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.
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	Edb<, 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<, Edb<↑	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	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, 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 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, 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 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	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	?
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.