geting: in Theor 
—— v= ae OT ir Os he the -vi- 
EQUAL. 
RE ERT ER RI cea 
ansehen, i=1, b= 4-20, andthe rest as before; and 
=—T6 
brations of pA; and SP = 17 is the ratio of EDA. And 
in order to obtain the ratios of these bi-equal thirds, we 
have in Theorem IIE. first, t=1, 6=20, &. nd [= 61 
4% (30040) 1200 15 
5x 3004-20’ 152019 =! PA, as before; and se- 
4% (300 x 54-20) 
cond, 1=2, &e, and5 = aoa 2542 oxGFH) 544 
poelere a= 
hare hen go, by inverting the vibrations of the nots 
already found : thus, 30 = {35 also 300x-= = 480,the 
Ge ND, cht art 
ship’s two new thirds, mentioned in our article Br-EQuAL 
Third, ~The minor sixth CpA is = 7. 
aig dace canal! Lae to tune his Lord- 
RI-EQUAL Quints, in the 
nisjer’ tia Ce Be On the mili venereal 
partly above and below it;) we have, ¢=3, ~ = 
= 5 and N = 180; and Theorem I. gives 
_180x (sues) 
2 or 3.1578947 flat beats 
7 edi of idee ites consecutive fifths 
» Se xxvii. p.13. In Theorem II. make 
and the rest as before, and v= 
get, 
ri © 19 
180%3— x (140) _s10 
= 268.4210526 vibrations 
a 
per 1” of the note D. "See Phil Sepang aye 
In like manner, when i=2, we get 1070 — 401.0526816 
vibrations of the note A. If the jules of these three 
_ trisequal quints be required, they may be had from 
180% 19 
6100 
; and the vibra- 
127 
Theorem III. ; or from their vibrations thus, 
Be et 5100 _ 85 
tions of e sing 128 wehave oe icy 
=vib. of Ac. 
Example 3. If three equal-beating major thirds be re- 
quired to be tuned in an’ octave above tenor-cliffC, we 
pn lia 
Tr = 11.80323 times sharp, the rate of each of 
their beating. ee ae =1, &. and V= 
VOL, IX, PART I. 
193 
240 x 5 OO a g4e0 
gE 
and 2d, when imate, v== =F 81.0998, those of 
61 
7qand 
gy Tespectively 5 and the values of these three new thirds 
97 
pA. The ratios of these two notes will be founa OL 
come out, CE=— Ep A = — and ) Ac=—. 
Example 4, If four equal-beating parkiuon be want- 
ed in the octave as above, we have t= ey. 8 ee 
m 625 
rien se 20x (+ x 625 —1296) 
N=240; = = 
2 eck 2164-1804-150+4- 125 
11040 Lessa cot 
"Sr? =16.45305 beats flat per 1” ; and putting 
t=1, &c, in Theorem II. we have van = 
284.70939 vibrations of PE ; also by making ‘=2, and 
f=8, we get ch =338,36066 =) G, and a 
402.74217=A vibrations. The notes themselves will 
be found pE ou sap Ga sand A= SU and the four 
equating minor thirds wil be, as fallow, _ 
ChE = Top PEGS FagiG A= and Ae= gee. 
if, nen at the use of our $d Theo- 
thirds, we put (=4, b= 
5x (240x216—0 Xx (96-430-425) ) 
x (2164-180-4 1504-125) 
Eingis's Suppose it were wird to calculate 
up Spe, were ree of 12 succes~ 
sive fifths within 7 octaves above tenor cliff C, we have 
vane < Ls jug and N=e40; and in the 1st 
240 x (128 x 4096 —531441) 
Theorem we find b= T7747 4 118098 4.787324, &c 
= — 949944 _ __s.0554021, the flat beats sek 1° ok 
1 
each of these di quints, Make t=1, &c. 
uodeci-equal 
and, in Theorem IT. we have, 
343344 
240 X S—s 60 
v= 5 =f, 168" = 358,3722989, 
the vibrations of G ; Sogou hE we get 
— 565204080 _ ; 
vimanas ers all the other notes of ‘such a 
may readily be found, and the ratio of each 
In a letter (which the writer has before him) from 
QB 
‘Equal- 
beating. 
= er = St2 B08, he wf Bs 
