
indeed in afcending from the Unilon, but 
in defcending they are contrary ; for to 
the Oftave that greater Sixth is a leffer 
Third, and the greater Third is a leffer 
Sixth ; which leffer Third and Sixth can- 
not well be call’d Flat, being in a Sharp 
Key ; Flat and Sharp therefore do not well 
diftinguifh them in general ; the leffer 
Third from the Oftave being fliarp, and 
the greater Sixth flat. So, from the Fifth 
defcending by Thirds, if the firft be a mi- 
nor Third, it is fharp, and the other being 
a m^jor Third, cannot be faid to be flat. 

The other Diftinflion of them, viz. by 
Major and Minor ^ is more proper, and 
does well exprefs which of them we mean. 
But ftill the common and confiifed Name 
of Thirds if the Diftinftion of inajor and 
minor be not always well remember’d, is apt 
to draw young Praftitioners, who do not 
well confider, into another Error. I would 
therefore call the greater Third (as the 
Greekd do ) Ditone^ i, e. of two whole 
Tones \ and the Third minor ^ Trihemito^ie 
or Sefqtiitone^ as confifling of three half- 
Tones, (or rather of a Tone and half a 
Tone) and this would avoid the mention’d 
Error which I am going to defcribe. 


It 


6i Of Concords. * 

I T IS a Rule in compofing Confort Mu- . 
fick, that it is not lawful to make a Move- 
ment of two Unifons, or two Eighths, or 
two Fifths together ; nor of two Fourths, 
ymlefs made good by the addition of Thirds 
in another Part : But we may move as 
many Thirds or Sixths • together as we 
pleafe. Which laft is falfe, if we keep to 
the fame fort of Thirds and Sixths ; for 
the two Thirds differ one from another in 
like manner as the Fourth differs from the 
Fifth : For in the fame manner as the 
Eighth is divided into a Fifth and Fourth, 
fo is a Fifth into a Third ma]or and Third 
minor. Now call them by their right 
Names, and, I fay, it is not lawful to 
make a Movement of as many Dito?iesy 
or of as many Sefrjuitones as you pleafe : 
And therefore when you take the Liberty 
fpoken of, under the general Names of 
Thirds y it will be found that you mix Di- 
tones and Trihemitonesy and fo are not con- 
cern’d in the aforefaici Rule ; and fo the 
Movements of Sixths wull be made with 
mixture and interchanges of 6th 7najor and 
6th rninory which is fafe enojugh. 

Y E T, I confefs, there is a little more 
Liberty in moving Trihemitones^ and Di- 
tones ^ as likewife either of the Sixths, than i 

there is in moving Fourths or Fifths ; and 

the 


Of Concords 




the Ear will bear it better. Nay, there is 
neceffity, in a gradual Movement of Tliirds, 
to make one Movement by two Jrihenii- 
tones together in every Fourth and Fifth, 
or Fourth disjunfl:; that is, twice in 
fafon^ or, at leaf!:, in two Fifths ; as in 
Gamut Key proper. The natural Afcent 
will be Ut Re Mi Fa Sol La : Now, to 
thefe join Thirds in Natural Afcent, and 
then they will be Mi Fa Sol La Fa Sol. 


in other Cliffs, but with fome variation, 
according to the Place of the Hemitone. 


Here and arc two Tahe- 


mitones fucceeding one another, and you 
cannot well alter them without difor- 
dering the Afcent, and diflurbing the Har- 
mony; becaufe, where there is a Hernia 
tone^ the Tone below join’d to it, makes a 
Trihemito?ie ; and the next Tone above it, 
join’d to it, makes the fame. Thus you 
fee the neceffity of moving two Trihemi^ 
tones together, twice in "'Diafafon^ or a 
9th, in progreffion of Thirds, in Diatonic 
Harmony, but you cannot well go fur<» 
ther. 

Now, there is Reafoa why t\wo Trihc'> 
mitones will better bear it, becaufe of their 
different Relations, by which one TrihemF 



And thus it will be 


F 


toJie 


6 4 Of Concords. 

tone is better diftinguifh’d from another, 
than one Oftave, or one Fifth, or one 
Fourth from another. 

In a Third mJnor^ which hath two De- 
grees or Intervals, confilting of a Tone and 
H^mitone^ the Heniitone may be placed 
either in the lower Space, and then gene- 
rally is united to his Third nujor (which 
makes the Complement of it to a Fifth) 
downward, and makes a fharp Key ; or 
dfe it may be placed in the upper Space, 
and then generally takes his third rn^jor 
.above, to make up the fifth upward, and 
conftitute a flat Key. And thus a Tritone 
is avoided both ways. I fay, if the Hemi- 
tone in the Third 'minor be below, then 
the Third 7na]or lies below it, and the 
Air is flaarp. If the Hemito?ie be above, 
then die Third inrjor lies above, and the 
Air is flat. And thus the two minor 
Thirds join’d in confequence of Move- 
ment, are differenc’d in their Relations, 
confequent to the place of the He^nitone ; 
which Variety takes off all Naufeoufnefs 
from the Movement, and renders it Iweet 
and pleafant. 

You cannot fo well and regularly make 
a Movement of Dito?ies^ tho’ it may be 
done fometimes, once or twice, or more, 

in 


0 / Concords. <5 5 

I 

m a Bearing Paffage (in like manner as 
you may romctimcs ufc Difcords ) to give, 
after a little grating, a better Relifli. I'he^ 
Skilful Artift may go farther in the ufe of 
'Ihirds and Difcords than is ordinarily al- 
low’d. 

I might enlarge this Chapter, by fet- 
ting down Examples of the Lawful and 
Unlawful Movements of Thirds major and 
minor ^ and of the Ufe of Difcords ; but, 
as I faid before, my Defign is not to treat 
of Compofition : However, you may caft 
your Eye upon thele following Inftances, 
and your own Obfervation from the beiL 
Mafters will furnifh you with the reft. 


Lawful Movement 
of Thirds, Mix’d. 


Unlawful Movement 
of Thirds Alajor, 


66 Of Concords. 

That the Reader may not incurr any 
Miftake or Confufion, by leveral Names 
of die fame Intervals, I have here fet 
them down together, with their Rations^ 


8th 

9th jM/rjnr. 
7th Jidinor, 
6tli lidajor, 
6tli Adinor, 
5 th 


Oclave, Diapafon. 
Hsptacliord Major. 
Heptachord Minor. 
Hexachord Major, 
Hexachord Almor. 
DiapentCj Pentachord. 


5th Falfe ( in de-1 

fed ) j^Semidiapente. 

4th Falfe (in ex-1 . 
cefs ) j^Tritone. 


4th 
3d Major 


Diaceffaronj Tetrachord. 
' Ditone. 

K Sefquitone. 


itone. "0 
Tnhemitone. > 
itone, 3 


3d Minor 

I 1 ^ Semiditone 

id jVZ/t;. or Whole . 

ISotQ Major '' T^oixq Major 

2d Min, or Whole > 

Isiote Minor STonc Minor 


J 


Note Greater 
Half Note Fefs 


’ Semi- 


2d Leajl^or Half-”^Hemi.7 

u\ 

n 


tone 



Max. 


Min. 


j J if Minim. 
tone Minor. 


Maj Q 


Quarter- Note 

Difference be- 
tween Tone 
M'jor., and 
Tone Minor, 


T Hemi- 
^ Sem 

^ Diefis Chromatic. 
i Diefis Maj 


lor. 


f Diefis Enharmonic. 1 
t Dicfis Minor. J 

Comma. Comma Majus . 
Schifm, 


} 


2 

to 1 

15 

8 

9 

5 

5 

5 

8 

5 

3 

2 

64 

45 

45 

32 

4 

3 

5 

4 

6 

5 

9 

8 

10 

9 

16 

15 

25 

24 

ii8 

125 

81 

80 


Note, VVhenever I mention Diefis without dihindion, I 
mean Diefis Adlnor, or En]iarmon:c: And wlicn I fo 
mention Commas I mean Comma Mnjus^ or Schifm, 


I 


of Concords. 67 


I fliould next treat of TDifcords^ but be- 
caufe there will intervene lb much Ufe of 
Calculation^ it is needful that (before I 
go further) I premife fome account of 
Proportion in General, and apply it to 
Harmony. 


H E R E A S it hath been faid before, 


That Harmonick Bodies and Mo- 
tions fall under Numerical Calculations, 
and the Rations of Concords have been 
already affign’d ; it may feem neceffary 
here (before we proceed to fpeak of Dif- 
cords ) to fhew the Manner how to cal- 
culate the Proportions appertaining to 
Harmonick Sounds : And for this I fliall 
better prepare the Reader, by premifing 
Ibmething concerning Proportion in Ge- 
neral. 

W E may compare ( /. e. amongft them- 
felves) either (i.) Magnitudes^ ( fo they 
be of the fame kind;) or (2.) the Grnvi- 
tatio7is y Motions y Velocities y 'DurationSy 
Sounds y &c. from tl ence arifing ; or fur- 


C H A P. V. 


of Proportion 5 and apply’ d to Harmony. 



ther. 


68 Of Proportion. 

ther, if you plcafe, the Numbers them- 
felves, by which the Things compar’d are 
explicated. And if thefe fliatl be unequal, 
we may then confider, either, how 

much one of them exceeds the other ; or, 
Secondly^ after what manner one of them 
jtands related to the other, as to the Qiio- 
tient of the Antecedent (or former Term) 
divided by thcConfequent (or latter Term:) 
‘Which Qriotient doth expound, denomi- 
nate, or ihew, how many times, or how 
much of a time or times, one of them 
doth contain the other. And this by the 
Greeks is call’d aoq/®-. Ratio ; as they are 
wont to call the Similitude^ or Equality of 
Ratio’s tivcLKoyU^ Afialogie^ Froportiou^ or 
Erof art i Quality : But Cuftom, and theSenfc 
afTifting, will render any over-curious Ap- 
plication of thefe Terms unneceffary. 

From thefe two Confiderations lad 
mention’d there are wont to be deduced 
three forts of Proportion, Arithmetical^ Gee-- ’ 
iuetricalj and a mix’d Proportion refulting 
from thefe two, call’d Harmo 7 iicaL 

1. oArithmetical^ when tliree or more 
Numbers in Progreffion have the fame Dif- 
ference •, as, 2, 4, 6, 8, or difeontinued, 
as 2, 4, 6 \ 14, 16, 18. 


2. GeO” 


of Proportion. 



2. Geometrical^ when three or more 
Numbers have the fame Ration, as 2,4,8, 
i5j 32; or difeontinued, 2,4*, 64,128. 

Laftly, Harmonica]^ ( partaking of both 
the other) when three Numbers are fo or- 
derM that there be the fame Ration of the 
Greatert to the Lealt, as there is of the 
Difference of the two Greater to the Diffe- 
rence of the two Lefs Numbers : As in 
thefe three Terms, 3, 4, 6, the Ration of 
6 to 3 (being thegreateft and leaft Terms) 
is Duple. So is 2, the Difference of 6 and 4 
(the two greater Numbers) to i, the Dif- 
ference of 4 and 3 (the two lefs Numbers) 
Duple alfo. This is Proportion Harmoni- 
cal, which Diapafon, 6 to 3, bears to Dia- 
pente 6 to 4, and Diateffaron 4 to 3, as its 
mean Proportionals. 

Now for the Kinds of Rations rhoff 
properly fo call’d, /. e. Geometrical : Firft 
obferve, that in all Rations the former 
Term or Number (whether greater or lefs) 
is always call’d the Antecedent^ and the 
other following Number is call’d the CV?.- 
fequent. If therefore the Antecedent be 
the greater Term, then the Ration is either 
Multif>lex^ Superparticular^ Snferpartient^ 
QV (what is compounded of thefe) JStdli- 

F 4 p lt oc 


70 Of Proportion. 

flex Siifer’^articulaYf oi Multif lex Safer- 
fartie?2t, 

1. Multlt'lex \ as Duple, 4 to 2 ; Triple, 
6 to 2 ; Quadruple, 8 to 2. 

2. Stifev'^ articular ; as, 3 to 2, 4 to 3, 
5 to 4, exceeding but by one aliquot part, 
and in their Radical or lead: Numbers, al- 
ways but by one *, and thefe Rations are 

Sefqui altera^ Sefquitertia (or Super- 
tertia) Sefquiquarta^ (or Super quart a) See. 
Note^ that Numbers exceeding more than 
by one, and but by one aliquot part, may 
yet be Sup erp articular^ if they be not ex- 
prefs’d in their Radical, /. e. leaft Num- 
bers ; as 1 2 to 8 hath the fame Ration as 
3 to 2 ; /. e. Superparticular ^ tho’ it feem 
not fo till it be reduced by the greatefl: 
Common Divifor to its Radical Numbers 
3 to 2. And the Common Diviforf/. e. the 
Number by which both the Terms may 
feverally be divided) is often the Diffe- 
rence between the two Numbers ; as in 
12 to 8, the Difference is 4, which is the 
Common Divifor. Divide 12 by 4, the 
Quotient is 3; divide 8 by 4, the Quotient 
is 2 ; fo the Radical is 3 to 2. Thusalfo 
1 5 to I o divided by the Difference 5, gives 
3 to 2 ; yet, in 16 to 10, 2 is the Common 
Divifor, and gives 8 to j, being Superpar- 

tient. 


of Proportion. 71 

tient. But in all Suferparticuliir Rations, 
whofe Terms are thus made larger by be- 
ing multiply’d, the Difference between the 
Te rms is always the greateft Common Di- 
vifor; as in the ’foregoing Examples. 

The third kind of Ration is Superpar^ 
tient^ exceeding by more than One, as 5 
to 3, which is call’d SuperhipartiensTertias 
(or TV/^) containing 3 and 7-; 8 to y, 
Supertripartiens Quintus^ 5 audr* 

The fourth Multiplex Superp articular^ 
as 9 to 4, which is duple, and Sefquiguarta^ 
1 3 to 4, which is triple, and Sefquiquarta. 

The fifth and 1 aft is Multiplex Superpar^ 
tient^ as 1 1 to 4 ; duple, and Supertripar- 
tiens Quartos, 

When the Antecedent is lefs than the 
Confequent, viz, when a lefs is compar’d 
to a greater, then the fame Terms ferve 
to exprefs the Rations, only prefixing Suh 
to them ; as, Submultiplex ^ Sulfuperparti^ 
cular ( or Subparticular ) Subfuperp art lent 
(or Subpartient') &c. 4 to 2 is Duple^ 2 to 

4 is Subduple, 4 to 3 is Sefquitertia\ 3 to 
4 is Subfefquitertia *, 5 to 3 is Superbipar- 
tiens Tertias ; 3 to 5 is Subfuperbipartiens 
Tertias,^ &c. 


This 


72 


of Proportion. 

Th IS flaort Account of Proportion was 
neceffary, becaufe almoft all the Philofophy 
of Harmony confifts in Rations, of the 
Bodies, ot the Motions, and of the In- 
tervals of Sound, by which Harmony is 
made. ^ 

And in fearching, ftating, and compa- 
ring the Rations of thefe, there is found 
fo much Variety, and Certainty, and Faci- 
lity ot Calculation, that the Contempla- 
tion of them may feem not much lefs de- 
lightful than the very Hearing the good 
Mufick it felf, which fprings from this 
Fountain. And thofe who have already 
an aifeftion for Mufick cannot but find it 
improv’d and much enhaunc’d by this plea- 
fan t recreating Chace ( as I may call it ) 
in the large Field of Harmonic Rations and 
Troportionsj where they will find, to their 
great Pleafure and Satisfaftion, the hidden 
Caufes of Harmony ( hidden to moft, even 
to Praftitioners themfelvesj fo amply dif- 
cover’d and laid plain before them. 

All the Habitudes of Rations to each 
other are found by Multiplication or Divi- 
fion of their Terms ; by which any Ration 
is added to, or fubfirafted from another : 
And there may be ufe of ProgrdTion of 


of Proportion. 75 

Rations, or Proportions, and of finding a 
or Mediety between the Terms of 
any Ration : But the main Work is done 
by Addition and Subrtraftion ot Rations; 
which, tho’ they are not perform’d like 
Addition and SubftraQion of Simple Num- 
bers in Arithmetick, but upon Algebraick 
Grounds, yet the Praxis is mofl ealie. 

O M E Ration is added to another Ra- 
tion, by multiplying the two antecedent 
Terms together ; /. e. the Antecedent of 
one of the Rations by the Antecedent of 
the other (for the more eafe they fliould 
be reduced into their leaft Numbers or 
’Perms) and then the two Confequent 
Terms in like manner. The Ration of the 
ProduQ: of the Antecedents, to that of the 
Produft of the Confequents, is equal to the 
other two added or join’d together. Thus 
(for Example) add the Ration of 8 to < 5 ; 
A (in Radical Numbers) 4 to' 3, to the 
Ration of 12 to 10; /. e, 6 to 5, 
the Produft will be 24 and ij'; 4 | 3 

/. f. 8 to 5. You may fet ’em thus, ^ 5 

and multiply 4 by 6, they make 

24, which fet at the bottom; 
then multiply 3 by 5, they make 
15, which likewiie let under, and you have 
24 to 15; which is a Ration compounded 

of 


74 P/ Proportion. 

of the other two, and equal to them both* 
Reduce thefe Produfls, 24 and 15, to their 
lead Radical Numbers, which is, by divi- 
ding as far as you can find a common Di- 
vifor to them both (which is here done 
by 3) and that brings them to the Ration 
of 8 to j'. By this you fee, that a Third 
wAnor^ ^ to 5, added to a Fourth, 4 to 3, 
makes a Sixth 7 ninor^ 8 to 5. If more Ra- 
tions are to be added, fet them all under 
each other, and multiply the firft Antece* 
dent by the fecond, and that ProduQ: by 
the third, and again that ProduQ: by the 
Fourth, and fo on *, and fo in like manner 
the Confequents. 


This Operation depends upon the Fifth 
Propofition of the Eighth Book of Euclid ; 
where he fhewSj that the Ration ot Plain 
Numbers is compounded of their Sides. 
See thefe Diagrams : 




Now 


of Proportion. 75 

Now compound thefe Sides. iTake 
for the Antecedents, 4 the greater Side 
of the greater Plane, and 3 the grea- 
ter Side of the lefs Plane, and they inul- 
tiply’d give 12 : Then take the remaining 
two Numbers 3 and 2, being the lefs Sides 
of the Planes (for Confequents) and they 
give 6. So the Sides of 4 and 3, and of 
3 and 2, compounded (by multiplying the 
Antecedent Terms by themfelves, and the 
Confequents by themfelves) make 12 to 6, 
/. e. 2 to I ; which being apply’d, amounts 
to this ; Ratio Sefqtii altera^ 3 to 2, added 
to Ratio Sefquitertia 4 to 3, makes Duple 
Ration, 2 to i. Therefore 'J)ia^e 7 ite added 
to DiateffaroJi^ makes Dia^afon. 

