The musical tone scale tends to be ''stretched" ([104], p. 220-222). More rigorously one
should say that it is the tone scale's intonation that is
stretched - where stretch means that low tones are tuned
flat, high tones, sharp. The reference against which this can be
said is standard equally tempered intonation. When the tones of
the scale, in ascending steps of semitones, are given numbers n
such that n = 0 denotes the lowest conceivable C, i.e., C0
(about 16.4 Hz), the tone A4 gets the number n = 57,
and the standard (i.e., non-stretched) intonation is defined by
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Here fA4 defines the absolute frequency level of the intonation. According to international standards it should be fA4 = 440 Hz. When the departures of the tone frequencies from these values (negative for n < 57, positive for n > 57), are smoothly distributed, the intonation still is equally tempered. It is equally tempered and stretched.
The best known case of scale stretch is that of the piano. While (on a concert grand piano) in the middle three octaves (about A3 to A5) the deviations of tone frequencies from standard equally tempered intonation are negligible, they rapidly increase for tones descending and ascending outside that region. The lowest and highest A's (A1, A7) are about 2% flat and sharp, respectively.
By listening tests it has been shown that stretched tuning of the piano indeed is preferable to non-stretched intonation (e.g. Martin & Ward 1961a). So - at least with respect to the particular type of tones produced by the piano - it was early evident that it was not desirable to correct for the stretch by any conceivable means. Indeed, today's electronic digital pianos frequently have scale stretch deliberately built in to improve their quality and the naturalness of sound.
Theoretical analysis and measurements on piano strings yielded a plausible explanation of piano-scale stretch. The eigenfrequencies of a real string are not exactly harmonics. Due to stiffness of the string, the second, third etc. part tones are systematically shifted to higher frequencies than would correspond to the second, third, etc. harmonic of the first (Schuck & Young 1943a). As piano tones essentially are free oscillations, their Fourier components are decaying oscillations with the eigenfrequencies. The explanation of scale stretch then emerges from considering what happens when simultaneous octaves are played. When the first part tone of the higher tone is tuned to twice the frequency of the first part tone of the lower, the frequency of the lower tone's second part tone will be slightly different from the higher tone's first part tone, and this gives rise to beats. Moreover, the lower tone's fourth part tone will be slightly higher in frequency than the higer tone's second part tone, and so on. To minimise beats in simultaneous octaves, the piano tuner is more or less forced to give the higher tone a frequency slightly higher than twice the lower tone's. So it is a combination of piano-string physics (stiffness for transversal oscillation) and auditory phenomena (perception of beats and unpleasantness of them) which explains - at least to a considerable extent - the stretch of the piano scale (cf. Schuck & Young 1943, Rasch & Heetvelt 1985a, Lattard 1993a).
As this explanation of piano-scale stretch makes occurrence or absence of scale stretch dependent on whether or not the individual tones of the instrument are true harmonic complex tones, one should expect that instruments whose tones are truly harmonic should not have a stretch of the tone scale. True harmonic complex tones emerge from steady periodic oscillations such as those of bowed strings, wind instruments, organ pipes, and the singing voice. The only keyboard type of instrument (i.e. with a fixed tuning) that produces steady periodic tones is the organ (pipe or electronic). Indeed, the intonation of pipe organs exhibits very little stretch, if any. (Yet pipe-organ experts say that it is much better to tune pipes of the high region sharp than to tune flat.) And in most types of electronic organ (beginning with the classical electro-mechanical Hammond organ) there cannot be any stretch at all, because the technique of tone production is such that octaves are exactly in a 1:2 frequency ratio.
So, on first sight - and if the above explanation were the whole story - it would appear that - where intonation is concerened - the piano were exceptional among the musical instruments. However, it turns out that stretching of the tone scale is very common in musical performance. Solo instruments such as the wind and string families, as well as singers, tend to play sharp in the high pitch region. And in the orchestra the bass string players are often advised to avoid tuning their instruments sharp but instead rather to tune slightly flat. These tendencies are clearly visible in the results of statistical frequency measurements on solo performances by expert players on the violin, flute, and oboe (Fransson et al. 1970a). Although these instruments produce truly harmonic complex tones, a stretch of the tone scale was found that resembles that of the piano - with the only exception that even the middle octaves were not unstretched. So, in fact, the aforementioned non-stretched keyboard instruments (the organs) turn out to be the exception rather than the rule.
