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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, P11/9)
| 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 ==
Line 27: Line 179:
|-
|-
! Tenney
! Tenney
| CTE: ~10/9 = 188.8535 {{c}}
| CTE: ~10/9 = 188.8535{{c}}
| CWE: ~10/9 = 188.8544 {{c}}
| CWE: ~10/9 = 188.8544{{c}}
| POTE: ~10/9 = 188.8548 {{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"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
Line 43: Line 254:
|-
|-
! Tenney
! Tenney
| CTE: ~10/9 = 188.7849 {{c}}
| CTE: ~10/9 = 188.7849{{c}}
| CWE: ~10/9 = 188.7755 {{c}}
| CWE: ~10/9 = 188.7755{{c}}
| POTE: ~10/9 = 188.7744 {{c}}
| POTE: ~10/9 = 188.7744{{c}}
|}
|}


Line 58: 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 158: 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
|  
|  
|}
|}
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