159edo/Interval names and harmonies

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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. It should be noted that since 159edo does a better job of representing the 2.3.11 subgroup than 24edo, some of the chords listed on the page for 24edo interval names and harmonies carry over to this page, even though the exact sets of enharmonics differ between the two systems.

Table of 159edo intervals
Step Cents 5 limit 7 limit 11 limit 13 limit 17 limit Interval Names Notes
0 0 1/1 P1 Perfect Unison D The root of any chord.
1 7.5471698 225/224 243/242 196/195, 351/350 256/255 R1 Wide Prime D/ Useful for slight dissonances that convey something less than satisfactory, however it can also add to the bandwidth of a sound.
2 15.0943396 ? 121/120, 100/99 144/143 120/119 rK1 Narrow Superprime D↑\ Useful for slight dissonances that create more noticeable tension.
3 22.6415094 81/80 ? ? 78/77 85/84 K1 Lesser Superprime D↑ Useful for appoggiaturas, and acciaccaturas, and quick passing tones.
4 30.1886792 64/63 56/55, 55/54 ? 52/51 S1, kU1 Greater Superprime, Narrow Inframinor Second Dt<↓ Useful for various types of subchromatic gestures, as well as for appoggiaturas, and acciaccaturas, and quick passing tones.
5 37.7358491 ? 45/44 ? 51/50 um2, RkU1 Inframinor Second, Wide Superprime Edb>, Dt>↓ By default, this interval is a type of paradiatonic quartertone and is used in much the same way as 24edo's own Inframinor Second.
6 45.2830189 ? ? ? 40/39 192/187 kkm2, Rum2, rU1 Wide Inframinor Second, Narrow Ultraprime, Semilimma Eb↓↓, Dt<\ This interval is particularly likely to be used as a cross between an Ultraprime and an Inframinor Second.
7 52.8301887 ? 33/32 ? 34/33 U1, rKum2 Ultraprime, Narrow Subminor Second Dt< By default, this interval is a type of parachromatic quartertone and is thus used in much the same way as 24edo's own Ultraprime.
8 60.3773585 28/27 ? ? 88/85 sm2, Kum2, uA1 Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime Eb↓\, Dt> Although this interval can act as a leading tone, it can also act as a trienstone- that is, a third of a tone- since it's one third of the Ptolemaic Major Second.
9 67.9245283 25/24 ? ? 26/25, 27/26 ? km2, kkA1 Greater Subminor Second, Diptolemaic Augmented Prime Eb↓, D#↓↓ Although this interval frequently acts as the Diptolemaic Chroma, it can also act as a trienstone- that is, a third of a tone- since it's one third of the Pyth Major Second.
10 75.4716981 ? ? ? 160/153 Rkm2, rKuA1 Wide Subminor Second, Lesser Sub-Augmented Prime Eb↓/, Dt<↑ This interval acts as a type of semitone, however, whether it's a diatonic or chromatic semitone depends on the situation.
11 83.0188679 21/20 22/21 ? ? rm2, KuA1 Narrow Minor Second, Greater Sub-Augmented Prime Eb\, Dt>↑ Not only does this interval serve as a type of leading tone, but it should be noted that six of these add up to a Perfect Fourth.
12 90.5660377 256/243, 135/128 ? ? ? ? m2, kA1 Pythagorean Minor Second, Ptolemaic Augmented Prime Eb, D#↓ As the approximation of both the Pythagorean Minor Second and the Ptolemaic Augmented Prime, this interval is used accordingly.
13 98.1132075 ? 128/121 55/52 18/17 Rm2, RkA1 Artomean Minor Second, Artomean Augmented Prime Eb/, D#↓/ This interval is one of two in this system that are essential in executing the frameshift cadence; it is also the closest approximation of the 12edo semitone found in this system.
14 105.6603774 ? ? ? 17/16 rKm2, rA1 Tendomean Minor Second, Tendomean Augmented Prime D#\, Eb↑\ As the closest approximation of the seventeenth harmonic, this interval is used accordingly.
