I wanted to do my own ranking of all the degrees of the 31 pitch classes of my tuning system (by using 31 equal temperament, which is an extended meantone system that contains quarter tone-type intervals and approximates 11-limit just intonation well). However, I'm working with an irrational temperament rather than just ratios, and I want to use a more scientific understanding of consonance and dissonance than merely looking at numerators and denominators.
This is where Boston-based guitarist and music theorist Paul Erlich comes in. He came up with the idea of harmonic entropy (see also this). (Of course Boston has its own established microtonal music scene.)
Using the Scala tuning program, I calculated the entropies (dissonances) of the pitches of 31 equal. In order from lowest to highest (not counting the unison):
- 31. perfect octave
- 18. perfect fifth
- 1. semi-augmented prime / diesis ("quarter tone")
- 13. perfect fourth
- 23. major sixth
- 10. major third
- 8. minor third
- 30. semi-diminished octave
- 25. augmented sixth (~ 7th harmonic)
- 15. augmented fourth
- 21. minor sixth
- 26. minor seventh
- 7. augmented second (~ 7/6 minor third)
- 5. major second
- 27. neutral seventh
- 6. semi-augmented second
- 12. semi-diminished fourth
- 16. diminished fifth
- 19. semi-augmented fifth
- 11. diminished fourth
- 9. neutral third
- 28. major seventh
- 22. neutral sixth
- 14. semi-augmented fourth (~ 11th harmonic)
- 20. augmented fifth
- 4. neutral second
- 24. diminished seventh
- 17. semi-diminished fifth
- 3. minor second / diatonic semitone
- 29. diminished octave
- 2. augmented prime / chromatic semitone
Edit: not only is the entropy function a good determiner of dissonance, it also gives an idea of how far a pitch can deviate from what the mind thinks is, for example, a semitone, as opposed to just an out-of-tune unison. This may explain why the semi-augmented prime (in JI, about 36/35) ranks higher than the much more obvious perfect fourth (4/3).
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