SuBSTR ACTION of One Ration from 
another greater is perform’d in like man- 
ner by multiplying the Terms; but this is 
done not Laterally^ as in Addition, but 
Crojjwife ; by multiplying the Antecedent 
of the former ( /. e, of the greater) by the 
Confequent of the latter, which produceth 
a new Antecedent ; and the Conlequent of 
the former by the Antecedent of the latter, 
which gives a new Confequent. And 
therefore it is ufually done by an Oblique 
Dccuflation of the Lines. Lor Example, 
If you would take 6 to 5 out of 4 to 3, 

you 


7 6 Of Proportion, 

you may fet them down as in the Mat- 
gin : Then 4 multiply’d by 
4 3 makes 20, and 3 by ^ gives 18 : 

X So 20 to 18, /. e, 10 to 9, is the 
Remainder. That is, fubftraQ: a 
6 5 Third Mi?ior out of a Fourth, 
20.18. and there will remain a Tone 
10. 9. Minor* 

Multiplication of Rations is the 
fame with their Addition, only kis notw^ont 
to be of divers Rations, but of the fame, 
being taken twice, thrice, or oftener, as you 
jpleafc. And as before in Addition you added 
divers Rations by multiplying them, fo here 
in Multiplication you add the fame Ration 
to it felf, after the fame manner, viz* by 
multiplying the Terms of the fame Ration 
by themfelves ; /. e, the Antecedent by it 
ielf, and the Confequeat by it felf, (which 
in other Words is to multiply the fame by 2) 
and will, in the Operation, be to fquare 
the Ration hrft propounded (or give the 
Second Ordinal Power, the Ration firlt gi- 
ven being the Firif Power or Side.) And 
to this Prod Lift, if the Simple Ration flaall 
again be added (after the lame manner as 
before) the Aggregate will be triple ot the 
Ration firll given ; or the Produft of that 
Ration imiltiplykl by 3, viz* the Cube, or 
'I'hird Ordinal Power. Its Biquadrate^ or 

Fourth 


of Proportion. 77 

Fourth Power, proceeds from multiplying 
it by 4, and fo fuccelTively in order as far 
as you pleafe you may advance the Powers. 
For inlfance, the Duple Ration, 2 to i, be- 
ing added to it felf, dupled, or multiply’d 
by 2, produced! 4 to i, (the Ration 
drtiple) : And if to this, the fiiil again be 
added, T which is equivalent to multiply- 
ing that faid firft by 3) there will arife the 
Ration or 8 to i. Whence the 

Ration 2 to i being taken for a Root,^ its 
Duple 4 to I will be the Square, its 1 riple 
8 to I the Cube thereof, as hath been 
faid above. And, to ufe another Inftance, 
To duple the Ration of 3 to i, it mufl be 
thus fquadd ; 3 by 3 gives 9 : 2 by 2 gives 
4 •, lo the Duple or S^quare of 3 to x is 9 
to 4. Again, 9 by 3 is 27, and 4 by 2 is 8, 
fo the Cubic Ration of ^ to 2 is 27 to 8. 
Again, to find the Fourth Power, or 
dratey (/. e. fquarM Square) 27 by 3 is 81, 
8 by 2 is 16 y fo 81 to 16 is the Ration of 
3 to 2 quadrupled, as ’tis dupled by the 
Square, tripled by the Cube, To ap- 
ply this Inftance to our prefent purpofe, 
3 to 2 is the Ration of Diapentey or a Fifth 
in Harmony ; 9 to 4 is the Ration of twice 
Diape7itey or a Ninth (viz. 'T>iapafon with 
Tone Mojor) ; 27 to 8 is the Ration of 
thrice Diapente^ or three Fifths, which is 
^Diapafon with Six M^jor (?;/>. 1 3^^ jvjrjor) 

The 


7 8 Of Proportion^ 

The Ration of 8 1 to 1 6 makes four Fifths, 
i. e. Dif-diafafon^ with two Tones Majorj 

e. a Seventeenth Major ^ and a Comma of 
8ito8o. 

To divide any Ration, you muft take 
the contrary Way, and by extrafting of 
thefe Roots refpeftively, Divifion by their 
Indices will be perform’d. £. gr. To di- 
vide it by 2, is to take the Square Root of 
it ; by '3, the Cubic Root ; by 4, the Biqua- 
dratick, iffc. Thus to divide 9 to 4 by 2, 
the Square Root of 9 is 3, the Square Root 
of 4 is 2 ; then 3 to 2 is a Ration juft half 
fo much as 9 to 4. 

From hence it will be obvious to any 
to make this Inference ; That Addition and 
Multiplication of Rations are (in this cafe) 
one and the fame thing. And thefe Hints 
will be fulBcient to luch as bend their 
Thoughts to thefe kinds of Speculations, 
and no great ^Trefpafs upon thofe that do 
not. 

T H E Advantage of proceeding by the 
Ordinal Powers, Square, Cube, ijfc. (as is 
before mention’d) may be very ufelul where 
there is^Occafion of large Progrelfions ; as, 
to find (for Example) how many Com- 
ma’s are contain’d in a Tone Major ^ or other 

Inter vaU 


of Proportion* 79 

Interval ; let it be, How many are in D/> 
pafon ? Which mufl' be done by multiply- 
ing Comma’s, /. adding them, till you 
arrive at a Ration equal to OBave^ (if that 
be fought) viz. Duple : Or elfe by dividing 
the Ration of Diapafon by that of a Com- 
ma, and finding the Quotient; which may 
be done by Logarithms. And herein I 
meet with fome Differences of Calcula- 
tions. 

Mersennus finds, by Ills Calculation, 
58^ Comma’s, and fomewhat more, in an 
OBave : But the late Nicholas Mercator^ 
a Modefl: Perfon, and a Learned and Judi- 
cious Mathematician, in a Manufcript of 
his, of which I have had a Sight, makes 
this Remark upon it ; In folve?ido hoc Pro- 
blemate aherrat Merfemms : And he, work- 
ing by :ihe Logarithms, finds out but 3^5, 
and a little more ; and from thence has de* 
duced an ingenious Invention of finding 
and applying a leaft Common Meafure to 
all Harmonic Intervals, not precifely per- 
feft, but very near it. 

Supposing a Comma to be part of 
Diapafon; for better Accommodation ra- 
ther than according to the true Partition 
ITT, which T3 he calls an Artificial Comma, 
not exafl:, but differing from the true Na- 

G rural 


8o Of Proportion. 

tural Comma about tv part of a Comma, 
and rvV-v of Diapafon ( which is a Diffe- 
rence imperceptible ) then the Intervals 
within Diapafon will be meafur’d by Com- 
ma’s according to the following Table ; 
which you may prove by adding tv/o, 'or 
three, or more of thefe Numbers of Com- 
ma’s, to fee how they agree to conftitute 
thofe Intervals, which they ought to make ; 
and the like by fubftrafting. 


Intervals 

o 

TT 

Intervals 

e 

TT 

Comma 

I 

^th 

22 

Diefts 

2 

Tritone 

26 

^emit. Mhim 

O 

D 

Semidiaj^ente 

27 

Semit. Medim}i 

4 

jth 

31 

Semite Majus 

5 

6^^ Minor 

36 

Semite Maximm 

6 

(T>th ]\/jTj.or 

39 

Tone Minor 

8 

Minor 

45 

Tone Major 

9 

17 

Major 

48 

3^ Minor 
3^ Major 

Ociave 

53 


This I thought fit, on this Occafion, 
to impart to the Reader, having Leave fo 
to do from Mv* Mercator s Friend, to whom 
he prefented the faid Manufeript. 

H E R E I may advertife the Reader, that 
it is indifferent whether you compare the 

greater 


of Proportion. 8 i 

S^reater Term ot an Harmonic Ration to 
the Ids, or the lets to the greater ; L e. 
whether of them you place as Antecedent, 

€. gr, 3 to 2, or x to 3 ; becaiife in Har-^ 
monies tlie Proportions of Lengths ot 
Chords, and of their Vibrations, are reci- 
procal or counter-changed : As the Length 
is encreas’d, fo the Vibrations are in the 
fame proportion decreas’d ; e co 7 itr{:;. 

If therefore (as in Diafe?ite) the length of 
the Unifon String be 3, then the length 
(^cateris farihis) of the String, which in 
afcent makes D 2 aj 7 e 7 ite to that Unifon muft 

be 2, or — : Thus the Ration of Dia-ieriie 

is 2 to 3 in refpcf!: of the. Length of it, 
compar’d to the Length of the Unifon 
String. 

Again, the String 2 vibrates thrice in 
the fame Time that the String 3 vibrates . 
twice ; and thus the Ration of Diafeiite^ in 
refpeft of Vibrations, is 3 to 2 : So that 
where you find in Authors fometimes the 
greater Number in the Rations fet before 
and made the Antecedent, fometimes fet 
after and made the Coniequent, you muft 
underhand in the former, the Ration of 
their Vibrations ; and in the latter, the 
Ration of their Lengths ; which comes all 
to one. 

G X Or^ 


Of Proportion. 

Or, you may underftand the Unifon to 
be compar’d to Diapeftte above it, and 
the Ration of Lengths is'3 to 2, of Vibra- 
tions 2 to 3, or elfe Diapente to be com- 
par’d to the Unifon, and then the Ration 

Lengths is 2 to 3, of Vibrations 3 to 2. 
^ his is true in fingle Rations, or if one 
Ration be compar’d to another ; then the 
two greater Terms rnuft be rank’d as An- 
tecedents ; or otherwife, the two leffer 
Terms. 

The Difference between Arithmetical 
and Geometrical Proportion is to be well 
lieeded. An Arithmetical mean Proportion 
is that which/ has equal Difference to the 
Antecedent and Coniequent Terms of thofe 
Numbers to which it is the Mediety, and 
is found by adding the Terms, and taking 
half the Sum. Thus between 9 and i, 
which added together make 10, the Me- 
diety is 5; being Ecjuidifferent fiom 9 and 
from 1 •, which Difference is 4 ; As 5 ex- 
ceeds I by 4*, fo likewife 9 exceeds 3 by 4. 
And thus in Arithmetical Progreffion 2, 4, 
6, 8*, where the Difference is only conli- 
der’d, there is the fame Arithmetical Pro- 
portion between 2 and 4, 4 and 6, 6 and 8; 
and between 2 and d, and 4 and 8 : But in 
Geometrical Proportion, where is confider’d 
not tlie Numerical Difference, but another 
Habitude of the Terms, viz- how many 

times, 


Of Proportion. 8} 

times, or how much of a time or times, 
one of them doth contain the other ( as 
hath been explain’d at in the begin- 
ning of this Chapter.) There the Mean 
Proportional is not the fame with Arithme- 
tical, but found another way *, and equidif- 
ferent ProgrefGons make different Rations. 
The Rations (taking them all in their lead 
Terms) exprefs’d by lefs Numbers, being 
greater than thofe of greater Numbers, I 
mean in Proportions Jufer farticuU.r^ &c. 
where the Antecedents are greater than the 
Confequents, (as, on the contrary, where 
the Antecedents arc lefs than the Confe- 
quents,’ the Ratio's of lefs Numbers are 
lefs than Rat id's of greater.) The Me- 

diety of p to i is not now y, but 3 ; 3 ha- 
ving the fame Ration to i as p has to 3, 
(as p to 3, fo 3 to i) viz^ triple. And fo 
in ProgrelTion Arithmetical, of Terms ha- 
ving the fame Differences ; if confider’d 
Geometrically, the Terms will all be com- 
prehended by unequal Rations. The Dif- 
terences of 2 to 4, 4 to 6, 6 to 8, are equal, 
but the Rations are unequal ; 2 to 4 is lefs 
than 4 to ( 5 , and 4 to 6 lefs than 6 to 8. 
As on the contrary, 4 to 2 is grater than 
6 to 4, and 6 to 4 greater than 8 to 6: 
For 4 to 2 is duple, 6 to 4 but Sefpiialtera 
(one and a half only, or 4-) and 8 to 6 is 
no more than Sefquitertia^ (one and a third 

G 3 part, 


§4 Of Proportion. 

part, or y) which fliews a confiderable la- 
equality of their Rations. In like manner 
6 to 2 is triple ; 8 to 4 is but duple, and 
yet their DiiFerences equal. Thus the 
mean Rations comprehended in any grea- 
ter Ration divided Arithmetically, /. e. by 
equal Differences, are unequal to one ano- 
ther, conliderM Geometrically. Thus 2,3, 
4, 5, 6, if you confider the Numbers, make 
an Arithmetical Progreffion : But if you 
confider the Rations of thofe Numbers, as 
is done in Harmony, then they are unequal, 
every one being greater or lefs (according 
as you proceed by Afcent or Defcent) than 
the next to it. Thus, in this Progreffion, 
(underftanding, together with the Ratio’s, 
the Intervals themfelves, as is before pre- 
mifed ) 2 to 3 is the greateft, being Dia- 
fente ; 3 to 4 the next, Diateffaron ; 4 to 5 
ftill Ids, viz. Ditone \ 5 to 6 the leaft, be- 
ing SefrpntGne. Or, if you defeend, 6 to 5 
lead: ; 5 to 4 next,gj)V. Thefe are the mean 
Rations comprehended in the Ration ot 
6 to 2, by which Diafafon cum Diapente^ 
ora divided into the aforefaid In- 

tervals, and meafured by them, viz. as 
is 6 to 2, {liz. triple) fo is the Aggregate 
of all the mean Rations within that Num- 
ber, 6 to 5, 5 to 4, 4 to 3, and 3 fo 2 : Or 
6 to 5, 5 to 2 ; or 6 to 4, 4 to x *, or 6 to 3, 
3 to 2. I'he Aggregates of thele are equal 
to (5 to 2, ^v:::. tfipie. 1 HiS 


of Proportion. 



Th I s is premifed in order to proceed 
to what was intimated in the Yoregoing 
Chapter. 

Taking notice firft of this Procedure, 
peculiar to Harmonics, viz. to make Pro- 
greflion or Divifion in Arithmetical Propor- 
tion in refpeft of the Numbers ; bur to 
confider the things number’d according to 
their Rations Geometrical. And thus Har- 
monic Proportion is faid to be compounded 
of Arithmetical and Geometrical. 

You may find them all in the Divifion 
of the Syrtem of DiapaJo?i into T^iafente 
and Diatejjaron^ i, e. 5^^ and 4^^^, afcending 
from- the (jnifon. 

If by Diapente firft, then by 2, 3, 4, 
Arithmetically. If firft hy DiatejJ'aron^thm 
by 3, 4, 6, Harmonically. And thefe Ra- 
tions confider’d Geometrically, in relation 
to Sound, there is like wife found Geome- 
trical Proportions between the Numbers 

3 6,4 to 3, 2. 

The Ancients therefore owning only 
gth, jth, ^th, Pqj^. fiixiple Confonant Inter- 
vals, concluded them all within the Num- 
bers ot 12, 9, 8, 6, which contain’d them 

G 4 all, 


of Proportion. 

all : viz, 12 to 6, Diapafon ; ii to 8, Dia- 
pente\ 12 to 9, Diatejjaron] 9 to 8, Tone. 
And which fervM to exprefs the three kinds 
of Proportion, viz. Harmonical, between 
12 to 8, and 8 to d ; Arithmetical, between 
12 to. 9, and 9 to 6; and Geometrical, be- 
tween 12 to 9 and 8 to 6 ; and between 
12 to 8, and 9 to 6 . It was faid therefore, 
that MercuriuTs Lyre was ilrung with four 
Chords, having thofe Proportions, 6, 8, 9,1 2. 
Gaj]'e 7 id. 

I intimated, that 1 would here more 
largely explain that ready and eafie Way 
of finding and meafuring the mean Ra- 
tions contain’d in any of thofe Harmonic 
Rations given, by transferring them out 
of their Prime or Radical Numbers into 
greater Numbers of the fame Ration. By 
dupling (not the Ration, but the Terms of 
it; ftill continuing the fame Ration) you 
will have one Mediety ; as, 2 to i dupled 
is 4 to z ; and you have 3 the Mediety. 
By tripling you will have two Means ; 
2 to I tj-ipled is 6 to 3, containing 3 Ra- 
tions ; 6 to 5, 5 to 4, 4 to 3 ; and to ftill 
more, the more you multiply it. 

N o w obferve, firft, that any Ration 
M^ikiplex or Snperpartient foi' by tranU 
ferrinn; it out of its Radical Numbers made 

like 


of Proportion. 87 

like Suferfartient') contains fo 
f articular Rations, as there are Units in 
the Difference between the Antecedent 
and the Confequent. Thus in 8 to 4 
(being 2 to i transferr’d by quadrupling) 
the Difference is 4, and it contains 4 Sti- 
perf articular Rations, viz. 8 to 7, 7 to 6, 

6 to 5, and 5 to 4 ; where tho’ the Pro- 
greflion of Numbers is Arithmetical, yet 
the Proportions of Excefs are Geometrical 
and Unequal. The Sup erf articular Ra- 
tions exprefsM by lefs Numbers being grea- 
ter (as hath been faid) than thofe that con- 
fill of greater Numbers; 5 to 4 is a greater 
Ration than 6 to 5, and 6 to 5 greater 
than 7 to 6, and 7 to 6 than 8 to y, as a 
Fourth part is greater than a Fifth, and a 
Fifth greater than a Sixth, ^c. But ia 
this Inlfance there are two Rations not ap- 
pertaining to Harmonics, viz. 8 to 7, and 

7 to 6 . 

Secondly therefore, you may mate un- 
equal Steps, and take none but Harmonic 
Rations, by felefting greater and fewer 
intermediate Rations, tho’ fome of them 
compos’d of feveral Superparticulars’'^ pro- 
vided you do not difeontinue the Rational 
ProgrelTion, but that you repeat Ifill the 
laff Confequent, making it the next Ante- 
cedent ; as if you meaiure the Ration of 

8 to 4 


88 Of Proportion. 

8 to 4, by 8 to 6 and 6 to 4, or by 8 to 5 
and 5 to 4, or’ by 8 to 6, and 5 to 5, and 

5 to 4 ; in thefe three ways the Rations 
are all Harmonical, and are refpeftively 
contain’d in, and make up the Ration of 
8 to 4. Thus you may meafure, and di- 
vide, and compound moll harmonic Rations 
without your Pen. 

T o that End I would have my Reader 
to be very perfeQ: in the Radical Numbers 
which exp refs the Rations of the feven firft 
(or uncompounded) Confonants, viz. Dia- 
fafon^ 2 to I ; ""Diafente^ 3 to 2 ; DiatejJ'a- 
ron^ 4 to 3 ; Ditofie^ 5 to 4 ; Triheniitoney 

6 to 5 ; Hexachordon Ma]ws^ 5* to 3 ; Hexa- 

chordo 7 i Minuys^ 8 to 5 ; and likewife of the 
Degrees in Diatonick Harmony, viz. Tone 
Major ^ 9 to 8 ; Tone Mhior^ 10 to 9 ; He- 
rnitone Major ^ 1 6 to i 5 ; and the Ditferen- 
CCS of thole Degrees ; Hemito 7 ie Greatefi^ 
27 to 25 ; He rnitone Mi 7 ior^ 25 to 24 ; 

Comma^ox Schijm^ 81 to 80 •, Die/is Enhar- 
monic ^ ii8 to 125. 

O F other Hemitones I fliall treat in the 
Eighth Chapter. 

N.o w if you would divide any of the 
Confonants into two Parts, you may do it 
by the Mean or Mediety of the twoRadi^ 

cal 


/ 


Of Proportion. 89 

cal Numbers, if they have a Mean ; and 
where they have not, (as when their 
tio^s are Suf erf articular) you need but 
duple thofe Numbers, and you will have 
a Mean (one or more.) Thus duple the 
Numbers of the Ration of DiafafoUy 2 to i, 
and you have 4 to 2 ; and then 3 is the 
Mean By which it is divided into two un- 
equal, but proper and harmonical parts, 
viz. 4 to 3, and 3 to 2. After this man- 
ner Diafafon^ 4 to 2, comprehends 4 to 3, 
and 3 to 2 : So 'Diafente^ 6 to 4, is 6 to 
and y to 4: TDitone^ 10 to 8, is 10 to 9, 
and 9 to 8 ; fo Sixth major ^ 5 to 3, is 5 to 4, 
and 4 to 3. 

