MUSIC AND SCIENCE. 
185 
be in substantial agreement. Various old writers give the 
rules in vogue among Greek theorists; in the last century 
Amiot described the Chinese rules, while in the last few 
decades the rules of Arab, Hindu, Japanese, and Siamese 
musicians have been made accessible. The most familiar 
rules, as is well known, depend on that law of vibrating 
strings which is followed by a violinist in his lingering — 
namely, that the frequency of vibration of parts of any 
stretched string is inversely as the length of the parts, pro- 
vided the tension does not change. Our latest rule, histor- 
ically derived from one of the many Greek and Arab rules 
by subdividing the whole tone-steps, so giving twelve steps to 
the Octave, is embodied on the neck of a guitar or mandolin ; 
here it is obvious that the successive stopping points as 
marked by frets get closer and closer together as the pitch 
rises. All musicians know that this number of notes, twelve, 
is found confusingly great for ordinary playing, and know 
the principles by which the player selects certain notes for 
any tune. But this multiplicity of notes has an important 
bearing on all studies on non-harmonic music made by har- 
monic musicians ; for every sound within the compass of the 
instrument comes very near to some one of the twelve notes 
and may readily be represented thereby, owing to the diffi- 
culty the hearer has in estimating deviations from the famil- 
iar series and in noting them down. The results of this ap- 
proximation are to mask all deviations from the twelve-tone 
piano-scale, whether intentionally or accidentally made, and 
to make it appear to musicians, first, that nearly all the music 
of the world is performed substantially in our scale; and 
second, that any other theoretical scales, such as those found 
among Orientals, or described by our European ancestors, are 
merely mathematical jugglery and of as little significance as 
proposals for a change that occasionally appear in modern 
musical or scientific journals. 
“But before going further it must be recognized that the 
word ‘scale’ has many meanings. Perhaps the lowest and 
loosest is, — the series of sounds used in any musical perform- 
ance, arranged in order of pitch. Another is, — the series of 
24— Bull. Phil. Soc., Wash., Vol. 15. 
