Keemun: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
+ infobox
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,015 edits
No edit summary
Tags: Mobile edit Mobile web edit
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Infobox regtemp
{{Infobox regtemp
| Title = Keemun
| Title = Keemun
| Subgroups = 2.3.5.7
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[49/48]], [[126/125]]
| Comma basis = [[49/48]], [[126/125]] (7-limit) <br>[[49/48]], [[56/55]], [[100/99]] (11-limit)
| Edo join 1 = 15 | Edo join 2 = 19
| Edo join 1 = 15 | Edo join 2 = 19
| Mapping = 1; 6 5 3
| Mapping = 1; 6 5 3 -2
| Generators = 6/5 | Generators tuning = 316.8 | Optimization method = CWE
| Generators = 6/5 | Generators tuning = 317.6 | Optimization method = CWE
| MOS scales = [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
| MOS scales = [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
| Pergen = (P8, P12/6)
| Odd limit 1 = 7 | Mistuning 1 = 17.8 | Complexity 1 = 7
| Odd limit 1 = 7 | Mistuning 1 = 17.8 | Complexity 1 = 7
| Odd limit 2 = 11 | Mistuning 2 = 27.3 | Complexity 2 = 15
}}
}}
'''Keemun''' is an [[extension]] of the [[hanson]] temperament, and [[tempering out|tempers out]] [[49/48]], [[56/55]], and [[100/99]] in the [[11-limit]]. This means it uses the same simple mappings for the 7th and 11th harmonics as [[orgone]]. Unfortunately, the optimal tunings for the 3rd and 5th harmonics is substantially flatter than that for the 7th & 11th ones, requiring you to compromise one set for the other. The edos that support keemun in their patent vals are [[4edo]], [[15edo]], [[19edo]], and [[34edo]], with [[49edo|49d]] and [[64edo|64bde]] coming closer to balancing the errors equally.
'''Keemun''' is an [[extension]] of the [[hanson]] temperament, and [[tempering out|tempers out]] [[49/48]], [[56/55]], and [[100/99]] in the [[11-limit]]. This means it uses the same simple mappings for the 7th and 11th harmonics as [[orgone]]. Unfortunately, the optimal tunings for the 3rd and 5th harmonics is substantially flatter than that for the 7th & 11th ones, requiring you to compromise one set for the other. The edos that support keemun in their patent vals are [[4edo]], [[15edo]], [[19edo]], and [[34edo]], with [[49edo|49d]] and [[64edo|64bde]] coming closer to balancing the errors equally.
Line 23: Line 25:
! colspan="3" | 13-limit ratios
! colspan="3" | 13-limit ratios
|-
|-
! Keemun<br>(4 & 19)
! Keemun <br>(4 & 19)
! Kema<br>(15 & 19)
! Kema <br>(15 & 19)
! Kumbaya<br>(4 & 15)
! Kumbaya <br>(4 & 15)
|-
|-
| 0
| 0
Line 35: Line 37:
|-
|-
| 1
| 1
| 317.576
| 317.555
| 6/5
| 6/5
|  
|  
Line 42: Line 44:
|-
|-
| 2
| 2
| 635.151
| 635.109
| 10/7, '''16/11'''
| 10/7, '''16/11'''
|  
|  
Line 49: Line 51:
|-
|-
| 3
| 3
| 952.727
| 952.664
| '''7/4''', 12/7
| '''7/4''', 12/7
| 22/13
| 22/13
Line 56: Line 58:
|-
|-
| 4
| 4
| 70.302
| 70.219
| 21/20, 25/24, 33/32, 36/35
| 21/20, 25/24, 33/32, 36/35
|  
|  
Line 63: Line 65:
|-
|-
| 5
| 5
| 387.878
| 387.773
| '''5/4''', 14/11
| '''5/4''', 14/11
| '''16/13'''
| '''16/13'''
Line 70: Line 72:
|-
|-
| 6
| 6
| 705.453
| 705.328
| '''3/2'''
| '''3/2'''
|  
|  
Line 77: Line 79:
|-
|-
| 7
| 7
| 1023.029
| 1022.882
| 9/5, 20/11
| 9/5, 20/11
|  
|  
Line 84: Line 86:
|-
|-
| 8
| 8
| 140.605
| 140.437
| 12/11, 15/14
| 12/11, 15/14
| 14/13
| 14/13
Line 91: Line 93:
|-
|-
| 9
| 9
| 458.180
| 457.992
| 9/7, 21/16
| 9/7, 21/16
|  
|  
Line 98: Line 100:
|-
|-
| 10
| 10
| 775.756
| 775.546
| 25/16
| 25/16
| 20/13
| 20/13
Line 105: Line 107:
|-
|-
| 11
| 11
| 1093.331
| 1093.101
| 15/8
| 15/8
| 24/13
| 24/13
Line 112: Line 114:
|-
|-
| 12
| 12
| 210.907
| 210.656
| 9/8
| 9/8
|  
|  
Line 119: Line 121:
|-
|-
| 13
| 13
| 528.482
| 528.210
| 15/11
| 15/11
|  
|  
Line 126: Line 128:
|-
|-
| 14
| 14
| 846.058
| 845.765
| 18/11
| 18/11
| 21/13
| 21/13
Line 133: Line 135:
|-
|-
| 15
| 15
| 1163.634
| 1163.320
| 27/14
| 27/14
| 25/13
| 25/13
Line 140: Line 142:
|-
|-
| 16
| 16
| 281.209
| 280.874
|  
|  
| 15/13
| 15/13
Line 147: Line 149:
|-
|-
| 17
| 17
| 598.785
| 598.429
| 45/32
| 45/32
| 18/13
| 18/13
Line 154: Line 156:
|-
|-
| 18
| 18
| 916.360
| 915.984
| 27/16
| 27/16
|  
|  
Line 161: Line 163:
|-
|-
| 19
| 19
| 33.936
| 33.538
| 81/80
| 81/80
|  
|  
Line 167: Line 169:
|  
|  
|}
|}
<nowiki>*</nowiki> In 11-limit POTE tuning, octave reduced
<nowiki>*</nowiki> In 11-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: ~6/5 = 317.3927{{c}}
| CWE: ~6/5 = 316.8293{{c}}
| POTE: ~6/5 = 316.4727{{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: ~6/5 = 317.5063{{c}}
| CWE: ~6/5 = 317.5546{{c}}
| POTE: ~6/5 = 317.5756{{c}}
|}
 
