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Line 2: Line 2:
| Title = Xenial
| Title = Xenial
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); <br>[[126/125]], [[162/161]], [[169/168]], [[171/170]], [[221/220]], [[231/230]], [[256/255]] (23-limit)
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); <br>[[126/125]], [[162/161]], [[169/168]], [[171/170]], [[208/207]], [[221/220]], [[231/230]] (23-limit)
| Edo join 1 = 19 | Edo join 2 = 70
| Edo join 1 = 19 | Edo join 2 = 70
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]]
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], <br>[[19L 13s]], [[19L 32s]], [[19L 51s]]
| Pergen = (P8, P11/3)
| Pergen = (P8, P11/9)
| Odd limit 1 = 7 | Mistuning 1 = 4.6 | Complexity 1 = 51
| Odd limit 1 = 7 | Mistuning 1 = 4.60 | Complexity 1 = 51
| Odd limit 2 = 9 | Mistuning 2 = 6.27 | Complexity 2 = 51
| Odd limit 3 = 17 | Mistuning 3 = 8.90 | Complexity 3 = 70
| Odd limit 4 = 23 | Mistuning 4 = 8.96 | Complexity 4 = 70
}}
}}
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.


See [[Starling temperaments #Xenial]] for more technical data.
See [[Starling temperaments #Xenial]] for more technical data.
== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.000
| [[1/1]]
|-
| 1
| 188.775
| [[10/9]], [[19/17]], [[28/25]]
|-
| 2
| 377.551
| [[56/45]]
|-
| 3
| 566.326
| [[18/13]], [[32/23]]
|-
| 4
| 755.102
| [[17/11]], [[20/13]]
|-
| 5
| 943.877
| [[19/11]], [[26/15]]
|-
| 6
| 1132.653
| [[23/12]], [[27/14]]
|-
| 7
| 121.428
| [[15/14]]
|-
| 8
| 310.204
| [[6/5]]
|-
| 9
| 498.979
| [[4/3]]
|-
| 10
| 687.755
| [[40/27]]
|-
| 11
| 876.530
|
|-
| 12
| 1065.306
| [[13/7]], [[24/13]]
|-
| 13
| 54.081
| [[26/25]], [[33/32]]
|-
| 14
| 242.857
| [[23/20]]
|-
| 15
| 431.632
| [[9/7]], [[23/18]]
|-
| 16
| 620.408
| [[10/7]]
|-
| 17
| 809.183
| [[8/5]]
|-
| 18
| 997.959
| [[16/9]], [[23/13]]
|-
| 19
| 1186.734
|
|-
| 20
| 175.510
|
|-
| 21
| 364.285
| [[16/13]], [[26/21]]
|-
| 22
| 553.061
| [[11/8]]
|-
| 23
| 741.836
| [[23/15]]
|-
| 24
| 930.612
| [[12/7]]
|-
| 25
| 1119.387
| [[40/21]], [[44/23]], [[48/25]]
|-
| 26
| 108.163
| [[16/15]], [[17/16]]
|-
| 27
| 296.938
| [[19/16]]
|-
| 28
| 485.714
|
|-
| 29
| 674.439
| [[34/23]]
|-
| 30
| 863.265
| [[38/23]], [[23/14]]
|-
| 31
| 1052.040
| [[11/6]], [[46/25]]
|-
| 32
| 40.815
| [[36/35]], [[46/45]], [[50/49]]
|-
| 33
| 229.591
| [[8/7]]
|-
| 34
| 418.366
| [[32/25]]
|}
<nowiki/>* In 23-limit CWE tuning


