159edo/Interval names and harmonies

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↑, 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↓, 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	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.