Musical "micro-scales" and "micro-melodies."

Originally published in The Journal of Psychology, pages 149-56, 1940.

Heinz Werner, Wayne County Training School

An optical figure remains unchanged under two conditions: (a) when shifted along parallel lines, or (b) when reduced or enlarged in proportion. As to auditory forms, it is a well-known fact that a melody remains the same if moved up or down the scale. But the question now arises whether there is also a plasticity of hearing comparable to that of the visual field, where a proportionate reduction in size leaves the gestalt unaffected. The problem, more specifically formulated, is this: if the elementary intervals of the tempered scale, the semi-tones, are proportionately reduced, will these micro-intervals acquire the character of a new scale comparable to our ordinary chromatic scale? Is it possible to use such a scale for the construction of melodies similar in character to those of our own scale?

The possibility of constructing a micro-scale of this sort is based on a peculiar fact, one which demonstrates the extraordinary flexibility of our hearing. A subject, when presented with a relatively small interval of-- let us say-- .12 of a semi-tone, at first may hear a very slight, indefinite difference between the two tones, or even no difference at all. But when the same interval is repeated a great many times with the subject deliberately focusing his attention on the task of discerning a clear-cut interval, there will almost invariably be reported an apparent enlargement of the objectively constant interval. Usually there is a maximum in this subjective augmentation: the maximum is reached when this small interval has acquired the subjective character of a semi-tone. The time necessary for the attainment of this maximum increase varies greatly with the individual, but so far not one individual has been encountered who did not hear the semi-tone-like quality after more or less repetition.

We have attempted to demonstrate this phenomenon more objectively by some sort of measuring process. In order to define this subjective enlargement of an objectively constant, repeated interval, it was compared with another variable interval, higher or lower on the scale, and presented singly, i.e., without repetition. In this way the subjective size of the repeated constant interval may be defined by determining the size of the variable single interval appearing equal to it. We may offer an example to illustrate exactly what this means: The subject is presented with two tones of a value of 782 and 788 cycles, that is, two tones separated by a constant interval of six cycles. The variable interval is lower on the scale, its basic tone being 721. Now, when the constant interval was presented singly, once at a time, it appeared equal to a variable interval of approximately six cycles. When, however, the constant interval was presented six times in repetition, the unrepeated variable interval had to be considerably larger in order to appear equal. Varying with the subject this variable interval had to be enlarged 8, 11, 17, 18, or 22 cycles in order to approximate the repeated interval of six cycles.

The results of this experiment may be briefly summarized as follows: (a) The enlargement is demonstrable with all five subjects. This means: the variable singly presented interval always had to be larger than the repeated interval. (b) The absolute degree of augmentation is greatly dependent on the number of repetitions of the constant interval. In our experiments the constant interval was repeated two, four, and six times respectively. The more repetitions, the greater the increase. The increase is negatively accelerated, i.e., the nearer a maximum, the smaller the increment of increase. (c) The augmentation depends, furthermore, on the size of the constant interval. Four constant intervals were used here, of 3, 4, 5, and 6 cycles each. And, as one would expect, the greater the constant interval, the greater the absolute augmentation. The relative difference, however, between the augmentations with respect to the different constant intervals grows smaller the larger the number of repetitions. In other words, no matter whether the interval is one of 3, 4, 5, or 6 cycles, all these intervals become increased convergently towards the limit of a maximal interval.  This maximum is apparently the semi-tone. (d) There are considerable individual differences in the S's. One S, after six repetitions, showed only a slight augmentation ranging from 3-6 cycles, whereas the most highly plastic individual went up to 21-28 cycles, or almost to a minor semi-tone in our normal scale. This different behavior in the experiment seems to have a certain diagnostic value with respect to the varying degree of plasticity with which the S's enter the new musical system of micro-tones to be discussed later. (e) Ascending intervals exhibit a more marked tendency to augmentation than descending intervals.

In summary, these results prove that, even with a relatively small number of repetitions, the tendency to augment is more or less noticeable in all cases. The results seem to be best interpreted in terms of gestalt psychology. The tendency to augment a less differentiated, indefinite interval in the direction of greater difference is obviously the expression of a general tendency known as the "law of precision." In further experiments which shall be presently described, experiments concerned with the development of a micro-musical scale, it was found that augmentation usually ceases when the constant interval has acquired the subjective character of a semi-tone. This also may be interpreted in terms of a precision tendency, because a semi-tone is a simple and definite element in the realm of tonal distance.

