Books › A descriptive catalogue of the musical instruments recently exhibited at the Royal Military Exhibition, London, 1890 …, 1891 › scan 259
Scan 259
A descriptive catalogue of the musical instruments recently exhibited at the Royal Military Exhibition, London, 1890 …, 1891
10 lines. Read by computer on 1 October 2026. Printed page 239. See the scan on archive.org.
1interval of twelve perfect fifths, and the working out may be written down thus:—
3Octaves { \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{128} \) }
5Fifths { \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{4096}{531441} = \frac{1}{129.746} \) }
7The difference between the two results, i.e., between \( \frac{1}{128} \) and \( \frac{1}{129.746} \), is known as the Pythagorean comma, the Pythagorean or theoretical Greek scale having been based upon such a succession of perfect fifths. A similar difference would evidently show itself on working through perfect fifths downwards, \( c – f – b \flat \), &c., to \( d \flat \flat \); therefore, the notes \( d \flat \flat \) and \( b \sharp \), when derived from \( c \) through perfect fifths, are respectively flatter and sharper than \( c \); the amount will be best appreciated by comparing the numbers of vibrations, \( c \) being taken as 512. We would then have \( d \flat \flat = 505.1 \), \( c = 512 \), and \( b \sharp = 519 \). As key-board instruments have had almost universally only twelve keys to the octave, the key or digital used for \( c \) has had to do duty also for \( d \flat \flat \) and \( b \sharp \), and the necessity for some compromise led to the introduction of temperament. In the modern or equal temperament the accumulated error is distributed over the twelve fifths, so that each fifth is tuned the twelfth part of a Pythagorean comma flat. This mis-tuning of the fifths is barely appreciable in practical music, but the error of the thirds, major and minor, is considerable, and is very perceptible on fixed toned instruments, such as the organ and harmonium. Equal temperament has come into general use only during the last fifty years; it was preceded by the mean-tone temperament, which prevailed throughout Europe for centuries.
9This temperament—the mean-tone—was based upon maintaining the true ratio of the major third 4:5, and neglecting the sub-division of the same into two unequal tones, the major and the minor, as required in just intonation: substituting for these, two equal or "mean" tones. The error of flatness in the fifths with this temperament is more than twice as great as in the "equal," but the most
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