Keemun extensions: Difference between revisions

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Created page with "{{Breadcrumb|Keemun}} Keemun has various competing extensions to the 11- and 13-limit. The following temperaments are discussed in this article: * '''Tridecimal keemun''' (15 & 19) – tempering out 49/48, 56/55, 91/90, 100/99; * '''Kimah''' (15f & 19) – tempering out 49/48, 56/55, 65/64, 100/99; * '''Kumbaya''' (15 & 19f) – tempering out 40/39, 49/48, 56/55, 66/65; * '''Qeema''' (15e & 19) – tempering out 45/44, 49/48, 78/77, 126/125;..."
 
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* '''Tridecimal keemun''' (15 & 19) – tempering out 49/48, 56/55, 91/90, 100/99;  
* '''Tridecimal keemun''' (15 & 19) – tempering out 49/48, 56/55, 91/90, 100/99;  
* '''Kimah''' (15f & 19) – tempering out 49/48, 56/55, 65/64, 100/99;  
* '''Kimah''' (15f & 19) – tempering out 49/48, 56/55, 65/64, 100/99;  
* '''Qeema''' (15e & 19) – tempering out 45/44, 49/48, 78/77, 126/125;
* '''Kumbaya''' (15 & 19f) – tempering out 40/39, 49/48, 56/55, 66/65;  
* '''Kumbaya''' (15 & 19f) – tempering out 40/39, 49/48, 56/55, 66/65;  
* '''Qeema''' (15e & 19) – tempering out 45/44, 49/48, 78/77, 126/125;
* '''Darjeeling''' (15 & 19e) – tempering out 49/48, 55/54, 66/65, 77/75.  
* '''Darjeeling''' (15 & 19e) – tempering out 49/48, 55/54, 66/65, 77/75.  


