User:Aura/Aura's introduction to 159edo

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While large EDOs like 159edo have a bit of a learning curve compared to smaller systems, the fact that this system sports not only consistency to the 17-odd-limit, but also both additional options for imitating the pitch-hue palettes of smaller tuning systems such as 10edo, 12edo, 13edo, 14edo, 17edo, 19edo, 22edo, 24edo and 31edo among others, and, the ability to perform sophisticated harmonic and melodic maneuvers that involve substituting intervals that are one or two steps away from the equivalents of traditional musical intervals in such a way as to fly under the radar of many listeners, all while having a step size that is above the average listener's melodic JND, makes this system well worth the price of admission- particularly for those who can use or make synthesizers and other digital-system-based instruments.

Intervals and Notation

159edo contains all the intervals of 53edo, however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. Furthermore, just as with 24edo can be thought of as essentially having two fields of 12edo separated by a quartertone, 159edo can be thought of as having three fields of 53edo, each separated from the others by a third of a 53edo step on either side. This even lends to 159edo having its own variation on the Dinner Party Rules—represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns 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.

Table of 159edo intervals
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 Prime D/ 0 0
2 15.0943396 rK1 Narrow Superprime D↑\ -10 -10
3 22.6415094 K1 Lesser Superprime D↑ -10 -3
4 30.1886792 S1, kU1 Greater Superprime, Narrow Inframinor Second Edb<, Dt<↓ -10 3
5 37.7358491 um2, RkU1 Inframinor Second, Wide Superprime Edb>, Dt>↓ -9 10
6 45.2830189 kkm2, Rum2, rU1 Wide Inframinor Second, Narrow Ultraprime Eb↓↓, Dt<\ -9 10
7 52.8301887 U1, rKum2 Ultraprime, Narrow Subminor Second Dt<, Edb<↑ -9 10
8 60.3773585 sm2, Kum2, uA1 Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime Dt>, Eb↓\ -8 10
9 67.9245283 km2, RuA1, kkA1 Greater Subminor Second, Diptolemaic Augmented Prime Eb↓, D#↓↓ -8 9
10 75.4716981 Rkm2, rKuA1 Wide Subminor Second, Lesser Sub-Augmented Prime Eb↓/, Dt<↑ -7 9
11 83.0188679 rm2, KuA1 Narrow Minor Second, Greater Sub-Augmented Prime Eb\, Dt>↑ -7 9
12 90.5660377 m2, kA1 Pythagorean Minor Second, Ptolemaic Augmented Prime Eb, D#↓ -6 10
13 98.1132075 Rm2, RkA1 Artomean Minor Second, Artomean Augmented Prime Eb/, D#↓/ -6 10
14 105.6603774 rKm2, rA1 Tendomean Minor Second, Tendomean Augmented Prime D#\, Eb↑\ -5 10
15 113.2075472 Km2, A1 Ptolemaic Minor Second, Pythagorean Augmented Prime D#, Eb↑ -5 10
16 120.7547170 RKm2, kn2, RA1 Wide Minor Second, Artoretromean Augmented Prime Ed<↓, Eb↑/, D#/ -5 9
17 128.3018868 kN2, rKA1 Lesser Supraminor Second, Tendoretromean Augmented Prime Ed>↓, D#↑\ -6 8
18 135.8490566 KKm2, rn2, KA1 Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime Ed<\, Eb↑↑, D#↑ -7 6
19 143.3962264 n2, SA1 Artoneutral Second, Lesser Super-Augmented Prime Ed<, Dt#<↓ -8 5
20 150.9433962 N2, RkUA1 Tendoneutral Second, Greater Super-Augmented Prime Ed>, Dt#>↓ -7 6
21 158.4905660 kkM2, RN2, rUA1 Lesser Submajor Second, Retrodiptolemaic Augmented Prime Ed>/, E↓↓, Dt#>↓/, D#↑↑ -6 8
22 166.0377358 Kn2, UA1 Greater Submajor Second, Ultra-Augmented Prime 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 Second, 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, Wide Superprime 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

5-limit diatonic music

Although 159edo inherits 53edo's close approximations of both the 5-limit Zarlino scale and the 3-limit diatonic MOS, these are not the only scales that one can use for fixed-pitch diatonic music even in that system. In fact, both of them are less than optimal for many facets of traditional Western classical music, seeing as for the most part, each of the traditional diatonic modes has its own optimized scale in the 5-limit. The reason for this is that in non-meantone 5-limit systems, one inevitably has to deal with the ~40/27 wolf fifth and the ~27/20 wolf fourth, and there's only two ideal positions in the scale for those intervals to be situated in fixed-pitch diatonic music. For major modes, the ~27/20 wolf fourth is ideally situated between the ~5/4 major third and the ~27/16 major sixth, while for the minor modes and the diatonic blighted mode Locrian, the ~40/27 wolf fifth is ideally situated between the ~6/5 minor third and the ~16/9 minor seventh. Not only that, but it also pays to distinguish the Pythagorean major and minor thirds from their Ptolemaic counterparts as both types of major and minor third have distinct roles to play in the 5-limit diatonic music.

