Monkey

Monkey

Subgroups	2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis	875/864, 5120/5103 (7-limit); 100/99, 243/242, 385/384 (11-limit) 100/99, 144/143, 243/242, 385/384 (13-limit)
Reduced mapping	⟨1; 4 9 -15 10 -2]
ET join	41 & 48
Generators (CWE)	~10/9 = 175.6 ¢
MOS scales	6L 1s, 7L 6s, 7L 13s, 7L 20s
Ploidacot	tetracot
Minimax error	9-odd-limit: 6.68 ¢; 13-limit 21-odd-limit: 12.8 ¢
Target scale size	9-odd-limit: 27 notes; 13-limit 21-odd-limit: 34 notes

The monkey temperament is one of the 7-limit extensions of tetracot, the 5-limit temperament tempering out the tetracot comma (20000/19683), and is naturally a full 13-limit temperament.

In addition to the tetracot comma, monkey tempers out 875/864, making it a keemic temperament. It also tempers out 5120/5103, making it a hemifamity temperament, so the septimal comma is equated with the syntonic comma. At 7 generator steps, this diesis-sized interval also represents 40/39, 45/44, 55/54, 65/64, 66/65, and 121/120 in the 2.3.5.11.13-subgroup version of tetracot, and divides the chromatic semitone in four. The same interval is now used to bridge septimal intervals with Pythagorean intervals alike.

Additionally, the generator can be taken to represent 21/19, which gives us an extension for prime 19 at -12 generator steps.

See Tetracot family #Monkey for technical data.

Interval chain

In the following tables, odd harmonics 1–13 and their inverses are in bold.

#	Cents*	Approximate ratios
0	0.0	1/1
1	175.6	10/9, 11/10
2	351.2	11/9, 16/13
3	526.9	15/11
4	702.5	3/2
5	878.1	5/3
6	1053.7	11/6, 24/13
7	29.4	40/39, 45/44, 55/54, 64/63
8	205.0	9/8
9	380.6	5/4
10	556.2	11/8, 18/13
11	731.8	20/13, 32/21
12	907.5	22/13
13	1083.1	13/7, 15/8
14	58.7	25/24, 27/26, 33/32, 36/35
15	234.3	8/7, 15/13
16	409.9	33/26
17	585.6	45/32, 88/63
18	761.2	25/16, 54/35
19	936.8	12/7
20	1112.4	40/21
21	88.1	22/21
22	263.7	75/64, 81/70
23	439.3	9/7
24	614.9	10/7
25	790.5	11/7
26	966.2	225/128, 243/140, 256/147
27	1141.8	27/14

* In 13-limit CWE tuning, octave reduced

Tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 175.6758 ¢	CWE: ~10/9 = 175.6622 ¢	POTE: ~10/9 = 175.6588 ¢

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 175.5978 ¢	CWE: ~10/9 = 175.5750 ¢	POTE: ~10/9 = 175.5703 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 175.6185 ¢	CWE: ~10/9 = 175.6217 ¢	POTE: ~10/9 = 175.6224 ¢

Tuning spectrum

Edo generator	Eigenmonzo (unchanged-interval)*	Generator (¢)	Comments
	11/10	165.004
1\7		171.429
	11/9	173.704
	13/7	174.746
	11/6	174.894
7\48		175.000	Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
	11/8	175.132
	11/7	175.300	11-odd-limit minimax
	7/4	175.412
	7/6	175.428
	9/7	175.438
	3/2	175.489
6\41		175.610	15-odd-limit diamond monotone (singleton)
	15/14	175.694
	7/5	175.729	7-, 9-, 13- and 15-odd-limit minimax
	13/11	175.899
	15/8	176.021
	5/4	176.257	5-odd-limit minimax
	13/9	176.338
5\34		176.471	Upper bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
	15/13	176.516
	5/3	176.872
	13/10	176.890
	13/12	176.905
4\27		177.778	27de val
	15/11	178.984
	13/8	179.736
3\20		180.000	20cdde val
	9/5	182.404

* Besides the octave