User:Aura/On 159edo Music Theory (Part 1)

From Xenharmonic Reference

Of all the multiples of 53edo, 159edo is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

Intervals and Notation

This page or section deals with proposed concepts. The terminology and concepts used in it are developed by one person or a small group and may lack widespread adoption.

159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step, however, the additional harmonies need naming, especially for those who seek to use 159edo as a means of extending the theory and praxis of Medieval, Neo-Medieval, and traditional Western Classical music.

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159edo also has its own variation on the dinner party rules— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.
159edo Interval Names and Compatibility Ratings
Step Cents Interval and Note names Compatibility rating
SKULO-based interval names Pythagorean-commatic-based interval names SRS notation Harmonic Melodic
0 0 P1 Perfect Unison D 10 10
1 7.5471698 R1 Wide Unison D/ 0 0
2 15.0943396 rK1 Narrow Superunison D↑\ -10 -10
3 22.6415094 K1 Lesser Superunison D↑ -10 -3
4 30.1886792 S1, kU1 Greater Superunison, Narrow Inframinor Second Edb<, Dt<↓ -10 3
5 37.7358491 um2, RkU1 Inframinor Second, Wide Superunison Edb>, Dt>↓ -9 10
6 45.2830189 kkm2, Rum2, rU1 Wide Inframinor Second, Narrow Ultraunison Eb↓↓, Dt<\ -9 10
7 52.8301887 U1, rKum2 Ultraunison, Narrow Subminor Second Dt<, Edb<↑ -9 10
8 60.3773585 sm2, Kum2, uA1 Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison Dt>, Eb↓\ -8 10
9 67.9245283 km2, RuA1, kkA1 Greater Subminor Second, Diptolemaic Augmented Unison Eb↓, D#↓↓ -8 9
10 75.4716981 Rkm2, rKuA1 Wide Subminor Second, Lesser Sub-Augmented Unison Eb↓/, Dt<↑ -7 9
11 83.0188679 rm2, KuA1 Narrow Minor Second, Greater Sub-Augmented Unison Eb\, Dt>↑ -7 9
12 90.5660377 m2, kA1 Pythagorean Minor Second, Ptolemaic Augmented Unison Eb, D#↓ -6 10
13 98.1132075 Rm2, RkA1 Artomean Minor Second, Artomean Augmented Unison Eb/, D#↓/ -6 10
14 105.6603774 rKm2, rA1 Tendomean Minor Second, Tendomean Augmented Unison D#\, Eb↑\ -5 10
15 113.2075472 Km2, A1 Ptolemaic Minor Second, Pythagorean Augmented Unison D#, Eb↑ -5 10
16 120.7547170 RKm2, kn2, RA1 Wide Minor Second, Artoretromean Augmented Unison Ed<↓, Eb↑/, D#/ -5 9
17 128.3018868 kN2, rKA1 Lesser Supraminor Second, Tendoretromean Augmented Unison Ed>↓, D#↑\ -6 8
18 135.8490566 KKm2, rn2, KA1 Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison Ed<\, Eb↑↑, D#↑ -7 6
19 143.3962264 n2, SA1 Artoneutral Second, Lesser Super-Augmented Unison Ed<, Dt#<↓ -8 5
20 150.9433962 N2, RkUA1 Tendoneutral Second, Greater Super-Augmented Unison Ed>, Dt#>↓ -7 6
21 158.4905660 kkM2, RN2, rUA1 Lesser Submajor Second, Retrodiptolemaic Augmented Unison Ed>/, E↓↓, Dt#>↓/, D#↑↑ -6 8
22 166.0377358 Kn2, UA1 Greater Submajor Second, Ultra-Augmented Unison Ed<↑, Dt#<, Fb↓/ -5 9
23 173.5849057 rkM2, KN2 Narrow Major Second Ed>↑, E↓\, Dt#>, Fb\ -4 10
24 181.1320755 kM2 Ptolemaic Major Second E↓, Fb -3 10
25 188.6792458 RkM2 Artomean Major Second E↓/, Fb/ -3 10
26 196.2264151 rM2 Tendomean Major Second E\, Fb↑\ -2 10
27 203.7735849 M2 Pythagorean Major Second E, Fb↑ -2 10
28 211.3207547 RM2 Wide Major Second E/, Fd<↓ -1 10
29 218.8679245 rKM2 Narrow Supermajor Second E↑\, Fd>↓ -1 10
30 226.4150943 KM2 Lesser Supermajor Second E↑, Fd<\, Fb↑↑, Dx -1 9
