1000000031

Fla' 3d and -- 8,h

Sharp 3d and -- 5,h

1'la[ 3d and -- 5[h

Sharp 3d and Sharp 6th

l'lal

3d and FI.. 6[h

6th and-- 4[h

FI.t 61h and -- 4,h

founded with lhe opeo fhing. or bars

no~c:,

are concordant

all together; and thereforc produce harmony.

Example of

...

(\Vo

concordiog with (he apeo firing or

bafs. No 4.

•

Next foHows ao example of three concording with

lhe

b. fs . No. 5.

Having (hus difcovered lhe concords, thei r aróer and

propartions;

wonh

remarking, that lhe

firn

concord,

8th,

which arifes from

lhe rooft

limpie

di",ifioD

of a

line, is the man- perfeél: concord;

l he

5th is

lhe D(Xl

ptrfcEt: cODcord; and fa of lhe

Ten.

in lhe arder they

havo boon found by Ibe divifroo of ,he lIring. For [he

naLure

and

perfeétion of lhe 4th, accouDted

fome a

very imperfell concord. (hall be explained in

lhe

corolla–

ries af the demODHratioo

lhe harmony. in

Pan

IL on

praétice.

The 8[h, .nd 5,hs ,hen are _caJled [he perfeél concord,.

The 3d, and 6,h. imperfeél concords. The 4th, of •

middle nature belween the others.

may

be called an

irn–

proper concord; for Ihis rearan, that with Ihe 6th, with

which it

alwa.ys

accompaniéd

harmony, though

make perfeét harrnony with the giveo note, yet they

change the chord into that of the 4th to that note.

L ikewife ,ho 6,h. wbe,hor joined wilh Iho 3d or 4,b

to the given note, tho'

make perfe(t hannooy with ei·

ther,

ye!

,hey chaoge ,he chord into tha! of ,he 61h or

4th to the fame note.

Hence ,be reafon why the 61h. are more imperfetl con·

cordl ,bao ,he 3<\"

From ,ho order and perfetlion of ,he concord, thus dif·

covered; we deduce the foJlowing corollary¡,.

C O Ro

lII.

The moll porfeél harmony is Iha! which

will bo produced by ,he pcrfrtl concords, namely, ,he 3d,

5[h, and 8,h. Thus No. 6.

From the foregoing corollary. we are able to give

jufi

defiaitiun of harmony.

HA RbtONY

con(¡(ls in one

cenain ínvariabls proportioo

dillance of four founds

performed a! ,he fame, inllan[ of [ime, and

moll

plea~ng

the ear.

Thde proportions of [he firll feries are called

~mple

concorda . If the notes of

fecond feries be adde:d

the

lira

oflave, the propon ían

é\ny

IWO

concording note:s

compounde:d with the o[tave retains the name lnd nature

of the fimple

~oncord;

as a te:nth, l ompouDded of ao oc–

tave and third, is calle:d

third

a twelft-h , compounded

of an o(tave and 5th. is called a 6fl h

a fj(teenth, com–

pounded of tWO

o(tavts,

is called an oét.ave, or doublc:

oélave:. And fo

third feries.

'Thcre are: the: compouod corlcords.

AH o[her proponions fou nded togelher are harO, and dif·

agreeable lo the: e:arí and are: for Ibis rearon caBed dircords.

From the compoundin, and dividing lhe proportions

de:live:red, nOl only the harmonical intcrvals are compute:d,

bu, Ihe tlifcQras lik_wife.

And

this

Ihe

fuJJowinc

calcuJ:aions. dcmonflr;tc.

Th~

proponion

the

o~hvc

is Ihe proponion uf Ihe

.t¡th

and 5th : fo r, by cOOlpounding } {=r!"l , or ..;.

the

proportion of Ihe oltave: .

A~!.in,

it is Ihe proportion of the (harp

3d and

Aat

6th:

for,

{ ~=~.~

l.? .;.

ils

lo\~en

terDlS.

J\gain, (he

n,u

3d and

fharp

6th :

for,

.f.f-·H-

or .}

the proportion of

lhe

oau C'.

Now

(jnee

lhe 4th

and 5th, the

and 6th, a.s alfo

lhe 2d and

7th,

compouoded, makc lhe: oltavc: ; thac

is,

aoy two numllers makin2

the middle term or note

being repeated, or common tO

bOlh ;

it foJlo\\'s, that to

faH a 4,h or rife a 5,h, as alfo ' o faH a 3d or rife a

6 th, aad to fal! a 2d or Jife a 71h, anO Ihe contrary, ano

fwers (he fame: purpofe of harmony; for they meet

the

oé"tave.

• This obfervation will be of great ufe in fetting the

bafs, and fjguriog the fame, by producing that varicty

and cont rary motion demonnrated necdr.try in the 4th

axiom.

Proporlion o(lh.

51h.

The proportion of,he 5,h i, ,he propoTlion of ,he

fh.rp

3d and flat 3d ; for by cOOlpounding

.~-

the

fefqui.heul and known proportion of ' hCSlh.

Pr.porlion of lh. Sharp

61h.

The proportion of ,he fharp 6,h i, ,he compound pro–

portions of 1he founh and fharp 3d; for

i=;'~,

.~.

Of ,he

fl..

6,h, ,he proponion is of ,he 41h . nd fla' 3d ;

for

-t

~=M

or {-.

By the fa me manner of compounding are found the

proportions of the concords of the

;ds;

which (hall be

fhe:wn when we fhall have got lhe tone! .and Cemitone:s ;.

which,

i1S

being difcords

arife

divirung the harmoaic

proport~ns

as follow,.

PROPORTION5

of the

DlSeoRD s

prorud.

Proportion

the Crea/u r one.

The propon ion of the greuer tone is the difFerence or

,he 4,h .nd 5,h; for

4-)

{(=~

,he propoltioo of ,he

grcater lone:.

Prop.rlion o( Ih.

L~(for

T on•.

T he

propon ion

the lelfer tone

Ihe difTerence: of

the Slh and fharp 61h; for

+=~o

Iho proportioo of ,he

Ie:lfer tone.

Proportioll o{ the SUlJilone.

The

proportion of the Cc:mitone is the difFerence of

the (harp 3d and

4111;

for ~)

i-=

~ }

Ihe proponion of

the

femitone.

H aving

no\'J

the propprtions of lhe tones and femitone:s,

we are enabled tO prove the: proportion o( (he fe:-llilone ,

flat

2d and Olarp 71h

,he 81h; as likewife alJ ,he

rcmaining proponions, whcther dircord or cODcord: For,

1he

5th and

fharp

3d,

{-,

give

the gre:ater 7th; and

lhe lharp 9th

and

femitone

'-J

-:-t={

~-~

in ilS lowen terma

{. the proportion of tht: oélave.

00;

'f he

proponion. of the {harp 3d is

lhat

of the

grealt.:..C