Xenial: Difference between revisions

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Line 6: Line 6:
| 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 = 4.6 | Complexity 1 = 51
| Odd limit 2 = 9 | Mistuning 2 = 6.3 | Complexity 2 = 51
| Odd limit 3 = 17 | Mistuning 3 = 8.9 | Complexity 3 = 70
| Odd limit 4 = 23 | Mistuning 4 = 9.0 | 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.
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| 13/11
| 13/11
| 188.623
| 188.623
|  
| 13-odd-limit minimax
|-
|-
|  
|  
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| 3/2
| 3/2
| 188.672
| 188.672
|  
| 9, 15 and 17-odd-limit minimax
|-
|-
|  
|  
| 11/9
| 11/9
| 188.685
| 188.685
|  
| 11-odd-limit minimax
|-
|-
|  
|  
| 19/13
| 19/13
| 188.687
| 188.687
|  
| 19, 21 and 23-odd-limit minimax
|-
|-
|  
|  
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| 7/6
| 7/6
| 188.880
| 188.880
|  
| 7-odd-limit minimax
|-
|-
|  
|  
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| 5/4
| 5/4
| 189.040
| 189.040
|  
| 5-odd-limit minimax
|-
|-
|  
|  

Revision as of 00:30, 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: 4.6 ¢;
9-odd-limit: 6.3 ¢;
17-odd-limit: 8.9 ¢;
23-odd-limit: 9.0 ¢
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  ¢
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
9/5 182.404
13/10 186.447
5 ⧵ 32 187.500 32cddefgh val
Lower bound of 7-odd-limit diamond monotone
23/12 187.720
13/9 187.794
23/13 188.208
8 ⧵ 51 188.235 51cdh val
Lower bound of 9-odd-limit diamond monotone
23/18 188.291
17/11 188.409
13/12 188.452
15/14 188.492
13/8 188.546
11 ⧵ 70 188.571 Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
7/5 188.593
17/13 188.605
21/20 188.621
13/11 188.623 13-odd-limit minimax
23/14 188.648
17/16 188.652
23/21 188.654
17/12 188.657
17/9 188.660
3/2 188.672 9, 15 and 17-odd-limit minimax
11/9 188.685 11-odd-limit minimax
19/13 188.687 19, 21 and 23-odd-limit minimax
11/6 188.689
23/15 188.6959
11/8 188.6963
23/20 188.711
21/17 188.738
19/18 188.747
17/14 188.748
21/11 188.758
14 ⧵ 89 188.764 19, 21 and 23-odd-limit diamond monotone (singleton)
19/12 188.766
11/7 188.773
17/15 188.782
21/16 188.791
21/19 188.793
19/16 188.797
17/10 188.806
19/14 188.811
15/11 188.814
7/4 188.823
11/10 188.846
23/17 188.851
19/15 188.854
7/6 188.880 7-odd-limit minimax
19/10 188.891
15/8 188.913
23/22 188.922
23/19 188.975
9/7 189.006
21/13 189.036
5/4 189.040 5-odd-limit minimax
19/11 189.239
13/7 189.308
5/3 189.455
3 ⧵ 19 189.473 Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
15/13 190.452
23/16 190.575
19/17 192.558
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