183edo

← 182edo

183edo

184edo →

Prime factorization	3 × 61
Step size	6.55738 ¢
Fifth	107\183 (701.639 ¢)
Semitones (A1:m2)	17:14 (111.5 ¢ : 91.8 ¢)
Consistency limit	17
Distinct consistency limit	17

The 183 equal divisions of the octave (183edo), or the 183(-tone) equal temperament (183tet, 183et) when viewed from a regular temperament perspective, divides the octave into 183 equal parts of about 6.56 cents each, a size close to 243/242, the rastma.

Theory

183edo is notable as a higher-limit system, distinctly consistent in the 17-odd-limit, or the no-19 no-31 33-odd-limit. It has especially low errors in all prime limits from 11 to 29, although its bad rendering of 19 makes it fail to be consistent in the 19-odd-limit. It is however a strong no-19's 29-limit system with the addition of an essentially perfectly accurate prime 43.

As an equal temperament, 183et tempers out the schisma in the 5-limit. In the 7-limit, it tempers out porwell, 6144/6125, cataharry, 19683/19600 and mirkwai, 16875/16807. In the 11-limit, it tempers out 540/539, 1375/1372, 3025/3024, 5632/5625, and 8019/8000; in the 13-limit, 351/350, 676/675, 729/728, 1001/1000, 1573/1568, 2080/2079, 4096/4095, 4225/4224, and 6656/6655; in the 17-limit 442/441, 561/560, 715/714, 936/935, 1089/1088, and 1156/1155; and in the 19-limit 456/455. It is the optimal patent val for 13- and 17-limit mirkat, the 72 & 111 temperament, and an excellent tuning for the rank-3 temperaments madagascar and borneo. It allows essentially tempered chord including ratwolfsmic chords, swetismic chords, squbemic chords, sinbadmic chords, and lambeth chords in the 13-odd-limit, in addition to island chords in the 15-odd-limit.

It is even stronger if 7 is left out of the picture. As a no-7 temperament, it tempers out 5632/5625, 8019/8000, 676/675, 4225/4224, 6656/6655, 936/935, 1089/1088, and 1377/1375.

Prime harmonics

Approximation of prime harmonics in 183edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.32	+0.57	+1.67	-0.50	-1.18	-0.04	-2.43	+1.23	-0.07	+2.51
	Relative (%)	+0.0	-4.8	+8.7	+25.4	-7.6	-18.0	-0.6	-37.1	+18.8	-1.1	+38.2
Steps (reduced)		183 (0)	290 (107)	425 (59)	514 (148)	633 (84)	677 (128)	748 (16)	777 (45)	828 (96)	889 (157)	907 (175)

Approximation of prime harmonics in 183edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-2.16	-2.83	-0.04	-3.21	-1.37	+3.12	-2.13	-0.62	-2.65	+1.72	+2.68
	Relative (%)	-33.0	-43.2	-0.6	-49.0	-20.9	+47.6	-32.5	-9.4	-40.4	+26.2	+40.8
Steps (reduced)		953 (38)	980 (65)	993 (78)	1016 (101)	1048 (133)	1077 (162)	1085 (170)	1110 (12)	1125 (27)	1133 (35)	1154 (56)

Subsets and supersets

Since 183 factors into primes as 3 × 61, 183edo contains 3edo and 61edo as its subsets.

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 183edo. Prime harmonics are in bold.

As 183edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 183edo

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/9, 18/11	0.133	2.0
11/6, 12/11	0.183	2.8
15/8, 16/15	0.256	3.9
3/2, 4/3	0.316	4.8
11/8, 16/11	0.498	7.6
13/9, 18/13	0.552	8.4
5/4, 8/5	0.572	8.7
9/8, 16/9	0.631	9.6
13/11, 22/13	0.685	10.4
15/11, 22/15	0.754	11.5
13/12, 24/13	0.868	13.2
5/3, 6/5	0.887	13.5
11/10, 20/11	1.070	16.3
7/5, 10/7	1.094	16.7
13/8, 16/13	1.183	18.0
9/5, 10/9	1.203	18.3
15/14, 28/15	1.410	21.5
15/13, 26/15	1.439	21.9
7/4, 8/7	1.666	25.4
13/10, 20/13	1.755	26.8
7/6, 12/7	1.982	30.2
11/7, 14/11	2.164	33.0
9/7, 14/9	2.297	35.0
13/7, 14/13	2.849	43.5

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-290 183⟩	[⟨183 290]]	+0.0996	0.100	1.52
2.3.5	32805/32768, ⟨10 23 -20]	[⟨183 290 425]]	−0.0157	0.182	2.78
2.3.5.7	6144/6125, 16875/16807, 19683/19600	[⟨183 290 425 514]]	−0.1601	0.296	4.51
2.3.5.7.11	540/539, 1375/1372, 5632/5625, 8019/8000	[⟨183 290 425 514 633]]	−0.0993	0.291	4.44
2.3.5.7.11.13	351/350, 540/539, 676/675, 1375/1372, 4096/4095	[⟨183 290 425 514 633 677]]	−0.0295	0.308	4.70
2.3.5.7.11.13.17	351/350, 442/441, 540/539, 561/560, 1375/1372, 4096/4095	[⟨183 290 425 514 633 677 748]]	−0.0240	0.286	4.36

• 183et has lower absolute errors in the 13-, 17-, 19-, and 23-limit than any previous equal temperaments, after 130, 171, 161, and 159, respectively. In the 13-, 19-, and 23-limit it is superseded by 190g. In the 17-limit, where it is the strongest, by 217.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	10\183	65.57	27/26	Luminal
1	16\183	104.92	17/16	Septendesemi
1	17\183	111.48	16/15	Stockhausenic
1	38\183	249.18	15/13	Hemischis
1	58\183	380.33	56/45	Quanharuk
1	59\183	386.89	5/4	Grendel
1	76\183	498.36	4/3	Helmholtz
1	77\183	504.92	104976/78125	Countermeantone
3	21\183	137.70	13/12	Avicenna
3	24\183	157.38	35/32	Nessafof
3	28\183	183.61	10/9	Mirkat
3	38\183 (23\183)	249.18 (150.82)	15/13 (12/11)	Hemiterm
3	76\183 (15\183)	498.36 (98.36)	4/3 (200/189)	Term / terminator
61	38\183 (2\183)	249.18 (13.11)	13750/11907 (?)	Promethium

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

Music

birdshite stalactite

• from meticulous clutter (2022)
  • "cursed owl windchimes" – Spotify | Bandcamp | YouTube – madagascar[19] in 183edo tuning
  • "octopus bones" – Spotify | Bandcamp | YouTube – madagascar[19] in 183edo tuning and parakleismic[19] in 61edo tuning
• "lemon drizzle" from tropical nosebleed (2023) – Spotify | Bandcamp | YouTube