The most prominent case of affinity of
tones is *octave affinity*, where *affinity* to
some extent is synonynous to *similarity*. In the context of
tonal music, the affinity of tones an octave apart is so
pronounced that it is also termed octave *equivalence*.
Consequently, in the notation of tonal music tones which are one
or several octaves apart are given the same letter symbol (C, D,
E, etc.). Octave equivalence is an auditory (i.e., subjective)
phenomenon.

The auditory sense of octave equivalence is universal (Burns & Ward 1978a, 1982a). The majority of individuals of any population are able to tell whether or not two successive tones are in the particular relationship of octave affinity. The sense appears to exist already in infancy (Demany & Armand 1984a). And most individuals who are presented with a musical tone can hum, or whistle, or adjust on an oscillator, another tone such that the latter is an octave above or below the former. The ability of adjusting two successive tones into the octave relationship is so accurate that one can measure the precise frequency ratio of sine tones, and of harmonic complex tones, that yields an optimal octave. (The accuracy of matching in fact is almost as high as that of matching two successive tones for identity of pitch.) Such measurements reveal that - on the average - the frequency ratio is not necessarily exactly 1:2. In particular, a pronounced tendency to ''stretch'' the octave is observed; i.e., for perceiving an optimal octave relationship a frequency ratio is adjusted that systematically exceeds 2:1 (octave stretch).

However, there are limitations to octave equivalence. Imagine, for instance, that a simple melody, well-known by listeners, is played such that successive tones never are played on the same instrument. Rather, the first tone is played, e.g., on a trumpet, the second on a trombone, the third on a clarinet, and so on. In such a performance not only the timbre of each tone will differ from the previous one but there will also occur octave offsets. This kind of performance does not only sound funny but reveals that recognition of the melody is much harder than in a ''natural'' performance, i.e., on one single instrument. Another ''real-life example'' is the construction of a melody from the barks of different dogs - which ordinarly is done just for fun (of humans, not dogs), but demonstrates the same effect. Serious experiments on recognition of distorted melodies were carried out, e.g., by Deutsch (1972a), and Dowling & Hollombe (1977a). Simple melodies were randomly split into different octaves. As opposed to the above examples, no timbre variations were applied; all tones of the melodies were of the same type (sine tones, and harmonic complex tones, respectively). So in the test melodies each of the tones had the correct ''chroma'' (see topic definition of pitch), but the temporal ''pitch contour'' was drastically distorted. The result was that the melodies, although well known to the listeners, became almost unrecognizeable. Evidently, for recognition of a melody temporal pitch contour is a much stronger clue than chroma. Octave equivalence obviously is of little or no help to the auditory system when a melody is split into different octaves. Equivalence turns out to be severely limited.

Another limitation is apparent from the results of experiments
in which listeners were asked to give estimates of the *similarity*
of successive pairs of tones. On first sight one may expect that
such estimates should yield distincly higher similarity ratings
for tone pairs that form an octave interval than for other
intervals. For piano tones (i.e., harmonic complex tones with
basically the same timbre) this turns out to be true (Parncutt 1989a), though the
difference between similarity estimates for the octave as
compared to other intervals is not dramatic. In Parncutt's data,
the ''peak ratio'' of the distribution of similarity ratings for
the octave as compared to other intervals is, roughly, 1.2. For *sine
tones*, similarity is an almost monotonic function of pitch
distance (Kallman 1982a),
i.e., the similarity ratings decrease when the interal gets
wider, and the distribution does not show any significant peak at
the octave interval.Yet some faint indication of octave affinity
can be seen in Kallman's data: The *slope* of the
similarity-distance distribution is distincly flatter (i.e.,
about 0) at a distance of one octave.

The results of Parncutt and Kallman can readily be explained
by taking into account that essentially it is *sensory*
octave affinity that can be involved in those similarity ratings
(see topic affinity of tones). As
sensory octave affinity is based on commonality of multiple
pitches in harmonic complex tones, sine tones do not have any
sensory octave affinity. Therefore, the similarity of sine tones
essentially is governed by the width of the frequency interval -
just as found by Kallman
(1982a).

Yet another limitation in octave equivalence was demonstrated
by Thurlow & Erchul
(1977a), who investigated the ability of listeners to
recognize *multiple* octaves, i.e., tones in larger pitch
distances. It was found that this ability differed drastically
between listeners. Moreover, they found little or no evidence for
similarity of tones in multiple-octave intervals, which is well
in line with the aforementioned findings of Parncutt and Kallman.

It is thus apparent that, in the context of other sensations of tone, octave equivalence is not quite a prevalent feature but is a comparatively subtle phenomenon. Moreover, it is apparent that octave equivalence is less pronounced for sine tones than it is for harmonic complex tones.

To understand octave equivalence, one should first of all be
aware of its two different main aspects. The first aspect is
existence of a pitch-interval template in the auditory system,
that enables a listener to match successive tones of any kind to
the particular pitch interval of an octave. The second aspect is *sensory*
affinity of tones being in an octave interval, i.e., a kind of
similarity that is based on commonality of pitches. The
explanation of octave equivalence, based on this concept, is
given in the topic affinity of tones.
Briefly, the explanation is, (1) that the exceptional character
of the octave interval is determined by the multiple pitches
elicited by harmonic complex tones, in particular, the human
voice; and (2) that the auditory pitch template corresponding to
the octave interval in each individual develops from frequent
exposure to harmonic complex tones with variable oscillation
frequencies.

Finally, it should be noted that the above explanation is supported by the phenomenon of octave stretch. The explanation given for octave stretch actually implies the above explanation of octave equivalence. When in the explanation of octave equivalence the effects of pitch shifts are taken into account, it turns out that the width of the pitch interval that corresponds to an octave indeed must be stretched relative to the width that would emerge without existence of pitch shifts.

Author: Ernst Terhardt terhardt@ei.tum.de - Feb. 13, 2000