T H o’, from what was now obferv’d, 
you may divide any of the Confonants into 
intermediate Parts, yet when you divide 
thefe three following, viz. Sixth minor^ 
T)iateJJaroUy and Trihemitone^ you will find 
that thofe Parts into which they are divi- 
ded, are not all fucli Intervals as are har- 
monical. The Sixth minor^ whofe Ration 
is 8 to 5*, contains in it three Means, viz. 
8 to 7, 7 to 6, and 6 to 5 5 the laft where- 
of only is one of the harmonick Intervals, 
of which the Sixth minor confilfs, viz. Tri~ 
hemitone ; and to make up the other Inter- 
val, viz, DiateJJaron^ you muft take the 
other two, 8 to 7, and 7 to 6 ^ which be- 


90 Of Proportion. 

ing added (or, which is the fame thing, 
taking tlie Ratio oi their two Extream 
Terms, ^ that being the Sum of all the in- 
termediate ones added ) you have 8 to 6, 
^ or (in the leaft Terms) 4 to 3. Again, 
Diateffaron^ in Radical Numbers 4 to 3; 
being (if thofe Numbers are dupled) 8 to 6, 
gives for his Parts 8 to 7, and 7 to 6 ; which 
Rations agree with no Intervals that are 
Harmonick ; therefore you muft take the 
Ration of Diatejjaron in other Terms, which 
may afford fuch Harmonick Parts. And to 
do this, you muft proceed farther than 
dupling (or adding it once to it felf ) for 
to this Duple you muft add the firft Radi- 
cal Numbers once again (which in effeft 
is the fame with tripling it at firft ) viz. 
4 and 3, to 8 and 6 ; and the Aggregate 
will be a new, but equivalent, Ration of 
Diatejjaron-, viz. 12 to And this gives 
you three Means, ix to ii, and ii to 10; 
both Unha rmonical ; but which together 
are, as was ftiewM before, the fame with 
3 2 to 10 ( or 6 to 5 ) Trihemitone ; and 
3 o to 9 Tone minor ; and are the two Har- 
monical Intervals of which Diatejjaron con- 
fifts, and which divide it into the two 
neareft equal Harmonick Parts. Laffly, 
Trihemitone, ov T\md mmor, 6 to 5-, or 
^hofe Numbers being dupled) 12 to 10, 
ttWes 12 to II, and 11 to 10, which are 
^ Un- 


of Proportion. pi 

Unharmonlcal Radons ; but tripled (after 
the former manner^ 6 to 5 gives 18 to 15*, 
which divides it felf (as before) into 1 8 to 
16, Tone 7najor\ and 16 to 15, He7nitone 
major* 

Thus, by a little Praftice, all Harmo- 
nick Intervals will be moft eafily meafur’d, 
by the lelTer Intervals compriz’d in them. 
Now, for Exercife fake, take the Meafures 
of a greater Ration : Snppofe that of 16 to 3 
be given as an Harmonick Syilem. To find 
what it is, and of what Parts it confifts; 
firft find the grofs Parts, and then the more 
minute. You will prefently judge, that 16 
to 8 (being a Part of this Ration) is T)/a~ 
fajon\ and 8 to 4 is likewife Diafafon: 
'Phen 16 to 4 is Dljdtafafon^ or a Fifteenth, 
and the remaining 4 to 3 is a Fourth. So 
then 16 to 3 is 'Di/d^-pafon and D/atejJh- 
ron ; /. e, an Eighteenth ; 16 to 8, 8 to" 4, 
and 4 to 3. 


9 2 of Proportion. 

But, to find all the Hatmonick Intervals . 
within that Ration ( for we now confider 
Rations as relating to Harmony) take this 
Scheme. , 

\ 

' 1 6 to 3 contains. 


In Radicals. 


|l6 to 15, 
15 to 12, 

5 to 4, 

HemitoJie. 

Ditone. 


12 to 10, 

6 to 5, 

Trihemito?ie. 


10 to 
9 to 85 
8 to 6, 

4 to 3, 

To7ie Minor- 
To?ie Major. 
DiateJJ'aron. 


6 to 5, 
5 to 4, 
4 to 3 ) 


Trihetnito77e. 

Ditofte. 

DiateJJdron. 


Tot. 16 to 3 

Difdiafafon cum DiateJJ'aron. 


Or thus, 
In Radicals. 


16 to TO, 

8 to 5, 

6^^ Minor. 

10 to 65 

5 to 3, 

6^^^^ Major. 

6 to 4, 

3 to 2, 


4 to 3, 


4th 

Tot. 16 to ^ 

Eijihteenth. 


All 


of Proportion. 95 

All thefe Intervals thus put together 
are comprehended in> and make up, the 
Ration of 1 6 to 3 , being taken in a conjuna 
Series of Rations. 


But otherwPfe, within this compafs of 
Numbers are contain’d many more Expref- 
fions of Harmonick Ration. Ex> gr* 


Radicals^ Radicals* 


1 

to 

15, 




12 

to 


2 

to 

I. 

1 

to 


4 

to 

3- 

12 

to 

4, 

3 

to 

I. 

1 16 

to 

10, 

8 

to 

5- 

12 

to 

3, 

4 

to 

I. 

1 

to 

8, 

z 

to 

I. 

10 

to 





1 

to 


8 

to 

3* 

10 

to 

8, 

5 

to 

4- 

II 6 

to 

4, 

4 

to 

I. 

10 

to 


5 

to 

3* 

116 

to 

3- 




10 

to 

5, 

X 

to 

I. 

15 

to 

12, 

s 

to 

4 . 

9 

to 

8, 




15 

to 

10, 

3 

to 

2 . 

9 

to 


3 

to 

2 . 

15 

to 

5, 

3 

to 

I. 

9 

to 

3, 

3 

to 

I. 

15 

to 

3, 

5 

to 

I. 

8 

to 

6, 

4 

to 

3- 

H 

to 

7) 

2 

to 

I. 

8 

to 





1 12 

to 

10, 

6 

to 

J- 

8 

to 

4> 

2 

to 

I. 

112 

to 

9, 

4 

to 

3- 

6 

to 




112 

to 

8, 

3 

to 

iJ 

Vid. Pag. 67 . 




And DOW I fuppofe the Reader better 
prepar’d to proceed in the remainder of this 
Difeourfe, where we lEall treat of Dif cords. 


CHAP. 


94 


Of Vif cords and Degrees. 


CHAP. VI. 


Of Difcords and Degrees. 


L L Habitudes of one Chord to ano- 


ther, that are not Concords^ (fuch as 
are before defcrib’d ) are Difcords ; which 
are or may be innumerable, as are the mi- ' 
nute Tenfions by which one Chord may 
be made to vary from it felf, or from ano- 
ther. But here we are to confider only 
fuch Difcords as are ufeful ( and in truth 
neceflary} to Har^yiony^ or at leaft relating 
to it, as arc the Differences found between 
Harniofiich Intervals. 

And thefe apt and ufeful Difcords are 
either fimple uncompounded Intervals, fuch 
as immediately follow one another, afcen- 
ding or defcending in the Scale of Mufick; 
as, Ut^ Re^ Mi^ Fa^ SoJ^ La, Fa, Sol, and are 
call’d Degrees : Or el(e greater Spaces or 
Intervals compounded of Degrees inclu- 
ding or skipping over fome of them, as all 
the Concords do, Ut Mi, Ut Fa,^ Ut Sol, 8cc. 
And fucii are the Difcords of which we 



now 


of Dif cords and Degrees ^ 9 5 

now treat, as principally tlie Tritone^ Falf^ 
Fifth, and the two Sevenths, Major and 
Minor^ if they be not ratlier among the 
Degrees, gjfc. For more Perfpicuity, I 
lhall treat of them feverally, viz- of De- 
grees^ oi Difcords^ and oi 'Differences. 


And firft of Decrees. 

Degrees are uncompounded Inter- 
vals, which are found upon eight Chords^ 
and in feven Spaces, by which an imme- 
diate Afcent or Defcent is made from the 
Unifon to the Ociave or Diapajon'^ and 
by the fame Progreflion to as many Otiaves 
as there may be Occafion. Theie are dif- 
ferent, according to the ditferent. Kinds of 
Mulic, VIZ. Enharmonic^ Chromatic^ and 
Diatonic^ and the feveral Colours of the 
two latter : ( all which I fliall more con- 
veniently explain by and by); but of 
thefe now mention’d, the Diatonic is the 
moll proper and natural Way : The other 
two, if for Curiofities fake we confider 
them only by running the Notes of an 
Otiave up or down in thefe Scales, feem 
rather a Force upon Nature ; yet here- 
in probably might lie the Excellency 
ot the ancient Greeks : But we now 

ufe only the Diatonic Kind , intermix- 
ing here and there fome^ of the Chro- 

H matic^ 


9 6 Of pif cords and Decrees. 

maticy (and more rarely fome of the 
harmonic:^ And our Excellency feems to 
lie in moft artificial Compofing, and join- 
ing feveral Parts in Symphony or Conlbrt ; 
which they cannot be fuppos’d to have ef- 
fected, at leaft in fo many Parts as we or- 
dinarily make, becaufe ( as is generally 
affirm’d of themj they own’d no Concords 
befides Eighth, Fifth, and Fourth, and the 
Compounds of thefe. 

E. Kjrcher (cited alfo by Gajjendm 
out any Mark of DilTent ) is of Opinion, 
that the ancient Greeks never ufed Con- 
fort Mufic, e. of different Parts at once, 
but only Solitary, for one fingle Voice or 
Inftrument ; and, that Guido Aretinm firft 
invented and brought in Mufic of Sym- 
phony or Confort, both for the one and 
the other. They apply’d Inftruments to 
Voice, but how they were managed, he 
muft be wifer than I that can tell. 

This Way of theirs feems to be more 
proper (by the elaborate Curiofity and 
Nicety of Contrivance of Degrees, and by 
Meafures rather than by harmonious Con- 
fonancy, and by long-ftudied Performance) 
to make great Impreffions upon the Fan- 
cy, and operate accordingly, as fome Hifto- 
ries relate : Ours more fedately affefts the 

Un- 


of Di/cords and ^egrees. 97 

Underflanding and Judgment, from the 
judicious Contrivance and happy Compo- 
(ition of Melodious Confoit. The One 
quietly, but powerfully, atfeffs the Intel- 
left by true Harmony ; the Other, chiefly 
by the Rythnm^ violently attacks and hur- 
ries the Imagination. In fine, upon the 
natural Grounds of Harmony (of which I 
have hitherto been treating) is founded 
the Diatonic Mufic ; but not fo, or not fa 
regularly, the Chromatic or Enharmonic 
Kinds. Take this following View of 
them. 

The Ancients afcended from the Z7;//- 
[on to an OBave by two Syftemes of Te- 
trachords or Fourths. Thefe were either 
Conjunft, when they began the Second 
Tetrachord at the Fourth Chord, viz- with 
the lafl: Note of the firil Tetrachord, and 
which being fo join’d, conftituted but a 
Seventh ; and therefore they affumed a 
Eofie beneath the Unijon (which they there- 
fore call’d Eroflamlanome7ws) to make a fulj 
Eighth. 

O R elfe the two Tetrachords were clif. 
junft, the Second taking its beginning at 
the Filth Chord, there being always a 
Major between the Fourth and Fifth Chords. 
So the Degrees were immediately apply’d 

H 2 |:a 


9 8 Of t>lJcords and Degrees. 

to the Fourths, and by them to the Offave ; 
and were different according to the diffe- 
rent Kinds of Mufic. In the common Dia~ 
tonic Genurs the Degrees were Tone and 
Se?nitone ; Intervals more Equal and Eafy, 
and Natural. In the common Chromatic^ 
where the Degrees were Hemitones and 
Trihe^nitones.^ the Difference of fome of the 
Intervals was greater : But the greateft 
Difference, and confequently difficulty, was 
ill the Enhar?no?iic Kind, ufing only Diefis^ 
or quarter of a To7ie^ and Ditone., as the 
Degrees whereby they made the Tetra- 
chord. 

And to conftitute thefe Degrees, fome 
of them, viz. th^ Followers of Arifioxennsy 
divided a Tone Major into Twelve equal 
Parts, e. fuppofed it fo divided *. Six of 
which being the Hemitone, (inz. half of it) 
made a Degree oi Chro?natic Toni^um ; and 
Three of them, or a quarter, call’d Diejis^ 
a Degree Enharmo7iic. The Chromatic 
Fourth rofe thus, viz. from the firft Chord 
to the fecond was a Hemitone ; from the 
fecond to the third, a Hemitone \ from the 
Third to the Fourth, a Trihetnitone ; or as 
much as would make up a juft Fourth. 
And this laft Space ( in this cafe ) was ac- 
counted as well as either of the other, but 
one Degree or undivided Interval. And 


of D if cords and Degrees- pp 

they caird them Sp//s Intervals [ ] 

when two of thofe other Degrees put to- 
gether, made not fo great an Interval as 
one of thefe ; as, in the Enharmo?ik Tetra- 
chord, two Diefes were lefs than the re- 
maining Ditone ^ and in the common CV^r<?- 
matic^ two Hemito?ie Degrees were left 
than the remaining Trihemitone Degree. 

Then for the Enharmonic Fourth, the 
fil'd Degree was a Diejis^ or quarter of a 
Tone ; the fecond alfo Three of thofe 
Twelve Parts, viz. a Diejis ; the third a 
Ditone^ fuch as made up a juft Fourth. 
And this Ditone ( tho’ fo large a Degree ) 
being confider’d as thus placed (in the 
Enhar?nonic 'Fetrachord) was likewife to 
them but as one uncompounded or entire 
Interval. 

These were the Degrees Chromatic 
and Enharmo?iic : Tho’ they alfo might be 
placed otherwife, i. e. the greater Degree 
in thefe may change its place, as the Hemi- 
tone (the leis Degree) doth in the Diatonic 
Ge7iiis ; and from this Change chiefly arofe 
the feveral Moods, Dorian^ Lydian^ &c. 
From all which, their Mufic no doubt 
(tip’ it be hard to us to conceive) muflr 
aftord extraordinary Delight and Pleafure, 
n It djcl bear but a reafonable Proportion 

H 3 to 


9871 


lOO 


of i)tJcords and Degrees. 

to their infinite Curiofity and Labour. And 
as we may fuppofc it to have differM very 
much from that which now is, and for fe- 
veral Ages hath been ufed ; fo confequently 
we may look upon it as in a manner loft 
to us. 

I N profecution of my Defign, I am on- 
ly, or chiefly, to infift on the x other Kind 
of Degrees, which are moft proper to the 
Natural Explanation of Harmony, viz. the 
Degrees Diatomc y which are fo call’d, not 
becaufe they are all To^/es yhut becaufe moft 
of ’em, as many as can be, are fuch ; viz. 
in every Diafafon five Tones and two He- 
mito7ies. Upon thefe, I fay, I am to infift, 
as being, of thofe before mention’d, the 
moft Natural and RationaU 

Diwflion. 

But before we proceed, it may perhaps 
be a Satisfaftion to the Reader, after what 
has been faid, to have a little better Pro- 
fpeft of the ancient Greek Mufic, by fome 
general Account ; not of their whole Do- 
ftrine, but of that which relates to our 
prefent Subjefl:, viz. their Degrees, and 
Scales of Harmony, and Notes. 


E X R S T 


of Dif cords and Decrees. i o l 

First then, take out of Euclid the 
Degrees according to the three Genera \ 
which were, Enharmonic^ Chromatic^ and 
Diatonic:, which Kinds have fix Colours 
(as they call’d them). Er^clid^ Introd, Harm. 
pag. lo. 

The Enharmonic Kind had but one 
Colour, which made up its Tetrachord by 
thefe Intervals ; a Diejis (or quarter of a 
To?ie) then fuch another Diefis^ and alfo a 
Ditone incompofit. 

The Chromatic had three Colours, by 
which it was divided into MoJIe^ Sefcuflunjy 
and Toni mm. 

Molle^ in which the Tetrachord rofe 
by a Triental D.iefis (four of thofe twelve 
Parts mention’d before) or third part of a 
To72C ; and another fuch Diejis ; and an in- 
compofit Interval, containing a Tone and 
half, and third part of a Tone : And it was 
call’d Mollcj becaufe it hath the lead, and 
confequently moll enervated Spfs Intervals 
within the Chromatic Genm. 

2^^ Sefcuflu?n^ by a Hiefis which is^S^- 
quialtera to the Enharmonic Diefis^ and 
another fuch Diefis^ and an Incompofit In- 
terval of feven Diefes Quadrantal, viz. each 
being three Duodecimals of d,To 7 /e. 

H 4 


102 Of Dljcords and Degrees. 

Tonimm^ by a Hemito?te^ and Hemi- 
tone and Trihemitone ; and is call’d Toni^um 
becaufe the two vS/?;y} Intervals make ^.Tone. 
And this is the ordinary Chromatic. 

The Diatonick had two Colours ; it 
was Mode and Syntonum. 

Molle\ by a Hemito7ie^ and an Incom- 
pofit Interval of three Quadrantal ^Diefes 
and an Interval of five fuch Diefes. 

2 *^’ Syntonmn^ by a Hemitone and a Tone^ 
and a "tone. And this is the common Dia- 
'tG7lic. 

■0 

T o underflrand this better, I muft re- 
affume fomewhat which I mention’d, but 
not fully enough before. A To7ie is fup- 
pos’d to be divided into twelve leafi: parts, 
and therefore a Hanitone contains fix of 
thofe Duodecimal (or twelfth) parts of a 
Do7te ; a DieJisTrie?italis 'Diefis Q7{adra7i- 
tails 3 , the whole T)iatejJaron 30 . And the 
TDiateJJdro7i in each of the three Kinds wa.s 
made and perform’d upon four Chords, 
having three mean Intervals of Degrees, 
according to the following Numbers and 
Proportions of thofe thirty Duodecimal 
'parts, 

'I 

* . 


E7iharr. 




Enhamonky 


Chroinaticy 


Diatonic, 


by 3, and 3, and 24 
Molk, by 4, and 4, and 22. 
Hemiolion, '1 

or > by 4I5 and 4I, and 2 1 . 
Sefcuplum, 3 ' 

Toniaum, by 6 , and 6 , and 18, 
Mol/e, by 6 , and p, and 1 5 . 

Syntonum, by 6 , and 12, and 12, 


T o each of thefe Kinds, and the Moods 
of them, they fitted a perfefl: Syftem or 
Scale of Degrees to T>ifdiapafon ; as in the 
following Example taken out of Nichoma- 
chuo ; to which I have prefix’d our modern 
Letters. 


£. Nichomacho^ pag. 


A 

G 

F 

E 

D 

C 

B 


Nete HyferloJ^eon. 

Paranefe Hyper-1 ^ 

hoUon.- ^Enbam.Chro. Diat. 

Trite HyferhoUon. Enh, Chro, Diat. 
Nete T)iezeugme^ 


non. 

Paranete Diezeug- 
menon. 

Trite Diezeugme- 

noTi. 

Parajuefe* 



Enh. Chro. TDiat. 
Enb. Chro. Diat. 


D 1 Nete 


1 04 Of Difcords and Dep’ees. 


Nete Synemmenon. 

Taranete Synem-1 „ t ^7 r.- 

menon, ^ E/w» Chro> Dtat^ 

Trite Synemme7ion, 

Mefe. 

Lichanos Mefon. 

Paryfate Mefo7/, 

Hyp ate Mefon. 

Lychanos Hypaton. 

Parypate Hypaton. 


Enh* Chro. T)iat. 

Enh. Chro, Diat. 
Enh. Chro. Diat. 

Enh. Chro. Diat. 
Enh. Chro. Diat. 


A I Trojlamhafiomems. 


In this Scale of Difdiapafon you fee the 
Mefe is an OHave below the Nete Hyfer- 
lolaon, and an OEiave above the Projlarn^ 
lanomenos : And the Lichanos^ Parypate, 
Parenete^ and Trite^ are changeable ^ as 
upon our Inftruments are the Seconds, and 
Thirds, and Sixths, and , Sevenths : The j 
Proflambanomenos, Hypate^ Hefe, Para7nefe, i 
and Nete, are immutable ; as are the Uni- 
fon, Fourths, Fifths, and Oftaves. 

Now from the feveral Changes of thefe 
Mutable Chords chiefly arife the feveral 
Moods (fome call’d them Tones) of Mu- 
fic, of which Euclid fets down Thirteen ; 
to which were joyn’d two more, viz. Hy- 

peraolian^ - 


of Dif cords and Degrees. i o 5 

'per ^zoUan zxiA Hyf^rlycliau , and afterwards 
Six more were added. 

I fhall give you, for a Tafte, EuclicTs 
Thirteen Moods. 

Euclid- p. 19. 

HyfermixoJydms^ five Hyperphrygim. 
Mixolydim acutior.^ five Hyperiaftim. ■ 
Mixoly dim gravior^ fi v e Hyper dori m . 

Lydim acutior. 

Lydias gravi or ^ five /Eolim. 

Phrygias acutior- 

Phrygius graviory five I aft i us.. 

Dorius. ^ 

Hypolydius acutior. 

Hypolydius gravior^ five Hypoxolius. 
Hypophrygius acutior. 

Hypophrygius gravior^ five Hypoiaftius. 

