User:Aura/On 159edo Music Theory (Part 2)
From Xenharmonic Reference
It is recommended that one read Part 1 prior to reading this article
Now that we have covered the intervals of 159edo in depth, it's time we begin looking at how to use them, starting with trines.
Trines
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Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory- specifically they're the result of two notes forming an octave, along with a third note for stark contrast, being played simultaneously, which is how 3-limit harmony naturally works.
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In addition to the 3-limit trines, there are such things as dissonant trines, in which the third note is something other than a perfect fourth or perfect fifth away from the doubled root- in fact the third note can be anything from a paraminor fourth to a paramajor fifth relative to the root.
The individual intervals that constitute trines serve as the backbone of not only the triads of harmony, but the tetrachords of melody as well. Both triads and tetrachords will be covered in later installments of this series. For now, it pays to go over which three-note structures can serve as trines as well as their names.
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Most of the names given here are unique to me at present, but some, such as the otonal perfect and utonal perfect are more objectively descriptive.
| Name | Notation (from D) | Steps | Approximate JI | Notes |
|---|---|---|---|---|
| Otonal Perfect | D, A, D | 0, 93, 0 | 2:3:4 | This is the first of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony |
| Utonal Perfect | D, G, D | 0, 66, 0 | 1/(2:3:4) | This is the second of two trines that can be considered fully-resolved in Medieval and Neo-Medieval harmony |
| Otonal Archagall | D, G\, D | 0, 65, 0 | 64:85:128 | This trine is the first of two that makes sense when imitating the feel of 22edo |
| Utonal Archagall | D, A/, D | 0, 94, 0 | 1/(64:85:128) | This trine is the second of two that makes sense when imitating the feel of 22edo |
| Bass-Up Marvelous | D, A\, D | 0, 92, 0 | 75:112:150 | This trine is the first of two that makes sense when imitating the feel of 19edo |
| Treble-Down Marvelous | D, G/, D | 0, 67, 0 | 1/(75:112:150) | This trine is the second of two that makes sense when imitating the feel of 19edo |
| Narrow Supernaiadic | D, G↓\, D | 0, 62, 0 | 16:21:32 | This dissonant trine is common in essentially tempered chords |
| Wide Subcocytic | D, A↑/, D | 0, 97, 0 | 1/(16:21:32) | This dissonant trine is common in essentially tempered chords |
| Subcocytic | D, A↑, D | 0, 96, 0 | 160:243:320 | This dissonant trine is common in essentially tempered chords |
| Supernaiadic | D, G↓, D | 0, 63, 0 | 1/(160:243:320) | This dissonant trine is common in essentially tempered chords |
| Wide Supernaiadic | D, G↓/, D | 0, 64, 0 | 25:33:50 | This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords |
| Narrow Subcocytic | D, A↑\, D | 0, 95, 0 | 1/(25:33:50) | This dissonant trine is on the outer edge of the diatonic range and is common in essentially tempered chords |
| Wide Naiadic | D, Gd<↑, D | 0, 61, 0 | 135:176:270 | This dissonant trine is among the more consistently complex |
| Narrow Cocytic | D, At>↓, D | 0, 98, 0 | 1/(135:176:270) | This dissonant trine is among the more consistently complex |
| Naiadic | D, Gd>/, D | 0, 60, 0 | 10:13:20 | This dissonant trine is relatively simple and thus expected to be rather common |
| Cocytic | D, At<\, D | 0, 99, 0 | 1/(10:13:20) | This dissonant trine is relatively simple and thus expected to be rather common |
