159edo

← 158edo

159edo

160edo →

Prime factorization	3 × 53
Step size	7.54717¢
Fifth	93\159 (701.887¢) (→31\53)
Semitones (A1:m2)	15:12 (113.2¢ : 90.57¢)
Consistency limit	17
Distinct consistency limit	17

159 equal divisions of the octave (abbreviated 159edo or 159ed2), also called 159-tone equal temperament (159tet) or 159 equal temperament (159et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 159 equal parts of about 7.55 ¢ each. Each step represents a frequency ratio of 2^1/159, or the 159th root of 2. The step size of this system is in between the sizes of 243/242 (the rastma) and 225/224 (the marvel comma).

Theory

A salient fact about 159edo is that 159 = 3 × 53, and thus, this system has both 3edo and 53edo as subsets- the former subset being shared with 12edo.

History

Despite being none other than the tripled superset of the famous 53edo, and hence, one would think, fairly easy to find, it is a wonder that the first person to show theoretical interest in it was Ozan Yarman- specifically in terms of extracting a voluminous subset for representing maqams- in late 2005 to early 2006.

Yarman started by dividing the 4/3 perfect fourth into 33 equal parts and continued the resultant 15.1 cent comma until just below the octave, only to then reinstate the final 79th degree with the octave so as to arrive at the "yegah-neva" partition. He then rotated the scale to begin on the Turkish "rast" note which he notated as C according to Sipürde Ahenk (C Ney), and thus the 22.8 cent larger singular comma previously on top now appears just below the Turkish neva note in the midst of his tuning scheme. Not long after proposing said 79-tone system, Yarman visited the now-deceased Ejder Güleç in İzmir who affixed mandals on Yarman's Qanun according to Yarman's instructions.

Gene Ward Smith was the first to point out that Yarman's scheme was a MOS of 159tet and had an 80-pitch twin. Yarman adopted this argument, because his approach and the related MOS subset out of 159tet was, for all intents and purposes, synonymous. Yarman has stated that he thinks he introduced his Qanun to his now-deceased supervisor in Istanbul Technical University Turkish Music State Conservatory sometime during late 2006, and she suggested that Yarman include the double-sharp mandals. At the time this information was added to this article, Yarman remembered that the Qanun in his doctorate defense of 2008 included the double-sharp mandals. The acceptance of his thesis was in June 2008.

Accordingly, it is no coincidence that the first records of 159edo on this Wiki from the days of Wikispaces concern said 79-tone subset related to the yarman temperament which had been proposed by Yarman as a tuning standard for Arabic, Turkish and Persian music. Based on the information given by Ozan Yarman himself, his elder colleague M. Ugur Kececioglu first utilized 159edo in his revamped 2011 release of the Notist score editor and therein allowed the Arel-Ezgi-Uzdilek accidentals to be bent by as little a detail as 1/3rd of a single step of 53edo, while also mapping AEU altogether to a suitable subset of 53edo to allow transpositions throughout.

Mappings and JI approximation quality

This system inherits its approximations of the 3rd, 5th, 13th, and 19th harmonics from 53edo, however, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this edo has a very accurate 11 and an only slightly less accurate 17. Furthermore, 159edo demonstrates 3-to-2 telicity, as despite being contorted in the 5-limit, it is the largest edo to temper out Mercator's comma in which said comma is less than half the size of a single edo step. This means, among other things, that there is a perfect match between the direct approximation and the more complicated traditional mapping for an octave-reduced stack of fifty-three tempered 3/2 perfect fifths – a complete circle of fifths for this edo.

159edo is consistent up to the no-17 29-odd-limit or the no-19 27-odd-limit as {19/17, 34/19} and {29/17, 34/29} exhaust the inconsistently mapped interval pairs in the 29-odd-limit. Thus its full 29-limit interpretation using the patent val is obvious. However, the direct approximation and the val mapping for intervals such as 49/32, 35/32, and 169/128 do not match, and as a result, 159edo can be thought of as having a perfunctory 7-limit that mainly serves to bridge to the 11-limit and divide the nearly just 3/2 into three, as well as a similarly perfunctory 13-limit that mainly serves to bridge to the 17-limit and to absorb complex combinations of 3 and 5.