31 233.9622642 SM2, kUM2 Greater Supermajor Second, Narrow Inframinor Third Fd<, Et<↓, E↑/ 0 9
32 241.5094340 um3, RkUM2 Inframinor Third, Wide Supermajor Second Fd>, Et>↓ -1 8
33 249.0566038 kkm3, KKM2, Rum3, rUM2 Wide Inframinor Third, Narrow Ultramajor Second, Semifourth Fd>/, Et<\, F↓↓, E↑↑ 0 8
34 256.6037736 UM2, rKum3 Ultramajor Second, Narrow Subminor Third Et<, Fd<↑ -1 7
35 264.1509434 sm3, Kum3 Lesser Subminor Third, Wide Ultramajor Second Et>, Fd>↑, F↓\ 0 7
36 271.6981132 km3 Greater Subminor Third F↓, Et>/, E#↓↓, Gbb -1 7
37 279.2452830 Rkm3 Wide Subminor Third F↓/, Et<↑ -1 8
38 286.7924528 rm3 Narrow Minor Third F\, Et>↑ 0 8
39 294.3396226 m3 Pythagorean Minor Third F -1 9
40 301.8867925 Rm3 Artomean Minor Third F/ 1 9
41 309.4339622 rKm3 Tendomean Minor Third F↑\ 4 10
42 316.9811321 Km3 Ptolemaic Minor Third F↑, E# 7 10
43 324.5283019 RKm3, kn3 Wide Minor Third Ft<↓, F↑/, Gdb< 4 9
44 332.0754717 kN3, ud4 Lesser Supraminor Third, Infra-Diminished Fourth Ft>↓, Gdb> 1 9
45 339.6226415 KKm3, rn3, Rud4 Greater Supraminor Third, Retrodiptolemaic Diminished Fourth Ft<\, F↑↑, Gdb<↑\, Gb↓↓ -1 8
46 347.1698113 n3, rKud4 Artoneutral Third, Lesser Sub-Diminished Fourth Ft<, Gdb<↑ 0 7
47 354.7169811 N3, sd4, Kud4 Tendoneutral Third, Greater Sub-Diminished Fourth Ft>, Gdb>↑ -1 7
48 362.2641509 kkM3, RN3, kd4 Lesser Submajor Third, Retroptolemaic Diminished Fourth Ft>/, F#↓↓, Gb↓ 0 8
49 369.8113208 Kn3, Rkd4 Greater Submajor Third, Artoretromean Diminished Fourth Ft<↑, Gb↓/ -1 9
50 377.3584906 rkM3, KN3, rd4 Narrow Major Third, Tendoretromean Diminished Fourth Ft>↑, F#↓\, Gb\ 3 9
51 384.9056604 kM3, d4 Ptolemaic Major Third, Pythagorean Diminished Fourth Gb, F#↓ 8 10
52 392.4528302 RkM3, Rd4 Artomean Major Third, Artomean Diminished Fourth Gb/, F#↓/ 4 10
53 400 rM3, rKd4 Tendomean Major Third, Tendomean Diminished Fourth F#\, Gb↑\ 1 9
54 407.5471698 M3, Kd4 Pythagorean Major Third, Ptolemaic Diminished Fourth F#, Gb↑ -1 9
55 415.0943396 RM3, kUd4 Wide Major Third, Lesser Super-Diminished Fourth F#/, Gd<↓, Gb↑/ 0 8
56 422.6415094 rKM3, RkUd4 Narrow Supermajor Third, Greater Super-Diminished Fourth F#↑\, Gd>↓ -1 7
57 430.1886792 KM3, rUd4, KKd4 Lesser Supermajor Third, Diptolemaic Diminished Fourth F#↑, Gd<\, Gb↑↑ -1 6
58 437.7358491 SM3, kUM3, rm4, Ud4 Greater Supermajor Third, Ultra-Diminished Fourth Gd<, F#↑/ 0 5
59 445.2830189 m4, RkUM3 Paraminor Fourth, Wide Supermajor Third Gd>, Ft#>↓ -1 3
60 452.8301887 Rm4, KKM3, rUM3 Wide Paraminor Fourth, Narrow Ultramajor Third Gd>/, F#↑↑, G↓↓ -2 1
61 460.3773585 UM3, rKm4 Ultramajor Third, Narrow Grave Fourth Gd<↑, Ft#< -4 -2
62 467.9245283 s4, Km4 Lesser Grave Fourth, Wide Ultramajor Third Gd>↑, G↓\ -7 -4
63 475.4716981 k4 Greater Grave Fourth G↓, Abb -6 -5
64 483.0188679 Rk4 Wide Grave Fourth G↓/ -4 0
65 490.5660377 r4 Narrow Fourth G\ 1 5
66 498.1132075 P4 Perfect Fourth G 9 10
67 505.6603774 R4 Wide Fourth G/ 1 8
68 513.2075472 rK4 Narrow Acute Fourth G↑\ -3 6
69 520.7547170 K4 Lesser Acute Fourth G↑ -5 5
70 528.3018868 S4, kM4 Greater Acute Fourth Gt<↓, G↑/, Adb< -3 5
71 535.8490566 RkM4, ud5 Wide Acute Fourth, Infra-Diminished Fifth Gt>↓, Adb> -2 5
72 543.3962264 rM4, Rud5 Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth Gt<\, G↑↑, Ab↓↓ -1 6
73 550.9433962 M4, rKud5 Paramajor Fourth, Lesser Sub-Diminished Fifth Gt<, Adb<↑ 0 7
74 558.4905660 RM4, uA4, Kud5 Infra-Augmented Fourth, Greater Sub-Diminished Fifth Gt>, Adb>↑ -2 5
