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Gamelismic clan: Difference between revisions

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The [[2.3.7 subgroup]] comma for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[Gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.


To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.  
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.  
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== Slendric ==
== Slendric ==
{{Main| Slendric }}
{{Main| Slendric }}
{{See also| No-fives subgroup temperaments #Slendric }}
[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


Line 39: Line 37:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~8/7 = 233.889
* [[CTE]]: ~2 = 1200.000, ~8/7 = 233.889
* [[POTE]]: ~2 = 1\1, ~8/7 = 233.688
* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.688


{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
Line 55: Line 53:
: [[gencom]]: [2 8/7; 896/891 1029/1024]
: [[gencom]]: [2 8/7; 896/891 1029/1024]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 234.381
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 234.381


{{Optimal ET sequence|legend=1| 36, 41, 87, 128 }}
{{Optimal ET sequence|legend=1| 36, 41, 87, 128 }}
Line 70: Line 68:
: sval mapping generators: ~17/12, ~8/7
: sval mapping generators: ~17/12, ~8/7


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.6155
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 233.616


{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg }}
{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg, 226ffg }}


== Rodan ==
== Rodan ==
Line 88: Line 86:
{{Multival|legend=1| 3 17 -1 20 -10 -50 }}
{{Multival|legend=1| 3 17 -1 20 -10 -50 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 234.417
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 234.417


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
: {{Monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
: {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5


[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
Line 99: Line 97:
{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}


[[Badness]]: 0.037112
[[Badness]] (Smith): 0.037112


=== 11-limit ===
=== 11-limit ===
Line 108: Line 106:
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.459
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.459


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: Eigenmonzo (unchanged-interval) basis: 2.11/9
: eigenmonzo (unchanged-interval) basis: 2.11/9


Algebraic generator: [[Algebraic number|positive root]] of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.


{{Optimal ET sequence|legend=1| 41, 46, 87 }}
{{Optimal ET sequence|legend=0| 41, 46, 87 }}


Badness: 0.023093
Badness (Smith): 0.023093


==== 13-limit ====
==== 13-limit ====
Line 128: Line 126:
Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.482
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.482


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
: Eigenmonzos (unchanged-intervals): 2, 13/9
: eigenmonzo (unchanged-interval) basis: 2.13/9


Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.
Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.


{{Optimal ET sequence|legend=1| 41, 46, 87 }}
{{Optimal ET sequence|legend=0| 41, 46, 87 }}


Badness: 0.018448
Badness (Smith): 0.018448


===== 17-limit =====
===== 17-limit =====
Line 147: Line 145:
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.524
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.524


Minimax tuning:
Minimax tuning:
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
: Eigenmonzos (unchanged-intervals): 2, 18/17
: eigenmonzo (unchanged-interval) basis: 2.17/9


{{Optimal ET sequence|legend=1| 41, 46, 87, 220dg, 307dgg }}
{{Optimal ET sequence|legend=0| 41, 46, 87, 220dg, 307dgg }}


Badness: 0.016743
Badness (Smith): 0.016743


==== Aerodactyl ====
==== Aerodactyl ====
Line 164: Line 162:
Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.639
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.639


{{Optimal ET sequence|legend=1| 5, 41f, 46, 133ff }}
{{Optimal ET sequence|legend=0| 5, 41f, 46, 133ff }}


Badness: 0.033986
Badness (Smith): 0.033986


=== Aerodino ===
=== Aerodino ===
Line 177: Line 175:
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.728
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.728


{{Optimal ET sequence|legend=1| 41e, 46 }}
{{Optimal ET sequence|legend=0| 41e, 46 }}


Badness: 0.054294
Badness (Smith): 0.054294


==== 13-limit ====
==== 13-limit ====
Line 190: Line 188:
Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.782
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.782


{{Optimal ET sequence|legend=1| 41ef, 46 }}
{{Optimal ET sequence|legend=0| 41ef, 46 }}


Badness: 0.035836
Badness (Smith): 0.035836


=== Varan ===
=== Varan ===
Line 203: Line 201:
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.145
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.145


{{Optimal ET sequence|legend=1| 36ce, 41 }}
{{Optimal ET sequence|legend=0| 36ce, 41 }}


Badness: 0.044937
Badness (Smith): 0.044937


==== 13-limit ====
==== 13-limit ====
Line 216: Line 214:
Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.089
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.089


