Affinity of tones


In tonal music, a number of fundamental auditory characteristics of tones play an important role, and one of these characteristics is affinity. Affinity means that tones may be perceived as similar in certain aspects; that in some respect a tone may be replaced by another one; and that one tone may even be confused with another one. These criteria apply practically only to two musical intervals, namely, the octave, and, to a lesser degree, the fifth. At least within the framework of tonal music, the affinity of tones being in an octave interval is so pronounced that this relationship is termed octave equivalence. For successive tones, affinity becomes apparent in the fact that repetition of a musical phrase in a "transposition" by an octave, or, to a lesser extent, by a fifth, is hardly perceived as a transposition at all - the transposition is often hardly noticed. So, indeed, affinity is a kind of similarity.

Paul Hindemith (1940a) wrote about these two intervals:

Es zeigt sich dann, daß allen Menschen bestimmte Klangverhältnisse von Natur aus vertraut sind. Beim Hören des Oktavintervalls empfindet selbst der Mensch niedrigster Kulturstufe den oberen Ton als das höherliegende Abbild des unteren. In allen bekannten Tonsystemen umfassen deshalb die Tonleitern mit geringen Ausnahmen den Raum der Oktave. Nächst der Oktave wird das Quintintervall als feste Marke im Bereiche der Tonschritte aufgefaßt. Das Begreifen seiner naturgegebenen Unabänderlichkeit wird allerdings dem ungeschulten Ohre um einiges erschwert: Die beiden Töne verschmelzen nicht zu dem völlig einheitlichen Klang, den die Oktave ergibt; der obere der beiden Intervalltöne ist nicht, wie bei der Oktave, nur die bloße höhere Wiederholung des unteren. Immerhin wirkt das Intervall der reinen Quinte doch so eindeutig und selbständig, daß wir ihm in den Tonleitern nahezu aller Systeme begegnen. Andere Intervalle (Terzen, Sexten, Sekunden, Septimen) lassen sich nicht so zweifelsfrei festlegen ...

In music, the affinity of tones is most significant for successive tones, i.e., in melodic phrases. In simultaneous sounding of tones that are in an octave- or fifth-interval, the affinity of these tones becomes apparent from the effect that "nothing essentially new" emerges from the sound. Indeed, the conventional theory of harmony appreciates neither the simultaneous octave nor the simultaneous fifth as dyads with an harmonic function.

For the theory of music, the outstanding characteristics of the octave and the fifth are highly significant, because these two intervals can be regarded as the origin of the chromatic tone scale. Indeed, as was basically shown by Pythagoras, the entire chromatic scale emerges "automatically" when the criterion is employed that the scale must include both the octave and the fifth interval above and below any tone that previously was determined.

The affinity of tones has two main aspects. Firstly, there appears to exist an auditory "sense" for these two particular intervals. Experimental verification of that sense can be accomplished by presenting successive tone pairs to a listener and ask him to adjust one of the tone frequencies such that the tones are in an octave- or fifth-relationship, respectively. This is briefly termed octave- and fifth-matching of tones, respectively. As it turns out, most listeners are able to do this, and the result, on the average, is a frequency ratio which is close to 1:2 and 2:3, respectively. Octave- and fifth-matching can be done with practically any type of tone, in particular, both with harmonic complex tones and with sine tones. Any listener who can do that experiment with consistent results, thereby reveals that he/she possesses an internal auditory template for the pitch interval that corresponds to an octave and fifth, respectively.

Although I am not aware of any matching experiments of the above kind with other intervals, I believe one can be sure that any person with some musical talent and training is able to match a few more intervals as well, in particular, the major and minor seconds, the major and minor thirds, and the fourth. Moreover, any person who is able to sing (or hum, or whistle) a song in correct intervals reveals thereby that he/she possesses internal auditory templates for practically any interval of the tone scale. It is very unlikely that all those templates have a "natural" origin. (I do not at all believe in the existence of a "relative pitch" faculty, based on auditory appreciation of integer-number frequency ratios per se.) Rather, it appears likely that the pitch-interval templates have been acquired, i.e., by passive and active "musical communication".