75 566.0377358 kkA4, RuA4, kd5 Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth Gt>/, G#↓↓, Ab↓ -3 4
76 573.5849057 rKuA4, Rkd5 Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth Gt<↑, Ab↓/ -2 4
77 581.1320755 KuA4, rd5 Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth Gt>↑, Ab\ 0 5
78 588.6792458 kA4, d5 Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth Ab, G#↓ -5 6
79 596.2264151 RkA4, Rd5 Artomean Augmented Fourth, Artomean Diminished Fifth G#↓/, Ab/ -9 7
80 603.7735849 rKd5, rA4 Tendomean Diminished Fifth, Tendomean Augmented Fourth Ab↑\, G#\ -9 7
81 611.3207547 Kd5, A4 Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth Ab↑, G# -5 6
82 618.8679245 kUd5, RA4 Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth Ad<↓, G#/ 0 5
83 626.4150943 RkUd5, rKA4 Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth Ad>↓, G#↑\ -2 4
84 633.9622642 KKd5, rUDd5, KA4 Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth Ad<\, Ab↑↑, G#↑ -3 4
85 641.5094340 rm5, Ud5, kUA4 Ultra-Diminished Fifth, Lesser Super-Augmented Fourth Ad<, Gt#<↓ -2 5
86 649.0566038 m5, RkUA4 Paraminor Fifth, Greater Super-Augmented Fourth Ad>, Gt#>↓ 0 7
87 656.6037736 Rm5, rUA4 Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth Ad>/, G#↑, Ab↑↑ -1 6
88 664.1509434 rKm5, UA4 Narrow Grave Fifth, Ultra-Augmented Fourth Ad<↑, Gt#< -2 5
89 671.6981132 s5, Km5 Lesser Grave Fifth Ad>↑, A↓\, Gt#> -3 5
90 679.2452830 k5 Greater Grave Fifth A↓ -5 5
91 686.7924528 Rk5 Wide Grave Fifth A↓/ -3 6
92 694.3396226 r5 Narrow Fifth A\ 1 8
93 701.8867925 P5 Perfect Fifth A 9 10
94 709.4339622 R5 Wide Fifth A/ 1 5
95 716.9811321 rK5 Narrow Acute Fifth A↑\ -4 0
96 724.5283019 K5 Lesser Acute Fifth A↑, Gx -6 -5
97 732.0754717 S5, kM5 Greater Acute Fifth, Narrow Inframinor Sixth At<↓, A↑/ -7 -4
98 739.6226415 um6, RkM5 Inframinor Sixth, Wide Acute Fifth At>↓, Bdb> -4 -2
99 747.1698113 Rm4, KKM3, rUM3 Narrow Paramajor Fifth, Wide Inframinor Sixth At<\, Bb↓↓, A↑↑ -2 1
100 754.7169811 M5, rKum6 Paramajor Fifth, Narrow Subminor Sixth At<, Bdb<↑ -1 3
101 762.2641509 sm6, Kum6, RM5, uA5 Lesser Subminor Sixth, Infra-Augmented Fifth At>, Bb↓\ 0 5
102 769.8113208 km6, RuA5, kkA5 Greater Subminor Sixth, Diptolemaic Augmented Fifth Bb↓, At>/, A#↓↓ -1 6
103 777.3584906 Rkm6, rKuA5 Wide Subminor Sixth, Lesser Sub-Augmented Fifth Bb↓/, At<↑ -1 7
104 784.9056604 rm6, KuA5 Narrow Minor Sixth, Greater Sub-Augmented Fifth Bb\, At>↑, A#↓\ 0 8
105 792.4528302 m6, kA5 Pythagorean Minor Sixth, Ptolemaic Augmented Fifth Bb, A#↓ -1 9
106 800 Rm6, RkA5 Artomean Minor Sixth, Artomean Augmented Fifth Bb/, A#↓/ 1 9
107 807.5471698 rKm6, rA5 Tendomean Minor Sixth, Tendomean Augmented Fifth A#\, Bb↑\ 4 10
108 815.0943396 Km6, A5 Ptolemaic Minor Sixth, Pythagorean Augmented Fifth A#, Bb↑ 8 10
109 822.6415094 RKm6, kn6, RA5 Wide Minor Sixth, Artoretromean Augmented Fifth Bd<↓, Bb↑/, A#/ 3 9
110 830.1886792 kN6, rKA5 Lesser Supraminor Sixth, Tendoretromean Augmented Fifth Bd>↓, A#↑\ -1 9
111 837.7358491 KKm6, rn6, KA5 Greater Supraminor Sixth, Retroptolemaic Augmented Fifth Bd<\, Bb↑↑, A#↑ 0 8
112 845.2830189 n6, SA5, kUA5 Artoneutral Sixth, Lesser Super-Augmented Fifth Bd<, At#<↓ -1 7
113 852.8301887 N6, RkUA5 Tendoneutral Sixth, Greater Super-Augmented Fifth Bd>, At#>↓ 0 7
114 860.3773585 kkM6, RN6, rUA5 Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth Bd>/, B↓↓, At#>↓/, A#↑↑ -1 8
115 867.9245283 Kn6, UA5 Greater Submajor Sixth, Ultra-Augmented Fifth Bd<↑, At#< 1 9
