@@ Line 1: / Line 1: @@
+{{Technical data page}}
 This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.
@@ Line 5: / Line 6: @@
 * For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
-* French decimal and trader are with ~2/1 period and ~5/4 generator. There is one-to-one correspondence between 2.5 subgroup and mapped intervals.
+* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
-* Didacus is with a ~28/25 generator, two of which give the ~5/4.
+* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
-* Ostara, movila and vengeance are with variety expressed generator, three of which give the ~5/2.
+* Insect has a ~55/32 generator, three of which give the ~5/1.
-* Insect is with a ~55/32 generator, three of which give the ~5/1.
+* Frostburn has a ~28/25 generator, four of which give the ~8/5.
-* Frostburn is with a ~28/25 generator, four of which give the ~8/5.
+Others have a more complex mapping of 5.
-Others are more far.
 == 2.5.7 temperaments ==
@@ Line 21: / Line 21: @@
 === Frostburn ===
-{{See also| Magic family #Quadrimage }}
+{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}
 [[Subgroup]]: 2.5.7
@@ Line 34: / Line 34: @@
 {{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
+[[Badness]] (Sintel): 0.886
 ==== 2.5.7.11 ====
@@ Line 46: / Line 48: @@
 Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
-{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
+{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
+Badness (Sintel): 0.463
 === Mabilic ===
-{{See also| Chromatic pairs #Mabilic }}
+{{See also| Chromatic pairs #Mabilic }}{{Main|Mabilic and trismegistus}}Given below is the no-three version of [[Mavila family#Armodue|armodue]], [[Mabila family#Semabila|semabila]], and [[Magic family#Trismegistus|trismegistus]]. It is the 7 &amp; 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
-Given below is the no-three version of [[Mabila family#Semabila|semabila]], or equivalently the no-threes version of [[Magic family#Trismegistus|trismegistus]]. It is the 7 &amp; 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
 [[Subgroup]]: 2.5.7
@@ Line 97: / Line 99: @@
 Comma basis: {{monzo|372 -159 -1}}
-Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
+Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]
 Optimal tuning (CTE): ~5/4 = 386.360
 {{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
+[[Badness]] (Sintel): 148.6
 ==== 2.5.7.11 subgroup ====
@@ Line 112: / Line 116: @@
 Optimal tuning (CTE): ~5/4 = 386.361
-{{Optimal ET sequence|legend=1|264, 733}}, ...
+{{Optimal ET sequence|legend=0|264, 733}}, ...
+Badness (Sintel): 52.150
 ==== 2.5.7.11.13 subgroup ====
@@ Line 123: / Line 129: @@
 Optimal tuning (CTE): ~5/4 = 386.361
-{{Optimal ET sequence|legend=1|1525, 1789}}, ...
+{{Optimal ET sequence|legend=0|1525, 1789}}, ...
+Badness (Sintel): 10.518
 === Bastille ===
 {{Main| Bastille }}
-Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].
+Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an [[wikipedia:Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].
 Subgroup: 2.5.7
@@ Line 139: / Line 147: @@
 {{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
+[[Badness]] (Sintel): 7224.3
 === Augment ===
@@ Line 172: / Line 182: @@
 [[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
-[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
+* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢
 {{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
+[[Badness]] (Sintel): 11.731
 ==== 2.5.7.11.13 subgroup ====
@@ Line 183: / Line 197: @@
 Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
-Optimal tuning (POTE): ~1664/1225 = 529.003¢
+Optimal tunings:
+* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
+* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
-{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
+{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
+Badness (Sintel): 3.415
 ==== 2.5.7.11.13.17 subgroup ====
@@ Line 194: / Line 212: @@
 Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
-Optimal tuning (POTE): ~1664/1225 = 529.003¢
+Optimal tunings:
+* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
+* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
+{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
-{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
+Badness (Sintel): 1.985
 ==== 2.5.7.11.13.17.19 subgroup ====
 Subgroup: 2.5.7.11.13.17.19
-Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
+Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]
 Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
-Optimal tuning (POTE): ~19/14 = 529.003¢
+Optimal tunings:
+* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
+* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢
+{{Optimal ET sequence|legend=0| 93, 338, 431 }}
+Badness (Sintel): 1.285
 === Tricesimoprimal miracloid ===
@@ Line 263: / Line 291: @@
 [[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
-=== Pakkanen (rank 3) ===
+=== Pakkanen ===
 [[Subgroup]]: 2.5.7.11
@@ Line 275: / Line 303: @@
 {{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
+[[Badness]] (Sintel): 0.573
+=== No-threes naiad ===
+{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}
+This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands [[Subgroup temperaments #Tridec|tridec]] and [[Subgroup temperaments #Naiadec|naiadec]].