=== Tuning spectrum ===
This tuning spectrum assumes undecimal keemun.
{| class="wikitable center-all left-4"
|-
! Edo <br>generator
! [[Eigenmonzo|Eigenmonzo <br>(unchanged interval)]]*
! Generator (¢)
! Comments
|-
| [[4edo|1\4]]
|
| 300.000
| Lower bound of 5- and 7-odd-limit diamond monotone
|-
|
| 7/5
| 308.744
|
|-
|
| 7/6
| 311.043
|
|-
| [[23edo|6\23]]
|
| 313.043
| 23d val
|-
|
| 15/14
| 314.930
|
|-
|
| 9/7
| 315.009
|
|-
|
| 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
|-
|
| 5/4
| 317.263
| 1/5-kleisma
|-
| [[34edo|9\34]]
|
| 317.647
|
|-
|
| 11/9
| 318.042
| 11-odd-limit minimax
|-
|
| 15/11
| 318.227
|
|-
|
| 11/10
| 319.285
|
|-
| [[49edo|13\49]]
|
| 318.367
| 49d 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
| 11b val, upper bound of 5- and 7-odd-limit diamond monotone
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==
Line 179: Line 345:
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Kleismic family]]
[[Category:Kleismic family]]
[[Category:Semaphoresmic clan]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 01:41, 18 April 2026

Keemun
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 49/48, 126/125 (7-limit)
49/48, 56/55, 100/99 (11-limit)
Reduced mapping ⟨1; 6 5 3 -2]
ET join 15 & 19
Generators (CWE) ~6/5 = 317.6 ¢
MOS scales 4L 3s, 4L 7s, 4L 11s, 15L 4s
Ploidacot alpha-hexacot
Pergen (P8, P12/6)
Minimax error 7-odd-limit: 17.8 ¢;
11-odd-limit: 27.3 ¢
Target scale size 7-odd-limit: 7 notes;
11-odd-limit: 15 notes

Keemun is an extension of the hanson temperament, and tempers out 49/48, 56/55, and 100/99 in the 11-limit. This means it uses the same simple mappings for the 7th and 11th harmonics as orgone. Unfortunately, the optimal tunings for the 3rd and 5th harmonics is substantially flatter than that for the 7th & 11th ones, requiring you to compromise one set for the other. The edos that support keemun in their patent vals are 4edo, 15edo, 19edo, and 34edo, with 49d and 64bde coming closer to balancing the errors equally.

This temperament was originally discovered by Dave Keenan and named by Herman Miller in 2006 after the Chinese black tea[1][2].

See Kleismic family #Keemun for technical data.

Interval chain

# Cents* 11-limit ratios 13-limit ratios
Keemun
(4 & 19) 		Kema
(15 & 19) 		Kumbaya
(4 & 15)
0 0.000 1/1
1 317.555 6/5 13/11, 16/13
2 635.109 10/7, 16/11 13/9
3 952.664 7/4, 12/7 22/13 26/15
4 70.219 21/20, 25/24, 33/32, 36/35 14/13
5 387.773 5/4, 14/11 16/13
6 705.328 3/2 20/13
7 1022.882 9/5, 20/11 24/13
8 140.437 12/11, 15/14 14/13 13/12
9 457.992 9/7, 21/16 13/10
10 775.546 25/16 20/13
11 1093.101 15/8 24/13 13/7
12 210.656 9/8 15/13
13 528.210 15/11 18/13
14 845.765 18/11 21/13 13/8
15 1163.320 27/14 25/13
16 280.874 15/13 13/11
17 598.429 45/32 18/13
18 915.984 27/16
19 33.538 81/80

* In 11-limit CWE tuning, octave reduced

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~6/5 = 317.3927 ¢ CWE: ~6/5 = 316.8293 ¢ POTE: ~6/5 = 316.4727 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~6/5 = 317.5063 ¢ CWE: ~6/5 = 317.5546 ¢ POTE: ~6/5 = 317.5756 ¢

Tuning spectrum

This tuning spectrum assumes undecimal keemun.

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
1\4 300.000 Lower bound of 5- and 7-odd-limit diamond monotone
7/5 308.744
7/6 311.043
6\23 313.043 23d val
15/14 314.930
9/7 315.009
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
5/4 317.263 1/5-kleisma
9\34 317.647
11/9 318.042 11-odd-limit minimax
15/11 318.227
11/10 319.285
13\49 318.367 49d 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 11b val, upper bound of 5- and 7-odd-limit diamond monotone

* Besides the octave

Music

Chris Vaisvil

References

  1. ↑ Dave Keenan's original write-up: 11 note chain-of-minor-thirds scale
  2. ↑ Yahoo! Tuning Group | Rich Holmes temperaments
Retrieved from "https://en.xen.wiki/index.php?title=Keemun&oldid=227969"