Hyp odor ius, 

O F thefe the moft grave, or loweft, wa 3 
the Hyperdorian Mood, the Profla?nbano- 
7ue?ios whereof was fix’d upon the loweft 
clear and firm Note, of the Voice or In- 
ftrument that was fuppos’d to be of the 
deepeft fettled Pitch in Nature, and adap- 
ted freely to exprefs it : And then all-along 
from Grave to Acute the Moods took their 
Afcent by Hemitonesy each Mood being a 

Hemi.. 


1 o (5 Of Vif cords and Degrees] 

Hemitone higher or more acute than the 
next under it. So that the TroJlamha?io- 
me no:: of the HyfermixoJydian Mood was 
juft an Eighth higher than that of the Hy- 
fodorian^ and the reft accordingly. 

Now each particular Chord in the pre- 
ceeding Scale had two Signs or Notes 
[ a^iiCicL ] by which it was charafterizM or 
defcrib’d in every one of thefe Moods re- 
fpeftively ; and alfo for all the Moods in 
the feveral Kinds of Mufic ; Enharmonic^ 
Chrofnatic^ and Diatonic ; of which two 
Notes, the upper was for reading [ hkln ] 
the lower for percuftion [ xpW ] one for 
the Voice, the other for the Hand. Con- 
fider then how many Notes they ufed ; 
,i8 Chords feverally for 13 Moods (or ra- 
ther 15, taking in the Hy{:er^eoIian and 
Hyferlydian^ which are all defer ib’d by 
JlypiusJ and thefe fuited to the three 
Kinds of Mufic. So many Notes, and fo 
appropriated, had the Scholar then to learn 
and conn who ftudied Mufic. Of thefe I 
will give you in part a View out of J- 
Jypins. 






Notes 




Of Difcords and Degrees'. 107 

Notes of the Lydia7t Mood In the 
'Diato?iic Ge?ius, 

7.n.B.$.C.P.M,I.© 

^.r.L.F.c.u,r] .<.v. 

.12 3 4 5 ‘6 7 .8 9 

r.ir.z.E.ir.-e-. x.m.i. 

10 II 12 13 14 15 16 17 18 

rr> n 7 T 76'/-<^itnperfeQ:,and 

I Troflamlanomenos. | 

yfa e jfa ^ ^ Gamma right. 

^ 4. zj ^ 4. f ^eta iraperfe£l:,anci 

3 ^aryfateHyfaton.i^ G~invetWd. 

4 HypatonDiato7ios, Phz\ and Digamma. 

5 Hyfate Mefoti^ Sigma^ and Sigma. 

6 TarypaU MefoTf.-- Rho^Sc Sigma invtrttd. 
Mefon Diatoms. — My^ and P/ drawn out. 
Me[e. — — M^,andL^w^^jacent 


$ Trite 


I o8 Of DiJ cords and Decrees * 

9 Trit^ Synemmenon, Lamlda 

C inverted. 

10 Syfiemmenon Diato- Gannna^ and Ny, 
nos. 

C n fquared, lying fu- 

I I JNctc Syn&jfi’ificnon.'^ pine upwards^ 

L, and 

12 ^aramefe. and P/ jacent. 

'J S Trite D^ezeu^menorA^ fquared, and P/ 

14 DiezengmenonDia-^Sifqm.VQd^ fupine, 

^oms. X and Xfta. 

' , jacent, and a 

1 5 Nete Diezeugmenon,^ carelefs Eta («) 

drawn out. 


K ^ looking down, 
t. y left half, 


16 Trite Hyperlolrcon. 

^ looking upward. 

17 HyperloJaon 

tonos. ■> an A- 

C cute above. 

Clotay and Lamhcla 

'ytZ .Ndte HyperhoJao?/. < jacent, with an 

t. Acute above. 


The Numeral Figures I have added under the Signs (or 
Alarks) only for Reference to the Names of the Notes 
fignified by tlwm, to fave deferibing them twice. 


Notes 


Of Difcords and Degrees. 1 09 

Notes of the MoUan Mood in the 
Diatonic Genus. 

li.X-a.x.T.c.o.K. I. 

1x345678 9 


Z . A.H.Z. A3K .-eiOJC. 

10 II 12 13 14 15 16 17 18 


CEta (H) imperfeS: 

1 EroJla 7 }ibanomenos.< averted, andEqua- 

C drate averted, 
r Delta inverted, and 

2 HyfateHyfaton.,hc:< Tau jacent, aver- 

C ted, iSc. 


(Ariflides (Pag. 91.) enumerates and de^ 
■fcribes all the Variations of every Letter ia 
the Greek Alphabet ; by which the Signs 
or Notes above mention’d, and thofe of 
the other Moods, were contriv’d out of 

themx 


1 1 o of Dif cords and Degrees. 

them. They are in all 91; including the 
Proper Letters : I fliall not defcribe, but 
only number them. 


Out of 


A 

were made 7 

N 

were made 2 

B 

z 

S 

z 

r 

7 

0 

2 

A 

4 

n 

7 

E 

3 

p 

z 

Z 

z 

s 

6 

H 

$ 

T 

4 

0 

z 

T 

3 

I 

4 


4 

K 

3 

X 

4 

A 

5 


z 

M 

5 


4 


49 


41 

91 


I fhall only add a Word or two con- 
cerning their ancient Ufe of the Words 
Diafiem and SyHe?}?. Diaflem fignifies an 
Interval or Space; Sy^hnij a Conjunftion 
or Compofition of Intervals. So that, ge- 
nerally fpeaking, an Octave^ or any other 
might be truly call’d a Diafiem^ 
and very frequently ufed to be fo call’d, 
where there was no occafion of Diflinflion. 
Tho’ a To?ie^ or, Hemito?ie^ could not be 
call’d a Sy^Jem ; for when they fpoke ftrifl:- 
ly, by a Diafiem they undeiilood only aa 

Incom- 


I I I 


of Dijcords and Degrees. 

Incompofit Degree, whether Diefts^ Hem?- 
tone/rone^ Sefyuitone^ or Ditone ; for the 
two laft were (bmetimes but Degrees, one' 
E?ihamo7iic^ the other Chro7nntic. By 
Sj'Hem they meant, a Comprehenfive Inter- 
val, compounded of Degrees, or of left 
or of both. Thus a To7/e was a 
Diafle 7 }j^ and Diatejjaron was a Sy'Eiem^ 
compounded of Degrees, or of a 3 ^ and a 
Deg ree. Diapajhn was a S)'Be7}i^ com- 

pounded of the leffer 

or and 6 ^^ ; or of a Scale of Degrees : 
And the Scale of Notes which they ufed, 
was their Greateft, or Perfe£tvS)'f/^’w. Thus 
with them, a 3 ^ Mojor^ and a 3 ^ Mi nor ^ in 
the Diatonic Ge?ius^ were ( properly fpeak- 
ing ) Syfiems ; the former being compoun- 
ded of two To7ies^ and the latter of three 
Hemito7ies^ or a Tone and He?nito7ie : But 
in the Enharmonic Kind, a Ditone was not 
a Syfiem^ but an Incompofit Degree ; which 
added to two Diefes^ made up the Diatef’^ 
faron : And in the Chror}iaticl^md^ a Tribe-' 
7nitone was the like ; being only an Incom- 
pofit Diafiem^ and not a Syjte 7 n. 

But to return from this Digrellion 
(which is not fo much to my Purpofe, as 
to gratifie the Reader’s Curiolity) and con- 
tinue our Difeourfe according to Nature’s 
Guidance, upon the Diatonic Degrees. It 

I was 


5 12 Of Dijconls and Decrees, 

was faid, that there are RveTo/^es and two 
Heniitoiies in every Diafafon. Now the 
rcafon why there muft be two Hemitones^ 
is, becaufe an 8^^ is naturally compofed of, 
and divided into and 4^^; and a Fifth is 
three Tories and a half; a Fourth VfJoTones 
and a half ; and the Afcent, by Degrees, 
rniifl: pafs by Fourth and Fifth ; which are 
always unchangeable, and keep the fame 
dilfance from U^tifon ; and a iuft To7ie 
jor of 9 to 8 always between them. There- 
fore the Diafafon has not an Afcent of fix 
Tories^ but of five To7tes and two Henntones^ 
one Hemito?ie being placed in each Fourth 
Disjunft ; in either of which Fourths, the 
Degrees fnay be alter’d by placing the He- 
inito7ie in the Firfl, or Second, or Third 
Degree of either. As, A/7, FJ^ SoJ^ La. 
La, MI, FJ, Sol. SoJ, La, Ml, FJ. If 
this be done in the former Tetrachord, then 
is chang’d the Second, or Third Chord ; if 
in the other DisjunO; Tetrachord, then the 
Sixth, or Seventh is chang’d : The Fourth 
and Fifth being liable and immutable, by 
them we naturally divide the Diapafo7i : 
The Second, Third, Sixth, and Seventh are 
alterable, ^sMhior, ^nAMrjor, according to 
the Place of the Hemito7ie. 

These To7ies and Hemito7ies thus pla- 
ced, are the Degrees or Notes by which 

an. 



an Afcent or Defcent is made from the 
[on to t\\^Otiave^ov thro’ any oxhtv Sy/iem^ 
giving all the Concords their juft Meafures 
or Rations ; and without which, we could 
neither Meafure, nor Divide, nor well 
Praftife, to learn the greater Intervals or 
Syfiems. 

As we Naturally by the Judgment of 
our Ear, own, and reft in the Ottave^ as tlie 
chief Conlbnant ; fo we do as Naturally 
(without Study or Skill in Miific) meafure 
the Syfiefn of a Diapafon by thefe Diatonic 
Degrees ; and can do no otherwife. We 
cannot with our Voice, without infinite 
Studv, frame to run up or down eight 
Notes, without fuch a Alixture of Tones 
and Hemitones ; and we do it eafieft when 
we avoid Tritones. We fee it in a Ring of 
Bells, of which the compleateft and moft 
pleafant is a Peal of Six ; which are 
beft forted to have the Hemitorie in the 
midft ; i. e. between the Third and Fourth, 
both in Afcending and Defcending-, and 
then there will be no Tritone : Ex. gr. La^ 
Sol, Fa, Mi, Re, Ut. Where all Afcents 
and Defcents are made by juft DiatejJ'a* 
rons. Ut, Re, Mi, Fa. Re, Mi, Fa, SoU 
Mi, Fa, Sol, La. Or downwards ; La, 


Sol, Fa, Mi. Sol, Fa, Mi, Re^ 
Re, Ut. 


Fa, Mi 


I a 




1 1 4 Of Di/cords and t)egree5. 

And this is fo Natural that it pleafeth 
all Ears ; and if they fliould be difpofed in 
any other Order, it would be fo difagree- 
able, that any Ruftick or unlearn’d Ear, of 
fuch as know not what a Trito^ie is, would 
be able to judge, and find a Diflike of it. 
But then, how much more, if the Ring 
of Bells were difposM by Chromatic or En- 
harmo 7 iic Degrees, conftituting the DiatejJ'a-^ 
rons r* how abfurd and uncouth it would 
appear ! The prafUfe of thofe kinds there- 
fore, and in fuch a manner, feems to be 
(as has been faid) a Violence upon Na- 
ture, and only for Curiofity. 

I N Diatonic Mufic there is but one fort 
of Hemitone amongft the Degrees, call’d 
Hemitone Major^ whofe Ration is i6 to 15 ; 
being the Difference, and making a Degree 
between a Tone Major and Third Mmor ; 
or between a Third and a Fourth. 

There are two forts of Tones ; viz. 
Major, and Minor. Tone Major ( 9 to 8 ) 
being the Difference between a Fourth and 
Fifth : And Tone Minor ( 10 to 9 ) which 
is the Difference between Third Minor 
and Fourth. But both the To?ies arifing 
(as hath been faid) out of the Partition of 
a Third Major, in like manner as 5^^ and 4^ 
do by the Partition of an 8'^: I may (with 

Ihb- 


•V 


Of D If cor cl f and T)egrecs> 1 1 y 

fubmiflion ) make the following Remark ? 
wherein, if I be too bold, or be miftaken, I 
fhall beg the Reader’s Pardon. 

Th e ancient Greek Matters found out 
the Tone by the Difference of a Fourth and 
Fifth, fubttrafting one from the other : But 
liad they found it alfo (and that more Na- 
turally) by the Divifion of a Fifth; ttrtt 
into a Ditone and Sefquitone^ and then by 
the like proper Divifion of a true Ditone 
( or Third Major ) into its proper Parts ; 
they mutt have found both To?ie Major and 
Tone Minor. E 7 tclid retts fatisfied, that /v- 
ter ftif er-f articular e no?i cadit Mediu 7 n. A 
fuper-particular Ration cannot have a Me- 
diety; viz. in whole Number : Which is 
true in its Radical Num.bers. But had he 
doubled the Radical Terms of a Super- 
particular, he might have found Mediums 
mott Naturally and Uniformly dividing the 
Syttems of Harmony; ex. gr. The Duple 
Ration 2 to i, as the Excefs is but by an 
Unity, has the Nature of Super-particular: 
but 2 to I, the Terms being dupled, is 4 
to^2 ; where 3 is a Medium, which divides 
it Into 4 to 3 (4^^) and 3 to 2 (5^^^) A«r 
gain, 3 to 2, dupling each Term, is 6 to 4; 
and in the fame manner gives the two 
1 birds, viz. 6 toy, Minor) and 5 to 4, 
Major). Likcwife the 3^ Major ^ y to 4, 

I 3 dupled 


1 1 6 Of Dif cords dnd Degrees. 

dupled as before, i o to 8, gives the two 
Tones'^ t. e. lo to To?ie Minor ^ and p to 8^ 

Tone Major* 

A K D it feems to be a Reafon why the 
Ancients did not difcover and ufe the Tone 
Minor^ and confequently not own the Di- 
tone for a Concord ; beoaufe they did not 
purfiie this Way of dividing the Syfiems. 
A\t\\d^ Euclid \i2id a fair Hint to fearch fur- 
ther, when he meafured the Diafafon by 
fix Tories [ Major~\ and found them to ex- 
ceed the Interval of Diafafon. 

H E Tythagoreans^ not ufing Tone JMT 
nor^ but two equal To 7 ies Major ^ in a Fourth, 
were forced to take a leffer Interval for the 
Hemito 7 ie ; which is call’d their Limma^ or 
Pythagorean Hemitone ; and, which added 
to thofc two Tones^ makes up the Fourth : 
’Fis a Comma lefs than Hemitone Major ^ 
(i 6 to 15) and the Ration of it is 156 to 

Yet wc find the later Greek Maflers, 
Ptolemy^ to take Notice of To 7 ie Minor \ 
and Ariflides Tilii 7 itilia 7 im^ to divide a Sej- 
quioPiave Tofie to 8/ by dupling the 
Terms of the Ration thereof into two He- 
mitofies ; 18 to 17, and 17 to 16. And thofe 

anain, by the fame Way; each into two 
^ Diefes • 


of Dlfcords and Degrees. 1 1 7 

D/efes ; 3^5 to 3 35 to 34 ; the Divifion of 

18 to 17, the lefs Heniitone : And 34 to 33, 
and 33 to 32 ; the Parts of 17 to 16, the 
greater Hemitone. But yet, none of thefc 
were the Complement of two SefquioBnve 
Tones to Diateffaron : but another Hemi- 
tone ^ whofe Ratio is about 20 to 19 ; not 
exaftly, but fo near it, that the DiiTerence 
is only 1 2 16 to 1215-, both which together 
make the Linma Pytbagoriarm. 

But I no where find, that they thus 
divided the Fifth and Third major^ but ra- 
ther feem’d to diflike this Way, becaiife of 
the Inequality of the Hemitones and Diefes 
thus found out ; and chofe rather to con- 
llitute their Degrees by the SejqmoHave 
Tone^ and thofe Duodecimal fupposM-equal 
Divifions of it. But to returua 

There are, you fee, three Degrees 
Diatonic \ viz. He?nito 7 te jnajor^Tone minor,, 
and To?ie major. The firft of thefe fome 
call Degree minor ; the fecond, Degree ma- 
jor \ the third. Degree maxim. Now thefe 
three forts of Degrees are properly to be 
intermix’d, and order’d, in every Afcent to 
an Eighth, in relation to the Key, or Uni- 
fon given, and to the AfteSions of that 
Key, as to Flat and Sharp, in our Scale of 
Mufic ; io, that the Concords may be all 

I 4 true, 


I 


1 1 8 0 / Dijcords and Decrees, 

true, and ftand in their own fettled Ration. 
Wherefore if you change the Key^ they 
mud be changed too ; which is the reafon 
why a Harpfichord, whofe Degrees are 
fixed ; or a fretted Inftrument, the Frets 
remaining fix’d, cannot at once be fet in 
Tune for all Keys : For, if you change the 
Key, you withal change the Place of Tone 
mtnor^ and To7ie fnajor^ and fall into other 
He?nitc7ies that are not proper T)iatonic 
Degrees, and confequently into falfe Inter- 
vals. 

You may fully fee this, if you draw 
Scales of Afcent fitted to feveral Keys ( as 
are here inferred ) and compare them. 
For an Example of this, Take the firft Scale 
of Afcent to Dhfafon [ I ] viz. upon C 
Key Proper, by DiatoTtic Degrees ; (making 
the firft to be Tone 7ninoTy as convenient 
for this Inftance) intermixing the Chroma- 
tic and other Hanitoites^ as they are ufual- 
]y placed in the Keys of an Organ ; e. 
run up an Eighth upon an Organ (tuned 
as well as you can) by Half-Notes, begin- 
ning at C ^olfa and you will find tliefe 
Meafures. The Proper Degrees flanding 
right, as they ought to be, being deferib’d 
by Breves ; the other by Sc7i;ihreves : The 
Breves reprefenting the Toties of tlie broad 
Gradual Keys of an Organ ; the Savihrtves 

renre- 

V ♦ 




of V'ljcords and fiegms. 1 1 9 

rcprcrcnting the narrow Uppci ICcysj winch 
are ufually call’d M'ffics. And let this 
be the firft Scale, and a Standard to the 

reft. 

Then draw a fecond Scale C II 3 
ning up an Eighth in like manner ; but let 
the Key, or Firft Note be D Sol re, with a 
Flat Sixth, on the fame Organ flanding 
tuned as before which Key is fet a Note 
(or Tofie Minor) higher than the former. 

Draw alfo a third Scale [ III ] for 
D Sol re Key with Sharps, viz. Third and 

Seventh Major \ i. e. F, and C, fliarp. 

1 

I N the Firft of thefe Scales, the Degrees 
( exprefsM by 'Breves ) are fet in good and 
natural Order. 

« 

I N the Second Scale (changing the Key 
from C to D j you w^ill find the Second, 
Fourth, and Sixth, Comma (8i to 8o) 
too much ; but between the Fourth and 
Fifth, a Tone Minor^ which fhould be al- 
ways a Tone Major. So, from the Fourth 
to the Eighth, is a Comma ftiort of 7)/^- 
\)ente , and from the Sixth, a Comma fhoit 
of Third And this, becaufe in this 

Scale the Degrees are mifplaced. 

Th? 


120 


Of Difcords and Degrees. 


The Third Scale rnakes the Second, 
Fourth, and Sixth, from the Unifon^ each 
a Comma too much; and from the OHave^ 
as much too little. In it, the third De- 
gree, between ^ F and G, is not the Pro- 
per Hemitone^ but tjie Greateft Hemitone^ 
2,7 to 25. And alh this, becaufe in this 
Scale alfo the Decrees are mifplaced ; and 
there happen ( as you may fee^ three Tones 
Minor ^ and but two Major; the deficient 
Comma being added to t\\e. Remit one. ^ 

I have added one Example more, of a 
Fourth Scale , [ IV ] viz. beginning at the 
Key t C ; with the like Order of Degrees 
as in the firft Scale (from the Note C t) 
upon the fame Inftrument, as it Hands 
tuned after the firlf Scale : And this will 
vaife the firft Scale half a Note higher. 

I N this Scale, all the Remit ones are of 
the fame Meafure with thole of the firft 
Scale refpedively. 

And the Intervals Ihould be the fame 
with thole of the firft Scale ; which has 
Third, Sixth, Seventh, Major. 

