357 / 868
M
u
s
fOl~nJs
lhe
5th) \I.,l::ch
t \\
o are lhe conflHucnt
inlerv.lls
of lhe c(l,I\!c.
And
lhe
fame di\,jfi'ln of
lhe
founds is
conn.\Ody
prc:·
fCl\fcd,
ir
lhe
J~ngth
of
.lO 0 (1.;\\c:
be c:qn311
y d.viol'J
be:–
t w('('n any
t \\'O
Hops.
A g1in, lhe
Ic::n~d\
of
3
5'h betwecn
t \l'O
fiops. or
nu' .nd Oop, cqtully di"idtd ; ,he hal f nex, ,lte bridge
~I \'es
the grctltcr 3d. th.: l>t her h:t1f lhe
ldr...:r
3d.
And
"galo,
lhe
l~nr:' h
,"lE the greater 3d, thus dividcd,
,f.iv:.s lhe grC:Hc:r ancl ldrc.. r tone.
AnJ
the grc:atc:r wnc',
lcng,h, equ.lly dil'ided ,
~
ns lhe gre"cr .nd lelfor fe·
mitone. Ar¡d Ihe
kn~th
of ,he
g leat <:1'
feOlironc, equally
dl\id«l,
gives
tht
fo:m
Is in proporuon
255
ano
4. 'file
grcatcr ¡olerl al OCIO:.!. nfXI
(he oridge,
:1ml
fa
continually.
H~ncc
lhe nC('lITiIY
or
the
gl t"ato:!r
and
Id:cr
tones and
Cl'mitones in mufie is
c\"id~nl
; anu
lhe trulh
confi rmc:J ,
which
is alTc.!rted
In lhe
2cl
corollary.
Now, in the rliatonic
Cc¡ le, whcrl.'foever
the femitones
lie. ,hat ¡s, whethl'r
th~
air be in a
f.a t
or fharp
key,
lhe
graver pan of the tone
will
be thr lerrer fc:maone,
and the acutcr the greu Cemitone ;
and
in tl'c ch rool.l–
ti~hich
. fceoc. by f<mitones, lhe ereater . r.d Iclfer
(emitones will, for the fame reaCon. fll cCeetl each other
alternate1y,
Wherefurc
I
if any feri es
of chrom.nic
notes
be removed a fc:mitone
hi~her
or lower; it
mu!l
h~ppen,
thu ,lte I,Ifer femi ,one will fuccced inlo
,he
plóCe of ,lte
greuer,
a.ndthe gre,uer
into
the place
of
che
letrer.
Hence
ditronances
\ViII
h;ppen
in the
c1ialOnic rcale. :ilS
beiDg compofed of the
fa
me nuterials with the chromatic,
if
lhe key be inj udicioufly changcd by tr.nfpofi lion. For,
as the diffonancc \\I¡1I oc cvídent. if the tt:tnl"po(jtion be
b y one femiten! : fo ,lte difproportion will Oill 'ppe.r,
if
, he remo,,1 be by any odd numbor of fe OlilOn"S wi,hin
,be complf. of lhe 4,1t .
A l the proportic ns of the concords
ha\'e
beeo cemon –
flrated from ,he dil'ifion of a line; fo are 'ltey likewife
t o be focod in lhe geometrical propon ion. of folid bo·
die5, aod ,herefore m. y be illuOrated by ,he f, me.
' Ve (hall begin wi,h lhe propon ion of , he 81h.
T he propon ion of ,he 8,h bcin¡ tite compound pro–
portion5 of , he 5'h Jnd 41h, i5, by coroll>r)' of lhe 3'I,h
propo(jtiOD of Archimedi!s,
i\S
lhe whoh: fu pe,
r.~ies
of a
r igh, crliader dcfcribed .bou, • fpltere, ' o ,he whole fu·
perficics of an f'c¡ uiI3t( ral
cyJinder
infcribed as
2
te
l .
F or, {he circumCcribcd is to the
fpheric
fupedicies as
12
;. lO
8
(by ,
2
Qf , hi5,) bu' ,lte
f~lteric
i5 ' o ,lte illfcribed
as 8 is
tO
6
uy
thill
prerent propofition: ther<:(ore the
circurnrcribtct
i-s
tO
the inCcriuc:d as
12
¡s tO 6, or
2
to
l .
In
hallllonic
tcrms thus
cxprdrl.:d:
(he 5th is (o a gi.
,'en note or l...cy as
t
2
is
(O
8; bDl lhe propon íon of the:
~,h,:5
. s 8 'o
6.
Tltcref~re,
,lte pro?on ion of ' he 8,h
u
T
"&
as 2 tO
t.
A gain, lhe
proponíon
of the
5th,
and the
neXl
har–
moni..
1
proport'on arifing ou, of , h: ¡,b, is heJu'ifully
illurtratl·d
iD
the admi r.¡olc proponían of th:
fpherr,
Tight cylindcr. aAd e:quilateral cone
ci rc~nlr":rlhcJ
ctuout
..eh o,her. Tbe LtU proronion
uein~
inl'tnteJ uy /\n
d rew
}'ac'l'Jct;
anel lh .lt of (he
1\'10
fi rU
by
ArchimcdcJ,
lO
dc!OIooll..,ed i. Ilts 'I5,h P'O¡>OfiUOD
iD
Tacquet',
EJ'
cltd.
e
K.
VI/<:
c.tnnol
forbc.lrtr",n:cli:'íng?l
h:r.gih
this
\l/o:-,d~r.
fuI
pr
op.Jf:llon,
:ilnd d ...
monfl ri. lio:'l
(,f
t:l!
fa.He;
lor
\hat
011
this projlon íon
ii
cr t:'~t,:J
tl.c
\\:H..