Notably, 159edo provides the optimal patent val for 11-limit guiron, 13-limit tritikleismic, the 13-limit rank-3 temperament portending, as well as the 17-limit rank-6 temperament tempering out 273/272. In addition to this, it also supports both forms of the yarman temperament, with a generator of 2\159 which can be taken as an approximate 105/104. Both have a mos of 79 or 80 notes to the octave, and have their optimal patent vals supplied by 159edo in 7-limit, 11-limit, 13-limit, 17-limit and even 19-limit forms. While the patent val supports both cartography and iodine temperaments, which are among the best 13-limit temperaments in the Mercator family, the 159d and 159f mappings support other members of this temperament family.

Prime harmonics

Approximation of prime harmonics in 159edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.00	-0.07	-1.41	-2.79	-0.37	-2.79	+0.70	-3.17	-1.86	-3.16	+2.13	-2.29
	Relative (%)	+0.0	-0.9	-18.7	-36.9	-5.0	-37.0	+9.3	-42.0	-24.6	-41.9	+28.3	-30.3
Steps (reduced)		159 (0)	252 (93)	369 (51)	446 (128)	550 (73)	588 (111)	650 (14)	675 (39)	719 (83)	772 (136)	788 (152)	828 (33)

Additional Properties of 159-tone equal temperament

The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic JND, and similarly this system can approximate the sounds of other systems such as 10edo, 13edo, 22edo and 31edo. Furthermore, the step size of 159edo is simultaneously above the average peak melodic JND and small enough to be well within the margin of error between Just 5-limit intervals and their 12edo counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another, even as it also allows one to also control the bandwidth of certain sounds. As a result of tempering out some of the commas, it allows essentially tempered chords including marveltwin chords, gentle chords, keenanismic chords, werckismic chords, and sinbadmic chords in the 13-odd-limit, in addition to island chords and nicolic chords in the 15-odd-limit.

MOSes and other scales

See also: List of MOS scales in 159edo

No less than five possible generators for the diatonic MOS Scale are supported by 159edo, though these different diatonic scales have different descendants. The 91\159 generator results in large and small scale steps at 23\159 and 22\159 respectively, making for a quasi-equalized diatonic scale, which can be extended to a siskontyttonic MOS, while the 95\159 results in large and small scale steps at 31\159 and 2\159 respectively, making for a version approaching collapsed, which can be extended to a reinatonic (5L 22s) MOS. The 92\159 generator results in large and small scale steps at 25\159 and 17\159 respectively, and this makes for a very meantone-like diatonic scale perfect for xenharmonic pieces that follow in the classical tradition, though this can be extended to a veljentyttonic (19L 26s) MOS. Conversely, the 94\159 generator results in results in large and small scale steps at 29\159 and 7\159 respectively, and this makes for a superpyth diatonic scale that is slightly harder and better than that of 22edo, which can be extended to an antireinatonic (22L 5s) MOS. Finally, the patent 93\159 generator results in the same diatonic MOS scale found in 53edo, which, despite now having competition from other possible generators, is still the go-to for those looking for something more akin to the classic Pythagorean tuning, as well as for those looking to deal with good approximations of related 5-limit scales, or even just the basic form of the pythamystonic (12L 29s) MOS.

In addition, 159edo has no less than four possible generators for the oneirotonic MOS scale, and of these, two of them are also supported by 53edo. The 60\159 generator results in large and small scale steps at 21\159 and 18\159 respectively, making for a distinctly ultra-soft scale, while the 63\159 generator results in large and small scale steps at 30\159 and 3\159 respectively, making for a distinctly ultra-hard scale. As for the remaining two generators, the 61\159 generator results in large and small scale steps at 24\159 and 13\159 respectively and comes the closest to any sort of basic form of this scale, however, the 62\159 generator is also a solid choice, and is also useful for at least one related non-MOS scale due to 62\159 approximating 21/16.

Furthermore, this EDO supports Wyschnegradsky's "Diatonicized Chromatic Scale" (11L 2s) with large and small scale steps at 13\159 and 8\159 respectively.

Intervals

See also: 159edo interval names and harmonies

The following table assumes 17-limit patent val ⟨159 252 369 446 550 588 650].

Intervals highlighted in bold are prime harmonics or subharmonics. In addition, intervals that differ from assigned steps by more than 50%, multiples of such intervals, and intervals of odd limit higher than 1024, are not shown. Furthermore, when multiple well-known intervals for a given prime-limit share a step size, they may share a cell in the chart; conversely, a "?" in the chart means that no known interval meets the criteria for inclusion. Note that no 5-limit intervals can be represented by degrees other than multiples of 3, so those entries are left blank.