But in this fourth Scale, the firft De- 

avee, from i C to ^ E, is T§ne major, and 
^ - Diejis ; 


I 2 I 


Of Dif cords and Decrees. 

Diefis ; as being compounded of 1 6 to 1 5, 
and 27 to 25. 

The Second Degree from Z' E to F, is 
Tone Minor \ therefore the Ditone^ made 
by thefe two Degrees, is too much by a 
Die (is ^ (12S to 125) and as much too little 
the Trihemitone^ from the Ditone to the 
Fifth. 

The Third Degree, from F to JF, is a 
Minor Hemit one ^ 25 to 24; which ( tho’ 
a wrong Degree) fets the DiateJJ'aron 
right. 

The Fourth Degree; from $ F to ^G, 
is Tone Major^ and makes a true Fifth. 

The Fifth Degree, from ^ G to B, is 
Tone major^ and Diejis ; fetting the Hexa^ 
chord (or Sixth) a "Diejis and Comma too 
much, or too high. It ought to have been 
Tone minor* 

The Sixth, from I B to C, is Tone 
minor ; too little in that place by a 
Comma, 

The Seventh, from C to $ C, is Hemi- 
tone Minor ; too little by a Diefis, And 
10, thefe ^wo laft Degrees are deficient by 

■a 


12 2 , Of Difcords and Degrees] 

a Diefis and Comma ; which Diefis and 
Comma being Redundant (as before) in the 
fifth Degree, are balanced by the deficien- 
cy of a Comma in the fixth Degree, and of 
a Diefis in the feventh : And fo the OBave 
is fet right. 


These Difagreements may be better 
viewM, if we fet together, and compare 
the Degrees of this IV Scale, and thofe of 
the I : Where we fhall find but one of all 
the feven Degrees, to be tlie fame in both 
Scales. 


Scale I. 

Degirees. „ ^ 

1^5 Tonejiiinor. 
Do 7 ie major. 
Hefuit^ major. 
4th, Done major. 
Tone minor. 
Tone major. 
Hemit. mc<jor. 


Scale IV. 

Tone maj. and Diefs. 
Tone minor. 

Hem it one minor. 
Tone major. 

Tone maj. and Diefis. 
Tone minor. 
Hemito 7 ie minors 


A N D thus ’twill fuccecd in all Inftru- 
mcftts, tuned in order by Hemit one s^ which 
are fix’d upon Strings; as Harp, ific. or 
Strings with Keys ; as C)rgan, Harpfichord, 
ific. or dillinguiflfd by Pretts ; as Lute, 
Viol, ific. for which there is no Remedy, 
but by fomc alterations of the Tune of the 

Strings 


I 


of Difcords and Degrees, 1 2 5 

Strings in the two former ; and of the Space 
of the Fretts in the latter ; as your prefent 
Key will require, when you change from 
one Key to another, in performing Mufical 
Compofitions. 

T ho’ the Voice, in Singing, being free, 
is naturally guided to avoid and corre£t 
thofe before defcrib’d Anomalies^ and to 
move in the true and proper Intervals : It 
being much eafier with the Voice to hit 
upon the right, than upon the aiiomalom QX. 
wrong Spaces. 

Much more of this Nature maybe 
found, if you make and compare more 
Scales from other Keys. You will flill find, 
that, by changing the Key, you do withal 
change and dilplace the Degrees, and make 
ufe of Improper Degrees, and produce In- 
congruous Intervals. 

For, infiead of the Proper Hemitone^ 
fome of the Degrees will be made of other 
fort of Hemitones ; amongft which chiefly 
arethefetwo: viz. Hemitone Maxim. 27, 
to 2j ; and Hemitone Minor ok Chromatic ^ 
2 j to 24. Which Hemitones conflitute and 
divide the two Tones ; viz. Tone majors 
9 to 8 : the Terms whereof tripled, are 
27 to 24* and give 27 to zj, and 25 to 24. 

The 


1 24 of DiJ cords and Degrees. 

The To^e minor likewife is divided into 
two Hemitones ; viz. Major ^ 1 5 to 15; and 
Minor ^ 25 to 24. 

These two ferve to meafure the Tones^ 
and are ufed alfo when you Divert into 
the Chromatic Kind. But the Hemito 72 e 
Degree in the Diatonic Genm^ ought al- 
ways to be Hemitone Major ^ 16 to ly ; as 
being the Proper Degree and Difference 
between Tone major and Trihe7nito7ie^ be- 
tween Ditone and a Fourth, between Fifth 
and Sixth minor ^ and alfo between Seventh 
7 najor and Odave^ 

Music would have feem’d much eaficr, 
if the Progreffion of Dividing had reach’d 
the Hemitones : I mean, if, as by dupling 
the Terms of Diafafon^ 4 to 2 ; it divides 
in 4 to 3, and >3 to 2 ; "DiateJJdron, and 
Diape7ite : And the Terms of Diaj>e7ite 
dupled, 6 to fall into 6 to 5, and 5 to 4, 
Third minor j and Third major \ and Ditone^ 
or Third 77/ajor; fo dupled, 10 to 8, falls 
into 10 to 9, and 9 to 8, Tone minor and 
Tone major : If, I fay, in like manner, the 
dupled Terms of Tone 7najor 18 to 16, thus 
divided, had given Ufeful and Propei //mf- 
to7/esj 18 to 17, and 17 to 16. But there 
are no fuch Hemito 72 es found in Harmony, 
and we are put to feek the Hefuitones out 

of 


of Difcords and Decrees. 1 2 5 

of the Differences of other Intervals ; as 
we lhall have more Occafion to fee, whea 
I come to treat of Differences, in Chap. 8. 


I may conclude this Chapter, by fliewlng 
how all Confonants, and other Concinnous 
Intervals, are Compounded of thefe three 
Degrees ; Tone ma]oY^ Tone minor^ and 
mito7ie ma]oY \ being feverally placed, as 
the Key iliall require. 


Tone Major ^ and) joyn’d, 
Hemitone Majovy } make 

Tone Major, and ) joyn’d. 
Tone Minor, 3 make 


Minors 


?3d Major. 


Tone Major, and^ • » t 7 

Tone Minor, ^ V 4th. 
Hemitone Major, ^ ® ^ 


2 Tones Major, 

I Tone Minor, 

I Hemitone Maj. 



joynd, 

make 




2 Tones Major, 

1 Tone Minor, 

2 Hemitones Maj, 



joyn’d, 

make 



6th Minou 


2 Tones Major, 

2 Tones Minor, 

I Hemitone Maj. 



ioyn’d, 

make 



6tl] Majgt] 


g Tones 


* i <5 of Difcords. 

3 T^ones Major, 9 . T 
1 Tone Minor, & ^ ■’°y" > C 7th Ai;«or. 
3 /*w/WBw M<y-. ^ \ 


3 Towfj Major, 

2 Minor, 

I HemhoneMaj, 



joyn’d, 

make 



7th Major, 


5 Major, 7 . , , 7 

2 Diapafon. 

2 Hemitones Maj, \ 3 

2 Major, \ joyn’d, \ Tritone, or 
I lone Minor, j make j falfe 4th. 


I lone Major, 

1 lone Minor, 

2 Hmiu Major, 



joynd, 

make 



Semidiapente, 
or falfe 5 th. 


C H A P. VII. 

Of Difcords. 

T7 E S I D E S the Degrees, vvhicli, tho’ 
they conftitute and compound all Con- 
cords, yet are reckon’d amongft Difcords ; 
becaufe every Degree is Dilcord to each 
Chord, to, or from which it is a Degree, 
either Afcending or Defcending, as being a 
Second to it : Befides theie, I fay, there 
are other Difcords, fome greater, and fome 

Icfs. 


of Di/cords. 1 17 

lefs. The lefs will be found amongft the 
Differences in the next Chapter *, and are 
fit, rather to be known as Differences, than 
to be ufed as Intervals. 

Th e greater Difeords are generally made 
of fuch Concords as, by reafon of mifpla- 
ced Degrees happen to have a Comma, or 
Diefts, or fometimes a Hemitone too much, 
or too little ; and fo become Difeords, moflb 
of them being of little Ufe, only to know 
them, for the better meafuring and re£li- 
fying the Syftems: Yet they are found 
amongft the Scales of our Mufic. 

Sometimes a Tone Major being 
where a Tone Minor ftiould have been pla- 
ced, or a Tone Minor inftead of a Tone Ma- 
jor ; fometime other Hemitones, getting the 
place of the T)iatonic Hemitone Ma]or, and 
ferving for a Degree, create unapt Difeor- 
ding Intervals : amongft which may be 
found at leaft two more Seconds, two more 
Thirds, two more Sixths, and two more 
Sevenths. In each of which, one is lefs^ 
and the other greater, than the true legi- 
timate Intervals, or Spaces of thofe Deno- 
minations \ as will be more explain’d in the 
enfuing Difeourfe. 



But 

t 


li 28 Of Dif cords. 

But befides thefe ( or rather amongft 
them, for I here treat of Degrees as Dif- 
cords ) there are two Difcords eminently 
confiderable, Tvitonc^ and Scnjidici'^ 
fente. The Tritone^ (or falfe Fourth) 
whofe Ration is 45 to 32, confifts of three 
whole Notes ; viz. two Tones Major^ and 
one Minor. The Semidiapente (or falfe 
Fifth) 64 to 45 ; is compounded of a Fourth 
^ndHemitone Major. 

And thefe two divide Diapafon^ 6 ^ to 
’32, by the Mediety of 45; And they di- 
vide it fo near to Equality, that in Praftice 
they are hardly to be diftinguifh’d, and 
may almoft pals for one and the fame : 
but in Nature, they are fufficiently diftin- 
guifh’d ; as may be feen both by their fe- 
veral Rations, and feveral Compounding 
Parts. 

I think we may reckon yths for Degrees, 
as well as among the greater Difcording In- 
tervals ; becaufe they are but Seconds from 
the OH:ave^ and ar<^ as truly Degrees De- 
fcending, as the Seconds are in Afcent : tho* 
they be great Intervals in relpeft of the 
Unifon^ and fuch as may be here regarded. 

These Difcords, the Tritone^ and Semi* 

diapente \ as alfo, the Seconds, and Sevenths, 

are 



are of very great ufe in Mufic, and add a 
wonderful Ornament and Pleafure to it, if 
they be judicioufly managed. Without 
them, Mufic would be much lefs grateful ; 
like as Meat would be to the Palate with- 
out Salt or Sawce. But, the further Con- 
fideration of this, and to give DireSions 
when, and how to ufe ’em, isnotmy Task^ 
but muft be left to the Matters of Compo- 
fition. 

Discords then, fuch as are more apt 
and ufeful ( Intervalla Concinna) are thefe 
which follow. 

2d Minors or, He mito 7 te Major ^ 16 to 
ad Major ; Tone Mi nor ^ i o to 9. 

2d Great eft ; Tone Major 9 to 8. 

7th Minor ; 5th Sc 3d Minor^ 9 to jr. 

7th Major ; 5th 8c 3d Major ^ i y to 8. 

Tritone ; 3 d Maj . & Tone Maj. 4 5 to 3 z • 

Semi di ape nte ; 4th & Hemit. Maj. 64 to 4 y. 

These are the Simple diflbnant apt 
Intervals within T)iapafon ; if you go a 
further Compafs, you do but repeat the 
fame Intervals added to Diapafon^ or TDtf- 
diapaj on^or Trif -diapaf &c. as, Ex. gr. 



A 


Of Difcords. 

Is T)iafafon with a 2d. 

T)iafafon with a 3d. 
KDiafafon with a4th^ or 
'^Diafafon cmn Diatejjaron, 

{ Diafafon with a 5 th, or 
Diapafon cum Diapente. 
T)ifdiapafon. 

Dif-diapafon cum Diapente. 
Trif-diapafon^ &c. 

Here, by the way, the Reader may 
take a little Diverfion, in praQifing to mea- 
fure the Rations of fome of thofe Intervals 
in the ’foregoing Catalogue of Difcords, by 
comparing them with Diapafon ; as thofe of 
the Sevenths^ which I leleft, becaufe they 
are the moft diftant Rations under T)iapa- 
fon ; viz. Seventh minor ^ 9 to 5 ; and Se^ 
venth major ^ 15 to 8. Now to find what 
Degree or Interval lies between thefe and 
Diapafon* 

Firft, 9 to 5 is 10 to j, wanting ib to 9 
{fCone minor*) Next, 15: to 8 is 16 to 8, 
wanting 16 to 15 {Hemitone major)) So 
the Degree between Sevatth minor and 
Diapafo 7 t^ is Tone minor and between 6'^’- 
venth major and Diapafon^ is Remit one 
major* 


A pth 

joth 

1 ith 

1 2 th 

15th 
19 th 
22 th 


Then 


of Dtfcords. 1 3 1 ' 

Then he may exercife himfelf in a Sur- 
vey of what Intervals are compriz’d in 
thofe feveral Sevenths, and of which they 
are compounded. 

First, 9 to 5 comprizeth 9 to 8, and 
8 to 5 : Or, 9 to 8, 8 to 6 , and 6 to 5. 
Next, i5to8contain 15 to 12, 12 to 10, 
10 to 9, and 9 to 8 : Or, 15 to 12, and 12 
to 8 : Or, 15 to 10, and 10 to 8, I 

fuppofe that the Reader, before this, is fo 
perfefl: in thefe Rations, that I need not 
lofe Time to name the Intervals exprefs’d 
by the Mean Rations, contain’d in the ’fore- 
going Rations of the Sevenths, which fliew 
of what Intervals the feveral Seve 7 iths are 
compounded. 

Besides thefe ( by reafon of Degrees 
wrong placed) there are two more Seveitths-, 
\_falfe Sevenths'] one, lefs than the true ones, 
and another greater. The leaft compoun- 
ded of two Fourths, whofe Ration is 16 
to 9, and wants a Comma o? Seventh minor, 
and a Tone major of Diapafo 7 t : The other 
is the greateft, call’d Set} 7 idia^afo 7 i, whofe 
Ration is 48 to 25 ; being a Diefis more 
than Seventh major, and wanting Hemitone 
7 ni 7 iordi DiapafoUp 



t 


Now, 


}^2 Of Di/cords. 

Now, firft, i 5 to 9 is i 6 to 8 (2 to i) 
wanting 9 to 8; /. e. wanting To^e major 
of Diafafon ; and contains 16 to 10 (8 to 5) 
and 10 to 9 ; Or,, 16 to 15, 15 to 12 (5 t04) 
ii2 to 10 (6 to 5) and 10 to 9. Next,«Sm/- 
diapafon 48 to 25, is 50 to 25, wanting 
to 48 ; /. e, 25 to 24 (viz. Hefnitone minor') 
of Diapafon. 

And the like happens, as hath been 
faid, to the other Intervals, which admit of 
major and ?ninor ; viz. Seconds^ Thirds^ and 
Sixths. The Fourth^ and Fifths and Eighth 
ought always to remain immutable *, tho’ 
they may fuffer too fometimes, and incline 
to Difcord, if we afcend to them by very 
wrong Degrees ; as you may fee in the 
II^ Scale in the ’foregoing Chapter ; where 
the Fourth having two To 7 tes major^ is a 
Comma too much. 

All thefe Intervals may be fubjefl: to 
more Mutations, by more abfurd placing 
of Degrees, or of Differences of Degrees ; 
but it is not worth the Curiofity to fearch 
farther into them : The Reader may take 
Pleafure, and fufficiently exercife himfelf, 
in comparing and meafuring thefe which 
are already laid before him. 


But 


Of Di/cords. 

But to return from this Digreflioii. 
There are many unapt Difcords, which may 
arife by continual Progreffion of the fame 
Concords ; /. e. by adding (for Example) 
a Fourth to a Fourth^ a Fifth to a F/fth^ &c. 
for ’tis obfervable, That only Diafafon ad- 
ded (as oft as you pleafe) to Diapajon^ ftill 
makes Concord : But any other Concord, 
added to it felf, makes Difcord. 

You will fee the Reafon of it, when you 
have confider’d well the Anatomy ( as I 
may call it) of the Conftitutive Parts of 
"Diapafon ; which contains, and is compos’d 
of feven Spaces of Degrees, or of Fourth and 
Fifths or of Thirds and Sixths^ or of Seconds 
and Sevenths ; which muft all keep their 
true Meafures and Rations belonging to 
them, and otherwife are eafily and often 
diforder’d. 

Then, confider Diapafon as conftituted 
of two Fourths disjunft, and a Tone mf\or 
between ’em. And this laft is moft need- 
ful to be very well confider’d ; as moft 
plainly (hewing the Reafons of thofe Ano- 
malies, or irregular Intervals, which are 
produced by changing the Key, and confe- 
quently giving a new and wrong Place to 
this odd Tone major ^ which ftands in the 

K 4 midlt 


't 34 Of Vif cords '. 

midft of Diapa/on^hQtwQm the two Fourths 
disjuna. 

Every fourth muft confift of one To 7 ie 
major^ one Tone minor^ and one Hemitone 
^najoTj as its Degrees, placing them in what 
Order you pleaft ; whofe Rations, added 
together, make the Ration of DiatejJ'aron. 
And of thefe fame Degrees contain’d in the 
Fourthly are made the two Thirds^ which 
conftitute the fifth. Tone major and Hetni- 
tone major make the lefs Third, oxTrihemi- 
tone ; Tone major and Tone fninor make the 
greater Third, ox Ditone ; Trihemito^ie and 
Ditone make piafente ; Trihe7nitone and 
Tone Minor (as likewife Dito?te and Heini- * 
tone major) make Diatejjaron. 

Now this To7ie Major^ that hands in the 
middle of Diafafon^ between the t\yo 
fourths y which it disjoins ; and the Degrees 
requir’d to the Fourths^ will not in a hxed 
Scale hand right, when you alter your Key, 
and begin your Scale of T)iapafon from 
another Note : For that which was t\\t Fifths 
will now be the Fourth^ or Sixths 8cc. and 
then the Degrees will be diforder’d, and 
create fome difcording Intervals. If you 
continue conjuntt Fourths^ there will be a 
Defeat of Tones Major ; if you continue con- 
jund. Fifths j there will be too many Tones 


Of Dif cords. 13 j 

Major in the Syftems produced. And if a 
Tone Major be found, where it ought to 
have been a Tone Minor ; or a Minor inttead 
of a Major ; that Interval will have a Com^ 
ma too much, or too little. And fo like- 
wife will from a wrong Hemitone be found 
the Difference of a Diejis. And thefe two, 
^ Comma and Diefts^ are fo often redundant, 
or deficient, according as the Degrees hap- 
jpen to be diforder’d or mifplaced; that 
thereby the Difficulties of fixing half-Notes 
' of a,n Organ in tune for all Keys, or giving 
the true Tune by Fretts, become fo infu- 
perable. 

You fee, that in every Space of an 
Eighth, there are to be three Tones major^ 
and two Tones minor ^ and two Hemitone s 
major : One Tone major between the Dia-- 
tejj'aron and Diapente, and a Tone major ^ 
a Tone minor ^ and Hemitone ?najor in each of 
the disjunfl: Fourths. 

These are the proper Degrees by 
which you fhould always Afcend or De- 
fcend thro’ Diapafon^ in the T)iatonic Kind j 
which Diafafon being the compleat Syftem^ 
containing all primary Simple Harmonic In- 
tervals that are ; (and for that reafon call’d 
Diapafon ;) you may multiply it, or add it 
Xo its felf as oft as you pleafe, as far as Voice 

or 


Of Di/cords. 

or Inftrument can reach, and it will ftiU be 
Concord, and cannot be diforder’d by fuch 
Addition ; becaufe every of them will con- 
tain (however placed) juft three To?^es 
majoYy two Tones minors and two Hemitones 
major. 


Whereas, if you add any other In- 
terval to itfelf, the Degrees will not fall 
right, and it will be Difcord, becaufe all 
Concords are compounded of unequal Parts, 
as hath been fhewn before ; and if you car- 
ry them in equal Progreflion, they will mix 
with other Intervals by incongruous De- 
grees, and thofe diforder’d Degrees will 
create a diffonant Interval. See the follow- 
ing Scheme of it. 


2 

2 


gds minor 
3ds 7 na]or 
qths 
5ths 

6ths minor 
6ths major. 