15 113.2075472 16/15 ? ? ? ? Km2, A1 Ptolemaic Minor Second, Pythagorean Augmented Prime D#, Eb↑ As the approximation of both the Pythagorean Augmented Prime and the Ptolemaic Minor Second, this interval is used accordingly; it is also one of two in this system that are essential in executing the frameshift cadence.
16 120.7547170 15/14 275/256 ? ?
17 128.3018868 ? ? 14/13 128/119
18 135.8490566 27/25 ? ? 13/12 ?
19 143.3962264 ? 88/81 ? ?
20 150.9433962 ? 12/11 ? ?
21 158.4905660 ? ? ? 128/117 561/512, 1024/935
22 166.0377358 ? 11/10 ? ?
23 173.5849057 567/512 243/220 ? 425/384
24 181.1320755 10/9 ? 256/231 ? ?
25 188.6792458 ? ? 143/128 512/459
26 196.2264151 28/25 121/108 ? ?
27 203.7735849 9/8 ? ? ? ?
28 211.3207547 ? ? 44/39 289/256
29 218.8679245 ? ? ? 17/15
30 226.4150943 256/225 ? 154/135 ? ?
31 233.9622642 8/7 55/48 ? ?
32 241.5094340 ? 1024/891 ? ?
33 249.0566038 ? ? ? 15/13 ?
34 256.6037736 ? 297/256 ? ?
35 264.1509434 7/6 64/55 ? ?
36 271.6981132 75/64 ? ? ? ?
37 279.2452830 ? ? ? 20/17
38 286.7924528 ? 33/28 13/11 85/72
39 294.3396226 32/27 ? ? ? ?
40 301.8867925 25/21 144/121 ? ?
41 309.4339622 ? ? 512/429 153/128
42 316.9811321 6/5 ? 77/64 ? ?
43 324.5283019 135/112 ? ? 512/425
44 332.0754717 ? 40/33, 121/100 ? 144/119, 165/136
45 339.6226415 ? ? ? 39/32 17/14
46 347.1698113 ? 11/9 ? ?
47 354.7169811 ? 27/22 ? ?
48 362.2641509 ? ? ? 16/13 21/17
49 369.8113208 ? ? ? 68/55
50 377.3584906 56/45 1024/825 ? ?
51 384.9056604 5/4 ? 96/77 ? ?
52 392.4528302 ? ? ? 64/51
53 400 63/50 121/96 ? ?
54 407.5471698 81/64 ? ? ? ?
55 415.0943396 ? 14/11 33/26 108/85
56 422.6415094 ? ? 143/112 51/40
57 430.1886792 32/25 ? ? ? ?
58 437.7358491 9/7 165/128 ? ?
59 445.2830189 ? 128/99 ? 22/17
60 452.8301887 ? ? ? 13/10 ?
61 460.3773585 ? 176/135 ? ?
62 467.9245283 21/16 55/42, 72/55 ? 17/13
63 475.4716981 320/243, 675/512 ? ? ? ?
64 483.0188679 ? 33/25 ? 45/34
65 490.5660377 ? ? ? 85/64
66 498.1132075 4/3 ? ? ? ?
67 505.6603774 75/56 162/121 ? ?
68 513.2075472 ? 121/90 ? ?
69 520.7547170 27/20 ? ? 104/77 ?
70 528.3018868 ? 110/81 ? ?
71 535.8490566 ? 15/11 ? ?
72 543.3962264 ? ? ? 160/117 256/187
73 550.9433962 ? 11/8 ? ?
74 558.4905660 112/81 ? ? ?
75 566.0377358 25/18 ? ? 18/13 ?
76 573.5849057 ? ? ? 357/256
77 581.1320755 7/5 ? ? ?
78 588.6792458 1024/729, 45/32 ? ? ? ?
79 596.2264151 ? ? ? 24/17
80 603.7735849 ? ? ? 17/12
81 611.3207547 729/512, 64/45 ? ? ? ?
82 618.8679245 10/7 ? ? ?
83 626.4150943 ? ? ? 512/357
84 633.9622642 36/25 ? ? 13/9 ?
85 641.5094340 81/56 ? ? ?
86 649.0566038 ? 16/11 ? ?