116 875.4716981 rkM6, KN6 Narrow Major Sixth Bd>↑, B↓\, At#> 4 9
117 883.0188679 kM6 Ptolemaic Major Sixth B↓, Cb 7 10
118 890.5660377 RkM6 Artomean Major Sixth B↓/ 4 10
119 898.1132075 rM6 Tendomean Major Sixth B\ 1 9
120 905.6603774 M6 Pythagorean Major Sixth B -1 9
121 913.2075472 RM6 Wide Major Sixth B/, Cd<↓ 0 8
122 920.7547170 rKM6 Narrow Supermajor Sixth B↑\, Cd>↓ -1 8
123 928.3018868 KM6 Lesser Supermajor Sixth B↑, Cd<\, Cb↑↑, Ax -1 7
124 935.8490566 SM6, kUM6 Greater Supermajor Sixth, Narrow Inframinor Seventh Cd<, Bt<↓, B↑/ 0 7
125 943.3962264 um7, RkUM6 Inframinor Seventh, Wide Supermajor Sixth Cd>, Bt>↓ -1 7
126 950.9433962 KKM6, kkm7, rUM6, Rum7 Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth Bt<\, Cd>/, B↑↑, C↓↓ 0 8
127 958.4905660 UM6, rKum7 Ultramajor Sixth, Narrow Subminor Seventh Bt<, Cd<↑ -1 8
128 966.0377358 sm7, Kum7 Lesser Subminor Seventh, Wide Ultramajor Sixth Bt>, Cd>↑, C↓\ 0 9
129 973.5849057 km7 Greater Subminor Seventh C↓, Bt>/, B#↓↓, Dbb -1 9
130 981.1320755 Rkm7 Wide Subminor Seventh C↓/, Bt<↑ -1 10
131 988.6792458 rm7 Narrow Minor Seventh C\, Bt>↑ -1 10
132 996.2264151 m7 Pythagorean Minor Seventh C, B#↓ -2 10
133 1003.7735849 Rm7 Artomean Minor Seventh C/, B#↓/ -2 10
134 1011.3207547 rKm7 Tendomean Minor Seventh C↑\, B#\ -3 10
135 1018.8679245 kM2 Ptolemaic Minor Seventh C↑, B# -3 10
136 1026.4150943 RKm7, kn7 Wide Minor Seventh Ct<↓, C↑/, Ddb<, B#/ -4 10
137 1033.9622642 kN7, ud8 Lesser Supraminor Seventh, Infra-Diminished Octave Ct>↓, Ddb>, B#↑\ -5 9
138 1041.5094340 KKm7, rn7, Rud8 Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave Ct<\, C↑↑, Ddb<↑\, Db↓↓ -6 8
139 1049.0566038 n7, rKud8 Artoneutral Seventh, Lesser Sub-Diminished Octave Ct<, Ddb<↑ -7 6
140 1056.6037736 N7, sd8 Tendoneutral Seventh, Greater Sub-Diminished Octave Ct>, Ddb>↑ -8 5
141 1064.1509434 kkM7, RN7, kd8 Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave Ct>/, C#↓↓, Db↓ -7 6
142 1071.6981132 Kn7, Rkd8 Greater Submajor Seventh, Artoretromean Diminished Octave Ct<↑, Db↓/ -6 8
143 1079.2452830 rkM7, KN7, rd8 Narrow Major Seventh, Tendoretromean Diminished Octave Ct>↑, C#↓\, Db\ -5 9
144 1086.7924528 kM7, d8 Ptolemaic Major Seventh, Pythagorean Diminished Octave Db, C#↓ -5 10
145 1094.3396226 RkM7, Rd8 Artomean Major Seventh, Artomean Diminished Octave Db/, C#↓/ -5 10
146 1101.8867925 rM7, rKd8 Tendomean Major Seventh, Tendomean Diminished Octave C#\, Db↑\ -6 10
147 1109.4339622 M7, Kd8 Pythagorean Major Seventh, Ptolemaic Diminished Octave C#, Db↑ -6 10
148 1116.9811321 RM7, kUd8 Wide Major Seventh, Lesser Super-Diminished Octave C#/, Dd<↓ -7 9
149 1124.5283019 rKM7, RkUd8 Narrow Supermajor Seventh, Greater Super-Diminished Octave C#↑\, Dd>↓ -7 9
150 1132.0754717 km2, RuA1, kkA1 Lesser Supermajor Seventh, Diptolemaic Diminished Octave C#↑, Db↑↑ -8 9
151 1139.6226415 SM7, kUM7, Ud8 Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave Dd<, C#↑/ -8 10
152 1147.1698113 u8, RkUM7 Infraoctave, Wide Supermajor Seventh Dd>, Ct#>↓ -9 10
153 1154.7169811 KKM7, rUM7, Ru8 Narrow Ultramajor Seventh, Wide Infraoctave C#↑↑, Dd>/ -9 10
154 1162.2641509 UM7, rKu8 Ultramajor Seventh, Narrow Suboctave Ct#<, Dd<↑ -9 10
155 1169.8113208 s8, Ku8 Lesser Suboctave, Wide Ultramajor Seventh Ct#>, Dd>↑ -10 3
156 1177.3584906 k8 Greater Suboctave D↓ -10 -3
157 1184.9056604 Rk8 Wide Suboctave D↓/ -10 -10
158 1192.4528302 r8 Narrow Octave D\ 0 0
159 1200 P8 Perfect Octave D 10 10