The latter notion suggests that existence of the internal auditory pitch templates for the octave and the fifth possibly is not quite that fundamental. While it is apparent that to the auditory system of an "ordinary" listener the octave is by far more familiar than, e.g., the minor third, the difference in familiarity of intervals probably is gradual rather than categorical.

Yet there are some psychophysical features of the octave and the fifth that make these two intervals outstanding amongst the entire collection of musical intervals. The octave and the fifth are outstanding in the sense that familiarity with them (manifest by existence of pertinent pitch templates) can be, and probably is, acquired from non-musical sounds. This brings us to the second main aspect of the affinity of tones.

The second aspect of tone affinity is sensory affinity. This term denotes the phenomenon that successive harmonic complex tones, i.e., only harmonic complex tones, are more similar to one another when their oscillation frequencies are in a 1:2 or 2:3 relationship, than when they are not. This kind of similarity, i.e., sensory affinity, emerges from the fact that the pitch of any complex tone is multiple (see topic definition of pitch). As a consequence of the multiplicity of pitch, and of the particular intervals that exist between the simultaneous pitches of any single harmonic complex tone, there occurs commonality of pitches when the oscillation frequencies of two successive harmonic complex tones are in a 1:2 or 2:3 ratio. Commonality means that one of the tones has some pitches in common with the other, and this immediately accounts for an enhanced similarity of the tones as compared to other frequency ratios, i.e., for sensory affinity.

In principle, this explanation of sensory affinity of the octave and fifth is in line with the arguments put forward by Helmholtz. However, Helmholtz, when referring to the multiplicity of pitch, had only the spectral pitches of harmonics in mind; he did not know about virtual pitch. By contrast, the present approach takes advantage of the fact that there are also virtual pitches. Inclusion of virtual pitches into the pitch patterns of harmonic complex tones indeed makes the effect of pitch commonality distincly more pronounced [86], [93], [104] p. 395-397.

Pitch commonality of harmonic complex tones is quite pronounced for a1:2 ratio of oscillation frequencies, i.e., when the tones are in an octave interval. Pitch commonality is less pronounced for tones in fifth and fourth intervals. And it is more or less negligible for any of the other intervals. This elucidates and explains the particular role played by the octave and the fifth, at least where sensory affinity is concerned.

Pitch commonality of harmonic complex tones can be experienced in real, non-musical life, namely in any situation where harmonic complex tones occur with variable oscillation frequencies. For humans living in societies, the most prominent and omnipresent source of harmonic complex tones is the voice. To become exposed to successive harmonic complex tones it is sufficient to be exposed to any kind of voiced utterances, i.e., normal speech, shouting, humming, and singing. Even for a non-socially living individual there still is his own voice. And even in a non-musical, primitive kind of communication by voice there is plenty of chances for a listener to be exposed to successive harmonic complex tones that, more or less accidentally and temporally, happen to be in an octave- or fifth-relationship, so that the sensory affinity of those tones can be apprehended. Obviously, there is a strong feedback effect: Once sensory affinity firstly had occurred by chance, and was noticed, this probably has stimulated people to "play around" with that effect, using their own voice.

The process of becoming familiar with octaves and fifths by apprehending sensory affinity even in a non-musical environment obviously is strongly supported by the fact that there are two sexes, i.e., that women's voices, on the average, are just about an octave higher than men's voices. For instance, when a woman responds to any kind of utterance of a man by mirroring his voice intonation, she invariably will do that about an octave higher in pitch. This obviously may with high probability cause apprehension of sensory octave affinity. So, besides the well-known kind of affinity that exists between men and women, there exists sensory octave affinity, as well.

From these considerations it becomes plausible (if not evident) that, and how, a "sense" for the octave- and fifth-intervals can develop in each individual, such that corresponding pitch-interval templates become "engraved" in the auditory system. This is how the sensory affinity of harmonic complex tones ultimately explains the existence of those templates. And, most important, this also explains why employment of those templates is not confined to harmonic complex tones, such that octaves and fifths can also be matched with sine tones. This is because those templates, once they have been installed, are pitch-interval templates which can be matched to the pitches of any type of sound, provided the latter has any pitch at all.


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


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