== Tunings ==
== Tunings ==
=== Norm-based 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 = 188.8535{{c}}
| CWE: ~10/9 = 188.8544{{c}}
| POTE: ~10/9 = 188.8548{{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 = 188.8295{{c}}
| CWE: ~10/9 = 188.8085{{c}}
| POTE: ~10/9 = 188.8028{{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 = 188.7987{{c}}
| CWE: ~10/9 = 188.7898{{c}}
| POTE: ~10/9 = 188.7875{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7811{{c}}
| CWE: ~10/9 = 188.7677{{c}}
| POTE: ~10/9 = 188.7655{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 19-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7828{{c}}
| CWE: ~10/9 = 188.7770{{c}}
| POTE: ~10/9 = 188.7762{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7849{{c}}
| CWE: ~10/9 = 188.7755{{c}}
| POTE: ~10/9 = 188.7744{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
Line 25: Line 269:
|  
|  
| 9/5
| 9/5
| 182.404
| 182.4037
|  
|  
|-
|-
|  
|  
| 13/10
| 13/10
| 186.447
| 186.4465
|  
|  
|-
|-
| 5 ⧵ 32
| 5 ⧵ 32
|  
|  
| 187.500
| 187.5000
| 32cddefgh val <br>Lower bound of 7-odd-limit diamond monotone
| 32cddefgh val <br>Lower bound of 7-odd-limit diamond monotone
|-
|-
|  
|  
| 23/12
| 23/12
| 187.720
| 187.7199
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 187.794
| 187.7941
|  
|  
|-
|-
|  
|  
| 23/13
| 23/13
| 188.208
| 188.2081
|  
|  
|-
|-
| 8 ⧵ 51
| 8 ⧵ 51
|  
|  
| 188.235
| 188.2353
| 51cdh val <br>Lower bound of 9-odd-limit diamond monotone
| 51cdh val <br>Lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 23/18
| 23/18
| 188.291
| 188.2910
|
|-
|
| 17/11
| 188.4094
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 188.452
| 188.4523
|  
|  
|-
|-
|  
|  
| 15/14
| 15/14
| 188.492
| 188.4918
|  
|  
|-
|-
|  
|  
| 13/8
| 13/8
| 188.546
| 188.5463
|  
|  
|-
|-
| 11 ⧵ 70
| 11 ⧵ 70
|  
|  
| 188.571
| 188.5714
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
|-
|-
|  
|  
| 7/5
| 7/5
| 188.593
| 188.5930
|
|-
|
| 17/13
| 188.6048
|  
|  
|-
|-
|  
|  
| 21/20
| 21/20
| 188.621
| 188.6213
|
|-
|  
|  
| 13/11
| 188.6230
| 13-odd-limit minimax
|-
|-
|  
|  
| 23/14
| 23/14
| 188.648
| 188.6483
|  
|  
|-
|-
|  
|  
| 17/16
| 17/16
| 188.652
| 188.6521
|  
|  
|-
|-
|  
|  
| 23/21
| 23/21
| 188.654
| 188.6537
|
|-
|
| 17/12
| 188.6572
|
|-
|
| 17/9
| 188.6601
|  
|  
|-
|-
|  
|  
| 3/2
| 3/2
| 188.672
| 188.6717
| 9, 15 and 17-odd-limit minimax
|-
|
| 11/9
| 188.6852
| 11-odd-limit minimax
|-
|
| 19/13
| 188.6872
| 19, 21 and 23-odd-limit minimax
|-
|
| 11/6
| 188.6891
|  
|  
|-
|-
Line 125: Line 409:
|  
|  
| 23/20
| 23/20
| 188.711
| 188.7115
|
|-
|
| 21/17
| 188.7379
|
|-
|
| 19/18
| 188.7467
|
|-
|
| 17/14
| 188.7480
|
|-
|
| 21/11
| 188.7584
|  
|  
|-
|-
| 14 ⧵ 89
| 14 ⧵ 89
|  
|  
| 188.764
| 188.7640
| 19, 21 and 23-odd-limit diamond monotone (singleton)
| 19, 21 and 23-odd-limit diamond monotone (singleton)
|-
|
| 19/12
| 188.7655
|
|-
|
| 11/7
| 188.7726
|
|-
|
| 17/15
| 188.7824
|
|-
|-
|  
|  
| 21/16
| 21/16
| 188.791
| 188.7909
|
|-
|
| 21/19
| 188.7932
|  
|  
|-
|-
|  
|  
| 19/16
| 19/16
| 188.797
| 188.7968
|
|-
|
| 17/10
| 188.8056
|
|-
|
| 19/14
| 188.8115
|
|-
|
| 15/11
| 188.8135
|  
|  
|-
|-
|  
|  
| 7/4
| 7/4
| 188.823
| 188.8235
|
|-
|
| 11/10
| 188.8463
|
|-
|
| 23/17
| 188.8511
|
|-
|
| 19/15
| 188.8537
|  
|  
|-
|-
|  
|  
| 7/6
| 7/6
| 188.880
| 188.8804
| 7-odd-limit minimax
|-
|
| 19/10
| 188.8909
|  
|  
|-
|-
|  
|  
| 15/8
| 15/8
| 188.913
| 188.9127
|
|-
|
| 23/22
| 188.9217
|
|-
|
| 23/19
| 188.9746
|  
|  
|-
|-
|  
|  
| 9/7
| 9/7
| 189.006
| 189.0056
|  
|  
|-
|-
|  
|  
| 21/13
| 21/13
| 189.036
| 189.0356
|  
|  
|-
|-
|  
|  
| 5/4
| 5/4
| 189.040
| 189.0404
| 5-odd-limit minimax
|-
|
| 19/11
| 189.2390
|  
|  
|-
|-
|  
|  
| 14/13
| 13/7
| 189.308
| 189.3085
|  
|  
|-
|-
|  
|  
| 5/3
| 5/3
| 189.455
| 189.4552
|  
|  
|-
|-
| 3 ⧵ 19
| 3 ⧵ 19
|  
|  
| 189.473
| 189.4737
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
|-
|-
|  
|  
| 15/13
| 15/13
| 190.452
| 190.4518
|  
|  
|-
|-
|  
|  
| 23/16
| 23/16
| 190.575
| 190.5752
|
|-
|
| 19/17
| 192.5576
|  
|  
|}
|}