{{Optimal ET sequence|legend=1| 36ce, 41 }}
{{Optimal ET sequence|legend=0| 36ce, 41 }}


Badness: 0.032284
Badness (Smith): 0.032284


== Guiron ==
== Guiron ==
Line 235: Line 233:
{{Multival|legend=1| 3 -24 -1 -45 -10 65 }}
{{Multival|legend=1| 3 -24 -1 -45 -10 65 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.930
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 233.930


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: {{Monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}


[[Badness]]: 0.047544
[[Badness]] (Smith): 0.047544


=== 11-limit ===
=== 11-limit ===
Line 255: Line 253:
: mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.931
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 233.931


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]
: Eigenmonzo (unchanged-interval) basis: 2.5
: eigenmonzo (unchanged-interval) basis: 2.5


{{Optimal ET sequence|legend=1| 36e, 41, 77, 118, 159, 277d }}
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}


Badness: 0.026648
Badness (Smith): 0.026648


=== 13-limit ===
=== 13-limit ===
Line 275: Line 273:
: mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.890
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 233.890


{{Optimal ET sequence|legend=1| 36e, 41, 77, 118 }}
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}


Badness: 0.028444
Badness (Smith): 0.028444


== Mothra ==
== Mothra ==
Line 298: Line 296:
{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
{{Multival|legend=1| 3 12 -1 12 -10 -36 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 232.193
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 232.193


[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
Line 304: Line 302:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 5, 26, 31 }}
{{Optimal ET sequence|legend=1| 5, 26, 31 }}


[[Badness]]: 0.037146
[[Badness]] (Smith): 0.037146


=== 11-limit ===
=== 11-limit ===
Line 320: Line 318:
{{Multival|legend=1| 3 12 -1 -8 12 -10 -23 -36 -60 -19 }}
{{Multival|legend=1| 3 12 -1 -8 12 -10 -23 -36 -60 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.031
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.031


{{Optimal ET sequence|legend=1| 5, 26, 31, 88, 150be, 181bee }}
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 150be, 181bee }}


Badness: 0.025642
Badness (Smith): 0.025642


==== 13-limit ====
==== 13-limit ====
Line 333: Line 331:
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.811
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 231.811


{{Optimal ET sequence|legend=1| 5, 26, 31, 57, 88 }}
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}


Badness: 0.023954
Badness (Smith): 0.023954


; Music:  
; Music:  
* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] ''(dead link)'' by [[Chris Vaisvil]] ([http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog post])
* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning]{{dead link}} by [[Chris Vaisvil]] ([http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog post])


=== Cynder ===
=== Cynder ===
Line 349: Line 347:
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.317
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 231.317


{{Optimal ET sequence|legend=1| 5e, 26, 57e, 83bce }}
{{Optimal ET sequence|legend=0| 5e, 26, 57e, 83bce }}


Badness: 0.055706
Badness (Smith): 0.055706


==== 13-limit ====
==== 13-limit ====
Line 362: Line 360:
Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 231.293
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 231.293


{{Optimal ET sequence|legend=1| 5e, 26, 57e, 83bce }}
{{Optimal ET sequence|legend=0| 5e, 26, 57e, 83bce }}


Badness: 0.034124
Badness (Smith): 0.034124


=== Mosura ===
=== Mosura ===
Line 377: Line 375:
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}


{{Multival|legend=1| 3 12 -1 23 12 -10 26 -36 12 68 }}
Wedgie: {{Multival| 3 12 -1 23 12 -10 26 -36 12 68 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.419
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.419


{{Optimal ET sequence|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}
{{Optimal ET sequence|legend=0| 31, 129, 160be, 191bce, 222bce, 253bcee }}


Badness: 0.031334
Badness (Smith): 0.031334


==== 13-limit ====
==== 13-limit ====
Line 392: Line 390:
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.640
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.640


{{Optimal ET sequence|legend=1| 31, 36, 67, 98 }}
{{Optimal ET sequence|legend=0| 31, 36, 67, 98 }}


Badness: 0.036857
Badness (Smith): 0.036857


== Gorgo ==
== Gorgo ==
Line 410: Line 408:
{{Multival|legend=1| 3 7 4 }}
{{Multival|legend=1| 3 7 4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 227.426
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~10/9 = 227.426