Table of 159edo intervals

Step	Cents	5 limit	7 limit	11 limit	13 limit	17 limit
0	0	1/1
1	7.5471698		225/224	243/242	196/195, 351/350	256/255
2	15.0943396		?	121/120, 100/99	144/143	120/119
3	22.6415094	81/80	?	?	78/77	85/84
4	30.1886792		64/63	56/55, 55/54	?	52/51
5	37.7358491		?	45/44	?	51/50
6	45.2830189	?	?	?	40/39	192/187
7	52.8301887		?	33/32	?	34/33
8	60.3773585		28/27	?	?	88/85
9	67.9245283	25/24	?	?	26/25, 27/26	?
10	75.4716981		?	?	?	160/153
11	83.0188679		21/20	22/21	?	?
12	90.5660377	256/243, 135/128	?	?	?	?
13	98.1132075		?	128/121	55/52	18/17
14	105.6603774		?	?	?	17/16
15	113.2075472	16/15	?	?	?	?
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		?	25/22	?	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	?	?	117/100	?
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		?	?	26/21	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	?	34/27
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	?	?	50/39	?
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		256/189	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		189/128	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	?	?	39/25	?
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	?	27/17
107	807.5471698		?	?	?	51/32
108	815.0943396	8/5	?	77/48	?	?
109	822.6415094		45/28	825/512	?	?
110	830.1886792		?	?	21/13	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	?	?	200/117	?
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		?	44/25	?	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

Notation

Main article: 159edo notation

Because of the complexity of 159edo, notation requires systems that make use of multiple extra pairs of accidentals. This is because at high EDOs, systems with only a single extra accidental pair become unwieldy due to the sheer number of such accidentals required for notating some pitches, which in turn results in high amounts of clutter on scores. So far, several notation systems addressing this problem have been proposed.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5.7	1029/1024, 10976/10935, 15625/15552	[⟨159 252 369 446]]	+0.411	0.413	5.47
2.3.5.7.11	385/384, 441/440, 4000/3993, 10976/10935	[⟨159 252 369 446 550]]	+0.350	0.389	5.15
2.3.5.7.11.13	325/324, 364/363, 385/384, 625/624, 10976/10935	[⟨159 252 369 446 550 588]]	+0.418	0.385	5.11
2.3.5.7.11.13.17	273/272, 325/324, 364/363, 375/374, 385/384, 3773/3757	[⟨159 252 369 446 550 588 650]]	+0.333	0.412	5.46
2.3.5.7.11.13.17.19	273/272, 325/324, 343/342, 361/360, 364/363, 375/374, 385/384	[⟨159 252 369 446 550 588 650 675]]	+0.385	0.409	5.42
2.3.5.7.11.13.17.19.23	273/272, 325/324, 343/342, 361/360, 364/363, 375/374, 385/384, 392/391	[⟨159 252 369 446 550 588 650 675 719]]	+0.388	0.386	5.11

• 159et has lower absolute errors in the 19- and 23-limit than any previous equal temperaments, past 152fg. The next equal temperament that does better in the 19-limit is 161, and in the 23-limit, 183.

Notably, while 159edo plays host to a number of fun temperaments like portent and minor minthmic, a number of microtemperaments have also been found hiding within its structure. This means that 159edo is well-balanced in terms of the possibilities for interesting temperament usage.

Commas

Assuming the patent val ⟨159 252 369 446 550 588 650 675 719], 159tet tempers out the following commas in the 23-limit.

