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, as well as the base representation of the Tonic.
1 7.5471698 225/224 243/242 196/195, 351/350 256/255 R1 Wide Prime D/ As the approximation of both the rastma and the marvel comma, this interval is 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↑\ As the approximation of the ptolemisma and the biyatisma, this interval is useful for slight dissonances that create more noticeable tension.
3 22.6415094 81/80 ? ? 78/77 85/84 K1 Lesser Superprime D↑ As the approximation of the syntonic comma and the Pythagorean comma, this interval is useful for appoggiaturas, acciaccaturas, and quick passing tones.
4 30.1886792 64/63 56/55, 55/54 ? 52/51 S1, kU1 Greater Superprime, Narrow Inframinor Second Edb<, Dt<↓ As the approximation of septimal comma and the telepathma, this interval is useful for various types of subchromatic gestures, as well as for appoggiaturas, 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 indeed, the 11-limit ratio this interval approximates is the namesake of 24edo's own Inframinor Second; however, in a higher-fidelity system such as this, one will notice that this syntactic second is actually noticeably narrower than 24edo's quartertone.
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; furthermore, as the name "Semilimma" suggests, this interval is one half of a Pythagorean Minor Second.
7 52.8301887 ? 33/32 ? 34/33 U1, rKum2 Ultraprime, Narrow Subminor Second Dt<, Edb<↑ By default, this interval is a type of parachromatic quartertone- specifically, the representation of the Al-Farabi Quartertone- and is thus used in much the same way as 24edo's own Ultraprime; what might be surprising is that five of these add up to this system's approximation of the Septimal Subminor Third.
8 60.3773585 28/27 ? ? 88/85 sm2, Kum2, uA1 Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime Dt>, Eb↓\ 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, rKU1, kkA1 Greater Subminor Second, Diptolemaic Augmented Prime Eb↓, Dt<↑\, 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 Pythagorean 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 due to it being the approximation of the Septimal Minor Semitone, 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 approximation of both the octave-reduced seventeenth harmonic and the interval formed from stacking two Ultraprimes, 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 ? ? RKm2, kn2, RA1 Wide Minor Second, Artoretromean Augmented Prime Ed<↓, Eb↑/, D#/ In addition to being the approximation of the Septimal Major Semitone, this interval is also one third of a Lesser Submajor Third in this system, and is thus used accordingly.
17 128.3018868 ? ? 14/13 128/119 kN2, rKA1 Lesser Supraminor Second, Tendoretromean Augmented Prime Ed>↓, D#↑\ In addition to its properties as a type of Supraminor Second, this interval is also one third of a Ptolemaic Major Third in this system and is thus used accordingly.
18 135.8490566 27/25 ? ? 13/12 ? KKm2, rn2, KA1 Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime Ed<\, Eb↑↑, D#↑ This interval is not only both two thirds of Pythagorean Major Second and the approximation of the Large Limma or Diptolemaic Limma in this system, but also a type of Supraminor Second, and is thus used accordingly.
19 143.3962264 ? 88/81 ? ? n2, SA1, kUA1 Artoneutral Second, Lesser Super-Augmented Prime Ed<, Dt#<↓ As one of two Neutral Seconds in this system, this interval is notable for being half of the Neo-Gothic Minor Third, though it is also sometimes used in much the same way as 24edo's own Neutral Second.
20 150.9433962 ? 12/11 ? ? N2, RkUA1 Tendoneutral Second, Greater Super-Augmented Prime Ed>, Dt#>↓ As one of two Neutral Seconds in this system, this interval is the one that most closely resembles the low-complexity JI Neutral Second, and thus, it is frequently used in much the same way as 24edo's own Neutral Second.
21 158.4905660 ? ? ? 128/117 561/512, 1024/935 kkM2, RN2, rUA1 Lesser Submajor Second, Diretroptolemaic Augmented Prime Ed>/, E↓↓, Dt#>↓/, D#↑↑, Fb↓ In addition to its properties as a type of Submajor Second, this interval is also one half of a Ptolemaic Minor Third in this system and is thus used accordingly.
22 166.0377358 ? 11/10 ? ? Kn2, UA1 Greater Submajor Second, Ultra-Augmented Prime Ed<↑, Dt#<, Fb↓/ In addition to its properties as the interval that most closely resembles the low-complexity JI Submajor Second, this interval serves as both the Ultra-Augmented Prime and as one third of a Perfect Fourth, and is used accordingly.
23 173.5849057 567/512 243/220 ? 425/384 rkM2, KN2 Narrow Major Second Ed>↑, E↓\, Dt#>, Fb\ While this interval is large enough to act as a type of whole tone, it is worth noting that two of these add up to the approximation of the low-complexity JI Neutral Third in this system.
