@@ Line 1: / Line 1: @@
-The '''stearnsmic clan''' tempers out the stearnsma, the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.
+{{Technical data page}}
+The '''stearnsmic clan''' tempers out the [[stearnsma]], the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.
 == No-five stearnsmic ==
@@ Line 12: / Line 13: @@
 {{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}
-[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.888
+[[Optimal tuning]]s:
+* [[WE]]: ~343/243 = 599.994¢, ~9/7 = 433.884¢
+* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 433.885¢
 {{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
+[[Badness]] (Sintel): 0.382
 === Overview to extensions ===
-The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|5]]. Pogo adds [[32805/32768]], supers [[5120/5103]], echidna [[1728/1715]], and hedgehog [[50/49]]. Those are strong extensions. The others are weak. Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Septisuperfourth adds [[6144/6125]] and splits the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three. Octoid adds [[4375/4374]] and splits the period in four. Decistearn adds [[3136/3125]] splits the period in five. They all have neat extensions to the 11-limit via tempering out both [[540/539]] and [[4000/3993]].
+The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|5]]. Pogo adds [[32805/32768]], supers [[5120/5103]], echidna [[1728/1715]], and hedgehog [[50/49]]. Those are strong extensions. The others are weak. Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Septisuperfourth adds [[6144/6125]] and splits the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three. Octoid adds [[4375/4374]] and splits the period in four. Decistearn adds [[3136/3125]] splits the period in five. They all have neat extensions to the 11-limit via tempering out both [[540/539]] and [[4000/3993]], as well as [[9801/9800]], so that [[11/10]] and [[9/7]] add up to the semioctave.
 Stearnsmic temperaments not listed include:
@@ Line 27: / Line 32: @@
 * ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
 Considered below are pogo, supers, stearnscape, garistearn and decistearn.
 == Pogo ==
-{{See also| Schismatic family #Pogo }}
+{{See also| Schismatic family }}
-The pogo temperament (94 &amp; 130) tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma.
+The pogo temperament (94 & 130) tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It is also the temperament that reconciles the difference between [[tertiaschis]] and [[countertertiaschis]], using its semi-octave period.
 [[Subgroup]]: 2.3.5.7
@@ Line 39: / Line 44: @@
 {{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
+: mapping generators: ~343/243, ~9/7
-{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~343/243 = 599.986¢, ~9/7 = 433.890¢
-[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
+* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 433.901¢
 {{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
-[[Badness]]: 0.079635
+[[Badness]] (Sintel): 2.015
 === 11-limit ===
@@ Line 55: / Line 61: @@
 Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
+Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~9/7 = 433.911{{c}}
-{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
+{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}
-Badness: 0.031857
+Badness (Smith): 0.031857
 === 13-limit ===
@@ Line 68: / Line 74: @@
 Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
+Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~9/7 = 433.911{{c}}
-{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
+{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}
-Badness: 0.017514
+Badness (Smith): 0.017514
 == Supers ==
@@ Line 81: / Line 87: @@
 {{Mapping|legend=1| 2 1 -12 2 | 0 3 23 5 }}
-{{Multival|legend=1| 6 46 10 59 -1 -106 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~343/243 = 599.806¢, ~9/7 = 434.078¢
-[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 434.218
+* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 434.203¢
 {{Optimal ET sequence|legend=1| 58, 94, 152 }}
-[[Badness]]: 0.092748
+[[Badness]] (Sintel): 2.347
 === 11-limit ===
@@ Line 98: / Line 104: @@
 Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.217
-{{Optimal ET sequence|legend=1| 58, 94, 152 }}
+{{Optimal ET sequence|legend=0| 58, 94, 152 }}
 Badness: 0.028240
@@ Line 111: / Line 117: @@
 Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.221
-{{Optimal ET sequence|legend=1| 58, 94, 152f }}
+{{Optimal ET sequence|legend=0| 58, 94, 152f }}
 Badness: 0.021645
@@ Line 124: / Line 130: @@
 Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181
-{{Optimal ET sequence|legend=1| 58, 94, 152f }}
+{{Optimal ET sequence|legend=0| 58, 94, 152f }}
 Badness: 0.021316
@@ Line 137: / Line 143: @@
 : mapping generators: ~2450/2187, ~567/500
-{{Multival|legend=1| 36 66 60 21 -6 -46 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2450/2187 = 199.998¢, ~567/500 = 216.940¢
-[[Optimal tuning]] ([[POTE]]): ~2450/2187 = 1\6, ~567/500 = 216.942
+* [[CWE]]: ~2450/2187 = 200.000¢, ~567/500 = 216.941¢
 {{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
-[[Badness]]: 0.090467
+[[Badness]] (Sintel): 2.289
 === 11-limit ===
@@ Line 152: / Line 158: @@
 Mapping: {{mapping| 6 3 2 6 11 | 0 6 11 10 9 }}
-Optimal tuning (POTE): ~121/108 = 1\6, ~567/500 = 216.950
+Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)
-{{Optimal ET sequence|legend=1| 72, 282, 354 }}
+{{Optimal ET sequence|legend=0| 72, 210e, 282, 354 }}
 Badness: 0.032096
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 540/539, 729/728, 1575/1573, 34398/34375
+Mapping: {{mapping| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}
+Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)
+{{Optimal ET sequence|legend=0| 72, 210ef, 282, 354 }}
+Badness: 0.0258
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573
+Mapping: {{mapping| 6 3 2 6 11 -6 5 | 0 6 11 10 9 26 18 }}
+Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)
+{{Optimal ET sequence|legend=0| 72, 210efg, 282, 354 }}
+Badness: 0.0154
 == Garistearn ==
@@ Line 167: / Line 199: @@
 {{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}
-{{Multival|legend=1| 0 94 0 149 0 -264 }}
+: mapping generators: ~1029/1024, ~5
-[[Optimal tuning]] ([[POTE]]): ~1029/1024 = 1\94, ~5/4 = 386.7805
+[[Optimal tuning]]s:
+* [[CTE]]: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.3137¢
+* [[CWE]]: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6637¢
 {{Optimal ET sequence|legend=1| 94, 282, 658d, 940dd }}
-[[Badness]]: 0.307043
+[[Badness]] (Sintel): 7.770
 === 11-limit ===
@@ Line 182: / Line 216: @@
 Mapping: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}
-Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.5968
+Optimal tunings:
+* CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
+* CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.4545¢
-{{Optimal ET sequence|legend=1| 94, 282, 376, 658de }}
+{{Optimal ET sequence|legend=0| 94, 282, 376, 658de }}
-Badness: 0.082255
+Badness (Sintel): 2.719
 === 13-limit ===
@@ Line 195: / Line 231: @@
 Mapping: {{mapping| 94 149 0 264 107 348 | 0 0 1 0 1 0 }}
-Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8141
+Optimal tunings:
+* CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
+* CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6570¢
-{{Optimal ET sequence|legend=1| 94, 282 }}
+{{Optimal ET sequence|legend=0| 94, 282 }}
-Badness: 0.045941
+Badness (Sintel): 1.898
 === 17-limit ===
@@ Line 208: / Line 246: @@
 Mapping: {{mapping| 94 149 0 264 107 348 166 | 0 0 1 0 1 0 1 }}
-Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8420
+Optimal tunings:
+* CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 385.9792¢
+* CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6751¢
-{{Optimal ET sequence|legend=1| 94, 282 }}
+{{Optimal ET sequence|legend=0| 94, 282 }}
-Badness: 0.027608
+Badness (Sintel): 1.406
 == Decistearn ==
@@ Line 227: / Line 267: @@
 : mapping generators: ~15/14, ~135/98
-{{Multival|legend=1| 30 20 50 -38 -5 60 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~15/14 = 119.975¢, ~135/98 = 553.845¢
-[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~135/98 = 553.810 (~36/35 = 46.190)
+* [[CWE]]: ~15/14 = 120.000¢, ~135/98 = 553.893¢
 {{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}
-[[Badness]]: 0.095543
+[[Badness]] (Sintel): 2.418
 === 11-limit ===
@@ Line 242: / Line 282: @@
 Mapping: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}
-Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)
+Optimal tunings:
+* WE: ~15/14 = 119.953¢, ~11/8 = 553.864¢
+* CWE: ~15/14 = 120.000¢, ~11/8 = 553.978¢
-{{Optimal ET sequence|legend=1| 50, 80, 130, 210e, 340ce }}
+{{Optimal ET sequence|legend=0| 50, 80, 130, 210e, 340ce }}
-Badness: 0.038556
+Badness (Sintel): 1.275
 === 13-limit ===
@@ Line 255: / Line 297: @@
 Mapping: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}
-Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)
+Optimal tunings:
+* WE: ~15/14 = 119.974¢, ~11/8 = 553.827¢
+* CWE: ~15/14 = 120.000¢, ~11/8 = 553.905¢
-{{Optimal ET sequence|legend=1| 50, 80, 130 }}
+{{Optimal ET sequence|legend=0| 50, 80, 130 }}
-Badness: 0.026895
+Badness (Sintel): 1.111
 === 17-limit ===
@@ Line 268: / Line 312: @@
 Mapping: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}
-Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.862 (~36/35 = 46.138)
+Optimal tunings:
+* WE: ~15/14 = 119.990¢, ~11/8 = 553.884¢
+* CWE: ~15/14 = 120.000¢, ~11/8 = 553.914¢
-{{Optimal ET sequence|legend=1| 50, 80, 130 }}
+{{Optimal ET sequence|legend=0| 50, 80, 130 }}
-Badness: 0.024234
+Badness (Sintel): 1.235
 === 19-limit ===
@@ Line 281: / Line 327: @@
 Mapping: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}
-Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.913 (~36/35 = 46.087)
+Optimal tunings:
+* WE: ~15/14 = 119.995¢, ~11/8 = 553.925¢
+* CWE: ~15/14 = 120.000¢, ~11/8 = 553.940¢
+{{Optimal ET sequence|legend=0| 50, 80, 130 }}
+Badness (Sintel): 1.109
+=== 23-limit ===
+By equating 16/13 with 69/56 and 85/69 (enabling 56:69:85 chords), we find [[23/16]] at 2 generators underneath 6 periods (which is the [[5edo]] fifth, [[10edo|6\10]]).
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475, 897/896
+Mapping: {{mapping| 10 2 14 5 30 37 27 24 36 | 0 3 2 5 1 0 3 4 2 }}
+Optimal tunings:
+* WE: ~15/14 = 119.997¢, ~11/8 = 553.925¢
+* CWE: ~15/14 = 120.000¢, ~11/8 = 553.933¢
-{{Optimal ET sequence|legend=1| 50, 80, 130 }}
+{{Optimal ET sequence|legend=0| 30dh, 50, 80, 130 }}
-Badness: 0.018232
+Badness (Sintel): 1.017
 [[Category:Temperament clans]]

Stearnsmic clan: Difference between revisions