The ability of the S's to learn gradually to perceive a small interval as a semi-tone provided the basis for the main experiment. The aim of this experiment was to develop a "micro-scale" and "micro-melodies." An electric oscillator connected with a keyboard was used for producing the tones. Each key of the keyboard corresponded to a tone in a "tempered micro-scale," the fundamental physical interval of which was seven cycles (the first tone 725, the second 732, etc.). Each of the 12 intervals was therefore reduced to less than one-sixth of a normal semi-tone (16.5 "cents"). This tempered micro-system was introduced to the S's step by step. After the introduction of the first semi-tone the interval above it had to be conceived as the next higher step, and so on.

The fundamental problem of this experiment is decidedly genetic in nature, that is, how does a micro-system, its intervals and its relations between intervals, develop? The objective and the subjective evidence from this experiment both suggest that the development occurs according to definite psychological laws. We may illustrate one of the laws found operative in the construction of the scale. We refer to it as the "law of increasing stabilization." As the subject first acquires them, the intervals are highly unstable. The micro-interval during the first sessions will have to be repeated constantly before it is heard as a semi-tone. From day to day, however, the number of repetitions necessary for attaining the semi-tone-like quality becomes fewer, until with the majority of my S's the interval can generally be heard as a half tone without the need of repetition. At this point the micro-interval is stabilized, but stabilized only under a particular condition. The condition is, that it be heard unaccompanied by other tones, and as an ascending step. The same interval will break down if presented in a reverse direction, that is, as a descending step. Therefore, if the subject should be presented with an ascending-descending interval, he will at first perceive the ascending phase as subjectively larger than the one which descends. At this stage of stabilization an interval is not simply dependent on the apprehension of the two single tones; it has rather a gestalt quality of its own, dependent on such conditions as upward or downward directions. This leads to the paradoxical statement frequently made by the S's that even though Tones 1 and 2 remain constant in character, nevertheless the interval separating these tones may audibly vary in size. All this seems to prove that it is not the position of the tones which is of primary importance and which creates the quality of the interval. It would seem that it is the quality of the interval which determines the position of the tones in the micro-system. Stabilization of the intervals is achieved when the semi-tones become independent of the patterns in which they are embedded. The same development towards stabilization repeats itself in a configuration where two intervals are used, one being above the other. This pattern occasions an accumulative effect. In the 3-tone pattern, 1-2-3, 2-3 seems bigger than 1-2. The middle tone, 2, paradoxically enough, quite often changes its subjective position when Tone 3 is heard after 1-2. Tone 2 has to be pinned down to a fixed position where it no longer fluctuates according to the influence of the surrounding tones. Stabilization of larger steps follows the same law. The step 1-3, equivalent to a whole tone in the normal scale, becomes stabilized only providing the semi-tone relations of the steps 1-2 and 2-3 are fixed, and only when the subject is prepared to hear that 1-3 equals 1-2 plus 2-3. A subject may, for instance, have attained stabilization in the case of Tone 3 in the pattern 1-2-3, but this stabilization may still break down when the great interval 1-3 is presented. It is usually by contrasting 1-2 and 1-3 that Tone 3 becomes gradually fixed in a definite place in the scale.

In summary, the experiment permits us to observe the development of a rationalized system under laboratory conditions. After a time a usable scale of micro-tones is perceptually constructed. That is, the originally labile tones and intervals acquire stable positions in tonal space, and the distance relations of all tones to each other become rationalized in terms of the micro-semi-tone as the elementary distance.

Beyond this the S's were introduced to the melodic patterns based on the micro-scale. Certain questions arise here which shall be briefly discussed. What objective and subjective evidence do we find in these experiments which entitle us to call these patterns melodies? What is the nature of these "micro-melodies?"

There are great individual differences in the ease and rapidity with which the subjects enter this micro-world of tone, and in the qualitative and quantitative perfection of discernment which they attain. For one S the scale did not develop above five steps (during a training period of over four months) and, even within this range, the melodies remained rather vague. The other four S's however, acquired a more or less accurate notion of the whole scale of 12 steps and were very certain about the patterns as clear, purely expressed melodic forms.

Since we know the importance of musical notation for the rationalization of musical systems in general, the tones were introduced, from the very beginning of the training period, as notes, designated by the numbers 1, 2, 3, . . . up to 13. One objective criterion for the successful stabilization of the micro-melodic pattern was, then, the correctness of the S's notation made after listening to the pattern. The correctness in notation as a criterion of stabilization increased steadily. Of course, since the observer is always subject to the strong influence of the normal musical system, a perfect stabilization could not be expected and, as a matter of fact, was never attained. A second criterion was the transposition of the melodic patterns up and down the scale. After a training period of 3-4 months the subjects were presented with Simple patterns consisting of eight tones (four beats). Transposed melodies were recognized as identical with the originals in practically every case. If one tone in the transposed pattern was shifted one step higher or lower than the others, the deviation was always detected, that is, providing this tone happened to be one of the principal notes in the melody. The deviation was not always, although rather frequently, observed if the tone were of a transitory nature.