{{Optimal ET sequence|legend=1| 5, 16, 21, 37b }}
{{Optimal ET sequence|legend=1| 5, 16, 21, 37b }}


[[Badness]]: 0.161799
[[Badness]] (Smith): 0.161799


=== 7-limit ===
=== 7-limit ===
Line 425: Line 423:
{{Multival|legend=1| 3 7 -1 4 -10 -22 }}
{{Multival|legend=1| 3 7 -1 4 -10 -22 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 228.334
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 228.334


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}


[[Badness]]: 0.060663
[[Badness]] (Smith): 0.060663


=== 11-limit ===
=== 11-limit ===
Line 438: Line 436:
Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.373
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 227.373


{{Optimal ET sequence|legend=1| 16, 21, 37b }}
{{Optimal ET sequence|legend=0| 16, 21, 37b }}


Badness: 0.049500
Badness (Smith): 0.049500


==== 13-limit ====
==== 13-limit ====
Line 451: Line 449:
Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.230
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 227.230


{{Optimal ET sequence|legend=1| 16, 21, 37b }}
{{Optimal ET sequence|legend=0| 16, 21, 37b }}


Badness: 0.032664
Badness (Smith): 0.032664


=== Spartan ===
=== Spartan ===
Line 464: Line 462:
Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 229.535
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 229.535


{{Optimal ET sequence|legend=1| 5, 16e, 21, 47c, 68bcce }}
{{Optimal ET sequence|legend=0| 5, 16e, 21, 47c, 68bcce }}


Badness: 0.062683
Badness (Smith): 0.062683


==== 13-limit ====
==== 13-limit ====
Line 477: Line 475:
Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 229.059
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 229.059


{{Optimal ET sequence|legend=1| 5, 16e, 21, 68bccef }}
{{Optimal ET sequence|legend=0| 5, 16e, 21, 68bccef }}


Badness: 0.047071
Badness (Smith): 0.047071


; Music
; Music
Line 497: Line 495:
{{Multival|legend=1| 3 2 -1 -4 -10 -8 }}
{{Multival|legend=1| 3 2 -1 -4 -10 -8 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.762
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 230.762


{{Optimal ET sequence|legend=1| 5, 16c, 21cc, 26ccc }}
{{Optimal ET sequence|legend=1| 5, 16c, 21cc, 26ccc }}


[[Badness]]: 0.062262
[[Badness]] (Smith): 0.062262


== Oncle ==
== Oncle ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Oncle]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 512: Line 510:
{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}
{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 232.498
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 232.498


{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}
{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}


[[Badness]]: 0.088384
[[Badness]] (Smith): 0.088384


== Archaeotherium ==
== Archaeotherium ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Archaeotherium]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 527: Line 525:
{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}
{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.258
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 230.258


{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc, 99bc }}
{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc, 99bc }}


[[Badness]]: 0.146306
[[Badness]] (Smith): 0.146306


== Clyndro ==
== Clyndro ==
{{See also| Pelogic family }}
{{See also| Mavila family }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 544: Line 542:
{{Multival|legend=1| 3 -9 -1 -21 -10 23 }}
{{Multival|legend=1| 3 -9 -1 -21 -10 23 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.469
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 226.469


{{Optimal ET sequence|legend=1| 5c, 11, 16 }}
{{Optimal ET sequence|legend=1| 5c, 11, 16 }}


[[Badness]]: 0.159179
[[Badness]] (Smith): 0.159179


=== 11-limit ===
=== 11-limit ===
Line 557: Line 555:
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.428
Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 226.428


{{Optimal ET sequence|legend=1| 5c, 11, 16 }}
{{Optimal ET sequence|legend=0| 5c, 11, 16 }}


Badness: 0.069703
Badness (Smith): 0.069703


== Miracle ==
== Miracle ==
{{Main| Miracle }}
{{Main| Miracle }}


''For the 5-limit temperament, see [[Ampersand comma#Ampersand]].
''For the 5-limit version, see [[Ampersand comma #Ampersand]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 578: Line 576:
{{Multival|legend=1| 6 -7 -2 -25 -20 15 }}
{{Multival|legend=1| 6 -7 -2 -25 -20 15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 116.675
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~15/14 = 116.675


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}
* [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}
: {{Monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}
: {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5/3
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: {{Monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
: {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5