87 656.6037736 ? ? ? 117/80 187/128
88 664.1509434 ? 22/15 ? ?
89 671.6981132 ? 81/55 ? ?
90 679.2452830 40/27 ? ? 77/52 ?
91 686.7924528 ? 180/121 ? ?
92 694.3396226 112/75 121/81 ? ?
93 701.8867925 3/2 ? ? ? ?
94 709.4339622 ? ? ? 128/85
95 716.9811321 ? 50/33 ? 68/45
96 724.5283019 243/160, 1024/675 ? ? ? ?
97 732.0754717 32/21 84/55, 55/36 ? 26/17
98 739.6226415 ? 135/88 ? ?
99 747.1698113 ? ? ? 20/13 ?
100 754.7169811 ? 99/64 ? 17/11
101 762.2641509 14/9 256/165 ? ?
102 769.8113208 25/16 ? ? ? ?
103 777.3584906 ? ? 224/143 80/51
104 784.9056604 ? 11/7 52/33 85/54
105 792.4528302 128/81 ? ? ? ?
106 800 100/63 192/121 ? ?
107 807.5471698 ? ? ? 51/32
108 815.0943396 8/5 ? 77/48 ? ?
109 822.6415094 45/28 825/512 ? ?
110 830.1886792 ? ? ? 55/34
111 837.7358491 ? ? ? 13/8 34/21
112 845.2830189 ? 44/27 ? ?
113 852.8301887 ? 18/11 ? ?
114 860.3773585 ? ? ? 64/39 28/17
115 867.9245283 ? 33/20, 200/121 ? 119/72, 272/165
116 875.4716981 224/135 ? ? 425/256
117 883.0188679 5/3 ? 128/77 ? ?
118 890.5660377 ? ? 429/256 256/153
119 898.1132075 42/25 121/72 ? ?
120 905.6603774 27/16 ? ? ? ?
121 913.2075472 ? 56/33 22/13 144/85
122 920.7547170 ? ? ? 17/10
123 928.3018868 128/75 ? ? ? ?
124 935.8490566 12/7 55/32 ? ?
125 943.3962264 ? 512/297 ? ?
126 950.9433962 ? ? ? 26/15 ?
127 958.4905660 ? 891/512 ? ?
128 966.0377358 7/4 96/55 ? ?
129 973.5849057 225/128 ? 135/77 ? ?
130 981.1320755 ? ? ? 30/17
131 988.6792458 ? ? 39/22 512/289
132 996.2264151 16/9 ? ? ? ?
133 1003.7735849 25/14 216/121 ? ?
134 1011.3207547 ? ? 256/143 459/256
135 1018.8679245 9/5 ? 231/128 ? ?
136 1026.4150943 1024/567 440/243 ? 768/425
137 1033.9622642 ? 20/11 ? ?
138 1041.5094340 ? ? ? 117/64 1024/561, 935/512
139 1049.0566038 ? 11/6 ? ?
140 1056.6037736 ? 81/44 ? ?
141 1064.1509434 50/27 ? ? 24/13 ?
142 1071.6981132 ? ? 13/7 119/64
143 1079.2452830 28/15 512/275 ? ?
144 1086.7924528 15/8 ? ? ? ?
145 1094.3396226 ? ? ? 32/17
146 1101.8867925 ? 121/64 104/55 17/9
147 1109.4339622 243/128, 256/135 ? ? ? ?
148 1116.9811321 40/21 21/11 ? ?
149 1124.5283019 ? ? ? 153/80
150 1132.0754717 48/25 ? ? 25/13, 52/27 ?
151 1139.6226415 27/14 ? ? 85/44
152 1147.1698113 ? 64/33 ? 33/17
153 1154.7169811 ? ? ? 39/20 187/96
154 1162.2641509 ? 88/45 ? 100/51
155 1169.8113208 63/32 55/28, 108/55 ? 51/26
156 1177.3584906 160/81 ? ? 77/39 168/85
157 1184.9056604 ? 240/121, 99/50 143/72 119/60
158 1192.4528302 448/225 484/243 195/98, 700/351 255/128
159 1200 2/1 P8 Perfect Octave D Reduplication of the root.
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