QJ 


5th, wanting Hemit. min, 
5 th, znd. Hemit. minor, 
j ^th, wanting major, 
6 ‘ 8th, and Tone major, 

8th, and Ditone Die fis, 
8th, and 4th, 


To which may be added, That 

2 Tones ?nin,\^ f Ditone^w 2 inting 3 ,Comma, 
2 Tones may S b\ Ditone ^ and a Comma, 


It was faid above. That Diapafon may 
be added to it felf as oft as you pleafe, and 


of f)i/cords. 1 5 7 

there will be no Diforder, becaufe every 
one of ’em will ftill retain the fame Degrees 
of which the firft was compos’d : ^ But it is 
not fo in other Concords ; of which I will 
add one more Example, becaufe of the Ufe 
which may be made of it. 

Make aProgreflion of ?oVivT)iapente^s^ 
and, as was fhew’d in the Fifth Chapter, it 
will produce 'Difdiafafon^ and two^Tones 
ma]or^ which is a with a Comma too 
much ; becaufe in that Space there ought 
to be juft feven Tones ma]or^ and five Tones 
minor ; whereas in four Fifths continued, 
there will be found eight Tones ma]or, and 
but four Tones minor : So that a Tone ma]or^ 
getting the Place of a Tone ininor^ there will 
be in the whole Syftem a Comma too much. 
One of thefe 7na)or Tones fhould have beeu 
a To7ie minor ^ to make theExcefs above Dif- 
diapafon a juft "Ditone. 

O N the other fide, if you continue the 
Ration of four Diatej]'aro7is^ there will be a 
Tone minor ^ inftead of a To7ie md)or ; and 
confequently a Comma deficient in conftitu- 
ting Diapafon and Sixth minor. For fince 
every Fourth muft confift of the Degrees of 
To77e minor ^ one To?te ma\or^ one Remit one 
7nd)or ; it follows, that if you continue four 
Fourths^ there will be four Tones mvior^ four 

Tones 


^ 3 ^ Of DiJ cords. 

Tones majorj and four Hemitones major: 
Whereas in the Interval of ^Diapafon with 
Sixth minor^ there ought to bo five Tones 
major, and but three tninor. 

By this you may fee the Reafon, why, 
to put an Organ or Harpfichord into more 
general ufeful Tune, you muft tune by 
Eighths and Fifths-^ making the Eighths 
perfeQ:,and the Fifths a little bearing down- 
ward ; /. e. as much as a quarter of a Co?n- 
ma, which the Ear will bear with in a Fifth, 
tho’ not in an Eighth. For Example, be- 
giriat C Fa at ; make C Sol fa nt 2 l perfeft 
Eighth to it, and G Sol re ut a bearing Fifth \ 
then tune a perfefl: Eighth to G, and a 
bearing Fifth at D La fol re ; and from 
thence downwards (that you may keep to- 
wards the middle of the Inflrument) a per- 
fefl; Eighth at D Sol re : And from thence 
a bearing Fifth up at A ; and from A, a 
perfect Eighth upwards, and bearing Fifth ' 
at E La mi. From E an Eighth down- 
wards ; and fo go on, as far as you are led 
by this Method, to tune all the middle part 
of the Infirument ; and at laft fill up all 
above, and below, by Eighths from thofe 
which are fettled in Tune, according to the 
Scheme annex’d ; obferving (as was faid ) 
to tune the Eighths perfeft, and the Fifths 
a little bearing flat;, except in the three 


of T)if cords. 1 jp 

laft Barrs of Fifths^ where the Fifths begin 
to be taken downward from C, as they 
were upwards in all before : Therefore, as 
before, the Fzfth above bore downward; 
fo here, the Fifth below mufl bear upward, 
to make a bearing Fifth : but that being 
not fo eafie to be judg’d, alter the Note 
below, till you judge the Note above to 
be a bearing Fifth to it. This will re£tifie 
both thofe Anomalies of Fifths Fourths: 

For the Fifth to the Unifon^ is a Fourth to 
the OFlave ; and what the Fifth lofeth by 
Abatement, the Fourth will gain : Which 
doth in a good Degree reftifie the Scale of 
the Inftrument. Taking Care withal, that 
what Anomalies will ftill be found in this 
Hemitonic Scale, may, by the Judgment 
of your Ear, in tuning, be thrown upon 
fuch Chords as are lead ufed for the Key ; 
as t G, h E, ifc. even which the Ear will 
bear with, as it doth with other Difcords 
in binding Paffages ; if fo, you clofe not 
upon them. But the other Difcords, fo 
ufed, are mod Elegant ; thefe only more 
Tolerable. 


CHAP. 


140 


of Differences. 


CHAP. VIII. 

Of Differences. 

A LL Rations and Proportions of In- 
equality, have a Difference between 
them, when compar’d to one another ; and 
confequently the Intervals, exprefs’d by 
thofe Rations, differ likewife. A Fifth is 
different from a Fourth^ by a To7te Ma]or ; 
frorn^ a Third Minor ^ by a Third Ma]or ; fo 
an Eighth from a Fifth, by a Fourth. Of 
the Compounding Parts of any Interval, one 
of them is the Difference between the other 
Part and the whole Interval. 

B u T I treat now of fuch Differences as 
are generally lefs than a Tone^ and create 
the Difficulties and Anomalies occurring in 
the two ’foregoing Chapters. I have the 
lefs to fay of them apart, becaufe I could 
not avoid touching upon them all-along^ 
’Twill only therefore be needful, to fet be- 
fore you an orderly View of them. And, 
firft, taking an Account of the true Harmo^ 
nic hitervals^ with their Differences, and the 
Degrees by which they arife ; ’twill be ea- 
fier to judge of the falfe Intervals^ and of 
what Concern they are to Harmony* 

Table 

V. 


of T>ife 


erences. 


141 


Table of true Diatonic Intervals within 
Diapafbn, with the Differences between 

them. 

' Their 


Rations. 


Their 
Differen- 
ces. 


Hemitone M^jor, 


16 to 1 51 

Tone Minor, 



10 to 9^ 

Tone Major, 



9 to 8 

3d Minor y "1 

f Tone Ma], ScHemit.Maj. 

6 to 5 

;d Ma]or., 

1 

Tone Maj. & Tone Min- 

5 to . 4 

4th. 

1 

3d minor ^Tone min%'j 




or 3d major & He- ^ 

4 to 3 


1 

mitone major. j 


•jth. 


4th, and Tone major 




or of the two 3ds. j 

3 to 2 

6th Minor ; 

Vm 

5 th, and Hemit. maj. 1 



0 

or 4th, and 3d min^ j 

8 to 5 

6th Major; 

<u 

5 th, and Tone minor T 




or 4th and 3d maj, j 

$ to 5 

7th Minor ; 

r 0 i 

p-i 

6 til maj. & Hem. max. y 



e 

or Othmin. & Tone ^ 

Q to K 


0 

u 

major ; or 5 th and C 


1 


3 d minor. 3 


7th Major ; 


6th maj. & Tone maj. > 

r ^ fr, S 



or $th, and 3d maj.f 

1 ^ wLI 0 

Diapafon ; 


7th, and 2d or 6th, ■) 




& 3d or $th, &4th./ 

i* LU X 

Tritone ; 


jd maj, and Tone maj. 

4^ to 32 

Semidiapenie;^ 

1 

l_4th, and Hemit. major. 

64 to 4$ 


to 24 

81 to 80 
16 to 
2$ to 24 

10 to 9 
9 to 8 

1 6 to 15 
2J to 24 

17 to 25 

2^ to 24 
16 to 15 


2048 
to 
202 ^ 


Thofe which arife from the Differences of Confonant In- 
tervals, are call’d Intevvatia Concinnay and properly apper- 
tain to Harmony : The reft are neceffary to be known, for 
making and underftanding the Scales of Mufick. 


Table 


142 


of Diffe 


erences. 


Table of falfe Diatonic Intervals^ caufed 
hy Improper Degrees ; with their Rations 
and*Differences from the true Intervals. 

This Mark ^ands for more *, — for Ufs, 


Rations. 


Differences 
from true* 


Trihemi-S J. 

fo e M tiemit. major, f 

/ Greateft ; Tone major , and i 
Hemit. max, f 


r Lefs ; 
Fourth ^ 


C Leaft ; z Toms minor. 
Ditone < 

Greateft ; 2 Toms major. 


2 Tones minor gr- \ 
for. i 


Fifth 


Hemit, major 
Greater , 2 Tones maj. 

Hemit. maj, J 


c Lefs ; Lefs 4th, and ^ 
y Tone major. } 

2 Greater ; Greater 4ch, and 
Tone maj. 


}z to Z7 
2-45 to 100 


81 to 80 — 
81 to 80 + 


too to 8i|8ito8o — 
8i to 64 8 1 to 80 -|- 


C Leaft ; ^thy and Hemit. 
Sixth < minor, 

^Greateft; 5th, and Tone 
major. 


} 

> 

> 


310 to 245 81 to 80 — 
Z7 to io^8i to 8o-|- 


40 to 27j8ito8o-*- 
I243 to 160 81 to 80 + 


Leaft ; 6th majory and 1 
Sev||ith ^ Hemit, major, j 

Greateft; 6th minory and 1 
3d minor. j 


z% to 
17 to 16 

16 to 9 
43 to Z5 


81 to 80 
81 to8o + 

to 80 — 
u8tou$-4- 


Here 


of Differences. 1 4 1 

He r e in this Account may be feen,how 
frequently the Covmia^ and the Die/tSj 
Abounding or Deficient, by reafonot Mif- 
placed Degrees, occafion Difcord in Har^ 
mo7iic Intervals. 

f 

The Comma by reafon ol ^ wvongTone^ 
7 . e. too much, when a To7ie Major hap- 
pens where there ought to be a To7ie Mi- 
noY\ or too little, when th^To^ze Minor is 
placed infiead of the Major. And the Diefis 
is Redundant, or Deficient, by reafon of a 
wrong Hemitone\ when the happens 

inftead of the Minor ^ or the contrary : the 
/Diefis being the Difference between them. 
And if Hetnltonimn Maximum get in the 
Place of Hemito7iimn Mrjus^ the Excefs will 
be 3.Comjna; if in the Place oi ' Hemitone 
Minor^ the Excefs will be Comma and 
Diefis. 

AND,thefe Anomalies are not Imagina- 
ry, or only Poffible, but are Real in anln- 
ftrument fix’d in Tune hy Hemito7ies\ as, 
Organ^ Harpfichord^ &c. And the Reader 
may find fome of ’em amongfl: thofe four 
Scales oi Diapafon\ in the Sixth Chapter; 
to which alfo more may be added : Out of 
the Firft of which, I have feletfed fome 
Examples, ufing the common Marks, as be- 
fore, for more ; and — for lejsov voa7i- 
ting. L From 


From 


T42 


0/Dife 


erences. 


' S C, to Z- E : ■[ Tone Major,-\-piefts ; or, 

t 3d Mtn. — Hennt. Min. 

iC to F- 3dA%V, + Diefis-, or, 
’ ’ ^ 4th, — Hemit. Minor. 

D, to G ; 4th, + Comma. 

Z.F to*F--[ 3 ‘^ Min.— Die f. Si Com. or 
j * ’ t Tone Min. -j- Hem. Min. 


hE to 4 G • 4*^^’ “ "T>iefis ; or, 

^AMaj. 4- Hemit. Min. 

*F tnhV.- ^ lAMa].-\-Dief.SiCom.ov 

* ’ ’ t 4th, — • Hem. Sulminim. 

4 F, to B ; 4th, ~\- Comma. , 

* G to i B • 4 ^T'j- + Diefis ; oi', 

* G to r • -[ 3d Major + Diejis ; or, 
^ C 4th, — Hemit* Minor* 

il B, to D \ '^AMin* — Comma. 


Next, take account of fome Differences 
which conftitute feveral Hemitones. 


Dif- 


a 

<L> 


o 

r—> 

U 

S-l 

tS 


Of Differences. 14 ^ 

ToneMajor^wi Maxim. 27 to 2^,’ 

Minor, f *' 

:^dA'fajor., and 4th. Hemit. MaJuSyi 6 to 15^ 

. "'"1 ■=»• 

I ^ jd Mimr, aiJ 5J 

iTones Major, md 4th {ot Limma) f' 2<.6 to 243. 

C Pytbagor. S 

C Apoiomej 2187 to 2048 ' 
and Limma,-^ or flemit, Med. with 
c Comma. 

To which may be added out of Mer^annuSy 

Hemit.Max/m. and j Hemitomum'^. ^ ^ 

Hemit. Minor. Minimum.} ^ 


<u 

<u 

> 

•a' 

-J-J 

CJ 


o j Tone Minor., and *\ 

liemitone Maxim. ) TIemiXo72ium.\ _ ^ 

" ^ or, ^ 

Hemit one Minor ^ \ 
and Comma. ^ 


• ^ 
Q 


Next, take a farther View of Differ 
rences.y moil of which arife out of the pre- 
ceeding Diff'ere 7 icesy by which you will bet- 
ter fee how all Intervals are Compounded 
and Differenced, and more eafily judge of 
their Meafurcs. 



Table 


• rs 


Difference between 


>44 


Of Vlfft 


erences. 


Table of more ^Differences. 

f ToTze Maj* and Tone Min. Comma. 

" Tone Maj. and Hem. Greateji. Hemitone Minor. 
Tone Maj. and Hem. Med iutn. Hemitozze Major. 
Tone Maj. and Hem. Vythag. ^ A-potome, 

Hem. Greateji^ and Hem.Maj. Comma. 

\\Hem.Greateft, 8 eBem.Mw. SCojrmm, zniLiefis-, 
^ ^viz. Hem. Minimum. 

j Hemit. Major., and Minor. Liefis. 

fiemit. Major, and Medium. { Jo viz. 

Hemit. Major, and Pffthag. Comma. 

Afoiome, and Hemit. ^^'^d^y^/^and “ 

J Apotome,2.ndi Hemit. Med. Comma. 

^ rT ‘*0*7 f ro//?7«^,and thc aforc- 

Apotome, HernA, Pythag.'^ faid Difference. 

Apotome,2cX\^Hemit. Minus. 2 Commas. 

Hemit. Medium, 2 i\\ 6 .P)tbag. 

j d Majlis, 2 cxA Minus, 

Hemit. Medium, and Minus. Comma. 

Hemi t. Pythag. and Minus. Comma M inus. 

^Somewhat 7 3^25 

Hemit. Minus, 2ind Die fis. \ more than > to 

( Comma, viz. j :^072. 

Hemit. Minus, and Comma. Pern. Suhminimum. 

r,. f. 3 ^ t Comma Minus, viz. 

Biefu, and Qomma. ^025. 

[ '\Com. Majus, and Com. Minus, 3 2803 to 3 2768; 


These 


of Differences. 1 4 j 

These Differefices ( with fome more ) 
are found between feveral other Intervals ; 
of which more Tables might be drawn, 
but I fhall not trouble the Reader with 
them. Having here fliewn what they are, 
he may ( if he pleafe ) exercife himfelf to 
examine Thefe by Numbers, and alfo find 
out Them ; and to fome it may be plea- 
fant and delightful : And, for thatReafon, 
I have the more largely infilled on this part 
of my Subjeft, which concerns the Mea^ 
fures^ Habitudes^ and Differences of Har-» 
m.onic Intervals. 


I fhall add one Table more, of the Parts 
of which thefe lelTer Intervals are com- 
pounded ; which will ftill give more 
Light to the former ; and is, in EffeB., the 
fame. 


L 3 Tons’ 


t4^ Of T)ijf a ‘ences. 


Tone Major con 
tains, & is com 


■ Tone Minor i and 
Comma. 


I 


Hem if one Maxim''. 
and Hemitone Min, 
z Hem. Min. 


TinunHpH of } Hemitone Maj.J Limma^y 
pounded 01 ^ ^ ^ j I D^ep. 

, ^ ^ f I Comma. 

Tnup Ml'- <f^ Maxim. J Hemit. Major, ( z Hemit. Min, 

'*'{^Hem. Suhmin^Hemit. Minor ^ \ Diejts. 

Hpm Ma f Hem. Aled. \ Hem.'2yxh,J Hemit. Min, 

Comma \DieJis. jjiComma^ s\T)'ief.^Com, 

TT S Hem.Med. \ Hem 'Pyth.J Hem. Min.jHem.Subm^ 

•^’l^Com- Min.^Comma, "\DieJis., ^Dief.SiCom, 

Hem. M,n.\ pthagoncum. 

n. <lJuterence between Coww^/kz^j«/j 


Hem.Med. 


{ 


Comma. } ’ ' • o' '"'’/i.'' 

and minus, viz. 32-805 to 32768. 

J Hemitone Minus. 


nem.¥ph.\commet Mimis. 

tt , r Hemit. Submin. f I^iejis, and 

Hem. J, 


J Comma, 

\ Comma Minus. 

_ f Comma Minus. 

Cmma. 


307?,. 


I think there fcarce needs an Apology for 
fome of thefe Appellations, in refpeft of 
Grammar. That I call He?n2toni7rm., and 
Hexachordon^ and Mmws *, fometimes 

Hemit one .^znA Hexachord^ major^ and minor* 
Thefe two laft Words are fo well adapted to 
our l.anguage, that there’s no E2tglijl7-ma7t 
but knows them. Therefore when I make 
Hemit o?te an Word, I take mdpr and 

7 ninor to be fo too, and fittcll to be join’d 
with it, without refped oi Gender. 


CHAP. 




CHAP. IX. 

Conclufton. 

T O conclude all. Bodies by Motion 
make Sound ; Sound, of fitly-conftitu- 
ted Bodies, makes Tune : Tune, by Swift- 
nefs of Motion is render’d more acute; 
by Slownefs more grave : in proportion to 
the Meafure of Courfes and Recourfes, of 
Tremblings or Vibrations of Sonorous Bo- 
dies. Thofe Proportions are found out by 
the Quantity and Affefliions of Sounding 
Bodies ; ex, gr. by the Length of Chords. 
If the Proportion of Length (ceteris pari* 
h</s) and confequently of Vibrations of fe- 
veral Chords, be commenfurate within the 
Number 6; then thofe Intervals of Tune 
are Confonant, and make Concord, the 
Motions mixing and uniting as they pafs : 
If incommenfurate, they make Difcord by 
the jarring and clafhing of the Motions. 
Concords are within a limited Number 
Difcords innumerable. But of them, thofe 
only here confider’d, which are ( as the 
Greeh term’d them) Co7icin7wu^^ 

apt and ufeful in Harmony : Or which, at 
leaftj are neceffary to be known, as being 

L 4 the. 


Conclujton. 

the Differences and Meafures of the other ; 
and helping to difcover the Reafon of 
Anomalies, found in the Degrees of Inftru- 
ments tuned by Hemitones. 

All thefe I have endeavour’d to ex- 
plain, with the manifeft Reafons of Con- 
lonancy and DifTonancy (the Properties of 
a "?end 7 dum giving much Light to it) fo as 
to render them eafie to be underftood by 
almoft all forts of Readers ; and to that 
end have enlarg’d, and repeated, where I 
might (to the more intelligent Reader) have 
compriz’d it very much fhorter. But I 
hope the Reader will pardon that, which 
could not well be avoided, in order to a 
full and clear Explanation of that, which 
was my Defign, viz. the Phlofophy of the 
Natural Grounds of Harmony. 

Upon the whole, you fee how Ratio- 
nally, and Naturally, all the Simple Con- 
cords, and the two Tones, are found and ^ 
demondrated, by Subdivifions of Diafafo7j. 

2 to I, i.e. 4 to 2 ; into 4 to 3, and 3 to 2. 

2 to I, i.c\ 6 to 3 ; into 6 to 5, and 5 to 3. 

2 to I, i.e. 3 to 4; into 8 to 5, and 5 to 4. 

2 to I, /.^-Mct05; intoiotoc^, 9 to 8, and 
8 to 5. 

In 


Conclujton, 


149 


In which are the Rations (in Radical, 
or Leaft Numbers) of the Oftave, Fifth, 
Fourth, I'hird Majorj Third Minor^ Sixth 
Ma]or^ Sixth Minor ^ and Tone Major ^ and 
Tone Minor, 

And then, aJl the Hemito?ies^ and Diejis^ 
and Comma^ are found by the Differences 
of thefe, and of one another • as hath been 
ihewn at large. 

