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Amity family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
Switch to Sintel's badness, WE & CWE tunings (2/)
Line 473: Line 473:
: mapping generators: ~343/240, ~7/6
: mapping generators: ~343/240, ~7/6


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.563{{c}}
* [[WE]]: ~343/240 = 600.3379{{c}}, ~7/6 = 260.5488{{c}}
* POTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.402{{c}}
: [[error map]]: {{val| +0.676 +1.127 +0.483 -4.519 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~7/6 = 260.4686{{c}}
: error map: {{val| 0.000 +0.388 -0.222 -6.014 }}


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 106d, 152d }}
{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60, 106d }}


[[Badness]] (Smith): 0.083601
[[Badness]] (Sintel): 2.12


=== 11-limit ===
=== 11-limit ===
Line 489: Line 491:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.653{{c}}
* WE: ~99/70 = 600.5039{{c}}, ~7/6 = 260.6119{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.393{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~7/6 = 260.4948{{c}}


{{Optimal ET sequence|legend=0| 14c, 32c, 46, 152de, 198, 244dee }}
{{Optimal ET sequence|legend=0| 14c, 32c, 46, 60e, 106de }}


Badness (Smith): 0.035504
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 504: Line 506:


Optimal tunings:
Optimal tunings:
* CTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.811{{c}}
* WE: ~55/39 = 600.3896{{c}}, ~7/6 = 260.7874{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.618{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~7/6 = 260.6927{{c}}


{{Optimal ET sequence|legend=0| 14cf, 32cf, 46 }}
{{Optimal ET sequence|legend=0| 14cf, 32cf, 46, 106deff }}


Badness (Smith): 0.030885
Badness (Sintel): 1.28


== Hamity ==
== Hamity ==
Line 518: Line 520:
[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


{{Mapping|legend=1| 1 8 19 15 | 0 -10 -26 -19 }}
{{Mapping|legend=1| 1 -2 -7 -4 | 0 10 26 19 }}
: mapping generators: ~2, ~14/9
: mapping generators: ~2, ~9/7


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.168{{c}}
* [[WE]]: ~2 = 1199.5499{{c}}, ~9/7 = 430.0579{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.219{{c}}
: [[error map]]: {{val| -0.450 -0.476 -1.657 +4.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 430.1958{{c}}
: error map: {{val| 0.000 +0.003 -1.222 +4.895 }}


{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}


[[Badness]] (Smith): 0.073956
[[Badness]] (Sintel): 1.87


=== 11-limit ===
=== 11-limit ===
Line 534: Line 538:
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}
Mapping: {{mapping| 1 -2 -7 -4 -3 | 0 10 26 19 18 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.220{{c}}
* WE: ~2 = 1200.2215{{c}}, ~9/7 = 430.2713{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2036{{c}}


{{Optimal ET sequence|legend=0| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=0| 14c, 39d, 53 }}


Badness (Smith): 0.042947
Badness (Sintel): 1.42


=== 13-limit ===
=== 13-limit ===
Line 552: Line 556:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.233{{c}}
* WE: ~2 = 1200.1379{{c}}, ~9/7 = 430.2658{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.216{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2227{{c}}


{{Optimal ET sequence|legend=0| 14cf, 39df, 53 }}
{{Optimal ET sequence|legend=0| 14cf, 39df, 53 }}


Badness (Smith): 0.029753
Badness (Sintel): 1.23


== Gamity ==
== Gamity ==
Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.
Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. It splits the interval of the grave major sixth in three.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 566: Line 570:
[[Comma list]]: 1029/1024, 1071875/1062882
[[Comma list]]: 1029/1024, 1071875/1062882


{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
{{Mapping|legend=1| 1 -2 -7 4 | 0 15 39 -5 }}
: mapping generators: ~2, ~320/189
: mapping generators: ~2, ~189/160


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~189/160 = 286.816{{c}}
* [[WE]]: ~2 = 1200.4434{{c}}, ~189/160 = 286.8930{{c}}
* POTE: ~2 = 1200.000{{c}}, ~189/160 = 286.787{{c}}
: [[error map]]: {{val| +0.443 +0.553 -0.591 -1.571 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/160 = 286.7955{{c}}
: error map: {{val| 0.000 -0.022 -1.288 -2.804 }}


{{Optimal ET sequence|legend=1| 46, 113, 159, 205d, 364d }}
{{Optimal ET sequence|legend=1| 46, 113, 159 }}


[[Badness]] (Smith): 0.125733
[[Badness]] (Sintel): 3.18


=== 11-limit ===
=== 11-limit ===
Line 582: Line 588:
Comma list: 385/384, 441/440, 1071875/1062882
Comma list: 385/384, 441/440, 1071875/1062882


Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}
Mapping: {{mapping| 1 -2 -7 4 8 | 0 15 39 -5 -19 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~33/28 = 286.813{{c}}
* WE: ~2 = 1200.3315{{c}}, ~33/28 = 286.8764{{c}}
* POTE: ~2 = 1200.000{{c}}, ~33/28 = 286.797{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/28 = 286.8002{{c}}


{{Optimal ET sequence|legend=0| 46, 113, 159, 205d, 364d }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness (Smith): 0.051111
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 597: Line 603:
Comma list: 325/324, 364/363, 385/384, 10985/10976
Comma list: 325/324, 364/363, 385/384, 10985/10976


Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}
Mapping: {{mapping| 1 -2 -7 4 8 8 | 0 15 39 -5 -19 -18 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.803{{c}}
* WE: ~2 = 1200.4248{{c}}, ~13/11 = 286.8906{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.789{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7912{{c}}


