ft.
4
f
\
, ♦ /»* -
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:[::
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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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