[[Tuning ranges]]:
[[Tuning ranges]]:
Line 592: Line 590:
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]
* 7-odd-limit diamond monotone and tradeoff: ~15/14 = [115.587, 116.993]
* 9-odd-limit diamond monotone and tradeoff: ~15/14 = [116.129, 116.993]


[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12
[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12
Line 599: Line 595:
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}


[[Badness]]: 0.016742
[[Badness]] (Smith): 0.016742


=== 11-limit ===
=== 11-limit ===
Line 610: Line 606:
{{Multival|legend=1| 6 -7 -2 15 -25 -20 3 15 59 49 }}
{{Multival|legend=1| 6 -7 -2 15 -25 -20 3 15 59 49 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.633
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.633


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
* 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]
: Eigenmonzo (unchanged-interval) basis: 2.9/5
: eigenmonzo (unchanged-interval) basis: 2.9/5


Tuning ranges:
Tuning ranges:
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]
* 11-odd-limit diamond monotone and tradeoff: ~15/14 = [116.129, 116.993]


Algebraic generator: Secor59
Algebraic generator: Secor59


{{Optimal ET sequence|legend=1| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}


Badness: 0.010684
Badness (Smith): 0.010684


==== Miraculous ====
==== Miraculous ====
Line 635: Line 630:
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.747
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.747


{{Optimal ET sequence|legend=1| 10, 21e, 31, 41, 72f, 113f, 185cff }}
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f, 113f, 185cff }}


Badness: 0.018669
Badness (Smith): 0.018669


===== 17-limit =====
===== 17-limit =====
Line 648: Line 643:
Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.769
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.769


{{Optimal ET sequence|legend=1| 10, 21e, 31, 41, 72fg, 113fgg }}
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg, 113fgg }}


Badness: 0.017084
Badness (Smith): 0.017084


==== Benediction ====
==== Benediction ====
Line 661: Line 656:
Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.574
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.574


{{Optimal ET sequence|legend=1| 31, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}


Badness: 0.015715
Badness (Smith): 0.015715


===== 17-limit =====
===== 17-limit =====
Line 674: Line 669:
Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.585
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.585


{{Optimal ET sequence|legend=1| 31, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}


Badness: 0.012537
Badness (Smith): 0.012537


==== Manna ====
==== Manna ====
Line 687: Line 682:
Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.739
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.739


{{Optimal ET sequence|legend=1| 31f, 41, 72, 185cf, 257cff }}
{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}


Badness: 0.017012
Badness (Smith): 0.017012


===== 17-limit =====
===== 17-limit =====
Line 700: Line 695:
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.727
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.727


{{Optimal ET sequence|legend=1| 31fg, 41, 72, 185cf, 257cff }}
{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}


Badness: 0.014680
Badness (Smith): 0.014680


==== Semimiracle ====
==== Semimiracle ====
Line 713: Line 708:
Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~15/14 = 116.624
Optimal tuning (POTE): ~99/70 = 600.000, ~15/14 = 116.624


{{Optimal ET sequence|legend=1| 10, 62, 72 }}
{{Optimal ET sequence|legend=0| 10, 62, 72 }}


Badness: 0.024622
Badness (Smith): 0.024622


===== 17-limit =====
===== 17-limit =====
Line 726: Line 721:
Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}


Optimal tuning (POTE): ~2 = 17\12, ~15/14 = 116.628
Optimal tuning (POTE): ~17\12 = 600.000, ~15/14 = 116.628


{{Optimal ET sequence|legend=1| 10, 62, 72 }}
{{Optimal ET sequence|legend=0| 10, 62, 72 }}


Badness: 0.016130
Badness (Smith): 0.016130


==== Hemisecordite ====
==== Hemisecordite ====
Line 739: Line 734:
Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.288
Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 58.288


{{Optimal ET sequence|legend=1| 41, 62, 103, 247c, 350bcde }}
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}


Badness: 0.025589
Badness (Smith): 0.025589


===== 17-limit =====
===== 17-limit =====
Line 752: Line 747:
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.261
Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 58.261


{{Optimal ET sequence|legend=1| 41, 62, 103 }}
{{Optimal ET sequence|legend=0| 41, 62, 103 }}


Badness: 0.022535
Badness (Smith): 0.022535


===== Semihemisecordite =====
===== Semihemisecordite =====
Line 765: Line 760:
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~27/26 = 58.288
Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.288


