Keemun: Difference between revisions

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{{Infobox regtemp
{{Infobox regtemp
| Title = Keemun
| Title = Keemun
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[49/48]], [[126/125]] (7-limit); <br>[[49/48]], [[56/55]], [[100/99]] (11-limit)
| Comma basis = [[49/48]], [[126/125]] (7-limit); <br>[[49/48]], [[56/55]], [[100/99]] (11-limit); <br>[[49/48]], [[56/55]], [[91/90]], [[100/99]] (13-limit)
| Edo join 1 = 15 | Edo join 2 = 19
| Edo join 1 = 15 | Edo join 2 = 19
| Mapping = 1; 6 5 3 -2
| Mapping = 1; 6 5 3 -2 14
| Generators = 6/5 | Generators tuning = 317.6 | Optimization method = CWE
| Generators = 6/5 | Generators tuning = 317.4 | 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)
| Pergen = (P8, P12/6)
| Odd limit 1 = 7 | Mistuning 1 = 17.8 | Complexity 1 = 7
| Odd limit 1 = 9 | Mistuning 1 = 17.8 | Complexity 1 = 15
| Odd limit 2 = 11 | Mistuning 2 = 27.3 | Complexity 2 = 15
| Odd limit 2 = 13 | Mistuning 2 = 27.3 | Complexity 2 = 19
}}
}}
'''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. It identifies the hemitwelfth as [[7/4]][[~]][[12/7]] and the two-generator step as [[13/9]][[~]][[16/11]], [[tempering out]] [[49/48]], [[56/55]], [[91/90]] and [[100/99]] in the [[13-limit]]. This means it uses the same simple mappings for the [[7/1|7th]] and [[11/1|11th harmonics]] as [[orgone]]. Unfortunately, the optimal tuning for the [[3/1|3rd]] and [[5/1|5th harmonics]] is substantially flatter than that for the 7th & 11th, requiring you to compromise one set for the other. [[Edo]]s that [[support]] keemun in their [[patent val]]s are [[4edo]], [[15edo]], [[19edo]], and [[34edo]], with [[49edo|49d]] and [[64edo|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 {{w|Keemun|Chinese black tea}}<ref>Dave Keenan's original write-up: [https://dkeenan.com/Music/ChainOfMinor3rds.htm ''11 note chain-of-minor-thirds scale'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_13712#13775 Yahoo! Tuning Group | ''Rich Holmes temperaments'']</ref>.
This temperament was originally discovered by [[Dave Keenan]] and named by [[Herman Miller]] in 2006 after the {{w|Keemun|Chinese black tea}}.<ref>[https://dkeenan.com/Music/ChainOfMinor3rds.htm ''11 note chain-of-minor-thirds scale''] by Dave Keenan</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_13712#13775 Yahoo! Tuning Group (archive) | ''Rich Holmes temperaments'']</ref>


See [[Kleismic family #Keemun]] for technical data. See [[Keemun extensions]] for a discussion on alternative 11- and 13-limit extensions.  
See [[Kleismic family #Keemun]] for technical data. See [[Keemun extensions]] for a discussion on alternative 11- and 13-limit extensions.  
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|}
|}


{| class="wikitable mw-collapsible mw-collapsed"
=== Tuning spectrum ===
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
This tuning spectrum assumes septimal keemun; for the tuning spectrum of tridecimal keemun, see [[Keemun extensions #Tuning spectra]].  
|-
! 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"
{| class="wikitable center-all left-4"
|-
|-
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|  
|  
| 315.789
| 315.789
| Lower bound of 9- and 11-odd-limit diamond monotone
| Lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
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|  
|  
| 316.981
| 316.981
| 53de val
| 53d val
|-
|-
|  
|  
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|  
|  
| 317.647
| 317.647
|
|-
|
| 11/9
| 318.042
| 11-odd-limit minimax
|-
|
| 15/11
| 318.227
|
|-
|
| 11/10
| 319.285
|  
|  
|-
|-
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| 318.367
| 318.367
| 49d val
| 49d val
|-
|
| 11/6
| 318.830
|
|-
|-
| [[15edo|4\15]]
| [[15edo|4\15]]
|  
|  
| 320.000
| 320.000
| Upper bound of 9- and 11-odd-limit diamond monotone
| Upper bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 7/4
| 7/4
| 322.942
| 322.942
|
|-
|
| 11/7
| 323.502
|
|-
|
| 11/8
| 324.341
|  
|  
|-
|-
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[[Category:Semaphoresmic clan]]
[[Category:Semaphoresmic clan]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
[[Category:Keemic temperaments]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 14:37, 2 November 2026

Keemun
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis 49/48, 126/125 (7-limit);
49/48, 56/55, 100/99 (11-limit);
49/48, 56/55, 91/90, 100/99 (13-limit)
Reduced mapping ⟨1; 6 5 3 -2 14]
ET join 15 & 19
Generators (CWE) ~6/5 = 317.4 ¢
MOS scales 4L 3s, 4L 7s, 4L 11s, 15L 4s
Ploidacot alpha-hexacot
Pergen (P8, P12/6)
Minimax error 9-odd-limit: 17.8 ¢;
13-odd-limit: 27.3 ¢
Target scale size 9-odd-limit: 15 notes;
13-odd-limit: 19 notes

Keemun is an extension of the hanson temperament. It identifies the hemitwelfth as 7/4~12/7 and the two-generator step as 13/9~16/11, tempering out 49/48, 56/55, 91/90 and 100/99 in the 13-limit. This means it uses the same simple mappings for the 7th and 11th harmonics as orgone. Unfortunately, the optimal tuning for the 3rd and 5th harmonics is substantially flatter than that for the 7th & 11th, requiring you to compromise one set for the other. 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. See Keemun extensions for a discussion on alternative 11- and 13-limit extensions.

Interval chain

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

# Cents* Approximate ratios
0 0.0 1/1
1 317.4 6/5
2 634.8 10/7, 13/9, 16/11
3 952.2 7/4, 12/7, 15/13
4 69.6 21/20, 25/24, 27/26, 33/32, 36/35
5 387.0 5/4, 14/11
6 704.4 3/2
7 1021.8 9/5, 20/11
8 139.3 12/11, 13/12, 15/14
9 456.7 9/7, 13/10, 21/16
10 774.1 25/16
11 1091.5 13/7, 15/8
12 208.9 9/8
13 526.3 15/11, 27/20
14 843.7 13/8, 18/11
15 1161.1 27/14

* In 13-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 ¢

Tuning spectrum

This tuning spectrum assumes septimal keemun; for the tuning spectrum of tridecimal keemun, see Keemun extensions #Tuning spectra.

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-odd-limit diamond monotone
9/5 316.799 1/7-kleisma
14\53 316.981 53d 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
13\49 318.367 49d val
4\15 320.000 Upper bound of 9-odd-limit diamond monotone
7/4 322.942
3\11 327.273 11b val, upper bound of 5- and 7-odd-limit diamond monotone

* Besides the octave

Music

Chris Vaisvil

References

  1. ↑ 11 note chain-of-minor-thirds scale by Dave Keenan
  2. ↑ Yahoo! Tuning Group (archive) | Rich Holmes temperaments
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