31 Tone Equal Temperament

Intervals: 3 : Diatonic Semitone

The diatonic semitone is the "half-step" of a diatonic scale, the familiar major or minor scales of western music. The diatonic semitone can be seen as the amount by which a stacked perfect fourth and major third fall short of one octave. This corresponds to the difference between the 15th and 16th harmonics. Unlike 12-ET, which treats all half-steps as the same, 31-ET distinguishes between the diatonic and chromatic semitones. In 31-ET, the diatonic semitone consists of 3 single steps of the scale. Because the whole tone in 31-ET consists of 5 steps and cannot be divided in half, the term "half step" is generally not used to refer to this interval in 31-ET.

Harmonically, the diatonic semitone usually functions as a dissonant interval, less dissonant than the chromatic semitone but more dissonant than seconds. However, it can be consonant. Because the diatonic semitone matches the 16:15 ratio more closely in 31-ET than 12-ET, the diatonic semitone has greater potential for being used in this consonant context in 31-ET than in 12-ET.

Root of the diatonic semitone

The diatonic semitone is particularly weak in terms of setting harmonic tone, but when the overall context supports it, it can contribute to the top note functioning as the root of a chord. An example would be stacking a major third or perfect fifth on top of a diatonic semitone. This phenomenon explains why a major seventh chord sounds so stable.

Septimal diatonic semitone

This interval is also a match to the septimal diatonic semitone, representing the 15:14 ratio in the harmonic series. 31-ET does not distinguish these two intervals, which are very close in size. The septimal diatonic semitone implies a root a septimal whole tone above the bottom note, or a diatonic semitone above the top note.

This interval may not seem important, and its relationship to its root may seem strange, but it becomes functional in 31-ET harmony, especially in tone clusters. For example, when two diatonic semitones are stacked (adding up to a septimal whole tone), the resulting chord provides a good match to the 16:15:14 harmonics in the harmonic series. This chord strongly implies the top note as root, as each of the three intervals in the chord implies the top note as root. This behavior is quite different from the behavior of semitones in 12-ET, in which two stacked semitones are more likely to imply the bottom note as root, because they add up to a whole tone very closely matching the 9:8 ratios, a harmonic interval that is stronger than the semitones and overrides any other sort of harmonic center they might provide.

This phenomenon is one example of the richness possible in harmony in 31-ET, as this chord of two stacked semitones is dissonant yet unambiguously defines a tonal center.

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