Interval Naming Terminology

The table above contains terminology that needs to be defined, and so it will be defined in this section, for cross-references, see List of interval regions and Adaptive diatonic interval names #Interval regions.

Narrow and Wide - These terms both refer to intervals that are respectively narrower or wider than the expected range of an interval class by a single EDO-step

Greater and Lesser - When there are two distinct intervals within the expected range of a given interval class, for example, two supermajor seconds, these terms will be used to distinguish the smaller and larger intervals respectively

Paramajor - Where there are multiple distinct intervals within the semiaugmented category for fourths and fifths, this term refers to the intervals just shy of half a chroma sharp of the perfect forms of these intervals; this term comes from adding the para- prefix (meaning alongside or resembling here) to Ivan Wyschnegradsky's own usage of the term major fourth in order to disambiguate it from other uses of the same term, while generalizing it to refer to fifths as well as fourths.

Paraminor - Where there are multiple distinct intervals within the semidiminished category for fourths and fifths, this term refers to the intervals just shy of half a chroma flat of the perfect forms of these intervals; this term comes from adding the para- prefix (meaning alongside or resembling here) to Ivan Wyschnegradsky's own usage of the term minor fifth in order to disambiguate it from other uses of the same term, while generalizing it to refer to fourths as well as fifths.

Infra-Augmented - When there are multiple distinct intervals within the semiaugmented category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma flat of the augmented forms of these intervals

Ultra-Augmented - When there are multiple distinct intervals within the sesquiaugmented category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma sharp of the augmented forms of these intervals

Infra-Diminished - When there are multiple distinct intervals within the sesquidiminished category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma flat of the diminished forms of these intervals

Ultra-Diminished - When there are multiple distinct intervals within the semidiminished category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma sharp of the diminished forms of these intervals

Artomean and Tendomean - When there are multiple distinct intervals situated between the Pythagorean and Ptolemaic versions of both major and minor intervals, these terms refer respectively to the narrower and wider intervals within those spaces

Usage and Functions of Specific Interval Ranges

This page or section deals with proposed concepts. The terminology and concepts used in it are developed by one person or a small group and may lack widespread adoption.