N o w, certainly, this is much to be pre- 
ferred before any Irrational Contrivance of 
expreffing the feveral Intervals. The Ari- 
lioxenian Way of dividing a To 7 ie \_Ma]or~\ 
into twelve Parts, of which 3 made a Die- 
fis^ 6 made Hemitone^ 30 made Diatejj'a- 
ron^ (as hath been faid ) might be ufeful 
as being eafier for pprehenfion of the In- 
tervals belonging to the three Kinds of 
Mufick ; and might ferve for a lead: com- 
mon Meafure of all Intervals (like Mer- 
cators artificial Comma) 72 of them being 
contain’d in Diapafon. 

B u t this Way, and fome other Methods 
of dividing Intervals equally, by Surd Nam- 
iers and t'raHions, attempted by fome mo- 
dern Authors ; could never conllitute true 
Intervals upon the Strings of an Inftrument, 

nor 


1 5 ^ Condnjton. 

nor afford any Reafon for the Caufes of 
Harmony, as is done by the Rational Way, 
explaining Confonancy by united Motions, 
and Coincidence of Vibrations. And tho’ 
they fupposM fuch Divifions of Intervals ; 
yet we may well believe, that they could 
not make them, nor apply ’em in tuning a 
Mufical Inftrument • and if they could, the 
Intervals would not be true, nor exaQ:. 
But yet, the Voice offering at thofe, might 
more eafily fall into the true Natural In- 
tervals. Ex. gr. The Voice could hardly 
exprefs the ancient Ditone of two Tories 
Major ; but, aiming at it, would readily 
fall into the Rational Confonant Ditone of 
5 to 4, confifting of Tone Major and Tone 
Minor. It may well be rejefted as unrea- 
fonable, to meafure Intervals by Irrational 
Numbers, when we can fo eafily difcover 
and affign their true Rations in Numbers, 
that are minute enough, and eafie to be 
underftood. 

I did not intend to meddle with the Ar- 
tificial Part of Mufick : The Art of Com- 
pofing, and the Metric and Rhythmical 
Parts, which give the infinite Variety of 
Air and Humour, and indeed the very Life 
to Harmony ; and which can make ilf//- 
fick^ without Intervals of Acutenefs and 
gravity, even upon a Drum *, and by which 

chiefly 


Conclujton. i 5 i 

chiefly tlie wonderful EflFefls of Mufick 
are perform’d, and the Kinds of Air diliin- 
guiOi’d ; as, Ahnand^ Cora?it^ 
which varioudy attack the Fancy of the 
Flearers ; lome with Sprightfulnefs, fome 
with Sadnefs, and others a middle Way ; 
Which is alfo improv’d by the Differences 
of thofe we call Flat, or Sharp Keys ; the 
Sharp, which take the Greater Intervals 
within Diafafon^ as Thirds, Sixths, and 
Sevenths Ma]or ; are more brisk and airy; 
and being affifted with Choice of Mea- 
fures lah fpoken of, do dilate the Spirits, 
and rouze ’em up to Gallantry and Magna- 
nimity. The Flat, confifting of all the lefs 
Intervals, contraQ: and damp the Spirits 
and produce Sadnefs and Melancholy. 
LafUy, a mixture of thefe, with a fuitable 
Rhythmmj gently fix the Spirits, and com- 
pofe them in a middle Way : Wherefore 
the Fir ft of thefe is call’d by the Greeks 
Diafialtic^ Dilating; the Second, 
Contrafting ; the Laft, Hefychiafiic, Ap- 
peafing. 

I have done what I defign'd, fearch’d 
into the Natural Reafons and Grounds, 
the Materials of Harmony ; not pretend- 
ing to teach the Art and Skill of Mu- 
fick, but to difcover to the Reader the 
Foundations of it, and the Reafons of the 


^ 5 ^ Conclujton. 

Anomalom l?ho37iomena^ which occur in 
the Scales of Degrees and Intervals - which 
tho’ it be enough to my Purpofe, yet is 
but a fmall (tho’ indeed the moft cer- 
tain, and, confequently, moft delightful ) 
Part of the Philofophy of Mufick; in 
which there remain infinite curious Dif- 
quifitions, that may be made about it ; 
as, what it is that makes Humane Voices, 
even of the fame Pitch, fo much to differ 
one from another? (For tho’ the DilFe- 
rences of Humane Countenances are vifi- 
ble, yet we cannot fee the Differences of 
Inftruments of Voice, nor confequently of 
the Motions and Collifions of Air, by 
which the Sound is made. ) What it is 
that conftitutes the different Sounds of 
the Sorts of Mufical Inftruments, and 
even fingle Inftruments? How the Trum- 
pet, only by the Impulfe of Breath, falls 
into fuch Variety of Notes, and in the 
Lower Scale makes fuch Natural Leaps 
into Confonant Intervals of Third, Fourth, 
Fifth, and Eighth. But this, I find, is 
very ingenioufly explicated by an honou- 
rable Member of the R. S. and publiih’d 
in the PhilofofhicalTranfalrtionsy N® ipj. 
Alfo how the Tuhe-Marine^ or Sea-Trum- 
pet ( a Monochord) fo fully expreffeth the 
Trumpet ; and is alfo made to render 
other Varieties of Sounds \ as, of a Vio- 


Conciujton. i 5 5 

I 

lin, and Flageolet, whereof I have been 
an’ Ear-witnefs ? How the Sounds of 
Harmony are receiv’d by the Ear ; and 
why fome Perfons do not love Mu- 

fick? 

As to this laft ; the incomparable 
Dr. mentions a certain Nerve in the 

Brain, which fome Perfons have, and fome 
have not. But further, it may be con- 
fider’d, that all Nerves are compofed of 
fmall Fibres ; Of fuch in the Guts of 
Sheep, Cats, ifc. are made Lute-Strings : 
And of fuch are all the Nerves, and a- 
mongfl: them, thofe of the Ear, compo- 
fed. And, as fuch, the latter are affefted 
with the regular Tremblings of Harmonic 
Sounds. If a falfe String ( fuch as I have 
before defcrib’d) tranfmit its Sound to 
the beft Ear, it difpleafeth. Now, if there 
be found Falfenefs in thofe Fibres, of which 
Strings are made, why not the like in 
thofe of the Auditory Nerve in fome Per- 
fons ? And then ’tis no Wonder if fuch 
an Ear be not pleas’d with Mufick, wdiofe 
Nerves are not fitted to correfpond with 
it, in commenfurate Impreflions and Mo- 
tions. I gave an Inftance, in Chaf. Ill, 
how a Bell-Glafs will tremble and eccho 
to its own Tune, if you hit upon it : And 
I may add, That if the Glafs fhould be 

irregu- 


1 54 Condujton. 

irregularly framed, and give an uncertain 
Tune, it would not anfwer your Trial. 
In fine. Bodies mull be regularly framed 
to make Harmonic Sounds, and the Ear re- 
gularly conftituted to receive them. But 
this by the by ; and only for a Hint of 
Enquiry. 

m 

I was faying, That there remain infinite 
Curiofities relating to the Nature of Har- 
mony, which may give the moft Acute 
Philofopher Bufinefs, more than enough, 
to find out ; and which, perhaps, will not ■ 
appear fo eafie to demonftrate and explain, 
as are the Natural Grounds of Confonaiicy 
and 'Di\Jona7icy. 

« 

After all therefore, and above all, by 
what is already difcover”’d, and by what 
yet remains to be found out ; we cannot 
but fee fufficienc Caufe to rouze up our 
beft Thoughts, to Admire and Adore the 
Infinite Wifdom and Goodnefs of Almighty 
God. His Wifdom, in ordering the Na- 
ture of Harmony in fo wonderful a man- 
ner, that it furpaffeth our Underftanding 
to ma,ke a through Search into it, tho’ (as 
I faid) we find fo much by Searching, as 
does recompenfe our Pains with Pleafure 
and Admiration, 


And 


Conclujton. i 5 5 

And his Goodnefs, in giying Mufick 
for the RefreHiings and Rejoycings of Man- 
kind ; fo that it ought, even as it relates to 
Common Ufe, to be an Inftrument of our 
Great Creator’s Praife, as H £ is the Foun- 
der and Donor of it. 

But much more, as ’tis advanc’d and 
ordain’d to relate immediately to his Holy 
Worfhip, when we Sing to the Honour 
and Praife of God. It is fo Elfential a 
Part of our Homage to the Divine Majefty, 
that there was never any Religion in the 
World, Pagan j ChnsUan^ or Me- 

humetan^ that did not mix fome Kind of 
Mufick with their Devotions ; and with 
Divine Hymns, and Inhruments of Mufick, 
fet forth the Honour of God, and cele- 
brate his Praife. Not only, Te decet H\'m- 
7 im in Sion^ (Pfal. 65.) but alfo — 

Sinq^ unto the Lord all the whole Earth. 

. (Pfal. 96.') 

A N D it is that which is incelTantly per- 
form’d in Heaven, before the Throne of 
God, by a General Confer t of all the Holy 
Angels and the Bleffed. 

In fhort, we are in Duty and Gratitude 
bound to blefs God, for our Delightful 

Refrejh- 


1 5 <5 Conclujlon. 

V 

RefreJImrents by the Ufe of Mufick ; but 
efpecially, in our Publick Devotions, we 
are oblig’d by our Religion, with Sacred 
Hymns and Anthems, to magnifie his Holy 
Name ; that we may at laft find Admit- 
tance above, to bear a Part in that Bleffed 
Confort, and eternally Sing Hallelujahs and 
Trifagions in Heaven. 


FINIS. 



^59 


RULES for Playing 

A 

THO KOW-B AS S, 


By the late Famous 

Mr. Godfry E^ller. 

M USICK confifts in Concord/, and Djfcordh, the Concords are Four, 
and of two kinds, viz. Psrfc6t and Impsrfcif ; the Pcrfcft arc 
the fytb and %th \ the fmpcrfeft the i,d and 6 th. The Difeords arc 
Three, viz- the 2 .i, 4 th, and 7th j the 9th being the fame with the 2 dy 
bur differently accompanyM. 

The Flat Imperfect ^th is ufed, either as a Concord or Difeord, but moll 
commonly like the latter. 

The following Scheme, (the Treble afeending by Semitones) (hews all 
Concords and Dijeords, as they if and with regard to the Bal>. 

By Chords IS meant either Concords or Difeords } by Semrto?jes is meant half 
Notes. 



There are other chords us’d fometimes, as the fiat 5 th, but thefe 
llrall be treated of hereafter. 

In common Chords which are the ^d, Uh, »r.d ^th avoid t!ie raking two 
$ths QT two Sths, together, not being allow’d cither in Playing or Com- 
pofition ; and the beft way to do it in playing, is to mov;i }our Hands 
contrary one to the other. 

When the firft common Chord you take is the 3^/, the next mnif be the 
^th and 8/^, and fo vies vorfa, as the folio wlrg bcheme wiu iilulfrate. 

M Ly.am' 


1 6o 


Rules for a Thorow-Bafs. 


Example of Common Chords C 8 
differenly taken* 5 

^3 




5 ? 

3 

3 

8 

8 

5 






1— 4 +- 






uziz^z? — 
I Y r' ^ 


I 


-G- 


■H' 


'B- 


:n 

f 


3 


44 > 


-B- 


lu 

G 


■P 




i 




V 


: M 




U-_ 


I 


i^ 5 «- 


^ The Sixth may be taken with the third and 
eighth, iu full Playing the following feveral 
waj s. 


' 3 

8 

3 

5 

6 

6 

6 

3 

3 

3 

8 

8 


IxampU of Common Chotds and Sixes, taktn the feveral mays above 

mentioned* 

J J I 


'E5^^ziz& 1 f f P^-|=^_ - 

e-»3j^rprf::(;^7erf:=?:l:a;fE^^g=Eg 


:o 


i 






1 


i id— H I i:i__ri 


_1. 








0 L_ 


L- 


t- 




\ 





J 

r 

■ 

Z15 

.7^ 

-a 


■B- 


— 


’B' 


1-4 


1 -: 

1 


— 

— 




I 


“h" 




On any Note where nothing Is mark’d, common Chords are play’d. 
In Sixes mull be obferv’d that when the Bafr is low, and requires a natu- 


ral flat 6th, you muft play two fixes and one third ; if the Baf is high 
and requires a natural fiat 6th, play two thirds and one fixth ,* if the 


play 


11 


3 

6 

8 

3 

6 

8 


Alfo in DIvifions w'here a fixth is required, Inficads 
of two thirds, or two fixes, play the fame. 


pat ui juaip lijaiK u uvci ur unucr any lu \.**'^ — 

fat or farp third to be play’d : A flat or flarp^ mark’d before a Note or 
figure, lignifics elite Note or figure to be play’d flat or fharp, 

EK§m“ 



RuJes for a Thorow-Bafs. 



All Keys are known to be flat or fl>ar^, not by the flats or plac 4 

I at the beginning of a Lcflon; but by the third above the Key, for if your 
’ Third is Flat, the Ke^ is Flat ; if your Third is Sharp, the Key is Sharp. 

All Jharp Notes naturally require fiat Thirds, all flat Notes reqjirc 
[ [harp Thirds ^ the fame Rule hold as to Sixes. 

B, E, and are naturally ft)arp Notes in an open Key ; F, C, and G 
are naturally Hat Notes in an open Kty. 

Difeords are prepared by Concords, and refolved into Concords, which 
are brought in when a part lies dill, and are fometimes uied in contrary 
Motion. 

There arc three forts of or full Clofes, as when the Bajs falls a 

5th, or rifesa4th, viz the Common Cadence ; the 6th and 4th Cadence ; 
1 and the great (or fulled) Cadence. Each of thefe may be accompany^d 
! 3 dIffercHt ways ; as will befeen by the following Examples. 


The Common Cadence 


{ 


8 

S 

4 


by 

#3 


4 #3 
8 by 

5 5 


4 ^3 

8 by 



-teSiit 


1 ■' 
4-! 

‘-r-j ^ 
1 . « 

4- 


r-4 



^-—0 


— Q-tt 




C8 8 by 

4 4 #3 

^ ^ 1. ^ 

The 6 th And ath Cadence. <65 1 

8 8 by 

4 4 #3 

^4 4 tj 

^ S 

8 8 by 



Rules for a Thorow-Bafs. 


The great Qadcucc. 




Tn ill Cs(??nces v;hatfoever, where the Biifs rifesa 4rh, or falls a cth,f 
Oblerve, that the 4th falls half a Note into a fharp Third, and the 8th 
a whole Note into a fiat 7th. 

There is another Cadence call’d the 7th and 6th Cadence, which is 

counted but a half Clole, and if the < 5 th is fiat, is never uftd for a finall 

C'ofc/ becau.rc it dees not fatisfy the Tar, like as when the falls a 

5th, cr rifes a 4th, tis often introduced in a piece of Mufick, as the' 

Air may require ; and when it ends any one part of a Piece, ’tis in order ( 

p begin a new Movement or Subjeft ; The 7ch and Jharp 6th may be ufed- 

for a final Clofe, if the Dciiga of the Compqfcr requires itj but ’tis very 

rarely done. 

^ / 


2^efolhi£i}Jng Example will jhew how both the -jth andhSth, and Jth and 

are us d. 



I 


Rtiks for a Thorow-Bafs. 


163 


Otrctve when a Difeerd happens in the higher N«te, leave the Cn6- 
cord out it 


C The Flat \ 6j 2 
and ^ 4\6 
/^fecond. T 2 j 46 

CThe Flat 
^th and 
1 6 th join’d. 


here inilcad of the 6 th, the C The Shar(> ^ 6 
Sth m»y be added, but 4?^ and S ^4 
then it ought to be mark’d- 1 fecond. C 2 

5|5 

Zs5k 




6 03 b5lhere the M is o-^ The perfeftC 6 
bS bj 3|mitccd unlefsicbc^ Sth and 6r^< 5 
3lb5l o'iinpafrmg Notes, cjoin’d. /5 


2 

6 


^4 

2 

6 


4T4 

here the ^th 
if one think 


fittoplayfull 
maybe added. 

The perfe£l fifth when joyn’d with a fixth Is ufed like a Difeord- 



The Sharp 
yh, when y 4 
the Bafs ^ ^ 

lies ftiil. 

The 9^» re- '^3 
folving into S 9 8 
the 8 r/;, { 5 


4!here to play t When the id 
2jfull thef/ri!^4and r^th 
^,']\ 6 th may be < mark’d abovei3l4 
added. one another. 


7[5 
4 7 


the fth Sharp 6th 
may be u led, but 
then it ought to 
be mark’d- 


^ |9 8 The and dr^\^j-|4 

5 15 Iwhen the Sr?/y 4’,(5U 

9 8' 5 'feends by degrees, i 2'4!6 


The 6/‘^and 4?^ 6|S 4 
when the Bafs ^ 4;6 8 
skips or lies rail. ( 6j4P 



a 


The Tth and 5 ^^ hap- r 7 
pening juft before < 5 
the Cadence Note, v 3 


1 liorow-Bafs, 


3[5 

7H 

5 I 7 


here inftead of the third, the ninth U 

nrirk-i «oughf 


The extreme Flat fevemh* r bj 
and Flat fifth happening juft-^ b$ 
before the Cadence Note, v 5 


3 

b5 

b7 


b5 



the extreme Flat feventh is 
the fame with the shin 
fixth. ^ 


The extreme 
Sharp fecoiid 
and fourth. 


{ 


#2 


<5 1^2 
6 


4 

#3 


the extreme Sharp fe 
cond is the fame dis' 
tance as the Flat third 


' fe- r 
dis-^ 
lird. V 


as follows. 




f-f 

¥ 


-9- 

I 

fi 

S 


• ' I 

n* — 
— ! — ‘ 


' ^' 1.1 1 t ^ ^ * 



The j.fh and ^th 
refolving Into 
the 3^/ and Sth. 


r 4 

9 

9 8 

^ [ 

The ^th and 7 th C 

9 S j 7 6 

3 

1 5* 

8 

5 

4^3 

refoJving into y 

1 

9 8 

i 

5 

4 > 

5? 8] 

the 3^1 and 8 th- ^ 

3 198 

7 6 



When the Bafr sfeends or defccncls one or two Notes, move } 
icend together. Afctt 


I- 



Rules for a Thoro\v-Ba(s. 


After a fixth where the eighth lies in the middle, you may either af* 
cend or dcfcend. 

Example, 


!y.fefest3:jEi4-E=jitt 




y-rrF-H 


iziz:| 

a— --0-- 

e| 


3z:3 

,_z^[ 

f- — p- 


1 

s> 

-p-4 


6 4 


^-pzJzt~iZ! 


«r •■' 

— f — Z — 


— — 1 

— E_i p-E- 

;=t"5 

=t=f:=t 

-E ^ — 


#J L 


zizf- -z Jr ^z|z j 

-A-' in--- •#^— ;ze.zi j ^z; zoz 



A^- 


6 b<i 

t?-:zzzzz:: 


-f-L « t » 




b 5 


4#3 


:tt 


t 


^--§z' 




t'DiUiizz 


Example of pafiug Notes in Common-time^ 

■pz^fesisg^f Sg PIf E® e%!Z 

ES^i ES;;t;g; 4 |-.gjE|^ 


o — 

" 

Q 

c:-: 




! 

r- 


— 15 ^ — fi 


p — p- 


+ .-iC 


t 1 !^ft--|-- 

^EZZpZZtt 



i66 


Rules for a Thorow-Bafs. 



Rules for a Thorow-Bafs. 


157 




Rules for a Thorow-Bafs. 


1 68 




Of Natural Sixes, 


Play common Chords on all Notes where the following Rules dontdl- 

reft you ocherwife. i . v 

' The natural Sixes In a Sharp Key are on the half Note below the K®y> 
the third above the Key> and on all extraordinary Sharp Notes out of the 
Key, if not to the contrary mark’d, or prevented by Cadences. 

The natural Sixes in a Flat Key are on the Note below the Key ; tne 
Note above the Key, and on all extraordinary Sharp Notes out ot tnc 
Key,- if not to the contrary mark’d, or prevented by Cadences. 