Scales and Harmony

For purposes of this section, we shall assume the tonic to be D-natural as in the interval chart above. Note that the following trines are available in 5-limit diatonic harmony.

Table of 159edo diatonic 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
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
Subagallic D, G↑, D 0, 69, 0 20:27:40 This dissonant trine is very likely to show up in non-meantone diatonic contexts
Supradusthumic D, A↓, D 0, 90, 0 1/(20:27:40) This dissonant trine is very likely to show up in non-meantone diatonic contexts

Ionian and Major

This mode- along the corresponding tonality- is optimized for 5-limit using a variation on the Didymian diatonic scale.

Table of Ionian notes and intervals
Interval Name Notation (from D) Steps from Tonic Function Corresponding JI
Perfect Unison D 0 Tonic 1/1
Pythagorean Major Second E 27 Supertonic (Bidominant) 9/8
Ptolemaic Major Third F#↓ 51 Mesodistomediant 5/4
Perfect Fourth G 66 Servient (Subdominant) 4/3
Perfect Fifth A 93 Dominant 3/2
Pythagorean Major Sixth B 120 Proximocontramediant (Tridominant) 27/16
Ptolemaic Major Seventh C#↓ 144 Distosubcollocant 15/8
Perfect Octave D 159 Tonic 2/1

As a consequence of this particular scale structure, you have the following basic chords...

Table of basic Ionian triads
Name Notation (from D) Steps Occur(s) on Scale Degree(s) Approximate JI Notes
Ptolemaic Major D, F#↓, A 0, 51, 93 I, V 4:5:6 This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
Ptolemaic Minor D, F↑, A 0, 42, 93 ↓III 1/(4:5:6) This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
Pythagorean Major D, F#, A 0, 54, 93 IV 1/(54:64:81) This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
Pythagorean Minor D, F, A 0, 39, 93 II 54:64:81 This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
Supradusthumic Pythagorean Minor D, F, A↓ 0, 39, 90 VI 27:32:40 This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
Greater Ptolemaic Diminished D, F↑, Ab↑ 0, 42, 81 ↓VII 45:54:64 This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
Table of basic Ionian tetrads
Name Notation (from D) Steps Occur(s) on Scale Degree(s) Approximate JI Notes
Ptolemaic Major Seventh D, F#↓, A, C#↓ 0, 51, 93, 144 I 8:10:12:15 This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
Ptolemaic Minor Seventh D, F↑, A, C↑ 0, 42, 93, 135 III 10:12:15:18 This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
Pythagorean Minor Seventh D, F, A, C 0, 39, 93, 132 II 54:64:81:96 This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
Pythagorean Major with Ptolemaic Major Seventh D, F#, A, C#↓ 0, 54, 93, 144 IV 64:81:96:120 This dissonant tetrad is an unconventional servient (subdominant) seventh for 5-limit music, and is one of a few diatonic tetrads to feature a wolf fifth
Ptolemaic Dominant Seventh D, F#↓, A, C 0, 51, 93, 132 V 36:45:54:64 This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~3/2 below its root
Supradusthumic Pythagorean Minor Seventh D, F, A↓, C 0, 39, 93, 132 VI 54:32:40:48 This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
Ptolemaic Half-Diminished D, F↑, Ab↑, C↑ 0, 42, 81, 135 VII 45:54:64:81 This dissonant tetrad is an option for imperfect half cadences in the 5-limit

With the above scale steps and chords, this variation of Ionian in 159edo is considerably more tonally stable than its more familiar counterpart from 12edo and 24edo. It is also one of only two traditional diatonic modes in which one is able to perform a complete circle progression.

Dorian

Unlike what is expected, the form of Dorian mode optimized for 5-limit is not symmetrical, owing to the need to maintain Rothenberg propriety.

Table of Dorian notes and intervals
Interval Name Notation (from D) Steps from Tonic Function Corresponding JI
Perfect Unison D 0 Tonic 1/1
Pythagorean Major Second E 27 Supertonic (Bidominant) 9/8
Ptolemaic Minor Third F↑ 42 Mesoproximomediant 6/5
Perfect Fourth G 66 Servient (Subdominant) 4/3
Perfect Fifth A 93 Dominant 3/2
Pythagorean Major Sixth B 120 Proximocontramediant (Tridominant) 27/16
Pythagorean Minor Seventh C 132 Subtonic (Biservient) 16/9
Perfect Octave D 159 Tonic 2/1

As a consequence of this particular scale structure, you have the following basic chords...