In addition to knowing the intervals in 159edo, it also pays to be able to get a general idea as to which intervals can be used for what purposes, though, for Medieval and Neo-Medieval music and their extensions, we have to begin by assuming the starting pitch class is the final, while for later Western Classical music and its extensions, we have to begin by assuming the starting pitch class is the tonic. From there, distance from the final or tonic in terms of both logarithmic distance as well as by approximated Just Intonation relations both govern the usage of 159edo's various intervals.

Au
Despite being largely derived from traditional schools of thought in the area of functional harmony, the terminology for these functions, along with the distinction between different variations of a given function, is largely unique to me at this point, though I do hope it gets adopted more in the future. I should mention that one reason for my differences in terminology in the area of functional harmony is that I have this idea that music can be hung from the high sounds downwards just as much as it can be built from the low sounds upwards, and the consideration of this angle requires the reconception of the nature of certain functions.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than what is called the fission boundary as per Dave Smey's video on the topic. Though there is a caveat which shall be discussed momentarily, the basic gist is that the fission boundary is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts- basically a cross between a trill and a vibrato. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than what Aura calls the gradient threshold, which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones. This brings up the aforementioned caveat about Dave Smey's video on the fission boundary- specifically, it should be noted that Dave Smey fails to distinguish between the fission boundary and the gradient threshold, and assumes the fission boundary to be at about 60 cents on average, rather than at about 35 cents, where it actually is.

Finally, 33\159 is the first interval to be larger than what is called the trill threshold as per Dave Smey's video on the topic. This point is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps. As a consequence of this, the trill threshold marks the boundary where intervals cease to cause crowding in chords.

Because both the size of 159edo intervals and their representation of Just intonation affect the functions of intervals, a list of JI intervals will be included in the chart below. Note that 159edo can only represent 5-limit intervals on every third step as it inherits 53edo's 5-limit representation, and that prime limits higher than 17 will not be shown on the chart due to inconsistencies.

