Monkey: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>FREEZE
No edit summary
+ more edo tunings in the spectrum
 
(10 intermediate revisions by 2 users not shown)
Line 1: Line 1:
See [[Tetracot_family#Monkey|Tetracot family]].
{{Infobox regtemp
[[Category:soft_redirect]]
| 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]] (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 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 [[2.3.5.11.13 subgroup|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'''.
 
{| class="wikitable center-1 right-2"
|-
! #
! 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
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 175.6758{{c}}
| CWE: ~10/9 = 175.6622{{c}}
| POTE: ~10/9 = 175.6588{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 175.5978{{c}}
| CWE: ~10/9 = 175.5750{{c}}
| POTE: ~10/9 = 175.5703{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 175.6185{{c}}
| CWE: ~10/9 = 175.6217{{c}}
| POTE: ~10/9 = 175.6224{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(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
|
|}
<nowiki/>* Besides the octave
 
[[Category:Monkey| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Keemic temperaments]]
[[Category:Hemifamity temperaments]]

Latest revision as of 10:21, 30 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 (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

Retrieved from "https://en.xen.wiki/index.php?title=Monkey&oldid=229064"