{{Optimal ET sequence|legend=1| 62, 144g, 206begg, 350bcdeggg }}
{{Optimal ET sequence|legend=0| 62, 144g, 206begg, 350bcdeggg }}


Badness: 0.046958
Badness (Smith): 0.046958


====== 19-limit ======
====== 19-limit ======
Line 778: Line 773:
Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}
Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~27/26 = 58.283
Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.283


{{Optimal ET sequence|legend=1| 62, 144gh, 206begghh }}
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}


Badness: 0.035057
Badness (Smith): 0.035057


====== 23-limit ======
====== 23-limit ======
Line 791: Line 786:
Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}
Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~27/26 = 58.283
Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.283


{{Optimal ET sequence|legend=1| 62, 144gh, 206begghhi }}
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}


Badness: 0.026421
Badness (Smith): 0.026421


==== Phicordial ====
==== Phicordial ====
Line 804: Line 799:
Mapping: {{mapping| 1 7 -4 1 17 4 | 0 -18 21 6 -45 -1 }}
Mapping: {{mapping| 1 7 -4 1 17 4 | 0 -18 21 6 -45 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 361.121
Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 361.121


{{Optimal ET sequence|legend=1| 103, 113, 216c }}
{{Optimal ET sequence|legend=0| 103, 113, 216c }}


Badness: 0.033198
Badness (Smith): 0.033198


===== 17-limit =====
===== 17-limit =====
Line 817: Line 812:
Mapping: {{mapping| 1 7 -4 1 17 4 8 | 0 -18 21 6 -45 -1 -13 }}
Mapping: {{mapping| 1 7 -4 1 17 4 8 | 0 -18 21 6 -45 -1 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 361.123
Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 361.123


{{Optimal ET sequence|legend=1| 103, 113, 216c }}
{{Optimal ET sequence|legend=0| 103, 113, 216c }}


Badness: 0.024705
Badness (Smith): 0.024705


=== Revelation ===
=== Revelation ===
Line 830: Line 825:
Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}
Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.277
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.277


{{Optimal ET sequence|legend=1| 10e, 21, 31 }}
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


Badness: 0.032946
Badness (Smith): 0.032946


==== 13-limit ====
==== 13-limit ====
Line 843: Line 838:
Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}
Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.268
Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.268


{{Optimal ET sequence|legend=1| 10e, 21, 31 }}
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


Badness: 0.029452
Badness (Smith): 0.029452


=== Hemimiracle ===
=== Hemimiracle ===
Line 856: Line 851:
Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 58.408
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 58.408


{{Optimal ET sequence|legend=1| 20, 21, 41, 144e, 185cee, 226cee }}
{{Optimal ET sequence|legend=0| 20, 21, 41, 144e, 185cee, 226cee }}


Badness: 0.059232
Badness (Smith): 0.059232


==== 13-limit ====
==== 13-limit ====
Line 869: Line 864:
Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}
Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 58.430
Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 58.430


{{Optimal ET sequence|legend=1| 20, 21, 41, 144eff, 185ceeff }}
{{Optimal ET sequence|legend=0| 20, 21, 41, 144eff, 185ceeff }}


Badness: 0.043151
Badness (Smith): 0.043151


=== Oracle ===
=== Oracle ===
Line 882: Line 877:
Mapping: {{mapping| 1 7 -4 1 3 | 0 -12 14 4 1 }}
Mapping: {{mapping| 1 7 -4 1 3 | 0 -12 14 4 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 541.668
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 541.668


{{Optimal ET sequence|legend=1| 11, 20, 31, 82e, 113e, 144ee }}
{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}


Badness: 0.042687
Badness (Smith): 0.042687


== Hemiseven ==
== Hemiseven ==
Line 897: Line 892:
{{Multival|legend=1| 6 29 -2 32 -20 -86 }}
{{Multival|legend=1| 6 29 -2 32 -20 -86 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~320/243 = 483.267
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~320/243 = 483.267


{{Optimal ET sequence|legend=1| 72, 77, 149, 221, 514bd, 735bcdd }}
{{Optimal ET sequence|legend=1| 72, 77, 149, 221, 514bd, 735bcdd }}


[[Badness]]: 0.056557
[[Badness]] (Smith): 0.056557


=== 11-limit ===
=== 11-limit ===
Line 910: Line 905:
Mapping: {{mapping| 1 4 14 2 -5 | 0 -6 -29 2 21 }}
Mapping: {{mapping| 1 4 14 2 -5 | 0 -6 -29 2 21 }}