159edo Interval Functions and JI Representation
Step Function JI Representation
Broad-Level Function Medium-Level Function Narrow-Level Function 5-Limit 7-Limit 11-Limit 13-Limit 17-Limit
0 Tonic 1/1
1 Supercommatic N/A 225/224 243/242 351/350, 352/351 256/255
2 Supercommatic N/A ? 100/99, 121/120 144/143 120/119
3 Superdietic 81/80 ? ? 78/77 85/84
4 Superdietic N/A 64/63 55/54, 56/55 ? 52/51
5 Supergradient N/A ? 45/44 ? 51/50
6 Supergradient ? ? ? 40/39 192/187
7 Supergradient N/A ? 33/32 ? 34/33
8 Supersanguinant N/A 28/27 512/495 ? 88/85
9 Supersanguinant 25/24 ? ? 26/25, 27/26 ?
10 Supersanguinant N/A ? ? ? 160/153
11 Supervicinant Supercollocant Proximosupercollocant N/A 21/20 22/21 ? ?
12 Supervicinant Supercollocant Proximosupercollocant 256/243, 135/128 ? ? ? ?
13 Supervicinant Supercollocant Proximosupercollocant N/A ? 128/121 55/52 18/17
14 Supervicinant Supercollocant Proximosupercollocant N/A ? ? ? 17/16
15 Supervicinant Supercollocant Distosupercollocant 16/15 ? ? ? ?
16 Supervicinant Supercollocant Distosupercollocant N/A 15/14 275/256 ? ?
17 Supervicinant Supercollocant Distosupercollocant N/A ? ? 14/13 55/51, 128/119
18 Supervicinant Superabrogant 27/25 ? ? 13/12 ?
19 Supervicinant Superabrogant N/A ? 88/81 ? ?
20 Supervicinant Superabrogant N/A ? 12/11 ? ?
21 Supervicinant Superabrogant ? ? ? 128/117 561/512, 1024/935
22 Supervicinant Superaequient N/A ? 11/10 ? ?
23 Supervicinant Superaequient N/A 567/512 243/220 ? 425/384
24 Supervicinant Superaequient 10/9 ? 256/231 ? ?
25 Supervicinant Supertonic N/A ? ? 143/128 512/459
26 Supervicinant Supertonic N/A 28/25 121/108 ? ?
27 Supervicinant Supertonic Bidominant 9/8 ? ? ? ?
28 Supervicinant Supertonic N/A 640/567 ? 44/39 289/256
29 Contravaricant Proximocontravaricant N/A ? 25/22 ? 17/15
30 Contravaricant Proximocontravaricant 256/225 ? 154/135 ? ?
31 Contravaricant Proximocontravaricant N/A 8/7 55/48 ? ?
32 Contravaricant Mesocontravaricant N/A ? 1024/891 ? ?
33 Contravaricant Mesocontravaricant ? ? ? 15/13 ?
34 Contravaricant Mesocontravaricant N/A ? 297/256 ? ?
35 Contravaricant Distocontravaricant N/A 7/6 64/55 ? ?
36 Contravaricant Distocontravaricant 75/64 ? ? 117/100 ?
37 Contravaricant Distocontravaricant N/A ? ? ? 20/17
38 Mediant Proximomediant N/A ? 33/28 13/11 85/72
39 Mediant Proximomediant Triservient 32/27 ? ? ? ?
40 Mediant Proximomediant N/A 25/21 144/121 ? ?
41 Mediant Proximomediant N/A ? ? 512/429 153/128
42 Mediant Mesoproximomediant 6/5 ? 77/64 ? ?
43 Mediant Mesoproximomediant N/A 135/112 ? ? 512/425
44 Mediant Mesoproximomediant N/A ? 40/33, 121/100 ? 144/119, 165/136
45 Mediant Mesomediant ? ? ? 39/32 17/14
46 Mediant Mesomediant N/A ? 11/9 ? ?
47 Mediant Mesomediant N/A ? 27/22 ? ?
48 Mediant Mesomediant ? ? ? 16/13 21/17
49 Mediant Mesodistomediant N/A ? ? 26/21 68/55
50 Mediant Mesodistomediant N/A 56/45 1024/825 ? ?
51 Mediant Mesodistomediant 5/4 ? 96/77 ? ?
52 Mediant Distomediant N/A ? ? ? 64/51
53 Mediant Distomediant N/A 63/50 121/96 ? 34/27
54 Mediant Distomediant Quadridominant 81/64 ? ? ? ?
55 Mediant Distomediant N/A ? 14/11 33/26 108/85
56 Superobstant Proximosuperobstant N/A ? ? 143/112 51/40
57 Superobstant Proximosuperobstant 32/25 ? ? 50/39 ?
58 Superobstant Proximosuperobstant N/A 9/7 165/128 ? ?
59 Superobstant Distosuperobstant N/A ? 128/99 ? 22/17
60 Superobstant Distosuperobstant ? ? ? 13/10 ?
61 Superobstant Distosuperobstant N/A ? 176/135 ? ?
62 Superobstant Distosuperobstant N/A 21/16 55/42, 72/55 ? 17/13
63 Pseudoservient 320/243, 675/512 ? ? ? ?
64 Pseudoservient N/A ? 33/25 ? 45/34
65 Pseudoservient N/A ? 968/729 ? 85/64
66 Servient 4/3 ? ? ? ?
67 Geminoservient N/A 75/56 162/121 ? ?
68 Geminoservient N/A ? 121/90 ? ?
69 Geminoservient 27/20 ? ? 104/77 ?
70 Geminoservient N/A 256/189 110/81 ? ?
71 Subient N/A ? 15/11 ? ?
72 Subient ? ? ? 160/117 256/187
73 Subient N/A ? 11/8 ? ?
74 Pseudosubient N/A 112/81 ? ? ?
75 Pseudosubient 25/18 ? ? 18/13 ?
76 Protosycophant N/A ? ? ? 357/256
77 Protosycophant N/A 7/5 ? ? ?
78 Antitonic 45/32, 1024/729 ? ? ? ?
