MUSIC AND SCIENCE. 
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lowest step chanced to be C-C # , the step an Octave higher 
would be about c-d. 
The same idea of equal linear spacing is very often applied 
to the holes in flutes, as one may see in any museum of in- 
struments, and the resulting scale is of the same type. It 
may also be applied to lengths of vibrating bars, and in these 
the steps of the scale will increase still more rapidly. 
One other principle of scale-building should be noted — 
that based on harmonics, which Helmholtz developed so 
admirably. It is familiarly known that a series of sounds 
can be produced from a simple bugle or trumpet, as C, G, c, e, 
g, etc. ; these are called “harmonics;” the same thing is easily 
proved to be the case with a stretched string or wire ; and it is 
also true that several of these sounds are ordinarily given off 
by the string simultaneously. If the lowest sound has, say, 
100 vibrations per second, the others will have very nearly 
200, 300, 400, etc,, vibrations; if another string is furnishing 
simultaneously the series 151, 302, 453, etc., there are two 
sounds, partial-tones, as they are called, 300 and 302, that are 
nearly in unison but not quite ; so they will beat and produce 
a roughness in the sensation ; by flatting the higher string to 
150 the beat will disappear and the ratio of the two vibra- 
tion-frequencies will become exactly 2:3; the musical inter- 
val is that of the Fifth. Taking other pairs of notes in the 
same way, it is easily seen that with a few other ratios for the 
fundamentals, the harmonics will be free from beats, as 3 : 4 
and 5:6; but with actual stringed instruments the largest 
numeral in any ratio is 9. If one tried to use this principle 
with stopped organ-pipes, the only numbers he could use 
would be 1, 3, and 5; in other words, the harmonic principle 
would not furnish a scale if one had only stopped organ-pipes. 
If he tried to work with bars the ratios of the partials to the 
fundamentals would be expressed in a complicated way, for 
the partial tones here are not harmonics. This principle of 
Helmholtz was probably not of much importance historically 
in the fixing of the old melodic scales; but it furnished an 
admirable justification of the scale in vogue at the date when 
he wrote. 