Optimal tuning (POTE): ~2 = 1\1, ~320/243 = 483.276
Optimal tuning (POTE): ~2 = 1200.000, ~320/243 = 483.276


{{Optimal ET sequence|legend=1| 72, 77, 149, 221e, 293de }}
{{Optimal ET sequence|legend=0| 72, 77, 149, 221e, 293de }}


Badness: 0.028467
Badness (Smith): 0.028467


=== 13-limit ===
=== 13-limit ===
Line 923: Line 918:
Mapping: {{mapping| 1 4 14 2 -5 19 | 0 -6 -29 2 21 -38 }}
Mapping: {{mapping| 1 4 14 2 -5 19 | 0 -6 -29 2 21 -38 }}


Optimal tuning (POTE): ~2 = 1\1, ~120/91 = 483.256
Optimal tuning (POTE): ~2 = 1200.000, ~120/91 = 483.256


{{Optimal ET sequence|legend=1| 72, 77, 149, 221ef }}
{{Optimal ET sequence|legend=0| 72, 77, 149, 221ef }}


Badness: 0.021900
Badness (Smith): 0.021900


=== 17-limit ===
=== 17-limit ===
Line 936: Line 931:
Mapping: {{mapping| 1 4 14 2 -5 19 21 | 0 -6 -29 2 21 -38 -42 }}
Mapping: {{mapping| 1 4 14 2 -5 19 21 | 0 -6 -29 2 21 -38 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/34 = 483.261
Optimal tuning (POTE): ~2 = 1200.000, ~45/34 = 483.261


{{Optimal ET sequence|legend=1| 72, 77, 149, 221ef }}
{{Optimal ET sequence|legend=0| 72, 77, 149, 221ef }}


Badness: 0.015701
Badness (Smith): 0.015701


== Unidec ==
== Unidec ==
Line 954: Line 949:
: mapping generators: ~177147/125000, ~10/9
: mapping generators: ~177147/125000, ~10/9


[[Optimal tuning]] ([[POTE]]): ~177147/125000 = 1\2, ~10/9 = 183.047
[[Optimal tuning]] ([[POTE]]): ~177147/125000 = 600.000, ~10/9 = 183.047


{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 2524, 2642, 2760, 2878b, …, 5002bc }}
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 2524, 2642, 2760, 2878b, …, 5002bc }}


[[Badness]]: 0.082423
[[Badness]] (Smith): 0.082423


=== 7-limit ===
=== 7-limit ===
Line 969: Line 964:
{{Multival|legend=1| 12 22 -4 7 -40 -71 }}
{{Multival|legend=1| 12 22 -4 7 -40 -71 }}


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~10/9 = 183.161
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 600.000, ~10/9 = 183.161


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}
: {{Monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}
: {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
* [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}
: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}


[[Badness]]: 0.038393
[[Badness]] (Smith): 0.038393


=== 11-limit ===
=== 11-limit ===
Line 993: Line 988:
* [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
* [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]
: Eigenmonzo (unchanged-interval) basis: 2.9/7
: eigenmonzo (unchanged-interval) basis: 2.9/7


{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}


Badness: 0.015479
Badness (Smith): 0.015479


==== Ekadash ====
==== Ekadash ====
Line 1,006: Line 1,001:
Mapping: {{mapping| 2 5 8 5 6 19 | 0 -6 -11 2 3 -38 }}
Mapping: {{mapping| 2 5 8 5 6 19 | 0 -6 -11 2 3 -38 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~10/9 = 183.187
Optimal tuning (POTE): ~99/70 = 600.000, ~10/9 = 183.187


{{Optimal ET sequence|legend=1| 26f, 46f, 72, 118, 190, 262df, 452cdef }}
{{Optimal ET sequence|legend=0| 26f, 46f, 72, 118, 190, 262df, 452cdef }}


Badness: 0.020381
Badness (Smith): 0.020381


==== Hendec ====
==== Hendec ====
Line 1,019: Line 1,014:
Mapping: {{mapping| 2 5 8 5 6 8 | 0 -6 -11 2 3 -2 }}
Mapping: {{mapping| 2 5 8 5 6 8 | 0 -6 -11 2 3 -2 }}


