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No-threes subgroup temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (4/4)
Line 612: Line 612:
[[Comma list]]: 5767168/5764801
[[Comma list]]: 5767168/5764801


{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}
{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~7/4 = 968.913{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}
: [[error map]]: {{val| -0.015 +0.051 -0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.9174{{c}}
: error map: {{val| 0.000 +0.091 +0.021 }}


{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}


Badness (Sintel): 0.031
Badness (Sintel): 0.0309


=== Score ===
=== Score ===
Line 629: Line 634:
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}


{{Mapping|legend=3| 1 0 0 1 3 1| 0 0 0 4 1 6 }}
{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~11/8 = 540.099{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}


{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents
[[Badness]] (Sintel): 0.368


=== Bossier ===
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
{{See also| Chromatic pairs #Bossier }}


Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.  
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].  


[[Subgroup]]: 2.7.11.13
[[Subgroup]]: 2.7.11.13


[[Comma list]]: [[1573/1568]], [[15488/15379]]
[[Comma list]]: 1573/1568, 15488/15379


{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
Line 652: Line 659:
: mapping generators: ~2, ~14/11
: mapping generators: ~2, ~14/11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~14/11 = 421.309{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}


{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
[[Badness]] (Sintel): 0.307


=== Voltage ===
=== Voltage ===
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.  
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 7<sup>1/4</sup> of 842.2065{{c}}, which is also the CTC (constrained Tenney–Chebyshevian) tuning.  


[[Subgroup]]: 2.7.13
[[Subgroup]]: 2.7.13


[[Comma list]]: [[28672/28561]]
[[Comma list]]: 28672/28561


{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}
{{Mapping|legend=2| 1 0 3 | 0 4 1 }}


{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~16/13
: mapping generators: ~2, ~13


[[Optimal tuning]]:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~16/13 = 357.677{{c}}
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
* [[TOP tuning|POTT]]: ~2 = 1200.000{{c}}, ~16/13 = 357.794{{c}} (1200 - 300 log<sub>2</sub>(7))
: [[error map]]: {{val| -0.217 -0.143 +0.991 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/8 = 842.2568{{c}}
: error map: {{val| 0.000 +0.201 +1.729 }}


{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents
[[Badness]] (Sintel): 0.115


=== Ultrakleismic ===
=== Ultrakleismic ===
Line 684: Line 695:


{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~[[17/14]]
: mapping generators, ~2, ~17/14


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 324.446{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1379{{c}}, ~17/14 = 324.3440{{c}}
: [[error map]]: {{val| +0.138 +4.482 -7.166 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 324.3738{{c}}
: error map: {{val| 0.000 +4.295 -7.460 }}


{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
[[Badness]] (Sintel): 0.460


=== Counterultrakleismic ===
=== Counterultrakleismic ===
Line 698: Line 715:
: mapping generators, ~2, ~17/14
: mapping generators, ~2, ~17/14


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 336.858{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}
: [[error map]]: {{val| -0.028 -0.240 +0.462 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}
: error map: {{val| 0.000 -0.205 +0.528 }}


{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
[[Badness]] (Sintel): 0.860


=== Shipwreck ===
=== Shipwreck ===
Line 710: Line 733:
: mapping generators, ~2, ~64/53
: mapping generators, ~2, ~64/53


[[Optimal tuning]]s ([[POTE]]): ~2 = 1200.000{{c}}, ~64/53 = 323.034{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6967{{c}}, ~64/53 = 323.1839{{c}}
: [[error map]]: {{val| -0.303 +0.119 +1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/53 = 323.1959{{c}}
: error map: {{val| 0.000 +0.762 +3.300 }}


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
[[Badness]] (Sintel): 0.224


=== Lovecraft ===
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
{{See also | Chromatic pairs #Lovecraft }}


Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  
Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out [[1352/1331]], and is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  


[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13


[[Comma list]]: [[1352/1331]]
[[Comma list]]: 1352/1331


{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
Line 728: Line 757:
: mapping generators, ~2, ~13/11
: mapping generators, ~2, ~13/11


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/11 = 279.318{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5223{{c}}, ~13/11 = 279.2064{{c}}
: [[error map]]: {{val| -0.478 +5.662 -4.341 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/11 = 278.9918{{c}}
: error map: {{val| 0.000 +6.666 -3.552 }}


{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}
{{Optimal ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
[[Badness]] (Sintel): 0.175


=== Blackbirds ===
=== Blackbirds ===
Line 741: Line 774:
[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13


[[Comma list]]: [[29282/28561]]
[[Comma list]]: 29282/28561


{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
Line 748: Line 781:
: mapping generators, ~13/11, ~11
: mapping generators, ~13/11, ~11


[[Optimal tuning]] ([[POTE]]): ~13/11 = 300.000{{c}}, ~11/8 = 546.660{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~13/11 = 299.9728{{c}}, ~11/8 = 546.6107{{c}}
: [[error map]]: {{val| -0.109 -5.033 +5.730 }}
* [[CWE]]: ~13/11 = 300.0000{{c}}, ~11/8 = 546.4664{{c}}
: error map: {{val| 0.000 -4.852 +5.939 }}


{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
[[Badness]] (Sintel): 0.668


=== Bluebirds ===
=== Bluebirds ===
Line 760: Line 797:
[[Subgroup]]: 2.11.13
[[Subgroup]]: 2.11.13


[[Comma list]]: [[265837/262144]]
[[Comma list]]: 265837/262144


{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
Line 767: Line 804:
: mapping generators, ~2, ~143/128
: mapping generators, ~2, ~143/128


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~143/128 = 182.368{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8795{{c}}, ~143/128 = 182.5017{{c}}
: [[error map]]: {{val| +0.880 -1.174 -2.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~143/128 = 182.4386{{c}}
: error map: {{val| 0.000 -4.002 -5.405 }}


{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
[[Badness]] (Sintel): 0.451


=== Yamablu ===
=== Yamablu ===
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Genchain mode numbering|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


[[Subgroup]]: 2.11.13.17.19
[[Subgroup]]: 2.11.13.17.19
Line 780: Line 821:
[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 209/208, 2057/2048, 83521/83486


{{Mapping|legend=2| 1 5 1 1 0 | 0 -4 7 8 11 }}
{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~17/13 = 462.9606{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}


{{Optimal ET sequence|legend=1| 13, 44, 57, 70 }}
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Badness]] (Sintel): 0.386


=== Mavericks ===
=== Mavericks ===
Line 794: Line 838:


{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~26/19 = 539.886{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8817{{c}}, ~26/19 = 539.9150{{c}}
: [[error map]]: {{val| -0.118 -1.156 +1.825 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/19 = 539.9280{{c}}
: error map: {{val| 0.000 -0.960 +2.127 }}


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.559


=== Yer (rank 3) ===
=== Yer (rank 3) ===
Line 805: Line 856:


{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.4457{{c}}, ~11/8 = 548.4934{{c}}, ~16/13 = 358.638{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
[[Badness]] (Sintel): 0.106


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"