Keemun: Difference between revisions

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== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–13 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | #
! #
! rowspan="2" | Cents*
! Cents*
! rowspan="2" | 11-limit ratios
! Approximate ratios
! colspan="3" | 13-limit ratios
|-
! Keemun <br>(4 & 19)
! Kema <br>(15 & 19)
! Kumbaya <br>(4 & 15)
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| '''1/1'''
|
|
|
|-
|-
| 1
| 1
| 317.555
| 317.4
| 6/5
| 6/5
|
|
| 13/11, '''16/13'''
|-
|-
| 2
| 2
| 635.109
| 634.8
| 10/7, '''16/11'''
| 10/7, 13/9, '''16/11'''
|
| 13/9
|
|-
|-
| 3
| 3
| 952.664
| 952.2
| '''7/4''', 12/7
| '''7/4''', 12/7, 15/13
| 22/13
| 26/15
|
|-
|-
| 4
| 4
| 70.219
| 69.6
| 21/20, 25/24, 33/32, 36/35
| 21/20, 25/24, 27/26, 33/32, 36/35
|
|
| 14/13
|-
|-
| 5
| 5
| 387.773
| 387.0
| '''5/4''', 14/11
| '''5/4''', 14/11
| '''16/13'''
|
|
|-
|-
| 6
| 6
| 705.328
| 704.4
| '''3/2'''
| '''3/2'''
|
|
| 20/13
|-
|-
| 7
| 7
| 1022.882
| 1021.8
| 9/5, 20/11
| 9/5, 20/11
|
|
| 24/13
|-
|-
| 8
| 8
| 140.437
| 139.3
| 12/11, 15/14
| 12/11, 13/12, 15/14
| 14/13
| 13/12
|
|-
|-
| 9
| 9
| 457.992
| 456.7
| 9/7, 21/16
| 9/7, 13/10, 21/16
|
| 13/10
|
|-
|-
| 10
| 10
| 775.546
| 774.1
| 25/16
| 25/16
| 20/13
|
|
|-
|-
| 11
| 11
| 1093.101
| 1091.5
| 15/8
| 13/7, 15/8
| 24/13
| 13/7
|
|-
|-
| 12
| 12
| 210.656
| 208.9
| 9/8
| '''9/8'''
|
|
| 15/13
|-
|-
| 13
| 13
| 528.210
| 526.3
| 15/11
| 15/11, 27/20
|
|
| 18/13
|-
|-
| 14
| 14
| 845.765
| 843.7
| 18/11
| '''13/8''', 18/11
| 21/13
| '''13/8'''
|
|-
|-
| 15
| 15
| 1163.320
| 1161.1
| 27/14
| 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
|
|
|
|}
|}
<nowiki>*</nowiki> In 11-limit CWE tuning, octave reduced
<nowiki/>* In 13-limit CWE tuning, octave reduced


== Tunings ==
== Tunings ==

Revision as of 08:53, 1 November 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. 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 ¢
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
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