Optimal tuning (POTE): ~91/64 = 1\2, ~10/9 = 183.198
Optimal tuning (POTE): ~91/64 = 600.000, ~10/9 = 183.198


{{Optimal ET sequence|legend=1| 26, 46, 72, 190ff }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}


Badness: 0.017707
Badness (Smith): 0.017707


===== 17-limit =====
===== 17-limit =====
Line 1,032: Line 1,027:
Mapping: {{mapping| 2 5 8 5 6 8 10 | 0 -6 -11 2 3 -2 -6 }}
Mapping: {{mapping| 2 5 8 5 6 8 10 | 0 -6 -11 2 3 -2 -6 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~10/9 = 183.196
Optimal tuning (POTE): ~17/12 = 600.000, ~10/9 = 183.196


{{Optimal ET sequence|legend=1| 26, 46, 72, 190ffg }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}


Badness: 0.011676
Badness (Smith): 0.011676


== Lagaca ==
== Lagaca ==
Line 1,047: Line 1,042:
{{Multival|legend=1| 18 -26 -6 -83 -60 59 }}
{{Multival|legend=1| 18 -26 -6 -83 -60 59 }}


[[Optimal tuning]] ([[POTE]]): ~3375/2401 = 1\2, ~15/14 = 122.027
[[Optimal tuning]] ([[POTE]]): ~3375/2401 = 600.000, ~15/14 = 122.027


{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}


[[Badness]]: 0.144345
[[Badness]] (Smith): 0.144345


== Necromanteion ==
== Necromanteion ==
Line 1,062: Line 1,057:
{{Multival|legend=1| 12 17 -4 -1 -40 -57 }}
{{Multival|legend=1| 12 17 -4 -1 -40 -57 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 541.779
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 541.779


{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c, 206bc, 237bc, 505bbccd }}
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c, 206bc, 237bc, 505bbccd }}


[[Badness]]: 0.117680
[[Badness]] (Smith): 0.117680


=== 11-limit ===
=== 11-limit ===
Line 1,075: Line 1,070:
Mapping: {{mapping| 1 7 10 1 17 | 0 -12 -17 4 -30 }}
Mapping: {{mapping| 1 7 10 1 17 | 0 -12 -17 4 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.729
Optimal tuning (POTE): ~2 = 1200.000, ~15/11 = 541.729


{{Optimal ET sequence|legend=1| 20ce, 31, 113c, 144c, 175c, 381bccdee }}
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c, 175c, 381bccdee }}


Badness: 0.053459
Badness (Smith): 0.053459


=== 13-limit ===
=== 13-limit ===
Line 1,088: Line 1,083:
Mapping: {{mapping| 1 7 10 1 17 1 | 0 -12 -17 4 -30 6 }}
Mapping: {{mapping| 1 7 10 1 17 1 | 0 -12 -17 4 -30 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.606
Optimal tuning (POTE): ~2 = 1200.000, ~15/11 = 541.606


{{Optimal ET sequence|legend=1| 20ce, 31, 51ce, 82cf, 113cf, 144cf }}
{{Optimal ET sequence|legend=0| 20ce, 31, 51ce, 82cf, 113cf, 144cf }}


Badness: 0.047015
Badness (Smith): 0.047015


== Restles ==
== Restles ==
{{ See also | Lesser tendoneutralic }}
{{See also| Lesser tendoneutralic }}
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,104: Line 1,100:
{{Multival|legend=1| 12 -19 -4 -58 -40 44 }}
{{Multival|legend=1| 12 -19 -4 -58 -40 44 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~315/256 = 358.5485
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~315/256 = 358.548


{{Optimal ET sequence|legend=1| 10, 77, 87, 164 }}
{{Optimal ET sequence|legend=1| 10, 77, 87, 164 }}


[[Badness]]: 0.108011
[[Badness]] (Smith): 0.108011


=== 11-limit ===
=== 11-limit ===
Line 1,117: Line 1,113:
Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 358.5713
Optimal tuning (POTE): ~2 = 1200.000, ~27/22 = 358.571


{{Optimal ET sequence|legend=1| 10, 77, 87, 164 }}
{{Optimal ET sequence|legend=0| 10, 77, 87, 164 }}


Badness: 0.054655
Badness (Smith): 0.054655


=== 13-limit ===
=== 13-limit ===
{{ See also | Lesser tendoneutralic }}
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 1,131: Line 1,126:
Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}
Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 358.5739
Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 358.574