The explanation for the universal tendency to stretch the musical scale emerges from taking pitch perception into account. Naturally, in evaluating musical intervals by ear, it is primarily pitch that communicates the necessary information to the cognitive auditory system. The fine intonation produced on violins, flutes, oboes, clarinets, trumpets, etc. is controlled by ear, i.e. in terms of pitches and pitch intervals in comparison to memorised templates. So it must be concluded that it is the internal standard pitch scale of a musician and listener that on the average is stretched. This has been documented by Ward (1954a) in a most direct manner, namely, by having absolute-pitch possessors adjust (on an electronic oscillator) each and every tone of the scale without any reference. The result indeed was a stretched tone scale. For the octave interval it is particularly easy and safe to verify that the corresponding internal pitch template indeed is stretched, and this does not require the faculty of absolute pitch. As the phenomenon of octave stretch practically is sufficient to enforce stretch of the entire tone scale, one may say that scale stretch is explained by octave stretch. And the psychophysical explanation of octave stretch in turn is included in the theory of virtual pitch [16] , [17], [18] , [22] , [38], [55], [56].
In summary, one may say that it is primarily the stretch of the internal pitch scale of the auditory system that is responsible for the universally observed tendency to stretch the musical tone scale. That tendency indeed is so pronounced that even solo instruments such as flutes and clarinets are built in such a way that they even without corrections by the player produce a stretched scale (cf. Meyer 1961a, 1969a, Coltman 1990a, Nederveen 1973a). This does not disprove the above primary explanation of the piano's scale stretch. Rather, the role of the piano among the other musical instruments is elucidated. The inharmonicity of piano strings which as such enforces stretched tuning turns out to be beneficial, as it enables reconciliation of two phenomenomena that with truly harmonic complex tones can hardly be reconciled: The preference of the ear for stretched successive pitch intervals on the one hand, and the occurrence of beats from the simultaneous sounding of simultaneous tones in stretched intonation, on the other. It is only the piano on which you can have both stretched and beat-free octaves.
These considerations further suggest that there cannot even theoretically exist such a thing as an optimal fixed intonation. As the occurrence of, and the disturbance by, beats depends on the complexity of the musical sound in every instant, and as the fine tuning of pitch and pitch intervals are subject to pitch shift effects which as well depend on the sound's degree of complexity, fine intonation must adapt to the sounds performed, i.e., it must be temporally variable [26] , [80] .
With a special tunable electronic organ we have performed listening tests on musical examples with normal, stretched, and even contracted intonation [28]. As a result it turned out that the preferred kind and amount of stretch depends on the type of music excerpt presented (solo melody vs. melody plus accompaniment vs. complex polyphonic sounds). This can be regarded as an experimental verification of the above conclusions.
On an electronic organ (which, as mentioned above, is ordinarily tuned in an unstretched tempered intonation) the ear's preference of stretched intonation can readily be demonstrated, as follows. Play a melody in a high-pitched register, accompanied by a simple bass-tone sequence in a low-pitched register, such that melody and accompaniment are several octaves apart. Evaluate by ear how well the accompaniment's intonation corresponds to that of the high-pitched melody (and vice versa). You will probably notice that the melody appears somewhat flat relative to the accompaniment although it is "mathematically correct". Now play the melody in a key one semitone sharp relative to the accompaniment. You may notice that this kind of "intonation", which corresponds to an oversized stretch of 6%, still is almost acceptable. To verify that this is not due to a failure of your auditory evaluation capabilities, do the same experiment with playing the melody one semitone flat. This will appear totally inacceptable - at least within the standards of conventional tonal music. From what was said about the intonation of the piano and about the size of octave stretch, one would predict that a strech of about 3% should be optimal for the above type of music sample. So, relative to aurally optimal intonation, the +1-semitone stretch is sharp by about the same amount as unstretched intonation is flat. Through many years I have played the above example to audiences of scientists, and of my students. When asked to rank the above three intonations in terms of acceptability, 60% of the listeners voted for [1) unstretched; 2) 1 ST stretch; 3) 1 ST contraction]. Fourty percent of the listeners voted for [1) 1 ST stretch; 2) unstretched; 3) 1 ST contraction]. All listeners pointed out that 1 ST contraction was totally inacceptable. This result fits well into the above consideration, i.e., that a stretch of about 3% would have been optimal (cf. the CD attached to [104]).
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