The partial-tone series provides us with the material of tones, but it gives in its totality a too complex sound and, above all, no scale. Measurements on split sounds of oboes and clarinets have shown that combination tones are built in arithmetic series. This knowledge made it possible by means of such series to form sonance series and selection scales out of the partial tones. An arithmetic series is formed by repeatedly adding the same difference d to an initial value a. This is called Sequence d on a and abbreviated d || a. As an example the Sequence 3 on 2:

3 || 2 → 2 5 8 11 14 17 20 23 26 29 32 …

Out of this sequence, you form a sonance series – e.g. up to the 32nd partial tone – and transpose the tones of this series into the range of one octave. Thus, you obtain an eight-step scale.

In case of a too small distance between two tones in the scale, so they can no longer be considered as independent degrees (e.g. less than ½ Limma), the first tone upward and the second tone downward, respectively, is omitted (corresponding to the melodic minor scale). As an example the Sequence 5 on 4 with 12 degrees: It includes too small distances between the 7th and 8th tone (partial tones 48, 49) and the 11th and 12th tone (partial tones 58, 59), respectively, and therefore it has 10 effective degrees.

The use of a single ekmelic scale naturally produces a tonal diatonic music which can be expanded by introducing tones that are foreign to the scale (chromaticism). The functional relations determining the harmonic progress also have to be derived from the partial tone series. By incorporating the ekmelic tones we gain a great number of quite new harmonious chords (in whole-numbered proportions) which we call "stable sounds". In the same way the melody shape will be substantially enriched, the more so as, besides novel sequences of tones, influences of non-European musical cultures are possible.

Select a Scale

Too small distances between two tones are red highlighted. In the upward and downward scale, the omitted tones are marked with — in front of the number of the respective other tone.

The note names specified here correspond to the German semitone and quarter-tone names in LilyPond, extended with names for sixth- and twelfth-tones.