{{Optimal ET sequence|legend=1| 10, 77, 87, 164 }}
{{Optimal ET sequence|legend=0| 10, 77, 87, 164 }}


Badness: 0.028187
Badness (Smith): 0.028187


== Quartemka ==
== Quartemka ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quartemka]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,148: Line 1,143:
{{Multival|legend=1| 21 32 -7 2 -70 -106 }}
{{Multival|legend=1| 21 32 -7 2 -70 -106 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 138.006
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 138.006


{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}
{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}


[[Badness]]: 0.152287
[[Badness]] (Smith): 0.152287


=== 11-limit ===
=== 11-limit ===
Line 1,161: Line 1,156:
Mapping: {{mapping| 1 4 6 2 3 | 0 -21 -32 7 4 }}
Mapping: {{mapping| 1 4 6 2 3 | 0 -21 -32 7 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 137.990
Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 137.990


{{Optimal ET sequence|legend=1| 26, 61, 87, 200, 287d, 487cdd }}
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d, 487cdd }}


Badness: 0.057307
Badness (Smith): 0.057307


=== 13-limit ===
=== 13-limit ===
Line 1,174: Line 1,169:
Mapping: {{mapping| 1 4 6 2 3 6 | 0 -21 -32 7 4 -20 }}
Mapping: {{mapping| 1 4 6 2 3 6 | 0 -21 -32 7 4 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 137.990
Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 137.990


{{Optimal ET sequence|legend=1| 26, 61, 87, 200, 487cdd }}
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 487cdd }}


Badness: 0.028393
Badness (Smith): 0.028393


== Tritriple ==
== Tritriple ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritriple]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,191: Line 1,186:
{{Multival|legend=1| 27 20 -9 -31 -90 -77 }}
{{Multival|legend=1| 27 20 -9 -31 -90 -77 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~864/625 = 559.295
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~864/625 = 559.295


{{Optimal ET sequence|legend=1| 15, 88, 103, 118, 339d }}
{{Optimal ET sequence|legend=1| 15, 88, 103, 118, 339d }}


[[Badness]]: 0.118640
[[Badness]] (Smith): 0.118640


=== 11-limit ===
=== 11-limit ===
Line 1,204: Line 1,199:
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 559.293
Optimal tuning (POTE): ~2 = 1200.000, ~242/175 = 559.293


{{Optimal ET sequence|legend=1| 15, 88, 103, 118, 339de }}
{{Optimal ET sequence|legend=0| 15, 88, 103, 118, 339de }}


Badness: 0.035350
Badness (Smith): 0.035350


== Widefourth ==
== Widefourth ==
Line 1,219: Line 1,214:
{{Multival|legend=1| 33 13 -11 -56 -110 -62 }}
{{Multival|legend=1| 33 13 -11 -56 -110 -62 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3125/2304 = 524.210
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3125/2304 = 524.210


{{Optimal ET sequence|legend=1| 16, 55b, 71, 87, 103, 190 }}
{{Optimal ET sequence|legend=1| 16, 55b, 71, 87, 103, 190 }}


[[Badness]]: 0.154117
[[Badness]] (Smith): 0.154117


=== 11-limit ===
=== 11-limit ===
Line 1,232: Line 1,227:
Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}
Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}


Optimal tuning (POTE): ~2 = 1\1, ~847/625 = 524.210
Optimal tuning (POTE): ~2 = 1200.000, ~847/625 = 524.210


{{Optimal ET sequence|legend=1| 16, 55be, 71, 87, 103, 190 }}
{{Optimal ET sequence|legend=0| 16, 55be, 71, 87, 103, 190 }}


Badness: 0.040785
Badness (Smith): 0.040785


=== 13-limit ===
=== 13-limit ===
Line 1,245: Line 1,240:
Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~65/48 = 524.209
Optimal tuning (POTE): ~2 = 1200.000, ~65/48 = 524.209


{{Optimal ET sequence|legend=1| 16, 55be, 71, 87, 103, 190 }}
{{Optimal ET sequence|legend=0| 16, 55be, 71, 87, 103, 190 }}


Badness: 0.021636
Badness (Smith): 0.021636


[[Category:Temperament clans]]
[[Category:Temperament clans]]
Retrieved from "https://en.xen.wiki/w/Gamelismic_clan"