Prime limit	Ratio	Monzo	Cents	Color name	Name
3	(52 digits)	[-84 53⟩	3.6150	Wa-53	Mercator's comma
5	(14 digits)	[-21 3 7⟩	10.0610	Lasepyo	Semicomma
5	15625/15552	[-6 -5 6⟩	8.1073	Tribiyo	Kleisma
5	(14 digits)	[9 -13 5⟩	6.1536	Saquinyo	Amity comma
5	(22 digits)	[24 -21 4⟩	4.1998	Sasaquadyo	Vulture comma
5	(28 digits)	[39 -29 3⟩	2.2461		Tricot comma
5	32805/32768	[-15 8 1⟩	1.9537	Layo	Schisma
5	(44 digits)	[-69 45 -1⟩	1.6613		Counterschisma
5	(36 digits)	[54 -37 2⟩	0.2924		Monzisma
7	1029/1024	[-10 1 0 3⟩	8.4327	Latrizo	Gamelisma
7	(12 digits)	[1 -1 -7 6⟩	6.8044		Triwellisma
7	10976/10935	[5 -7 -1 3⟩	6.4790	Satrizo-agu	Hemimage comma
7	(20 digits)	[16 -9 -8 6⟩	4.8507
7	(20 digits)	[-19 14 -5 3⟩	2.2792		Forge comma
7	(12 digits)	[-11 2 7 -3⟩	1.6283		Meter
7	(12 digits)	[-4 6 -6 3⟩	0.3254		Landscape comma
7	(58 digits)	[9 -28 37 -18⟩	0.0011	Satritribiru-athiseyo	Termite comma
11	4375/4356	[-2 -2 4 1 -2⟩	7.5349
11	385/384	[-7 -1 1 1 1⟩	4.5026	Lozoyo	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.9302	Luzozogu	Werckisma
11	6250/6237	[1 -4 5 -1 -1⟩	3.6047		Liganellus comma
11	(36 digits)	[-55 11 1 -1 11⟩	3.4902		Tritonoquartisma
11	4000/3993	[5 -1 3 0 -3⟩	3.0323	Triluyo	Wizardharry comma
11	19712/19683	[8 -9 0 1 1⟩	2.5488	Salozo	Symbiotic comma
11	(14 digits)	[16 -3 0 0 6⟩	2.0427	Tribilo	Nexus comma
11	(24 digits)	[-35 17 -1 0 3⟩	0.8751	Trila-trilo-agu	Triagnoshenisma
11	(28 digits)	[-34 28 0 0 -3⟩	0.7862	Quadla-trilu	Frameshift comma
11	3025/3024	[-4 -3 2 -1 2⟩	0.5724	Loloruyoyo	Lehmerisma
11	(18 digits)	[24 -6 0 1 -5⟩	0.5062	Saquinlu-azo	Quartisma
11	(14 digits)	[-1 -11 -1 0 6⟩	0.0889	Satribilo-agu	Parimo
13	2197/2187	[0 -7 0 0 0 3⟩	7.8980	Satritho	Threedie
13	325/324	[-2 -4 2 0 0 1⟩	5.3351	Thoyoyo	Marveltwin comma
13	364/363	[2 -1 0 1 -2 1⟩	4.7627	Tholuluzo	Minor minthma
13	13720/13689	[3 -4 1 3 0 -2⟩	3.9161
13	625/624	[-4 -1 4 0 0 -1⟩	2.7722	Thuquadyo	Tunbarsma
13	676/675	[2 -3 -2 0 0 2⟩	2.5629	Bithogu	Island comma
13	1575/1573	[0 2 2 1 -2 -1⟩	2.1998		Nicola
13	1001/1000	[-3 0 -3 1 1 1⟩	1.7304		Fairytale comma
13	10985/10976	[-5 0 1 -3 0 3⟩	1.4190		Cantonisma
13	43904/43875	[7 -3 -3 3 0 1⟩	1.1439		Punctisma
13	2080/2079	[5 -3 1 -1 -1 1⟩	0.8325	Tholuruyo	Ibnsinma
13	(12 digits)	[11 -9 3 0 0 -1⟩	0.8185	Sathutriyo	Phaotic comma
13	6656/6655	[9 0 -1 0 -3 1⟩	0.2601	Thotrilu-agu	Jacobin comma
13	(12 digits)	[-6 2 6 0 0 -3⟩	0.2093		Catasma
13	(12 digits)	[-6 6 -2 -1 -1 2⟩	0.0141	Lathotholurugugu	Chalmersia
17	15379/15300	[-2 -2 -2 1 0 3 -1⟩	8.9161
17	273/272	[-4 1 0 1 0 1 -1⟩	6.3532	Suthozo	Tannisma
17	375/374	[-1 1 3 0 -1 0 -1⟩	4.6228		Ursulisma
17	595/594	[-1 -3 1 1 -1 0 1⟩	2.9121		Dakotisma
17	715/714	[-1 -1 1 -1 1 1 -1⟩	2.4230		September comma
17	833/832	[-6 0 0 2 0 -1 1⟩	2.0796	Sothuzozo	Horizon comma
17	936/935	[3 2 -1 0 -1 1 -1⟩	1.8506		Ainos comma
17	2025/2023	[0 4 2 -1 0 0 -2⟩	1.7107		Fidesma
17	1089/1088	[-6 2 0 0 2 0 -1⟩	1.5905		Twosquare comma
17	1701/1700	[-2 5 -2 1 0 0 -1⟩	1.0181		Palingenetic comma
17	24576/24565	[13 1 -1 0 0 0 -3⟩	0.7751		Archagallisma
17	2431/2430	[-1 -5 -1 0 1 1 1⟩	0.7123		Heptacircle comma
17	(12 digits)	[-10 -5 0 0 4 0 1⟩	0.44522
17	12376/12375	[3 -2 -3 1 -1 1 1⟩	0.1399		Flashma
17	14400/14399	[6 2 2 -1 -2 0 -1⟩	0.1202	Sululuruyoyo	Sparkisma
19	343/342	[-1 -2 0 3 0 0 0 -1⟩	5.0547
19	361/360	[-3 -2 -1 0 0 0 0 2⟩	4.8023
19	513/512	[-9 3 0 0 0 0 0 1⟩	3.3780

