Harmony


In the present context, the term harmony denotes the most important of two major components of musical consonance. The major components of harmony, in turn, are affinity of tones, and root-relationship. The term harmony essentially addresses the same collection of phenomena as were by Helmholtz (1863a) termed Verwandtschaft der Klšnge (relationships between musical sounds). Likewise, harmony basically addresses the goals and methods of the conventional theory of harmony, i.e., to understand the organization and auditory effects of musical sounds, and to provide methods for organizing musical sounds such that they are appreciated by the ear.

Helmholtz (1863a) was the first who proposed a comprehensive, psychophysically based explanation of the origin of harmony. He demonstrated that, and why, any harmonic complex tone (such as the human voice) evokes multiple pitches, namely not only the main pitch that corresponds to the tone's oscillation frequency, but also a number of spectral pitches, i.e., of harmonics. He pointed out that on the basis of this notion one can plausibly explain that tonal music may have evolved in a cultural process.

According to this explanation, the origin of that process was melody, i.e., in the sense of just sequences of harmonic complex tones. Helmholtz argued that, when sequences of harmonic complex tones are produced (at first in a primitive and unorganized manner), the ear gets frequently exposed to the basic harmonic pitch intervals, i.e., those that are included in the lower harmonics (essentially, the major triad). The devices that he had in mind for early production of harmonic complex tones, were the human voice, blown horns, and, perhaps, plucked strings.

Helmholtz further suggested that on this line of development first a sense of tone affinity (i.e., octave and fifth relationship) should have evolved, including perhaps a sense for the major triad. This was sufficient to suggest that later on melodic performances were more and more confined to preferred pitch intervals, i.e., octaves, fifths, fourths, and thirds. From that achievement then it was only a relatively small step to the development of the pentatonic and the diatonic scales, and to the development of polyphonic performances, i.e., simultaneous tones in certain of those preferred intervals.

Helmholtz himself emphasized that his theory of harmony was speculative - as opposed to his explanations of the theory's psychophysical fundamentals. So, he was fully aware that the theory was based on several assumptions that were not - and perhaps cannot be - verified by experimental observations.

When it comes to evaluating Helmholtz's theory, it should at first be noted that a priori there is nothing wrong about speculative theories, because, as we have learned from Karl Popper, any scientific theory that is concerned with phenomena of real life inevitably is speculative. Such type of theory cannot be proven; it can only be disproven, i.e., by well-established observations that are in an unresolvable conflict with the theory.

So the first question is: Is there any well-established evidence today that is in conflict with Helmholtz's theory of harmony? I am not aware of such evidence. If Helmholtz turned out to have been wrong in certain details of music history, this would not at all affect his line of argument. In particular, there cannot be any doubt that Helmholtz was correct on presuming that tonal music has evolved in a cultural process that started much earlier than written history.

The next question then is: Is there any theory of harmony (in the sense of a psychophysical explanation of harmony) that can explain more observable facts than those envisaged by Helmholtz, and that such gets an even higher degree of plausibility? My answer is: Yes, there is my own theory - which in fact is merely an extension and complementation of Helmholtz's [22], [30], [31], [34], [63], [104] p. 397-405.

The key phenomenon for the modification of Helmholtz's theory (once again) is virtual pitch. This phenomenon was not known to (or, at least, not apprehended by) Helmholtz, and this is why his theory is weak in four items:

1) Helmholtz did not have a solid explanation of the main pitch of harmonic complex tones. While he proposed the "place principle" as a basic explanation of spectral pitch (which I believe is not questionable), and pointed out that a number of spectral pitches of harmonics can be "heard out" from a complex tone, he apparently believed that the main pitch (the "musical pitch") of harmonic complex tones is determined by the spectral pitch of the fundamental. However, already in his time it had become apparent that this view was not tenable (cf. Seebeck 1841a).

2) As a consequence of missing the phenomenon of virtual pitch, Helmholtz's explanation of tone affinity by commonality of pitches of harmonic complex tones, was confined to the spectral pitches of lower harmonics. Thus, the amount of commonality of pitches that he could envisage was scarce as compared to when virtual pitches are also included.

3) As another consequence, Helmholtz could not become aware of the fact that existence of virtual pitch, and how it is created in the auditory system, as such provides for the existence of templates in the auditory system of the basic harmonic pitch intervals (see topic virtual pitch). Inclusion of the principles of the virtual-pitch theory in fact makes Helmholtz's assumption dispensable, that at the beginning of music evolution there had to be "melody".

4) Finally, Helmholtz entirely missed the psychophysical basis of the root phenomenon whose explanation so readily suggests itself once the phenomenon of virtual pitch has been apprehended. Helmholtz explicitly rejected Rameau's concept of an "inferred" basse fondamentale, and he believed that the use and perception of fundamental notes in music was a cultural phenomenon, i.e., acquired as a kind of side effect of tonal music.

It should be apparent that, and how, these weaknesses of Helmholtz's theory are cured by incorporating virtual pitch. Doing so, Helmholtz's basic arguments get considerably strengthened, and the evolution of music is elucidated, providing some new insights.