+[[Subgroup]]: 2.5.7.11
+[[Comma list]]: 5021863/5000000
+{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
+: mapping generators: ~2, ~5, ~100/77
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.080¢, ~5 = 2786.820¢, ~100/77 = 454.618¢
+* [[CWE]]: ~2 = 1200.000¢, ~5 = 2786.740¢, ~100/77 = 454.590¢
+{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
+[[Badness]] (Sintel): 1.862
+==== 2.5.7.11.13 subgroup ====
+Subgroup: 2.5.7.11.13
+Comma list: 847/845, 1001/1000
+Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
+Optimal tunings:
+* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
+* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢
+{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
+Badness (Sintel): 0.179
+==== 2.5.7.11.13.17 subgroup ====
+Subgroup: 2.5.7.11.13.17
+Comma list: 170/169, 221/220, 847/845
+Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
+Optimal tunings:
+* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
+* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢
+{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
+Badness (Sintel): 0.438
 == Higher 2.5 temperaments ==
@@ Line 392: / Line 473: @@
 == No-threes-or-fives subgroup temperaments ==
 Temperaments discussed elsewhere include
-* Orgone → [[Orgonia #Orgone]]
+* Orgone → [[Orgonia #Orgone|Orgonia]]
-* Berylic → [[4th-octave temperaments #Berylic]]
+* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
-* 21-23-commatic → [[21st-octave temperaments #21-23-commatic]]
+* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
-* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic]]
+* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
-* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)]]
+* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]
 * etc.
+=== Amaranthine ===
+{{See also| No-fives subgroup temperaments #Chrysanthemum }}
+Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].
+[[Subgroup]]: 2.7.11
+[[Comma list]]: 5767168/5764801
+{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913
+{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
+Badness (Sintel): 0.031
+=== Score ===
+{{See also| Chromatic pairs #Score }}
+[[Subgroup]]: 2.7.11.13
+[[Comma list]]: 343/338, 847/832
+{{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 }}
+: [[gencom]]: [2 11/8; 343/338 847/832]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099
+{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
+[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents
+=== Bossier ===
+{{See also| Chromatic pairs #Bossier }}
+Bossier can be described as the 3 &amp; 17 in the 2.7.11.13 subgroup.
+[[Subgroup]]: 2.7.11.13
+[[Comma list]]: [[1573/1568]], [[15488/15379]]
+{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
+{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
+: [[gencom]]: [2 14/11; 1573/1568 15488/15379]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309
+{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
+=== Voltage ===
+Voltage is the 3 &amp; 7 temperament in the 2.7.13 subgroup.
+[[Subgroup]]: 2.7.13
+[[Comma list]]: [[28672/28561]]
+{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}
+{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
+: [[gencom]]: [2, 16/13; 28672/28561]
+[[Optimal tuning]]:
+* [[POTE]]: ~2 = 1\1, ~16/13 = 357.677
+* [[TOP tuning|POTT]]: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log<sub>2</sub>(7))
+{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents
 === Ultrakleismic ===
@@ Line 439: / Line 598: @@
 {{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
-=== Yer (rank 3) ===
+=== Lovecraft ===
-[[Subgroup]]: 2.11.13.17.19
+{{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.
+[[Subgroup]]: 2.11.13
+[[Comma list]]: [[1352/1331]]
+{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
+{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
+: [[gencom]]: [2 13/11; 1352/1331]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318
+{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
+=== Blackbirds ===
+{{See also | Chromatic pairs #Blackbirds }}
+Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.
+[[Subgroup]]: 2.11.13
+[[Comma list]]: [[29282/28561]]
+{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
+{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
+: [[gencom]]: [13/11 11/8; 29282/28561]
+[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660
+{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
+=== Bluebirds ===
+{{Distinguish| Bluebird }}
+{{See also| Chromatic pairs #Bluebirds }}
+[[Subgroup]]: 2.11.13
+[[Comma list]]: [[265837/262144]]
+{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
+{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
-[[Comma list]]: 209/208, 2057/2048
+: [[gencom]]: [2 143/128; 265837/262144]
-[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368
-[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
+{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
-{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
 === Yamablu ===
@@ Line 476: / Line 686: @@
 {{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
+=== Yer (rank 3) ===
+[[Subgroup]]: 2.11.13.17.19
+[[Comma list]]: 209/208, 2057/2048
+[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
+[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
+{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
 [[Category:Temperament collections]]
 [[Category:Subgroup temperaments]]

No-threes subgroup temperaments: Difference between revisions