In the 23-limit, with the 19-limit skipped, this system is known to temper out 392/391, 460/459, 507/506, 529/528, 897/896, 1105/1104, 1288/1287, 2024/2023, 2025/2024, and 2646/2645 among others.

Rank-2 temperaments

Note: 5-limit temperaments supported by 53et are not included.

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	2\159	15.09	121/120	Yarman I / yarman II
1	7\159	52.83	33/32	Quartkeenlig
1	11\159	83.02	21/20	Sextilififths
1	22\159	166.04	11/10	Tertiaschis
1	31\159	233.96	8/7	Guiron
1	38\159	286.79	13/11	Gamity
1	41\159	309.43	448/375	Triwell
1	64\159	483.02	160/121	Quarterframe
1	67\159	505.66	75/56	Marfifths
1	68\159	513.21	121/90	Trinity
1	74\159	558.49	112/81	Condor
3	4\159	30.19	55/54	Hemichromat
3	8\159	60.38	28/27	Chromat
3	22\159	166.04	11/10	Tritricot
3	33\159 (20\159)	249.06 (150.94)	15/13 (12/11)	Altinex / hemiterm
3	42\159 (11\159)	316.981 (83.02)	6/5 (21/20)	Tritikleismic
3	66\159 (13\159)	498.11 (98.11)	4/3 (35/33)	Term / terminal
53	31\159 (1\159)	233.96 (7.55)	8/7 (225/224)	Schismerc / cartography
53	121\159 (1\159)	913.21 (7.55)	441/260 (196/195)	Iodine

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

The songs below are written in 159edo, or, in approximations that differ from the actual 159edo by only fractions of a cent.

Dawson Berry

• Space Tour (2020)
• Welcome to Dystopia (2021)
• The Forest of Loss (2021)
• Life in a Brave New World (2024)
• Where the Heart Is (2024)
• Foes and Forgiveness (2024)

Ozan Yarman

• ENTIRE ORCHESTRAL 79-tone "Pesendide Fugue" (III. Selim & Ozan Yarman) – Ozan Yarman's (5-voice & 2-subject) Choral Fugue of Sultan III. Selim's Pesendide Ağır Semai in Ağır Aksak usul] (re-scored in accordance with Yarman's 79-tone tuning)
• Whitecap Visualizer assisted (CHOIR + Digitized Ottoman Singing VocalWriter + STRINGS only) – flow of the ENTIRE "Pesendide Fugue" in the 79-tone Qanun tuning
• 79'lu sistemde MISIRLI Udi İbrahim Efendi'nin çoksesli Acemaşiran Sazsemaisi -- Ozan Yarman – Adjemashiran Sazsemai of "Egyptian" Ud-player Ibrahim Efendi in 79 MOS 159-tET as polyphonalized by Ozan Yarman (2005-2022)
• 79-ton RAST KAR-I NATIK (Ozan Yarman -- thicc vokal ver.) – RAST KAR-I NATIK by Ozan Yarman in his 79-tone Qanun tuning

Instruments

Currently, there is an instrument under development by Erik Natanael called the "Neod", which utilizes this system. Although 53edo is the basis for most of the keys on this instrument, there are additional buttons which modify the pitch by a single step of 159edo.

Articles

• 79-Tone Tuning & Theory for Turkish Maqam Music – Ozan Yarman's dissertation
• Search For A Theoretical Model Conforming To Turkish Maqam Music Practice: A Selection Of Fixed-Pitch Settings From 34-tone Equal Temperament To The 79-tone Tuning – also by Ozan Yarman, gives a summary
• Letter to Ozan Yarman by Margo Schulter (permalink)