User:Aura/On 159edo Music Theory (Part 2)

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.

Au

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.

Au

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.

Table of 159edo Trines

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