Table of basic Dorian triads
Name Notation (from D) Steps Occur(s) on Scale Degree(s) Approximate JI Notes
Ptolemaic Minor D, F↑, A 0, 42, 93 I 1/(4:5:6) This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
Pythagorean Major D, F#, A 0, 54, 93 IV, bVII 1/(54:64:81) This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
Pythagorean Minor D, F, A 0, 39, 93 II, V 54:64:81 This dissonant triad is common in Western Classical, Medieval, and Neo-Medieval Harmony
Supradusthumic Ptolemaic Major D, F#↓, A↓ 0, 51, 90 b↑III 1/(27:32:40) This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
Lesser Ptolemaic Diminished D, F, Ab↑ 0, 39, 81 VI 1/(45:54:64) This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
Table of basic Dorian tetrads
Name Notation (from D) Steps Occur(s) on Scale Degree(s) Approximate JI Notes
Ptolemaic Minor with Pythagorean Minor Seventh D, F↑, A, C 0, 42, 93, 132 I 90:108:135:160 This dissonant tetrad is perhaps best considered a cross between a suspension and a triad owing to the dissonance of the wolf fifth
Pythagorean Major Seventh D, F#, A, C# 0, 54, 93, 147 bVII 128:162:192:243 This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
Pythagorean Minor Seventh D, F, A, C 0, 39, 93, 132 II, V 54:64:81:96 This dissonant tetrad is common in 3-limit music, but still has a role to play in 5-limit music
Supradusthumic Ptolemaic Major Seventh D, F#↓, A↓, C#↓ 0, 51, 90, 144 b↑III 240:256:320:405 This dissonant tetrad is one of a few diatonic tetrads to feature a wolf fifth
Ptolemaic Parallel Dominant Seventh D, F#, A, C↑ 0, 54, 93, 135 IV 1/(45:54:64:81) This tetrad is one of two main dominant-seventh-type chords in the 5-limit, tonicizing the note located at ~9/8 above its root
Ptolemaic Parallel Half-Diminished D, F, Ab↑, C 0, 39, 81, 132 VI 1/(36:45:54:64) This dissonant tetrad is an option for imperfect half cadences in the 5-limit

With the above scale steps and chords, this variation of Dorian in 159edo has a few noteworthy tricks- especially once one considers suspensions, which will have to be covered later in this document.

Phrygian

This mode is optimized for 5-limit using a variation on the left-handed Zarlino diatonic scale.

Table of Phrygian notes and intervals
Interval Name Notation (from D) Steps from Tonic Function Corresponding JI
Perfect Unison D 0 Tonic 1/1
Ptolemaic Minor Second Eb↑ 15 Distosupercollocant 16/15
Ptolemaic Minor Third F↑ 42 Mesoproximomediant 6/5
Perfect Fourth G 66 Servient (Subdominant) 4/3
Perfect Fifth A 93 Dominant 3/2
Ptolemaic Minor Sixth Bb↑ 108 Mesodistocontramediant 8/5
Pythagorean Minor Seventh C 132 Subtonic (Biservient) 16/9
Perfect Octave D 159 Tonic 2/1

As a consequence of this particular scale structure, you have the following basic chords...

Table of basic Phrygian triads
Name Notation (from D) Steps Occur(s) on Scale Degree(s) Approximate JI Notes
Ptolemaic Major D, F#↓, A 0, 51, 93 b↑II, b↑VI 4:5:6 This is the first of two triads that can be considered fully-resolved in Western Classical Harmony
Ptolemaic Minor D, F↑, A 0, 42, 93 I, IV, bVII 1/(4:5:6) This is the second of two triads that can be considered fully-resolved in Western Classical Harmony
Supradusthumic Ptolemaic Major D, F#↓, A↓ 0, 51, 90 b↑III 1/(27:32:40) This dissonant triad is one of two possible diatonic wolf triads in the 5-limit
Lesser Ptolemaic Diminished D, F, Ab↑ 0, 39, 81 V 1/(45:54:64) This dissonant triad is one of two possible diatonic diminished triads in the 5-limit
Table of basic Phrygian tetrads
Name Notation (from D) Steps Occur(s) on Scale Degree(s) Approximate JI Notes
Ptolemaic Minor with Pythagorean Minor Seventh D, F↑, A, C 0, 42, 93, 132 I 90:108:135:160 This dissonant tetrad is perhaps best considered a cross between a suspension and a triad owing to the dissonance of the wolf fifth
Ptolemaic Major Seventh D, F#↓, A, C#↓ 0, 51, 93, 144 b↑II 8:10:12:15 This tetrad is likely to decompose into a voicing variation on the corresponding triad if the same scale degree is used as a chord root multiple times in a row
Supradusthumic Ptolemaic Dominant Seventh D, F#↓, A↓, C 0, 51, 90, 132 b↑III 1/(45:54:64:80) This dissonant dominant tetrad has a different function than its more traditional counterpart

Notes

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