Conceptually, the most important change to Helmholtz's theory is that the two major components of harmony, i.e., affinity of tones, and root-relationship, both are given explanations that are of a more psychophysical nature than assumed by Helmholtz. In particular, the root phenomenon, which obviously is of particular importance for understanding polyphonic music - and of its development - turns out to be "natural" rather than "artificial". Natural means that the root phenomenon, as it is just virtual pitch, emerges from the basic challenges of auditory communication, in particular, from speech, such that its existence is not at all dependent on existence of tonal music.

Most concisely one may assert that both affinity of tones and root-relationship are primarily based on the fact that humans have a voice and that they communicate with it. Once in this way affinity of tones and root-relationship are given, the evolution of tonal music can be explained on the route suggested by Helmholtz, though more readily and with more plausibility. Instead of having to explain the root as a side effect of cultural development of music (as Helmholtz did), the root phenomenon must be regarded as a major primary psychophysical factor that has driven the evolution of music.

Regarding the theory of harmony - in the conventional sense -, the above insights indeed provide new and significant prospects. As the theory of harmony to a considerable extent is concerned with discussing the harmonic function of sounds, and as this is more or less equivalent to discussing the root-relationships of sounds, it is apparent that an efficient algorithm for the determination of roots makes a significant contribution to the theory of harmony. As a consequence of the nature of the root phenomenon, the theory of virtual pitch readily provides such an algorithm.

To take advantage of this insight, one must at first become aware that the virtual-pitch algorithm - just as the auditory system - operates on sounds as input, as opposed to notes (musical score). This is to say that, naturally, the number, values, and prominence of pitches elicited by a sound basically depend on the sound's Fourier spetrum, i.e., not just on one single parameter such as oscillation frequency. This makes apparent that the conventional theory of harmony, operating just with musical scores, is hazardous, as the tones that are symbolically denoted may be transferred into quite different types of sound when the score is performed.

Fortunately (for the theory of harmony), the pitch patterns elicited by harmonic complex tones are not too heavily dependent on the shape of the Fourier spectrum, i.e., timbre. So, if one presupposes - as a kind of general convention - that musical tones throughout are to be realized as harmonic complex tones, one can indeed represent them by note symbols, at least to a reasonable approximation.

Based on this convention, one can deduce from the theory of virtual pitch a root-finding algorithm that operates on the score - just as in conventional music theory [53]. The principle is, that all candidates of roots must be subharmonics of the spectral pitches elicited by the actual sound, and that the prominence of any root is enhanced by "subharmonic coincidence" (see topic virtual pitch). When one translates this concept into musical notation, one gets a simple algorithm that shall be explained by the following example.

Chord notes: c f a
same C F A
- 1 fifth F Bb D
- 1 mj 3rd Ab Db F
+ 1 mj 2nd D G B
- 1 mj 2nd Bb Eb G

The sample chord of which the root(s) are to be determined is written into the first line (2nd to 4th column, lowercase letters), i.e., c-f-a. The root candidates derived from the notes are written into the pertinent columns (uppercase letters) The first candidate corresponds to both the first and second subharmonics, i.e., it is just the same note (second line), because, by definition, octave-equivalent pitches are not distinguished. The second candidate (third line), corresponding to the third subharmonic, is obtained by stepping one fifth down from the chord note, as octaves are not distinguished. The third candidate (fourth line), corresponding to the 5th subharmonic (the fourth is omitted, due to octave-equivalence) is obtained by stepping one major third down. The fourth candidate (fifth line), correponding to the 7th subharmonic (the 6th is omitted, due to octave-equivalence to the third), is obtained by stepping either down by a minor seventh or stepping up one full tone (due to octave equivalence). The 8th subharmonic is omitted due to octave equivalence, such that the last candidate (6th line) corresponds to the 9th subharmonic, and it is obtained by either stepping up a 9th interval or stepping down one full tone (octave equivalence). As one can see in the table, there is one and only one candidate, namely F, that occurs in all three columns (full subharmonic match). This, by definition, indicates that F is the most prominent root of the chord.

As is apparent from this example, the determination of root candidates cannot fail for any type of chord. What may happen just is that there is no "full match", i.e., that there does not occur any candidate which is found in all columns, such that the root is less pronounced and more ambiguous than in a major triad such as above.

As another example let us consider a chord that by conventional theory has been regarded to be "at the borderline to atonality", i.e., the famous Tristan chord. Here is the corresponding table:

Tristan chord: f b d# g#
  F B D# G#
  A# E G# C#
  C# G B E
  G C# F A#
  D# A C# F#

The algorithm tells us that actually there is a full match for the root C#. Thus the chord, when considered in isolation, is far from being atonal. One can easily verify that the root C# indeed "makes sense", i.e., by playing it in the bass register along with the Tristan chord. So, the subharmonic matching algorithm has found out what one may as well explain in terms of the conventional theory: The Tristan chord (f-b-d#-g#) can be said to be a major 9th chord with root C#, of which the root note itself is missing.

In summary, I believe that with the above concept of harmony, and with the explanations outlined for tone affinity and root-relationship, the psychophysical foundations of harmony are laid.


Author: Ernst Terhardt terhardt@ei.tum.de - Mar 9 2000


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