H. W. Poole on the Musical Ratios. 295 
must be entirely _— from the combination which we have just ex- 
amined, ave therefore is an absolutely perfect consonance. 
There is eneetel strange in calling a combination harmo- 
nious because the rapidity of the discordant beats is so t sa, 
that they are inaudible. The fact is, that a chord is nearly in 
tune and begins to give agreeable effects when its beats are very 
slow. When the chord is more out of tune and the beats be- 
come rapid, and mingle in a flutter, whether of 132 a second or 
more, the ear ceases ‘to regard the sounds as a chord; but a 
cannot be said that “dissonance must be entirely absent; “0 
the contrary it is terribly present. 
Whether the twenty sounds given above be harmonious or 
not, it can be shown that this does not depend on the circum- 
stances that there are no coincidences less than 132. For 
twenty other sounds might be added at random utterly discor- 
dant, but not beating owe) than 192. For example, 265 : 
529, 266 : 530, etc., all having the difference of 264, And in 
all the examples given, the addition of the overtones does not 
effect the case, for if the fundamental tones have the desired 
difference of 132, the first harmonic is sure to have twice this 
difference, and the others still more. Buta point might have 
een made, which has been overlooked by Tyndall, in es cir- 
cumstance that these overtones make a “rough ” chor 
harmonious, See the major third, 4 : 5, 330—264= difference 
66, and the minor third, 5 : 6, 396 — 330=difference 66. These 
are called rough, because their differences are less that 132. 
But their first harm rmonics have a difference of 132, and all x 
the following differences are far beyond the assumed limits of . 
dissonance. But the whole thsory is untenable. 
mrad the effect is good when each note is accompanied by har- 
nics, the reason will be found in the simple relations Tehich 
ase and in the circumstance that what would be oe 
alone, may be clear when auxiliary sounds are pre 
After as sserting the equality of the musical pare in compo- 
sition, I admit the su Deaionity of the simpler ratios for modu- 
lations. A cha key aS made to _ fifth above or 
one. But modulations do re seem pac racticable by a third o or 
seventh, or the primes 5 and a change would sound 
e an abrupt gqandition. 
In rhythm again, no other primes than 2 and 3 have been 
found polable. A single composition has been ma de with jive — 
beats, or d equal divisions, i in 2 bar, but at Ve esires 
Be it 
is more satis eur a 