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|-
|-
! rowspan="3" | #
! rowspan="3" | #
! colspan="5" | Approximate ratios
! colspan="7" | Approximate ratios
|-
|-
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! rowspan="2" | 7-limit
! rowspan="2" | 7-limit
! colspan="3" | 13-limit extensions
! colspan="5" | 13-limit extensions
|-
|-
! Keemun <br>(15 & 19)
! Keemun
! Kimah <br>(4 & 19)
! Kimah
! Kumbaya <br>(4 & 15)
! Qeema
! Kumbaya
! Darjeeling
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| '''1/1'''
|
|
|  
|  
|  
|  
Line 34: Line 38:
| 316.8
| 316.8
| 6/5
| 6/5
|
|
|  
|  
| 13/11, '''16/13'''
| 13/11, '''16/13'''
|  
| 11/9, 13/11
|-
|-
| 2
| 2
Line 43: Line 49:
| 13/9, '''16/11'''
| 13/9, '''16/11'''
| '''16/11'''
| '''16/11'''
| 13/9
| '''16/11'''
| '''16/11'''
| 13/9, 22/15
|-
|-
| 3
| 3
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| 26/15
| 26/15
| 22/13
| 22/13
| 22/13, 26/15
|  
|  
| 26/15
|-
|-
| 4
| 4
| 67.3
| 67.3
| 21/20, 25/24, 36/35
| 21/20, 25/24, 36/35
| 33/32,
| 27/26, 33/32
| 33/32,
| 33/32
| 27/26
| 14/13, 33/32
| 14/13, 33/32
| 27/26, 88/45
|-
|-
| 5
| 5
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| 14/11
| 14/11
| 14/11, '''16/13'''
| 14/11, '''16/13'''
| 11/9
| 14/11
| 14/11
|
|-
|-
| 6
| 6
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|  
|  
|  
|  
| 22/15
| 20/13
| 20/13
|
|-
|-
| 7
| 7
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| 20/11
| 20/11
| 20/11
| 20/11
|
| 20/11, 24/13
| 20/11, 24/13
| 11/6
|-
|-
| 8
| 8
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| 12/11, 13/12
| 12/11, 13/12
| 12/11, 14/13
| 12/11, 14/13
| 13/12
| 12/11
| 12/11
| 11/10, 13/12
|-
|-
| 9
| 9
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| 13/10
| 13/10
|  
|  
| 13/10
|  
|  
| 13/10
|-
|-
| 10
| 10
| 768.3
| 768.3
| 25/16
| 25/16
|
| 20/13
| 20/13
|  
|  
|  
|  
| 11/7
|-
|-
| 11
| 11
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| 13/7
| 13/7
| 24/13
| 24/13
| 11/6, 13/7
|  
|  
| 13/7
|-
|-
| 12
| 12
| 202.0
| 202.0
| 9/8
| '''9/8'''
|  
|  
|  
|  
| 11/10
| 15/13
| 15/13
|
|-
|-
| 13
| 13
| 518.8
| 518.8
|  
| 27/20
| 15/11
| 15/11
| 15/11
| 15/11
|
| 15/11, 18/13
| 15/11, 18/13
| '''11/8'''
|-
|-
| 14
| 14
| 835.6
| 835.6
|  
| 45/28
| '''13/8''', 18/11
| '''13/8''', 18/11
| 18/11, 21/13
| 18/11, 21/13
| 11/7, '''13/8'''
| 18/11
| 18/11
| '''13/8'''
|-
|-
| 15
| 15
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|  
|  
| 25/13
| 25/13
|
|
|  
|  
|-
|-
| 16
| 16
| 269.3
| 269.3
|  
| 75/64, 81/70
| 13/11
| 13/11
| 15/13
| 15/13
|
|
|  
|  
|-
|-
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|  
|  
| 18/13
| 18/13
| '''11/8'''
|
|  
|  
|-
|-
Line 153: Line 191:
| 902.9
| 902.9
| 27/16
| 27/16
|
|
|  
|  
|  
|  
Line 160: Line 200:
| 19.8
| 19.8
| 81/80
| 81/80
| 45/44
| 45/44
|
| 45/44
| 33/32
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings (keemun)
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~6/5 = 317.5063{{c}}
| CWE: ~6/5 = 317.5546{{c}}
| POTE: ~6/5 = 317.5756{{c}}
|}
=== Tuning spectra ===
==== Tridecimal keemun ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo <br>(unchanged interval)]]*
! Generator (¢)
! Comments
|-
| [[4edo|1\4]]
|
| 300.000
| 4f val, lower bound of 5- and 7-odd-limit diamond monotone
|-
|
| 7/5
| 308.744
|
|-
|
| 7/6
| 311.043
|
|-
| [[23edo|6\23]]
|
| 313.043
| 23dff val
|-
|
| 15/14
| 314.930
|
|-
|
| 9/7
| 315.009
|
|-
|
| 13/7
| 315.609
|
|-
|
| 5/3
| 315.641
|
|-
| [[19edo|5\19]]
|
| 315.789
| Lower bound of 9- and 11-odd-limit diamond monotone
|-
|
| 9/5
| 316.799
| 1/7-kleisma
|-
| [[53edo|14\53]]
|
| 316.981
| 53de val
|-
|
| 3/2
| 316.993
| 5-, 7- and 9-odd-limit minimax, 1/6-kleisma
|-
|
| 15/8
| 317.115
| 2/11-kleisma
|-
|
| 13/10
| 317.134
|
|-
|
| 13/8
| 317.181
|
|-
|
| 5/4
| 317.263
| 1/5-kleisma
|-
|
| 13/12
| 317.322
|
|-
|
| 15/13
| 317.420
|
|-
| [[34edo|9\34]]
|
| 317.647
|
|-
|
| 11/9
| 318.042
| 11-, 13- and 15-odd-limit minimax
|-
|
| 13/10
| 318.076
|
|-
|
| 15/11
| 318.227
|
|-
|
| 11/10
| 319.285
|
|-
|
| 13/9
| 318.309
|  
|  
|-
| [[49edo|13\49]]
|  
|  
| 318.367
| 49df val
|-
|  
|  
| 11/6
| 318.830
|
|-
| [[15edo|4\15]]
|
| 320.000
| Upper bound of 9- and 11-odd-limit diamond monotone
|-
|
| 7/4
| 322.942
|
|-
|
| 11/7
| 323.502
|
|-
|
| 11/8
| 324.341
|
|-
| [[11edo|3\11]]
|
| 327.273
| 11bff val, upper bound of 5- and 7-odd-limit diamond monotone
|}
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced
<nowiki/>* Besides the octave


