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Modern Mathematics and Music*
Published online by Cambridge University Press: 03 November 2016
Extract
The pitch of a musical note is defined by its frequency, measured in cycles per second. The frequencies of pure musical tones form an infinite set of real numbers lying between the lower and upper limits of audibility. The notes of the pianoforte form a finite subset of this infinite ‘spectrum’ containing usually 88 members. Other instruments provide different subsets of available notes, these subsets being infinite in the case of the strings and the trombone, whereas in the case of most other instruments, we have finite subsets of discrete musical tones.
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- Copyright © Mathematical Association 1967
Footnotes
page note 204 *
The pith of an illustrated talk given to the Cardiff meeting of the Association.
References
page note 205 * Sir Malcolm Sargent once confessed to the writer that he found great difficulty in distinguishing the octave of a whistle.
page note 206 * See p. 213 the group of musical intervals, and the homomorphic mapping onto intervals less than the octave.
page note 207 * Here we cross, as Sir Walford Da vies so aptly put it, the “Musical Date-Line”
† Or equally well, those between F# and Bb inclusive, i.e., the black notes of the keyboard.
page note 208 * If the four strings of a violin, viola or cello are tuned to r interval between the highest and lowest string will be (3/2)3 whereas the interval of the major sixth should have a freque, 10/3 = 3.333. This discrepancy is of little moment in the case since the A string which is tuned first is not an outside string. the cello and viola, however, A is the top string, and tuning will render the C string (the bottom string) slightly flat. Ho seen from the ratio 3.375/3.333 = 1.0126; since a semitone represents a of about 6%, we see that the C string would be about one fifth of a semi flat. Sometimes, therefore a cello may prefer to tune its bottom string to C of the keyboard, or possibly of the bassoon. The difficulty is reduced by equal tempered system, whereby the same interval becomes 2 = 29/1 27/4 = 3.362.
page note 209 * See p. 213, the homomorphism of the intervals onto the intervals less than an octave.