{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}
{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}


Badness (Smith): 0.030297
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 612: Line 618:
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757


Mapping: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}
Mapping: {{mapping| 1 -2 -7 4 8 8 6 | 0 15 39 -5 -19 -18 -8 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.804{{c}}
* WE: ~2 = 1200.3287{{c}}, ~13/11 = 286.8731{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.795{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7957{{c}}


{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness (Smith): 0.022036
Badness (Sintel): 1.12


== Trinity ==
== Trinity ==
Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. It splits the acute minor tenth (~243/100, an octave plus acute minor third) in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1{{c}} flat), three of which makes acute minor third of ~243/200 with octave reduction.
Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. It splits the grave major sixth plus an octave in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1{{c}} flat), three of which makes acute minor third of ~243/200 with octave reduction.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 629: Line 635:
[[Comma list]]: 703125/702464, 1600000/1594323
[[Comma list]]: 703125/702464, 1600000/1594323


{{Mapping|legend=1| 1 8 19 46 | 0 -15 -39 -101 }}
{{Mapping|legend=1| 1 -7 -20 -55 | 0 15 39 101 }}
: mapping generators: ~2, ~125/84


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~168/125 = 513.180{{c}}
* [[WE]]: ~2 = 1199.9474{{c}}, ~125/84 = 686.7921{{c}}
* POTE: ~2 = 1200.000{{c}}, ~168/125 = 513.178{{c}}
: [[error map]]: {{val| -0.053 +0.295 -0.369 +0.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~125/84 = 686.8215{{c}}
: error map: {{val| 0.000 +0.367 -0.276 +0.144 }}


{{Optimal ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}
{{Optimal ET sequence|legend=1| 7d, …, 152, 311, 463, 774 }}


[[Badness]] (Smith): 0.119453
[[Badness]] (Sintel): 3.02


=== 11-limit ===
=== 11-limit ===
Line 644: Line 653:
Comma list: 3025/3024, 4000/3993, 19712/19683
Comma list: 3025/3024, 4000/3993, 19712/19683


Mapping: {{mapping| 1 8 19 46 18 | 0 -15 -39 -101 -34 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 | 0 15 39 101 34 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~121/90 = 513.181{{c}}
* WE: ~2 = 1199.9090{{c}}, ~125/84 = 686.7710{{c}}
* POTE: ~2 = 1200.000{{c}}, ~121/90 = 513.177{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~125/84 = 686.8218{{c}}


{{Optimal ET sequence|legend=0| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}
{{Optimal ET sequence|legend=0| 7d, …, 152, 311, 463, 774, 1237e }}


Badness (Smith): 0.031296
Badness (Sintel): 1.03


=== 13-limit ===
=== 13-limit ===
Line 659: Line 668:
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689


Mapping: {{mapping| 1 8 19 46 18 64 | 0 -15 -39 -101 -34 -141 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 | 0 15 39 101 34 141 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* WE: ~2 = 1199.9531{{c}}, ~52/35 = 686.7909{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8172{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311 }}
{{Optimal ET sequence|legend=0| 7df, …, 152f, 311 }}


Badness (Smith): 0.026418
Badness (Sintel): 1.09


=== 17-limit ===
=== 17-limit ===
Line 674: Line 683:
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619


Mapping: {{mapping| 1 8 19 46 18 64 -22 | 0 -15 -39 -101 -34 -141 61 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 | 0 15 39 101 34 141 -61 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* WE: ~2 = 1200.0347{{c}}, ~52/35 = 686.8342{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8146{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}
{{Optimal ET sequence|legend=0| 152f, 159, 311, 1092cdg }}


Badness (Smith): 0.025588
Badness (Sintel): 1.30


=== 19-limit ===
=== 19-limit ===
Line 689: Line 698:
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573


Mapping: {{mapping| 1 8 19 46 18 64 -22 53 | 0 -15 -39 -101 -34 -141 61 -114 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 | 0 15 39 101 34 141 -61 114 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* WE: ~2 = 1200.0406{{c}}, ~52/35 = 686.8382{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8152{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311 }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness (Smith): 0.018412
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 704: Line 713:
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104


Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 | 0 -15 -39 -101 -34 -141 61 -114 -104 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 -55 | 0 15 39 101 34 141 -61 114 104 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8324{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8148{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness (Smith): 0.014343
Badness (Sintel): 1.03


=== 29-limit ===
=== 29-limit ===
Line 719: Line 728:
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044


Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 72 | 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 -55 -85 | 0 15 39 101 34 141 -61 114 104 157 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8318{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8142{{c}}


{{Optimal ET sequence|legend=0| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
{{Optimal ET sequence|legend=0| 152fj, 159, 311, 781dh }}


Badness (Smith): 0.012038
Badness (Sintel): 1.01


== Amicable ==
== Amicable ==
Line 740: Line 749:
{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}
{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8831{{c}}
* [[CTE]]: ~2 = 1200.0000{{c}}, ~21/20 = 84.8831{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 84.880{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 84.880{{c}}


{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}
{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}
Line 937: Line 946:
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}


[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.909{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~48/35 = 547.909{{c}}


{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
Line 978: Line 987:
{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~11907/10000 = 307.915{{c}}
* [[CTE]]: ~2 = 1200.000{{c}}, ~11907/10000 = 307.915{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11907/10000 = 307.941{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~11907/10000 = 307.941{{c}}


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}
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