[[Category:Keemun]]
[[Category:Keemun]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]

Latest revision as of 13:04, 1 November 2026

< Keemun

Keemun has various competing extensions to the 11- and 13-limit. The following temperaments are discussed in this article:

  • Tridecimal keemun (15 & 19) – tempering out 49/48, 56/55, 91/90, 100/99;
  • Kimah (15f & 19) – tempering out 49/48, 56/55, 65/64, 100/99;
  • Qeema (15e & 19) – tempering out 45/44, 49/48, 78/77, 126/125;
  • Kumbaya (15 & 19f) – tempering out 40/39, 49/48, 56/55, 66/65;
  • Darjeeling (15 & 19e) – tempering out 49/48, 55/54, 66/65, 77/75.

See Kleismic family #Keemun, #Kimah, #Kumbaya, #Qeema, and #Darjeeling for technical data.

Interval chain

# Approximate ratios
Cents* 7-limit 13-limit extensions
Keemun Kimah Qeema Kumbaya Darjeeling
0 0.0 1/1
1 316.8 6/5 13/11, 16/13 11/9, 13/11
2 633.7 10/7 13/9, 16/11 16/11 13/9 16/11 13/9, 22/15
3 950.5 7/4, 12/7 26/15 22/13 22/13, 26/15 26/15
4 67.3 21/20, 25/24, 36/35 27/26, 33/32 33/32 27/26 14/13, 33/32 27/26, 88/45
5 384.1 5/4 14/11 14/11, 16/13 11/9 14/11
6 701.0 3/2 22/15 20/13
7 1017.8 9/5 20/11 20/11 20/11, 24/13 11/6
8 134.6 15/14 12/11, 13/12 12/11, 14/13 13/12 12/11 11/10, 13/12
9 451.5 9/7, 21/16 13/10 13/10 13/10
10 768.3 25/16 20/13 11/7
11 1085.1 15/8 13/7 24/13 11/6, 13/7 13/7
12 202.0 9/8 11/10 15/13
13 518.8 27/20 15/11 15/11 15/11, 18/13 11/8
14 835.6 45/28 13/8, 18/11 18/11, 21/13 11/7, 13/8 18/11 13/8
15 1152.4 27/14 25/13
16 269.3 75/64, 81/70 13/11 15/13
17 586.1 45/32 18/13 11/8
18 902.9 27/16
19 19.8 81/80 45/44 45/44 45/44 33/32

* In 7-limit CWE tuning, octave reduced

Tunings

11-limit norm-based tunings (keemun)
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~6/5 = 317.5063 ¢ CWE: ~6/5 = 317.5546 ¢ POTE: ~6/5 = 317.5756 ¢

Tuning spectra

Tridecimal keemun

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
1\4 300.000 4f val, lower bound of 5- and 7-odd-limit diamond monotone
7/5 308.744
7/6 311.043
6\23 313.043 23dff val
15/14 314.930
9/7 315.009
13/7 315.609
5/3 315.641
5\19 315.789 Lower bound of 9- and 11-odd-limit diamond monotone
9/5 316.799 1/7-kleisma
14\53 316.981 53de val
3/2 316.993 5-, 7- and 9-odd-limit minimax, 1/6-kleisma
15/8 317.115 2/11-kleisma
13/10 317.134
13/8 317.181
5/4 317.263 1/5-kleisma
13/12 317.322
15/13 317.420
9\34 317.647
11/9 318.042 11-, 13- and 15-odd-limit minimax
13/10 318.076
15/11 318.227
11/10 319.285
13/9 318.309
13\49 318.367 49df val
11/6 318.830
4\15 320.000 Upper bound of 9- and 11-odd-limit diamond monotone
7/4 322.942
11/7 323.502
11/8 324.341
3\11 327.273 11bff val, upper bound of 5- and 7-odd-limit diamond monotone

* Besides the octave

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