I::
fe?
rflruélJrt
of
h:umoni~
cllOllb,
AR C III ¡.I E. DE~'s
FROPOSITION
H .
THEORE~t .
An \!.p.:il.ttcral
cone circulTi(..:nl.:d
:;.bont
a (¡,here, anti
1
n gh t
cyllndcr In llk-c m.mnu cil ClI:T.·
fcn bcd
tlbout
lhe (ame f'phere, ar.d lhe la me
fphere
It–
fdf conlinu('
Ihe
r.'une PI(
port ion ,
10 \Vil,
Ihe k rquid t.
l c:ral, as
wd l
in
nfpeét
of the
l~I IJ it y .
as
of
Ih\! wholc:
(uper'icit's.
For,
by
32d
of this
hvuk,
lhe li1:" t
cy~i n
der ( )
encomp;¡fhng
the
fFhC"Tc.
is
te
Iht! fpherl!,
as
\Vea
in relpc:él
of
tullJ.lY,;.<:
uf
111t:
whole
fUpt,fit.:,es, a, 3 is
to
2 .
or
as 6
l O
4 ,
Ihn
oy
the foregoi'ng. the
cquda–
cCl'd l ¡;one ci lc\!lnfcribed abollt the fphl'rc.!1
tS
to Ihe
tpher~,
in bOlh ,lte Lid rcfrtlls,
os
9
tu
4-
Thercfo,., ,lte
lame cone is ' o ,he cylindcr, bOllt in refreél of folidi, y
anJ fu rflc,:,
jIl$
9
is
tO
6
\ Vherc:fore, tI.eíe
lIuee bQ–
dlf.5,
?.
cone. cylinder,
,:and
(pherc, are,
bCI\\o ixI
them·
felves, 35 ,he ntlmbers
9' 6
and
4,
3nd conf'qu cntly
continue ,lte fefqttialteral proponion,
Q:
E, D.
In
harmonic
tcrms
exp relT~d
IhllS :
Ihe
5th is
tO
the
key
:as
3 is
tO
2,
or :as
6
tO
4 :
but the
9th is
10
the key,
(,ha, is,
ti"
5,h ' o ,he 51h) as
9
i5 'o
4;
(for , he
od
and in 8th ';
i=1
:)
there(ore, lhe Carne 9th
i5
to the
5th as 9 is
l O
6. \oVhereforc
the(~
(h ree tones.
Ihe
9 th,
lhe
5th,
and
the key.
are,
oClwixt
themfc:!ves,
C:iS
tht!
numbers ') , 6, and 4 ; ar.d
con~eq ut'ndy
conl inue
üle
fefq .. ,Iteral proportion,
~
E, D.
Thercfore, , he proportions of ,he key, lhe 5,h .nd
¡ts
5th, being lhe
Carne fefqui.tlter:a l proportion continucd
J
are lhe rarne prop!)T[ions
as
that of the equil.lteral cone,
righ, c)'lioder and fpltere ; , he , wo fir O dcfcribed abo"t
, he fpltae,
Q..
E, D.
0 0
, he prAportioa of ,hefe ,hree is ereéled every olhe,
proportion of harmony; which \Ve (hall pur(ue one
elep
{urther,
forafmu d l
;5
thefe truths
wilI
be
moa
ma~irdt
and eO.blinlCd in lhe pratlice when IVe {hall hal'e deltver–
ed , he rules nf h"mony.
T he
5'1t Jtlided .rilh me,ie.lIy, or equ. lly, giH', ..
hath bttn Ih e\\'n, lhe f,re:lt cr 3d or
n eXI
perft(t
concord:
the fefqui.llteral proporuon
t O
which gives the greater
7th; for { .{.
gi,'cs
1.'
lhe grc:ater 7th, as
demooHrated
abo\ c.
H c::nce,
from
the (erquialteral
r rC\rorlion
thrice
rep~at
ed, n::tmcly, to Ihe kcy, its ph aad 3d, we
aTe
furndh–
ed ...·jth Ihe
pfrf~a
harOlony. or
ccnctHd.,.
of the
key
and
5th ;
tO
which
cvcry
harml1nic plopon"m.
~'!le H.:
e\"C"1'
foun i ,
ji
anal0l!0lls ; that is,
r ¡U I,tklO&
a t
ln~ nat ur~,
proponían.
Ino
nl ation of Ihe
ker
:ud
its 5th.
11 1S
wonh
r\.'01arking
in
I!lis
plA.: e,
lh.u lhe nlcm
::r5
llf
the(e
chords anfc'oul
of
(he
pr cvoltions,
as abo\( dcmontlr a
a
t~d,
by
lurns.
'The
k,y
bring (j11l
f'Tr~i(J
\\ i\h
one
P"oFoTlion ; anJ Ihen in
itS
tUro
ch e
5th
wl~h
the (¡¡meo.
' Yhcrc:fore (he mi;(ture of the
haTllCllly
of tI:e
tC'\'
anJ
5 th
15 rC;l rC't.!y
fl..·p.t ra.IJIt': A
.IlHh
,>h.icb wtll ..
l'unJ~od ,.
lhfcl,vcr id:l!' in the
r 1
~licc.
bOlh
10
th~
r.Jlls
l,f
h.,~nHlny. íl.nJ in {\'cry mll..:r
r.trt
of cOOlp·diuon . 0 0
\h.,
il fuundc!J lhe fullowins ""k'n!.