Monkey: Difference between revisions

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#redirect [[Tetracot family #Monkey]]
{{Infobox regtemp
| Title = 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);<br>[[100/99]], [[243/242]], [[385/384]] (11-limit)<br>[[100/99]], [[144/143]], [[243/242]], [[385/384]]<br>(13-limit)
| Edo join 1 = 41 | Edo join 2 = 48
| Mapping = 1; 4 9 -15 10 -2
| Generators = 10/9
| Generators tuning = 175.6
| Optimization method = CWE
| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]], [[7L 20s]]
| Odd limit 1 = 9 | Mistuning 1 = 6.68 | Complexity 1 = 27
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 12.8 | Complexity 2 = 34
}}
The '''monkey''' [[regular temperament|temperament]] is one of the [[7-limit]] [[extension]]s of [[tetracot]], the [[5-limit]] temperament [[tempering out]] the [[tetracot comma]] (15625/15552), and is naturally a full [[13-limit]] temperament.


In addition to the tetracot comma, monkey tempers out [[875/864]], making it a [[keemic temperaments|keemic temperament]]. It also tempers out [[5120/5103]], making it a [[hemifamity temperaments|hemifamity temperament]], so the [[septimal comma]] is equated with the [[syntonic comma]]. At 7 generator steps, this [[diesis (interval region)|diesis-sized]] interval also represents [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], and [[121/120]] in the [[2.3.5.7.13 subgroup|2.3.5.7.13-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'''.
{| class="wikitable right-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.00
| '''1/1'''
|-
| 1
| 175.62
| 11/10, 10/9
|-
| 2
| 351.24
| 11/9, '''16/13'''
|-
| 3
| 526.87
| 15/11
|-
| 4
| 702.49
| '''3/2'''
|-
| 5
| 878.11
| 5/3
|-
| 6
| 1053.73
| 11/6, 24/13
|-
| 7
| 29.36
| 55/54, 45/44, 40/39
|-
| 8
| 204.98
| 9/8
|-
| 9
| 380.60
| '''5/4'''
|-
| 10
| 556.22
| '''11/8''', 18/13
|-
| 11
| 731.85
| 20/13
|-
| 12
| 907.47
| 22/13
|-
| 13
| 1083.09
| 13/7, 15/8
|-
| 14
| 58.71
| 33/32, 27/26, 25/24
|-
| 15
| 234.34
| '''8/7''', 15/13
|-
| 16
| 409.96
|
|-
| 17
| 585.58
| 45/32
|-
| 18
| 761.20
|
|-
| 19
| 936.83
| 12/7
|-
| 20
| 1112.45
|
|-
| 21
| 88.07
|
|-
| 22
| 263.69
|
|-
| 23
| 439.31
| 9/7
|-
| 24
| 614.94
| 10/7
|-
| 25
| 790.56
| 11/7
|-
| 26
| 966.18
|
|-
| 27
| 1141.80
| 27/14
|-
| 28
| 117.43
| 15/14
|}
<nowiki/>* in 13-limit POTE tuning
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-3"
|-
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! Generator (¢)
! Comments
|-
| 11/10
| 165.004
|
|-
| 11/9
| 173.704
|
|-
| 14/13
| 174.746
|
|-
| 12/11
| 174.894
|
|-
| 11/8
| 175.132
|
|-
| 14/11
| 175.300
| 11-odd-limit minimax
|-
| 8/7
| 175.412
|
|-
| 7/6
| 175.428
|
|-
| 9/7
| 175.438
|
|-
| 4/3
| 175.489
|
|-
| 15/14
| 175.694
|
|-
| 7/5
| 175.729
| 7, 9, 13 and 15-odd-limit minimax
|-
| 13/11
| 175.899
|
|-
| 16/15
| 176.021
|
|-
| 5/4
| 176.257
| 5-odd-limit minimax
|-
| 18/13
| 176.338
|
|-
| 15/13
| 176.516
|
|-
| 6/5
| 176.872
|
|-
| 13/10
| 176.890
|
|-
| 13/12
| 176.905
|
|-
| 15/11
| 178.984
|
|-
| 16/13
| 179.736
|
|-
| 10/9
| 182.404
|
|}
[[Category:Monkey| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Tetracot family]]
[[Category:Keemic temperaments]]
[[Category:Keemic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]

Revision as of 10:28, 15 April 2026

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 (15625/15552), 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 2.3.5.7.13-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 0.00 1/1
1 175.62 11/10, 10/9
2 351.24 11/9, 16/13
3 526.87 15/11
4 702.49 3/2
5 878.11 5/3
6 1053.73 11/6, 24/13
7 29.36 55/54, 45/44, 40/39
8 204.98 9/8
9 380.60 5/4
10 556.22 11/8, 18/13
11 731.85 20/13
12 907.47 22/13
13 1083.09 13/7, 15/8
14 58.71 33/32, 27/26, 25/24
15 234.34 8/7, 15/13
16 409.96
17 585.58 45/32
18 761.20
19 936.83 12/7
20 1112.45
21 88.07
22 263.69
23 439.31 9/7
24 614.94 10/7
25 790.56 11/7
26 966.18
27 1141.80 27/14
28 117.43 15/14

* in 13-limit POTE tuning

Tunings

Tuning spectrum

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
11/10 165.004
11/9 173.704
14/13 174.746
12/11 174.894
11/8 175.132
14/11 175.300 11-odd-limit minimax
8/7 175.412
7/6 175.428
9/7 175.438
4/3 175.489
15/14 175.694
7/5 175.729 7, 9, 13 and 15-odd-limit minimax
13/11 175.899
16/15 176.021
5/4 176.257 5-odd-limit minimax
18/13 176.338
15/13 176.516
6/5 176.872
13/10 176.890
13/12 176.905
15/11 178.984
16/13 179.736
10/9 182.404
Retrieved from "https://en.xen.wiki/index.php?title=Monkey&oldid=227859"