View source for User:Aura/Aura's introduction to 159edo

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.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | Compatibility rating
|-
! [[SKULO interval names|SKULO]]-based interval names
! [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean-commatic]]-based interval names
! [[Syntonic-rastmic subchroma notation|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.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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 [[Wikipedia: Vi–ii–V–I|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]]. 

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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.

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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...

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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
|}

{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | 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
|}




[[Category:Method]]
[[Category:Approaches to tuning systems]]

== Notes ==
<references group="note" />