EUL 
experimental pi and the number answering to Euler’ 
this last is “O4R1S the | of string to the unison I, Scale. 
In like manner, if F 
i 
| 
i 
: 
E 
: 
a 
: 
ee 
Z 
; 
4 
Z 
Hi 
z 
id 
uh 
f 
a 
ae 
yet 
i i a7 
4 
= 
& 
: 
Hf 
it 
3 
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of cures by their sacredtouch.” See also the Edi | 
eh ee Journal, Vol. III. p. 185. 
VIL... See Tuzonocy. ; 
ULER’s Locarirums, for musical calculations, are 
musical, intervals. in their relations de« 
1 and .30103 respecti 
re Dare ly any, fon. If 
maimawenieneiie Fouthnhes atinictnontornions , were 
given, 68% € .076814 x .30108=.0231233, whose re~ 
cip. -9768767 is the common log. near enough for most 
wanted from Euler’s logs. since both of these are of the 
nature of reciprocal logs. the octave being expressed by 
612 and 1 in them 3 ively, we have only to mul- 
tiply Euler’s post By. 612,to obtain nearly their artificial 
as. The last example will stand thus: .076814 x 
612—47.010168, or 47, as in the schisma column (=) 
of Plate XXX: Vol. II. ; and it may be remarked here-~ 
on, that whenever the product approaches a whole num- 
ber, either above or below it, that such whole number 
is, in general, the proper number of artificial commas 
t. . 
eT eri Scar of musical intervals. It is proba- 
ble, we think, that M. Euler made the first, though 
an unsuccessful attempt, at the grand harmonic im- 
vement, (which Mr Liston has lately effected, by 
scale of perfect harmony, and his Evnarmonic Or- 
gan, which see), by endeavouring, on a very limited 
scale, to avoid tempered or imperfect concords, by means 
of more notes introduced in the octave than the 12 in 
common use. M. Euler extended these to 24, eight of 
which new notes were only a major comma higher than 
his notes CX, DX, E, FX, GX, A, Bp, and B ; and four 
others of them a minor comma lower than his notes D, 
F, G, and C: at the same time, that three notes of his 
original scale differ from Liston’s, viz. CX, Ep, and Bp, 
‘ . -__ 24 64 128. 128 
in having the’ ratios 25° 75 and a5 instead of Te 
Aas and iW which are Liston’s notes respectively. 
In order to facilitate the labours of those, who, like 
the gentleman alluded to in the last article, may be de- 
sirous of trying, either by calculation or experiment, 
the effect and extent of this scale in producing perfect 
harmonics, we have been at the of reducing M. 
Euler’s vingtquatreave scale, from the octave F to f, in 
which he has published it, to that of C to c, in which 
all musical are given in our work; and we have 
added in the columns of the following Table, the lengths 
of strings, and number of artificial commas, answering 
to each of Euler's notes ; and in the last column but one, 
we have set down the notes on Liston’s , (see the 
Phil. Mag. vol. xxxix. p. 419, or the Mont ag, vol. 
Xxxvi. p. 217,) whereon the several chords in this sys~ 
tem might be tried, and whence these notes might be 
aes mac hey ad gad weap oo ene 
w one’s while to e such aone, having a 
Tar es betiigtoncoes Ge eight genta motes..eah onather 
for supplying the four that are minor comma: flats, in 
pee. halla itty, ieizanliguepeeed 
are wanted in performance, in 
by M. Euler. The last column shews. the’ numeral 
value of the notes, major and minor, above C. 
8 artificial commas were ~~ ¥™ 