Latest revision as of 06:00, 9 May 2026

Xenial
Subgroups 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
Comma basis 126/125, 177147/175616 (7-limit);
126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230 (23-limit)
Reduced mapping ⟨1; -9 -17 -33 22 -21 26 27 -3]
ET join 19 & 70
Generators (CWE) ~10/9 = 188.8 ¢
MOS scales 6L 1s, 6L 7s, 13L 6s,
19L 13s, 19L 32s, 19L 51s
Ploidacot zeta-enneacot
Pergen (P8, P11/9)
Minimax error 7-odd-limit: 4.60 ¢;
9-odd-limit: 6.27 ¢;
17-odd-limit: 8.90 ¢;
23-odd-limit: 8.96 ¢
Target scale size 7-odd-limit: 51 notes;
9-odd-limit: 51 notes;
17-odd-limit: 70 notes;
23-odd-limit: 70 notes

Xenial is a rank-2 temperament that is generated by a sharpened minor whole tone of ~10/9, so that nine generators reach 4/3, 17 reach 8/5, 21 reach 16/13 and 33 reach 8/7 with octave reduction. It is also generated by dividing 11th harmonic into 22 equal parts, 17th harmonic into 26 equal parts, or 19th harmonic into 27 equal parts.

See Starling temperaments #Xenial for more technical data.

Interval chain

# Cents* Approximate ratios
0 0.000 1/1
1 188.775 10/9, 19/17, 28/25
2 377.551 56/45
3 566.326 18/13, 32/23
4 755.102 17/11, 20/13
5 943.877 19/11, 26/15
6 1132.653 23/12, 27/14
7 121.428 15/14
8 310.204 6/5
9 498.979 4/3
10 687.755 40/27
11 876.530
12 1065.306 13/7, 24/13
13 54.081 26/25, 33/32
14 242.857 23/20
15 431.632 9/7, 23/18
16 620.408 10/7
17 809.183 8/5
18 997.959 16/9, 23/13
19 1186.734
20 175.510
21 364.285 16/13, 26/21
22 553.061 11/8
23 741.836 23/15
24 930.612 12/7
25 1119.387 40/21, 44/23, 48/25
26 108.163 16/15, 17/16
27 296.938 19/16
28 485.714
29 674.439 34/23
30 863.265 38/23, 23/14
31 1052.040 11/6, 46/25
32 40.815 36/35, 46/45, 50/49
33 229.591 8/7
34 418.366 32/25

* In 23-limit CWE tuning

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 188.8535 ¢ CWE: ~10/9 = 188.8544 ¢ POTE: ~10/9 = 188.8548 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 188.8295 ¢ CWE: ~10/9 = 188.8085 ¢ POTE: ~10/9 = 188.8028 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 188.7987 ¢ CWE: ~10/9 = 188.7898 ¢ POTE: ~10/9 = 188.7875 ¢
17-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 188.7811 ¢ CWE: ~10/9 = 188.7677 ¢ POTE: ~10/9 = 188.7655 ¢
19-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 188.7828 ¢ CWE: ~10/9 = 188.7770 ¢ POTE: ~10/9 = 188.7762 ¢
23-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 188.7849 ¢ CWE: ~10/9 = 188.7755 ¢ POTE: ~10/9 = 188.7744 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
9/5 182.4037
13/10 186.4465
5 ⧵ 32 187.5000 32cddefgh val
Lower bound of 7-odd-limit diamond monotone
23/12 187.7199
13/9 187.7941
23/13 188.2081
8 ⧵ 51 188.2353 51cdh val
Lower bound of 9-odd-limit diamond monotone
23/18 188.2910
17/11 188.4094
13/12 188.4523
15/14 188.4918
13/8 188.5463
11 ⧵ 70 188.5714 Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
7/5 188.5930
17/13 188.6048
21/20 188.6213
13/11 188.6230 13-odd-limit minimax
23/14 188.6483
17/16 188.6521
23/21 188.6537
17/12 188.6572
17/9 188.6601
3/2 188.6717 9, 15 and 17-odd-limit minimax
11/9 188.6852 11-odd-limit minimax
19/13 188.6872 19, 21 and 23-odd-limit minimax
11/6 188.6891
23/15 188.6959
11/8 188.6963
23/20 188.7115
21/17 188.7379
19/18 188.7467
17/14 188.7480
21/11 188.7584
14 ⧵ 89 188.7640 19, 21 and 23-odd-limit diamond monotone (singleton)
19/12 188.7655
11/7 188.7726
17/15 188.7824
21/16 188.7909
21/19 188.7932
19/16 188.7968
17/10 188.8056
19/14 188.8115
15/11 188.8135
7/4 188.8235
11/10 188.8463
23/17 188.8511
19/15 188.8537
7/6 188.8804 7-odd-limit minimax
19/10 188.8909
15/8 188.9127
23/22 188.9217
23/19 188.9746
9/7 189.0056
21/13 189.0356
5/4 189.0404 5-odd-limit minimax
19/11 189.2390
13/7 189.3085
5/3 189.4552
3 ⧵ 19 189.4737 Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
15/13 190.4518
23/16 190.5752
19/17 192.5576
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