24 181.1320755 10/9 ? 256/231 ? ? kM2 Ptolemaic Major Second E↓, Fb As the approximation of the Ptolemaic Major Second, this interval is used accordingly, however, it is worth noting that in this system, two of these add up to the approximation of the thirteenth subharmonic; furthermore, it is also one the intervals in this system that are essential in executing any sort of variation on Jacob Collier's "Four Magical chords" from his rendition of "In the Bleak Midwinter".
25 188.6792458 ? ? 143/128 512/459 RkM2 Artomean Major Second E↓/, Fb/ This interval has surprising utility in modulating to keys that are not found on the same circle of fifths owing to both its size and its ease of access through octave-reducing stacks of approximated low-complexity JI intervals.
26 196.2264151 28/25 121/108 ? ? rM2 Tendomean Major Second E\, Fb↑\ This interval is created from stacking two of this system's closet approximation of the 12edo semitone, and thus, it is one of two intervals that come the closest to approximating the 12edo whole tone found in this system.
27 203.7735849 9/8 ? ? ? ? M2 Pythagorean Major Second E, Fb↑ This interval is the standard-issue whole tone in this system as it is one of two intervals that come the closest to approximating the 12edo whole tone, and the only one of the two that actually approximates the Pythagorean Major Second; furthermore, it is the whole tone that is used as a reference interval in diatonic-and-chromatic-style interval logic in this system as it pertains to both semitones and quartertones.
28 211.3207547 ? ? 44/39 289/256 RM2 Wide Major Second E/, Fd<↓ This interval is interesting on the basis that it is formed by stacking two instances of the octave-reduced approximation of the seventeenth harmonic.
29 218.8679245 ? ? ? 17/15 rKM2 Narrow Supermajor Second E↑\, Fd>↓ This interval is of note because it is utilized in approximations of the 17-odd-limit; what's more, it is also the whole tone in this system's superpyth diatonic scale, and in fact, two of these add up to the approximation of the Septimal Supermajor Third in this system.
30 226.4150943 256/225 ? 154/135 ? ? KM2 Lesser Supermajor Second E↑, Fd<\, Fb↑↑, Dx This interval can be interpreted as a type of second on the basis of it approximating the sum of the syntonic comma and the Pythagorean Major Second; it also appears in approximations of 5-limit Neapolitan scales as the interval formed from stacking two Ptolemaic Minor Seconds, making it double as a type of diminished third, and is likely the smallest interval in this system that can be used in chords without causing crowding.
31 233.9622642 8/7 55/48 ? ? SM2, kUM2 Greater Supermajor Second, Narrow Inframinor Third Fd<, Et<↓, E↑/ As the approximation of the octave-reduced seventh subharmonic- that is, the Septimal Supermajor Second- this interval is used accordingly; in fact, since three of these add up to a Perfect Fifth in this system, there are multiple ways this interval can be used in chords to great effect.
32 241.5094340 ? 1024/891 ? ? um3, RkUM2 Inframinor Third, Wide Supermajor Second Fd>, Et>↓ The 11-limit ratio this interval approximates is the namesake of 24edo's own Inframinor Third; however, in a higher-fidelity system such as this, one will notice that this is a syntactic third that sounds more like a second.
33 249.0566038 ? ? ? 15/13 ? kkm3, KKM2, Rum3, rUM2 Wide Inframinor Third, Narrow Ultramajor Second, Semifourth Fd>/, Et<\, F↓↓, E↑↑ This interval is particularly likely to be used as a cross between an Ultramajor Second and an Inframinor Third; furthermore, as the name "Semifourth" suggests, this interval is one half of a Perfect Fourth, and used in exactly the same way as 24edo's own Semifourth, right down to the low-complexity 13-limit interpretation.
34 256.6037736 ? 297/256 ? ? UM2, rKum3 Ultramajor Second, Narrow Subminor Third Et<, Fd<↑ The 11-limit ratio this interval approximates is the namesake of 24edo's own Ultramajor Second; however, in a higher-fidelity system such as this, one will notice that this is a syntactic second that sounds more like a third.
35 264.1509434 7/6 64/55 ? ? sm3, Kum3 Lesser Subminor Third, Wide Ultramajor Second Et>, Fd>↑, F↓\ As the approximation of the Septimal Subminor Third, this interval is used accordingly; what's more, due in part to both the keenanisma being tempered out and the fact that three of these add up to the Pythagorean Minor Sixth, this interval can be used to great effect in chords.
36 271.6981132 75/64 ? ? ? ? km3 Greater Subminor Third F↓, Et>/, E#↓↓, Gbb This interval can be interpreted as a type of third on the basis of it approximating result of subtracting a syntonic comma from a Pythagorean Minor Third; however, it most frequently appears in approximations of 5-limit Harmonic scales as the interval between the Ptolemaic Minor Sixth and the Ptolemaic Major Seventh, making it double as a type of augmented second.
37 279.2452830 ? ? ? 20/17 Rkm3 Wide Subminor Third F↓/, Et<↑ This interval is utilized in approximations of the 17-odd-limit, courtesy of acting as the fourth complement to the Narrow Supermajor Second.
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 or Tonic.