353 / 868
3
20
1\1
u
s
,rocal
tO
the fe ngths. T hJi
in
( WQ
ll rin~s
of
lhe
(ame
man er,
20d
(qual diamcters;
ir
oee be d.oublc
lhe
lenglh of lhe ol her. il will give h. lf lhe qUlek ncfs of
pulre$;
th;H
¡s,
hi\1t
t h~
number
of
pulr~s
.In
tI.lefa me
time : or, lhe Icngths be,ng
OIS 2 (O
J,
lhe VlbratlOns are
as
1
to
2.
O n
this
axiorn are
dCLnonflrated lhe onJcr
~Ild
proportions of
t..he concords, as follows.
Proporlion ofIhe Ofla"e.
e
Jl.---------- ---.---B
L et A
B.
a mur,eal flring.
be
divided equally in
t.
, nd
Hopt there:
e
B \Vi II [ou nd an oaave
fa
lhe whole a r
apen llri ng A B. Now,
e
B, A B, are as.
J
[O
2 : Ihere·
fore, lhe vibrations Me as 2 la 1 ; that 15, lhe propar
tion of lhe
oth.\'c
a r
diaparon
is
dou.le,a r 2 tO l .
Q.:.
E, D.
A_
Prcporlion
p
the 5th.
e
---·-- - B
L el A
B
be di.ided inlO three equal p.m, and (lopt
in
e: e
B
will found
a
51h tO the whole or open flring .
N
OW,
e
B
is to
A B
as
2
to
3: therefore tia
vibra–
lians are as
3
to 2;
tha(
¡S, lhe proportion of
the 5th,
or
diapecte, is fefq uialtcral, or
3
tO 2 .
Proportio" of the 4th.
e
A
·--B
Lct lhe flring bc flopt i.
C.
",hich is a 4th part of
,he \Vhole:
e
D
",ill found a 4th to the whole A
B
or o.
peo tl ring. Now,
e
B
is to
A B
as
3 10
4:
therefore
tJl~
vibratíoos are as
4
to
3;
or,
lhe
proponio
of the
..th, or d¡atefrarOD,
is
4
tO
3-
P",p>rlion of the¡¡'arp 3d.
e
A---
---
-
___B
Stop the fl ring in
e,
the 5'h pan:
e
B
will found
• greu er 3d 10
A B.
But
e
B
ísro
A B
as
4
lO 5. There–
{orc
the
vibrations are as
5 tO 4;
or,
(he
proponion
of
the fharp 3d is as 5 104 '
Proportion oí theJlat 3d.
e
A-----__·_B
SIOp in
e
lhe 61h pan :
e
B will found lhe lell'er or
S".l
ll~lrd.
!lUl,
6<.
Therefor. the pruportion of lhe
/lO(
Ililrd
15
as
6 to
S.
Proporlion of thegNat(r or ¡barp 6th.
e
'~----------------'----B
e
D
+lh,. of A
8
\ViII found lhe gr.ater 61h. Tlterefcre
lite propo. tlon uf lhe fllUp 6'¡¡, i,
as
5 10
3.
1
e
K,
Profortion oí Ihe
I"¡[a
or
jI,J
6th.
e
A---- - -----·----B
e
B, '.lhs of A B, found, the k ll'er 6th, Thererur.
lhe proportlon of the
flH
6th
is as
8
tO
5.
Ir
thefe dlvifions of [he Itring, whofe numcrators are
the
(aOle,
or
unlty,
be f".'t down
in
fraélions , in [hc
na –
tural order of numbers, t hus
~
i-TT'
ec.
and rcd uccd to
a commoo denominato r, l he hHOlonical proportions
\ViII
appear in the
(.unt
(ucceffion of concords,
as
im-cfligated
tiy
{ound in lhe divifion oí
the
fame line; and the n um.e·
rators, being by tbis red utl ion
as
whole numbers,
\Vdl
n and thus,
60. 40, 30, 24.
6c.
For the COOlmon deno–
minawr,
120.
anfwering
te
(he whole or open n ring; the
rela[it"e proponion of the fame
[O
lhe firfl: fraétion, and
the relatjve proponion between e3ch two fucceflivc: frac·
tioos,
will
expre(s the propon ioa of lhe harmonic chords.
T hus
'ti;'
{
o r 8th.
~~ }
or
5th_
*
i-
or
4th .
{~!r
or
Iharp
3d.
H~Dce
\Ve ddcover relative harmonic..l propor–
tion
in
numbers : whicb ¡s, As lhe
6rtl
is lO the third ;
fo is lhe uill'<rence of the fi rfl and feconu to the d,ffereDce
of the (ccond and third . For, r.ducíog lhe fir lt three
nUlllbers to t be Jowefl: terms, and invertlllg, they will be
2.3.6. NolV
2:
6::
1:
3.
Again, redueíng the fecond
thrce. theywillbt3'4. 6 . Now3:
6 ::
I :, .
'Vhere–
ever Ihi5 proponíon obtains, the numbers bear harmonical
or muGcal relatioo.
Furth~rJ
the reétangle, or q uotient
of the
lidl
and third num bers multiplied, being divíded
by the excef. of t\Vice the lirfl aboye the fecond, Gnds a
fou rth proponiona!. T hus
3, 4, 6 ,
&i ven as abovc;
3 multiplied by 6, gives 18: whieh divided by
2 ,
tbe
exct.fsof twice
3,
the
firlt
above
4
the fecond, gi\TC59.
the fourth muGc.1 prOpOrtíOD'!. T hus 3, 4 , 6,9: And
of thefe, lhe r,rtl is to the fourth, " lhe difference of the
fi rfl .nd feeond is
lO
the difference of the third .od
4
th .
So
3: 9::
t :
3.
T he harmonic proponíon o," th ree numbers
in
tbis
c:uural fu ccdJion of fraétions, extends as far
as
the chord
of the n.t 3d. ' Vilích thírd, bdng
.r
of the whole numo
bo!r, limils this equality of proportlon, feeing that the
numuer
7 is
no aliquot pan.
BUl
as to tbe fonrth pro.
portional, it cannot be found even from that number
which
exprell'es the f1t>. p 3el, which ís fli ll of fhorter extent.
This
lirniution of proponion lhen e); plains
lhe
extent
of
h armony, and likewifc hecomes the principie of the
(ame ;
as \ViII be {cen in the deúnition of harnlOny .
Hcnc~
it is evident, lhe remainins:! concords o f the
~Iarp
6th.,
w~ich
is
+,
ami of the
fht
6th, or -}, a rc out
IOcluded
10 tl1ls
equality of proponion.
Thcre are
the
concords, lheir oruer and proportion,;
any
one of which rounded togclher with th e opeo Hring,
is concordant \Vith it. anJ produces htlfOlooy.
E X:l.nlple of the names and ordL'r of lhe interva ls
in
concord " ilh the Open Itrinlt or ba{s. anJ. thc
[cllIitoncs
l.:onlJined in
cach,
!JlIlJic~
Plato,
No.
~.
A gain, n vo of lhefe concol
'u.mtintc rv~Js,
clnlcly, the
5 1h and -- 81h
SIt>rp
3d
. OU --
Sth
fl.11