Since the days of Pythagoras (or even earlier) the musical octave interval has been associated with the ratio 1:2. Until the 17th Century, that ratio was essentially referring to the lengths of two strings with the same stress that give tones which are heard as being in the particular relationship of octave equivalence. In Galileo's days it became known that tones are periodic oscillations of air pressure and that the pitch of tones is determined by oscillation frequency. So the meaning of the 1:2 ratio was generalized, now referring to the frequency ratio of periodic tones of any origin. Understandably, in ancient days it was not possible to determine the ratio to an accuracy of, say, a few percent. So it was not possible to tell whether the ratio is precisely 1:2. Besides, hardly anyone cared - at least until the beginning of psychophysics in the second half of the 19th century. Stumpf (1890a) appears to have been the first to report on the observation that when the frequency of a tone is adjusted such that the auditory sense of octave equivalence is satisfied in comparison with a fixed preceding tone, the frequency ratio of the tones tends to be slightly greater than 2:1.
With simultaneous periodic tones it is easy to adjust them very accurately to a 2:1 frequency ratio just "by ear": When the frequencies differ ever so slightly from that ratio, beats are heard. Accurate tuning to 2:1 thus is merely a matter of making the beats as slow as possible, i.e., by adjusting either of the two frequencies appropriately. However, that kind of "octave adjustment" is in no respect dependent on the perception of pitch; little or nothing can be learned from it about the pitch interval that corresponds to octave-equivalent tones. With respect to music it is most interesting to know precisely the width of the pitch interval pertinent to auditory octave equivalence.
Measurement of that interval is easy enough. The listener is presented with a pair of alternating, i.e., successive tones of which one frequency is fixed, the other, variable. The listener adjusts the variable tone until optimal octave equivalence is obtained. This requires, of course, that listeners somehow are familiar with the music-psychological concept of "octave", and that they possess a built-in pitch-interval template of the octave interval to which the pitches of the test tones can be matched. This is more or less synonymous with saying that the auditory sense of octave equivalence is required to exist. As it turns out, this criterion indeed is met by the vast majority of individuals, i.e., not only in trained musicians but also in individuals who are musically inactive. Even in populations outside the Western musical culture existence of the sense for octave equivalence was verified (Burns & Ward 1982a).
Another requirement to do the above octave-adjustment experiment is sufficient accuracy of auditory short-term memory for pitch. This criterion in fact is met by the auditory system. High accuracy of short-term memory for pitch is a basic and universal feature of any normal auditory system. This is why successive tones can be adjusted to octave equivalence with a precision that is essentially identical to the accuracy to which the pitch of single tones can be measured. Octave adjustment with tone pairs that are two or more octaves apart, however, appears to require some musical experience and talent (Thurlow & Erchul 1977a).
The octave-adjustment experiment yields two frequencies, f1,
and f2, which correspond to aurally estimated optimal
octave equivalence. Formally, octave stretch - or, in a more
general term, octave deviation - then is suitably defined
where f1 < f2 is presumed. Octave stretch is indicated by W > 0.
Depending on a number of parameters, values of W are obtained that, grossly, are in the range between 0 and a few percent (positive). It was shown by Walliser (1969b) that octave stretch is additive: When f2 was obtained as corresponding to the octave of f1, and f4 was in another experiment obtained as corresponding to the octave of f2, then a third experiment in which the listener adjusts f4 relative to f1 (i.e., spanning two octaves) yields a stretch that is the sum of those obtained in the first two single-octave tests.
Octave stretch with tones of musical instruments was described by Corso (1954b). The first comprehensive study of the phenomenon was carried out by Ward (1954a), who essentially used sine tones and determined octave deviations in a wide range of reference frequencies f1. Though the data obtained in the latter study demonstrate that, grossly, there is a tendency for octave stretch, they also show that, in an individual ear of an individual listener, the octave deviation W may at particular frequencies as well be systematically negative. As a function of reference frequency, the octave deviation shows an oscillating pattern that is comparable to that found in binaural diplacusis (cf. van den Brink 1970a,  p. 338). The pattern is characteristic for any particular ear of any particular listener, and it is reproducible (van den Brink 1977a). It may thus be concluded that the gross tendency for octave stretch is superimposed by more effects, in particular, by the oscillating deviations of the ''frequency-to-pitch'' characteristic of an ear that can be indirectly deduced from measurements of binaural diplacusis (van den Brink 1977a, ).
Such systematic details of the frequency characteristic of W can occur only if the experiments are done with sine tones, and if the tones are presented monaurally. This is plausible, as the oscillating structure of that characteristic is different for each ear, such that it is averaged out when both ears are used at once. Therefore, in binaural octave adjustments with sine tones, and in monaural or binaural adjustments with harmonic complex tones, these details are not expected to occur, and they are in fact not found in the results of, e.g., Walliser (19969b), Sundberg & Lundqvist (1973a), Dobbins & Cuddy (1982a), . What remains as a general effect is octave stretch. With harmonic complex tones, the octave stretch is generally smaller than with corresponding sine tones (Sundberg & Lindqvist 1973a, ).
The explanation of octave stretch virtually is included in the explanation of octave equivalence (see also topic affinity of tones). To explain the stretch of the octave one just has to take into account the phenomenon of pitch shifts. The key to explaining octave equivalence is the multiplicity of pitch of harmonic complex tones (see topic definition of pitch). Any harmonic complex tone evokes not only spectral pitches but also virtual pitches, of which the former occur above fundamental frequency, i.e. in harmonic positions, the latter below fundamental frequency, i.e. in subharmonic positions ( p. 313). Due to that kind of pitch multiplicity, harmonic complex tones whose oscillation frequencies are in a 1:2 ratio (or close to that) inevitably will have a number of pitches in common - which yields a kind of similarity. This is my explanation of octave equivalence. The explanation of octave stretch immediately follows from that, i.e., by taking into account that the intervals between the pitches of a harmonic complex tone must be expected to be stretched by pitch shifts. Due to this stretch of the pitch pattern, the best match of the two pitch patterns is obtained for an oscillation-frequency ratio that, on the average, somewhat exceeds the value 2:1 , , ,  p. 197.
Author: Ernst Terhardt firstname.lastname@example.org - Mar 10 2000
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