| Wide Cocytic | D, At<, D | 0, 100, 0 | 11:17:22 | This essentially tempered trine is very likely to be used as a basis for cocytic triads |
| Narrow Niadic | D, Gd>, D | 0, 59, 0 | 1/(11:17:22) | This essentially tempered trine is very likely to be used as a partial basis for suspended chords |
| Narrow Superdusthumic | D, Ad<↑, D | 0, 89, 0 | 128:189:256 | This dissonant trine is common in essentially tempered chords |
| Wide Subagallic | D, Gt>↓, D | 0, 70, 0 | 1/(128:189:256) | This dissonant trine is common in essentially tempered chords |
| Subagallic | D, G↑, D | 0, 69, 0 | 20:27:40 | This dissonant trine is very likely to show up in non-meantone diatonic contexts |
| Superdusthumic | D, A↓, D | 0, 90, 0 | 1/(20:27:40) | This dissonant trine is very likely to show up in non-meantone diatonic contexts |
| Narrow Subagallic | D, G↑\, D | 0, 68, 0 | 90:121:180 | This dissonant trine is on the outer edge of the diatonic range |
| Wide Superdusthumic | D, A↓/, D | 0, 91, 0 | 1/(90:121:180) | This dissonant trine is on the outer edge of the diatonic range |
| Wide Agallic | D, Gt<, D | 0, 73, 0 | 8:11:16 | This ambisonant trine is very likely to be used as a partial basis for suspended chords |
| Narrow Dusthumic | D, Ad>, D | 0, 86, 0 | 1/(8:11:16) | This ambisonant trine is very likely to be used as a basis for dusthumic triads |
| Dusthumic | D, Ad<\, D | 0, 87, 0 | 128:187:256 | This dissonant trine is common in essentially tempered chords |
| Agallic | D, Gt<\, D | 0, 72, 0 | 1/(128:187:256) | This dissonant trine is common in essentially tempered chords |
| Narrow Agallic | D, Gt>↓, D | 0, 71, 0 | 11:15:22 | This trine is very likely to be used as a partial basis for suspended chords |
| Wide Dusthumic | D, Ad<↑, D | 0, 88, 0 | 1/(11:15:22) | This trine is very likely to be used as a basis for dusthumic triads |
| Wide Subdusthumic | D, Ad<, D | 0, 85, 0 | 56:81:112 | This essentially tempered trine is likely to be used as a basis for subdusthumic triads |
| Narrow Superagallic | D, Gt>, D | 0, 74, 0 | 1/(56:81:112) | This essentially tempered trine is likely to be used as a partial basis for suspended chords |
| Subdusthumic | D, Ab↑↑, D | 0, 84, 0 | 9:13:18 | This essentially tempered trine is very likely to be used as a basis for subdusthumic triads |
| Superagallic | D, G#↓↓, D | 0, 75, 0 | 1/(9:13:18) | This essentially tempered trine is very likely to be used as a partial basis for suspended chords |
| Wide Superagallic | D, Gt<↑, D | 0, 76, 0 | 256:357:512 | This essentially tempered trine is very likely to be used as a partial basis for suspended chords |
| Narrow Subdusthumic | D, Ad>↓, D | 0, 83, 0 | 1/(256:357:512) | This essentially tempered trine is very likely to be used as a basis for subdusthumic triads |
| Narrow Hyperquartal | D, Gt>↑, D | 0, 77, 0 | 5:7:10 | This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords |
| Wide Hypoquintal | D, Ad<↓, D | 0, 82, 0 | 1/(5:7:10) | This ambisonant trine is very common as a basis for diminished chords, and is very likely to be used as a partial basis for suspended chords |
| Hyperquartal | D, G#↓, D | 0, 78, 0 | 32:45:64 | This trine is very likely to be used as a partial basis for suspended chords |
| Hypoquintal | D, Ab↑, D | 0, 81, 0 | 1/(32:45:64) | This trine is very common as a basis for diminished chords |
| Narrow Hypoquintal | D, Ab↑\, D | 0, 80, 0 | 12:17:24 | This trine is very common as a basis for diminished chords |
| Wide Hyperquartal | D, G#↓/, D | 0, 79, 0 | 1/(12:17:24) | This trine is very likely to be used as a partial basis for suspended chords |