When the either in a Flat, or a SW Key, afeends or dclccnds 
half a Note, Sixes are proper on thefirft Note, falling on thclecon , 

lefs prevented by a Cadence. , ^ 

When the Bafs cither in a Flat or Sharp Key, defeends With a common 

Chord by thirds ; Sixes are proper on the falling thirds. 

When the Bafs either in a Flat or a Key. afeends With a ^omwoii 

Chord by thirds ; Sixes are proper on the rifing thirds. In * ^ ^ l. 

third above the Key generally requires a lixth to prepare the Ca » 

fifth being repugnant to the half Note below the Key. 

Seldom two Notes «f«nd ot defecad but one of them h.th 

Sixth. 





Rules for a Thorow-Bafs. 


\6g 

Example of Natural Sixes and proper Cadences ht a fbarp 

Key. 





17P 


Rules for a Thorow-Bafs. 


^xampu of Natural Sixes attd proper Cadences f« a jfjj 
' Key. 






Rules for n Thorow-Baft. 


171 


Now all thefe Sixes mentioned cither m a Flat . 

cnly to be obfcrved in the Key you play in, but 
Talnces you are going into : And for the time you keep in that 
obfcrve the Rules for Sixes as tho’ you were in the Key your LefTon is 


^ Whcr^e'lhe‘6^/5 afeends a perfeft fourth, or defeends a perfeft fifth, 
Sixes are generally left. 


Other Rules for Sixes are where the Bafs moves by degrees downwards, 
then thefe Sixes may be play’d on every other Note, 


Example 



The Compofer fefpcclally In few parts) may Compofe as many Sixer 
cither afeending or defeending by degrees as they think fit, but then 
they ought to be marked. 



Now 


17* 


Rules for a Thorow-Bafs. 


Now here follows an Example where two ^ixes jir#. i 
.^.jnd thn defending becule th=y .re Ihort C. W fnft«d 

In a fiat Key Defcenditig, 




l3 =^- 

ll 



"id 

1- : 


A fcevding. 


(j 


iizzr 


=|d=zi|r|ii| 


44 


In a fijarp Key Defeendhig, 




i=d-Pr^ 

111 

) I 1 
1 1 1 

=S-tEk?-E-t=p 



’P|: 




6 6# ^ ^ 

iiiiiiiipip 




Afcending, 



Rules for a Thorow-Ba(s. 


When the B«fs liet ftill, the Seventh is generally refolved into the 
tfth, and the 9'S into the 8th The Example which follows, Ihews 
how Dilcords may be Reiolvcd feveral ways. 

Example. 






£ 2 ^ 6 ^ 6 9 6 ^4 


ZO 








by^degrees^**"’^'” '*'*'*" **** Defcends 




~Cz: 


Cl 


2 

-i "■ 








The Common v>ay. 




9- 


icizt 


P-4 


76 

=iP: 


S6 

p: 


U Q 


4 .. 


76 





::zD 


Natural and Artificial. 



All 


174 


Rules for a Thordw-Bafs- 



When the Bafs Afcends hy Degree:* 








56 ^ 




56 56 

:zq: 


0 — 


56 

0 


b5 




— *1 


^z^.rpri=pr^=l:;#:tp=ptt---- 

* * bS 



__'^6_^_j6__9« ^ 


Examfle cf att forU of Difcords in < f»t Key. 


S^7 


Ciziz:: 

zzzo:; 


-0--f ®z z 


zzDz:t“:-0 — 


3 6 76 , ' 



Rules for a TIiorow-Bafs. 179 




150 



Rules for a i liorow-isals. 




1 




r □ 

n -iS 


5 


□ 

"i n 



. < 1 




•' ■ 

L J 

J i 

t1^ ■ 



l (#5^)#3 

i » 

‘ 

a 

1 

87 6 
^5 “V(H) 





^ — J 

— ' 

' "ff 


:□ i 

Ih. — 1 — — — 


Where the Figures are fee in parenthefis, thofe I wou’d have only dropt - 
to fee off Playing* 


Example of all forts of Difeords in a farp Key. 



Rules for a TIiorow-Bafs 


i8r 





J 


0 2 


iSx Rules for a Thorow-Bafs. 





Rules for a Tliorow-Bafs. 



To make fome Chords eafie to your memory, you may obferve as 
follows: A common Chord to any Note makes a $d- -5th.and 8th. to the 
3d. above it, or 6th- btlow it. A common Chord makes a 4th. 6th. and 
bth. to the 5'rh. above it, or a 4th. below it. A common Chord makes 
aid. 5th. and 7th- to the 6ih. above it, or a 3d. below it. A common 
Chord makes a 4th. 6th. and 2d. to the 7th- above it, or a 2d. below 


A 2d. and 4th marked makes a common Chord to the Note above It, 
cbferving the 5th. perieft or imperfeft, according to the Key, as alfo an 
8th. 5d. and 6th. to the 4th. above it, or 5th- below it. A Sharp 7th. 
marked, where tbe Bafs lies ftill makes 8th. 3d. and Jharp 6th. to the 
Note above it, and 5th. 7th. and P^arp 3d. to the Note below it. An 
c.xtream P^arp 2d- and 4th. marked on a Flat Note, makes fharp sd. 5th. 
and 7th- on the half Note below it, as alfo a Jharp 6th. 8th. and 3d- to 
the Jharp 4th above it, or fiat 5th below it ; the fiat jth. and txtrenm fiat 
7th. marked on a Sharp Note, makes 3d. fiat 5ih- and 8th. to the 3d. 
above it, or the 6th. below it, as alfo an 8th. 5d. and to the fiat 
5th. above it, or Sharp 4th- below it- The 4th. or 9th. mark’d is the 
perfcrT 5th. 6th. and 3d. on the whole Note below it, and the fiat 5th. 
6th, and 3d, on the half Note below it, as alfo 3d. 7rh- and 9th to the 
3d. above it, or 6th. below it, the 9th. and 7th mark’d is the 5th. 9th. 
and 4th. on the 3d. below it, and the 6th. 3d. and perfect 5th* to the 
perfeft 4th. below it, or the 5th. above it, and 6ch. 3d. and fiat 5th. 
on the perfeft .4th. below it. 



9 

7 


4 

9 


9 

7 



• 




d 



I 


— 

- — 

r 

U -i 


-B- 


6 9 ^^ 

5 7 bS 

_i 




a 


'T 


II 




.. X 

. . 


fiitt 5th. and J)->arp 4th. the extream jharp 2d. and fiat 3d, The 
extream 7th, and jharp 6th- upon any fretted InOruinenrs, or Harp- 
freord. without Quarter Notes, arc the fame thing in diftance, yet the 
dillinfHon is as tollows 



184 


Rules for a Thorow-Bafs. 



There are fome other Chords of the fame kind, VU. the extream flat t 
4th being the fame as 3d. and the extream jharp 5 th. the fame I 

as a 6th. but f are only ufcful In three Parcs, and will . 
not admit a 4th* the diflinfticn is as follows. 





j- hj_ 




b-vth. ^3d. :^5th. b 5 th. :^5th. b^th. 


id- 55- 

pri 






This extream flat 4th* admits a 6ch. for the fecond Part and Is re. | 
folv’d into the third, and the 6th. into the Jlat 5th. Theextream Jhan I 
Sth. admits for th« fecond Part a third. 


Of Tranfpoftion. 


Before any one can pretend to Tranfe-pofe from one Key Into ano- 
ther, it is neceffary they Ihou’d know all the Flats and Sharps naturally 
belonging to all, at lealf the PrafUcable Keys. 





Note: The Keys which are mark’d with a Crofs undtfr them are feldom ufed. 


»■ 

I 

I 

* 



\ 

/sdditional flats and fliarps In Order, 

The reafon why I call the Flats and • 
Sharps One, Two, Three, dt'r. is becaufe 
where S is flat E may not, but 'yhcre ' 
E is flat B muft. The fame reafon holds 
good for J])arps, 


Rules for a 1 liorow-isais. 


185^ 


Next It is requlfite to be acquainted with the feveral CUjfs and their 
* Removes, and laft obfcrve, that all Vlat Keys muft be Tranfpos’d Into 
flat Keysj and into jharp ones : Which Keys are known according 
ro their 3d. which is cither Flat or Sharp. 

FfautCVif^s, CfolfautCYi&.CfolreutC- The Natural Key. 

6 7^6 6 S 


im 


FB-3: 



1 -» • 

■ _ ^ . 

■?=r: — 

:n 

I— . 



— 










A Note higher. 


A [harp 3d. higher. 


6 7^6 6 ,, 


.^3 


6 7 ^ 




A flat gd* higher. 

ZLZ'gl 

A 5th. higher. 


e __ ^ 


A 4th. higher. 

6 7 i «;6 6 


5 ^ 


■:± 

zd^z: 


-d 


4 ? 


A y^;zr/> 6th. higher. 

r.'iL^ . 6 . 7^6 6 


K-C:-~z:z 

5fr:Zi:I:d“ggi: 













tj; 


I— 

I 

-j — 

— -• 

[-0 

_g 

yiD 



A flat 7 th higher# 


A flat 6tli. higher- 

J 1 ? ^ ^ ^ 7 ^^ ^6 45 







I ID 


-0 


:r 


In a flat Key, the Natural. 

6 7 r, 6 


A 2d. lower. 


i l-Y-r 


hCipiZ ~ I P~^ ^ 

A fiat 5 d. lower. A Jh^rp 3d. lower. 


!«-,ZIZZZ- 'tz 




z^:; IgSgCpz^; ^z 2; :-z|: ;: dzE : : 




A 




A 4th. lower. 

6 16 ^ 6 


K7ues for TTiioro^Ssr 

A $th. lower* 


‘#-±rb:t±rl-i b-f 


-42.— 



— 

0 

s 

lb 

:i 



h ; 

it 


A Jharp 6ch, lower. 


A flat 6th. lower. 


r—~r- Ir- 'i -*^tt 




A /harp 7th. lower. 




r*" 


4*3 


6 76 ^ 6 

iic|£}aDj^:Bt:p:±| 


:□: 


A i^at 7 th. low’er, 

6 67 



2; 

'F 

r- . 


m 

— 


.-1 

p — i" 

11,— — 






2: 












U_d. 

[ntJZ 


t 


You are to obfcrvc what Flats or Sharp, belong to all the Keys, and 
imagine the Clif that puts you in the Key you have a mind to Play in; 
and what you find too high or tw’o low, according to the Compafs of 
the Inftrument you play on, you muft Tranfpofc an 8 ch. higher, or low- 
er which is eafie enough to be done* 


Of Difeords, hoip viany ways they may be prepar'^d aytd refoWd. 


The 4th. when joyn’d with a 5 th. or 6 th* and is generally refolv’d into 
the third, may be prepar’d by a 3d. 5 th. 6 th- or 8 th. 


The 4th- prepar’d by a 3d. and 
Refolv’d into a 3d. 



The 4th. prepar’d by a 5th and 
Refolv’d into a 3d. 



Rules for a Thorow-Baft. 



The 4th. prepar’d by a 6ch 
and Rcfclv’d iato a 3d. 



The 4th. prepar’d by 
8lh. and Refolv'd into a 



The 4th. on occafion; 
Refolv’d into a 6th. 



45 


The 4th Refolv’d into the^dzQ,:; 

3d. feveral times before yoU i;j^- "fj 

come to the Cadence, 





- 4 — 


4 3 


-44. 




tt- 


Thc 7th. may be prepar’d by a 3d. 5th* 6ch- 7th. or 8th. The 7th’ 
when the Bafs lies ftill Rjefolves into the 6th* and when the Ba[s falls 
Five Notes, or rifes four Notes it Refolvcs into the 3d. The 7th. 
fome times Refolves into a cth. and then it is in order to a Cadence ; 
fo that the Bafs rifes one Note. I have feen the 7th. Rcfolv’d into an 
8th. but it lounds fo like two Schs, chat it makes me utterly againft 
it. 




The 7th* prepar’d 
bv a 3d. and Rc-\ 
folv’d into a 6ch. 


r r The 7 th.prep,r’dj^r " 

1 il^y * Sth. and ReA^=;T*-*n 

j=q=|:: folv’d into » 6th. - 


P 


i8S 


Rules for a Thorow-Bafs. 




The 7th, prepar’d 
by a 6th. and Rc-\ 
iolv'd into a 6th- I 



Example of the 7th 
Refolving into the 3d. or| 
the 5 th- fome times. 





There 5s fome times two 7ths- Com* 
pos’d one after another, but it is call’dJ 
a Licence in Mufick and commonly 
in order co a Cadence* 



6 

__ 

f-r*- 

7 7 ^ 


P 


The 9th. Is generally prepar’d by a 3d- or a 5th- and it may be by a 6ch- 
or 8th. bur not fo naturally. The 9th when the Baft lies (Hll, Refolvcs 1 
into the 8th* The 9th- when the Baft falls a 3d. Rcfolves into a 3d. | 
The 9th- when the Baft rifes a 3d- Rcfolves into a 6th- The pth-mayjj 
Refolve into a 5th- but not fo naturally as the othcrj and then the Bajf 
rifes four Notes. 


The 9th. prepar’d 
by a 3d. and Re- 
folv’d into the 
8th- 



Thepth-prepar’i 
by the 5 th- am 
Refolv’d into th 
8th. 



^^^^y^T^Triiorow-Bals* 


w 




folv’d into the 
8 th. 


^ ..Rcfolv’dinto thc^ 

^ — 8 th. 




lip^ 


— tt 


of the 9th. Refolving into the ji .nd 6th- but rarely into a 5*.’ 

Example'* 






— . 

-p-1 





The Flat 4 th. and 2d- and Sharp 4th. ^d. is 

when the Bafhn a driving Note defcends a half 
Sharp 4 th. always Refolvcs into the 6th. as docs generally the Flat 4th* 
but fome times with the Flat 5th. the 2 d. Refolves into the 3d. 

Where the feveral driving Notes defeend by degrees. 

Example* 

* ^ 4 4 ' . ^ 

Si 6 i 4 4 4 ^4 


4 « , < 

S i h S 4 O 

- ” ^ Si b 5 


Another Example. 


liiliiigi* 



ipo Rtdes for a Tharow-Bafs, 

The 9th. And 7fh. mark’d above one another, may be prepar’d by 
the 3d a *d 8ch. and Rcfolv’d into the 8th and 6th. the Buj) lying ftill 
and ^ome times is artificially into the Flat 5th- and 3d- and the Baft 
falls a blat 5 th. 


Example- 



The 4th. and 9th mark’d one above another is beft prepar’d by the 3d. 1 
and 5th and Relblv’d into the 3d. and 8th. the Bafs lying ftill, feme r 
times artificially into the Flat 5th. and 3d. the Baft falling a 3d. and 
feme times into a 7th. the Baft rifing four Notes contrary to the ♦ 
lie- 


Example. 



The 4 rh. «nd 3(5- mark’J one above another 
wlun the Brf! afeends by degrees, the dih. and 4 th. with a ed. is com- 
monly us’d when the Bafs defeends by degrees. 


Example - 



The 


Rules for a Thorow-Bafs. 19 1 


The (5th, and 4th. where the 8th. is joyti’d is commonly msM when 
the Ba{s lies t\lll ia a Shsif^ Key, or when the Ba[s either dcfcciKis four 
Notes, or alcencls five Notes. Exxmple. 



The shetr^ 7th. when accompany’d with a 2d, and 4th. is us’d, when 
the Bafs lies ft ill in a Flat Key. Example* 



The extream Sharp 2d- and 4th. p^rnerally prepaies a Cadenct^ The ^th. 
and 7 ch- and the Flat 5th. and extream Flat 7th. arc generally the fore 
runners of a Cadence- Example^ 



>r 


192- Rtdes for a Thorow-Bafs. 

The perfe£^ 5th. and 6th. joyn’d is commonly us’d as the 7th. and 5th- 
before a Cadence^ as alfo when the Bajs defcends by 3 ds. 

The Flaf 5 th* may be joyn’d to any Sharp Note that requires a 6th. 
unlcfs it be contrary to the Key, or it be mark’d otherwife. 


$• 




The cKtre&m Sharp, and the extream Notes belonging naturally to ' 
either F/at or Sharp Key : The extream Sharp In a (harp Key, is the half i 
Note below the Key : The extream Sharp in a flat Key> is the Note I 
above the Key, unlel’s taken oif by an additional Sharp: The extream I 
Flat in a Sharp Key, is a 4th- above or the 5th> below the Key ; The 1 
extream Flat in a flat Key, is a 3d. below or a 6th. above the Key. 

The extream Sharp being too harfli, and the extream Plat too lufcious '■ 
onlcfs taken off by an additional Sharp or Flatf or what is excepted in the 
following Rules ought to be doubled. 

On either extream Sharp or Flat Note, or any extraordinary Sharp or 
Flat Note out of the Key, that requires a common Chord, you double 
the 8th. in Composition, or Playing four Parts. If the extream Sharp, • 
or an extraordinary lharp Note requires a natural Flat 6th. you leave out 
the 8th. in four parts, and Compofe, or Play two Sixes and one third, 
or two thirds, according as the Bafs is too high, or too low. , . « , 

If the extream Flat or any extraordinary flat Note requires adth.inltead 
of double Sixes, or double thirds, you may Compofe, or Play in four 

. a o t \ ^ 

Where the efittream Flat, or an extraordinary flat r » 3 6 

Note happens to make a 6ch. to any Note, never <6 8 3 

double that Sixth. 3 


Example in a Sharp Key. 





Rules for a Thorow-Bafs. 


IP3 


Example in a Flat Key. 







4 


Rules for a Thorow-Bafs. 


IP4 








44k; 


6 

4 


; j_:sLZii; : 






Same Lefov.s ivhere the F. avd the C. ClifFs Interfere one 
with the other. 





Rules for a Tliorow-Bafsi 





a 


Rules for a Thorow-Bals^ 

In thh Lejjon the G, C, and F. are aU m^d, 


ip6 







Ktdes for a Thorow-Bafs. 







Q, 2 


/ 


Rules for a Thorow-Bafs. 





Rules for a Thorow-Bafs. 







li 


200 


Rules for a Thorow-Bafs. 


6 

9 5_^S 


4 

Si 

i-Q 


4 4 4 

'z 6 2 


7 




%3 


-fefel 



I fhall here add fbme fhort LefTons by way of F^^gdJ^g. to 
make the whole work Compleat. 



*76 


TT 



6 

4 5i 76 ;^5V 


r^' 




;es±ee 


s 

4^9:3 


-d- 






t 


20t 


Rules for a Thorow-Bafs. 








203 


I 


Rules for a Thorow-Bafs. 


56 6 S6 6 _ 6 76 


7 # 




Tolt- 




4- Pjiks for a Thorow-Bafs. 




Rules for ct Thorow-Bafs. 4? 





4^?# 6 b5 _£ 7__ 


5:ti:p:[:: 




-t- 


■t- 

I 




— + 






G 2 


2o5 Rules for a Thoi*ow-Bafs, 

^ks for Tuning a Harpsicord or Spinet. 


Tune the C-fol-fa-ut by a Confort Fitch-pipe, 

1 1 — 1---1 — :-4-t4-a-f_o — a — : 


-Q- 4b- --- 4§: — 2- o 

r *- 0 - 4 — R - 4 — R— 4 ^R — 4 ± 4 fc^— a_U_ 


- 0 - 


{^ 0 - 4 -^— Wr-|— 

Q—.'fTrt • 0 - 


zDz: ;^ 8 ' 


-t — — —■ : 


- 0 - 
R- 4 - 0 - 
0 0 - 


H=zr- 

0 — 


s 


^- 0 - 


:zd: 

:zo: 


-g_| |Z ° 

z[:^zt:zzz; z:z?zp?z|zz:: 


4 ^ 0 - 


-toz: 


^:„z: :zDz 


_ 5 _ 4 -Q-tbQ-j- 0 -||Q-- 


-e- 


^zizailfeZ 




Obferve all the Sharp Thirds muft be as (harp as the Ear 
will permit 5 and all Fifths as flat as the Ear will permit. 

Now and then by way of Tryal, touch Unifon Third, Fifth, 
and Eights^ and afterward Unifon Fourth and Sixth. 



FINIS. 







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