79 Antitonic N/A ? ? ? 24/17
80 Antitonic N/A ? ? ? 17/12
81 Antitonic 64/45, 729/512 ? ? ? ?
82 Prototyrant N/A 10/7 ? ? ?
83 Prototyrant N/A ? ? ? 512/357
84 Pseudoimponent 36/25 ? ? 13/9 ?
85 Pseudoimponent N/A 81/56 ? ? ?
86 Imponent N/A ? 16/11 ? ?
87 Imponent ? ? ? 117/80 187/128
88 Imponent N/A ? 22/15 ? ?
89 Geminodominant N/A 189/128 81/55 ? ?
90 Geminodominant 40/27 ? ? 77/52 ?
91 Geminodominant N/A ? 180/121 ? ?
92 Geminodominant N/A 112/75 121/81 ? ?
93 Dominant 3/2 ? ? ? ?
94 Pseudodominant N/A ? 729/484 ? 128/85
95 Pseudoservient N/A ? 50/33 ? 68/45
96 Pseudoservient 243/160, 1024/675 ? ? ? ?
97 Subobstant Distosubobstant N/A 32/21 84/55, 55/36 ? 26/17
98 Subobstant Distosubobstant N/A ? 135/88 ? ?
99 Subobstant Distosubobstant ? ? ? 20/13 ?
100 Subobstant Distosubobstant N/A ? 99/64 ? 17/11
101 Subobstant Proximosubobstant N/A 14/9 256/165 ? ?
102 Subobstant Proximosubobstant 25/16 ? ? 39/25 ?
103 Subobstant Proximosubobstant N/A ? ? 224/143 80/51
104 Contramediant Distocontramediant N/A ? 11/7 52/33 85/54
105 Contramediant Distocontramediant Quadriservient 128/81 ? ? ? ?
106 Contramediant Distocontramediant N/A 100/63 192/121 ? 27/17
107 Contramediant Distocontramediant N/A ? ? ? 51/32
108 Contramediant Mesodistocontramediant 8/5 ? 77/48 ? ?
109 Contramediant Mesodistocontramediant N/A 45/28 825/512 ? ?
110 Contramediant Mesodistocontramediant N/A ? ? 21/13 55/34
111 Contramediant Mesocontramediant ? ? ? 13/8 34/21
112 Contramediant Mesocontramediant N/A ? 44/27 ? ?
113 Contramediant Mesocontramediant N/A ? 18/11 ? ?
114 Contramediant Mesocontramediant ? ? ? 64/39 28/17
115 Contramediant Mesoproximocontramediant N/A ? 33/20, 200/121 ? 119/72, 272/165
116 Contramediant Mesoproximocontramediant N/A 224/135 ? ? 425/256
117 Contramediant Mesoproximocontramediant 5/3 ? 128/77 ? ?
118 Contramediant Proximocontramediant N/A ? ? 429/256 256/153
119 Contramediant Proximocontramediant N/A 42/25 121/72 ? ?
120 Contramediant Proximocontramediant Tridominant 27/16 ? ? ? ?
121 Contramediant Proximocontramediant N/A ? 56/33 22/13 144/85
122 Varicant Distovaricant N/A ? ? ? 17/10
123 Varicant Distovaricant 128/75 ? ? 200/117 ?
124 Varicant Distovaricant N/A 12/7 55/32 ? ?
125 Varicant Mesovaricant N/A ? 512/297 ? ?
126 Varicant Mesovaricant ? ? ? 26/15 ?
127 Varicant Mesovaricant N/A ? 891/512 ? ?
128 Varicant Proximovaricant N/A 7/4 96/55 ? ?
129 Varicant Proximovaricant 225/128 ? 135/77 ? ?
130 Varicant Proximovaricant N/A ? 44/25 ? 30/17
131 Subvicinant Subtonic N/A 567/320 ? 39/22 512/289
132 Subvicinant Subtonic Biservient 16/9 ? ? ? ?
133 Subvicinant Subtonic N/A 25/14 216/121 ? ?
134 Subvicinant Subtonic N/A ? ? 256/143 459/256
135 Subvicinant Subaequient 9/5 ? 231/128 ? ?
136 Subvicinant Subaequient N/A 1024/567 440/243 ? 768/425
137 Subvicinant Subaequient N/A ? 20/11 ? ?
138 Subvicinant Subabrogant ? ? ? 117/64 1024/561, 935/512
139 Subvicinant Subabrogant N/A ? 11/6 ? ?
140 Subvicinant Subabrogant N/A ? 81/44 ? ?
141 Subvicinant Subabrogant 50/27 ? ? 24/13 ?
142 Subvicinant Subcollocant Distosubcollocant N/A ? ? 13/7 102/55, 119/64
143 Subvicinant Subcollocant Distosubcollocant N/A 28/15 512/275 ? ?
144 Subvicinant Subcollocant Distosubcollocant 15/8 ? ? ? ?
145 Subvicinant Subcollocant Proximosubcollocant N/A ? ? ? 32/17
146 Subvicinant Subcollocant Proximosubcollocant N/A ? 121/64 104/55 17/9
147 Subvicinant Subcollocant Proximosubcollocant 243/128, 256/135 ? ? ? ?
148 Subvicinant Subcollocant Proximosubcollocant N/A 40/21 21/11 ? ?
149 Subsanguinant N/A ? ? ? 153/80
150 Subsanguinant 48/25 ? ? 25/13, 52/27 ?
151 Subsanguinant N/A 27/14 495/256 ? 85/44
152 Subgradient N/A ? 64/33 ? 33/17
153 Subgradient ? ? ? 39/20 187/96
154 Subgradient N/A ? 88/45 ? 100/51
155 Subdietic N/A 63/32 108/55, 55/28 ? 51/26
156 Superdietic 160/81 ? ? 77/39 168/85
157 Supercommatic N/A ? 99/50, 240/121 143/72 119/60
158 Supercommatic N/A 448/225 484/243 700/351, 351/176 255/128
159 Tonic 2/1
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