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

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 ==
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, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed.  However, there's more.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than the '''fission boundary''', which 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.  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 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. 

Finally, 33\159 is the first interval to be larger than the '''trill threshold''', which 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 if all that weren't enough, 159edo 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.

{| class="wikitable"
|+159edo Interval Names and Compatibility Ratings
! rowspan="2" | Step
! rowspan="2" | Cents
! colspan="3" | Interval and Note names
! colspan="2" | 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 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
|}

== Trines ==
159edo has multiple types of trine.  Trines are important in the aspects of 159edo music theory derived from Medieval and Neo-Medieval music theory.  In fact, 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.

{| class="wikitable" 
|+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 are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Utonal Archagall
| D, A/, D 
| 0, 94, 0
| 1/(64:85:128)
| This trine is the second of two that are often used in the extended harmony of t<IV chords and is considered a dissonance
|-
| Bass-Up Marvelous
| D, A\, D 
| 0, 92, 0
| 75:112:150
| This dissonant trine is the first of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| Treble-Down Marvelous
| D, G/, D 
| 0, 67, 0
| 1/(75:112:150)
| This dissonant trine is the second of two that are formed from stacking identical approximations of the LCJI neutral third
|-
| 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
|}