How do the four S's regard this system with respect to the normal musical scale? There are differences in the individual response. When the scale as such was given chromatically, i.e., step by step, none of these subjects, providing that they were completely "in the system," found any distinction between the micro-scale and the normal piano scale. Individual differences came to the fore when the micro-melodies were compared to patterns of the normal scale. Two of the S's felt sure that no differences existed. One observer expressed the feeling that it was as if she were sitting in a puppet theatre. "If one keeps on looking at the puppets, after a while they acquire the full size of human beings !" The other two S's felt a certain difference. One of them thought they were true melodies, "but with something missing." "I think it is the harmonic relationship which I miss here," one subject remarked. The other observer found that the melodies lack one dimension; he compared them to two-dimensional pictures of three-dimensional objects.

I do not believe that these individual differences with respect to micro-melodies can be entirely derived from the varying ability of the subjects to enter into the system. I think, rather, that these differences may be attributed to the very character of the system itself. This scale is built up on the principle of dstance. It is therefore of exactly the same character as what in the history of music is known as a "melodic scale" devoid of any genuine harmonic chord relationships. Most musical systems outside the sphere of Western culture are constructed according to this same principle of tonal distance. (It is probable, therefore, that those observers who identify our micro-scale with the piano scale do not pay any particular attention to harmonic relationships, whereas the others who find "one dimension missing" actually do.) However, it would be quite erroneous to assume that such a scale based on the principle of tonal distance, and not on the principle of harmonics, should lack differential interval qualities such as the fifth or even the octave. Hornbostel, for instance, argues that in purely (Non-European) melodic systems definite melodic relationships exist which are somewhat analogous to the harmonic chords. These melodic relationships are formed according to the principle of similarity. That is, two tones of an octave, of a fifth and of a fourth, are melodically similar in a decreasing degree (1). Hornbostel's assumption is to a certain extent confirmed by our experiments. It was found that for the four S's the two tones 1 and 8, and 1 and 13 (fifth and octave of the normal system) acquired the character of similarity. One of the S's identified these intervals exactly with the octave and fifth of the piano scale. In a further experiment a comparison was made of the intervals 1-13, 1-8, 1-6. It was discovered that, generally speaking, the order of similarity is 1-13, 1-8, 1-6, that is, equivalent to the normal order of octave-fifth-fourth in the micro-scale. Judgments in the experiment of comparison were based on such qualities as "empty," in the case of 1-8, or on the "feeling of returning to the same point of departure" in the case of 1-13, etc. The other intervals of this scale acquired more or less definite qualities. Particularly clear to all S's from the beginning was also the difference between major and minor quality in the intervals 1-6 and 1-5 (equivalent to our major and minor third), the leading tone quality of the semi-tone below the first and the thirteenth note, etc.

In a final experimental series the S's listened to melodic patterns somewhat more complicated than those previously employed. These 12 patterns consisted of two phrases, eight beats each. Their equivalent in the normal scale sounded more or less well rounded with a semi-closure at the end of the first phrase. The S's were asked to evaluate each melody from the standpoint of musical qualities. The three S's engaged in the experiment behaved quite as if they were evaluating melodies in the normal scale, that is, they showed preference or rejection of certain melodies on the basis of reasons bringing into account formal construction or melodic content. They made such statements as the following: "This one is perfect, very melodious; this is rather artificial; this is too simple, too much like a child's melody; this is patriotic; this one is humorous, etc." Statements referring to the formal construction were of this sort: "This melody is a little too empty, since it is built altogether on the tones of a triad." "The two parts do not fit because one is major, the other is minor." "This is an excellent semi-closure leading of necessity to the second part." And so on. The same patterns were repeated three weeks later and a surprisingly high degree of consistency in the judgments and the preference of each subject was found. However, if the evaluations of the three subjects are compared with one another, a consistency with regard to preferences can hardly be expected. Nevertheless, it seems significant that amongst the 12 patterns presented there actually was a complete agreement as to the three patterns most preferred and also as to the three patterns definitely rejected.

In summary, the results of these experiments suggest that the characteristic of optical configuration to retain the same gestalt when reduced proportionately also applies to tonal configurations, provided that the patterns are based on the principle of distance alone and not on chord relationship. The remarkable plasticity of the ear as revealed in the experiments is obviously a basic factor in the process of rationalization which can be seen in the historical development of musical scales. This plasticity must, it seems, be responsible for the fact that the historical development has proceeded in so many divergent directions, creating musical systems very different in kind, but all satisfactory to the human ear.


1. Hornbostel, E.M., von. Musikalische Tonsysteme. In Handbuch der Physik (Ed. Geiger & Scheel). (Vol. VIII, 427.)

Wayne County Training School, Northville, Michigan