4 Evidence from practical music
4.1 Harmonic and inharmonic partials
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The sounds with which music is made are commonly categorised into families by the method of production: strings, woodwind, brass, percussion, voice are common divisions. From the point of view of the psycho-acoustician interested in the phenomenon of consonance, two larger categories are of interest: instruments in which vibrations are forced and those in which they are free.  The first category includes the voice, all wind instruments and bowed strings. When these instruments are played at a constant dynamic level, they produce a repeating wave-form. This can be represented as the sum of sine waves whose frequencies are in the ratio 1:2:3:4 ... (a Fourier series).  In the second category are percussion instruments, notably bells, and plucked and struck strings. These instruments have multiple modes of vibration, several of which are typically excited simultaneously by striking or plucking. These modes are weakly coupled, if at all (ie. little energy is transferred from one to another), and their frequencies are not, in general, in a simple mathematical relationship. Further sub-categorisations are of interest. Untuned percussion, such as snare drums and cymbals, have many different modes, so randomly related that listeners perceive no pitch to the sound which they produce. Many of the tuned percussion instruments, such as timpani and xylophone, have a single frequency which is so much stronger, or persists so much longer than the others that listeners do perceive a pitch, typically preceded by a percussive noise due to the other heavily damped modes. Also, when a sound has several strong components with frequencies approximately equally spaced, it is possible to perceive a pitch corresponding to their fundamental frequency (equal to the highest common factor of those present) even though no component of that frequency is present. The most interesting instruments in the present context are those, such as church bells, pianos and harpsichords, in which a small number of vibrational modes persist and can be separately identified by the trained listener.
4.2 The tuning of pianos and organs
The piano sound is produced by hammers striking long thin rods, some of which are loaded with a uniform wire winding. The modes of vibration of these long rods approximate fairly closely to those of a stretched flexible string, but the stiffness associated with their finite diameter is detectable and is taken into account by the piano tuner. Whereas the ideal, infinitely flexible stretched string would have free vibrations whose frequencies coincided with the harmonic series based on the lowest, whole string frequency, stiffness increases the frequency of the partials relative to the harmonic frequency. Once a piano tuner has established an equal tempered chromatic octave, his/her normal procedure (from which special purposes, such as to prepare a piano for playing with an electronically tuned electric bass guitar, may require departure) is to tune octaves above and below these notes by eliminating beats between the first partial (fundamental) of the upper note and the second partial of the lower one. On one of this writer's pianos, a baby grand, measurement of frequencies with an electronic tuner shows that this procedure produces a smooth enlargement of the intervals of about three cents over the seven octaves. This is consistent with an average enlargement of the ideal octave interval between the partials by about 1 part in 3000. Benade tabulates the calculated frequencies of the partials of a typical piano string with a fundamental of 261.63 Hz. (middle C), and gives a slightly higher figure, just under 1 part in 2000 for the sharpness of the second (octave) partial. This may be regarded as reasonable agreement between an average over the whole keyboard and a single value which may not be typical.
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