Meantone family: Difference between revisions
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The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]]. | |||
== Meantone == | == Meantone == | ||
{{Main| Meantone }} | {{Main| Meantone }} | ||
Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]]. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 17: | Line 20: | ||
: mapping generators: ~2, ~3 | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}} | ||
: [[error map]]: {{val| | : [[error map]]: {{val| +1.391 -3.519 +1.868 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}} | ||
: | : error map: {{val| 0.000 -5.304 +0.291 }} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | * [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | ||
: [[Eigenmonzo basis| | : [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
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{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }} | {{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.173 | ||
=== | === Overview to extensions === | ||
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at. | The second comma of the normal comma list defines which [[7-limit]] family member we are looking at. | ||
* | * Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo. | ||
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, | * Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo. | ||
* | * Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo. | ||
* | * Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo. | ||
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, | * Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, for an [[exotemperament]] never exactly well-tuned, and where 5edo is the only [[diamond monotone]] tuning, with a terrible 5-limit part. | ||
Those all have a fifth as generator. | Those all have a fifth as generator. | ||
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period. | * Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period. | ||
* Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two. | * Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two. | ||
* Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~8/7 generator, two of which give the [[4/3|fourth]]. | * Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~[[8/7]] generator, two of which give the [[4/3|fourth]]. | ||
* Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth. | * Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth. | ||
* Liese adds {{monzo| -9 11 0 -3 }} with a ~10/7 generator, three of which give the [[3/1|twelfth]]. | * Liese adds {{monzo| -9 11 0 -3 }} with a ~[[10/7]] generator, three of which give the [[3/1|twelfth]]. | ||
* Squares adds {{monzo| -3 9 0 -4 }} with a ~9/7 generator, four of which give the [[8/3|eleventh]]. | * Squares adds {{monzo| -3 9 0 -4 }} with a ~[[9/7]] generator, four of which give the [[8/3|eleventh]]. | ||
* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five. | * Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five. | ||
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Temperaments discussed elsewhere include | Temperaments discussed elsewhere include | ||
* ''[[Plutus]]'' → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]] | * ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]] | ||
* [[Godzilla]] → [[ | * [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]] | ||
* [[Mothra]] → [[Gamelismic clan #Mothra|Gamelismic clan]] | * [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]] | ||
* [[Mohaha]] → [[Rastmic clan #Mohaha|Rastmic clan]] | * ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]] | ||
The rest are considered below. | The rest are considered below. | ||
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{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }} | {{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}} | ||
: [[error map]]: {{val| | : [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}} | ||
: | : error map: {{val| 0.000 -5.299 +0.311 -2.264 }} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | ||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }} | : [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }} | ||
: [[eigenmonzo basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
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{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }} | {{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.347 | ||
=== Undecimal meantone (huygens) === | === Undecimal meantone (huygens) === | ||
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}} | {{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}} | ||
{{See also| | {{See also| Huygens vs meanpop }} | ||
Undecimal meantone maps the [[11/8]] to the double-augmented third (C–E𝄪), and tridecimal meantone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. | Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪), and tridecimal meantone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 128: | Line 127: | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }} | Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | * 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | ||
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}] | : projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 146: | Line 143: | ||
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents. | Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents. | ||
{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }} | |||
Badness: 0. | Badness (Sintel): 0.563 | ||
; Music | ; Music | ||
| Line 161: | Line 158: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
{{Optimal ET sequence|legend=0| 12f, 19e, 31 }} | |||
Badness: 0. | Badness (Sintel): 0.746 | ||
===== Meantonic ===== | ===== Meantonic ===== | ||
| Line 182: | Line 179: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.2376{{c}}, ~3/2 = 697.0954{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4563{{c}} | ||
{{Optimal ET sequence|legend=0| 12fg, 19eg, 31, 50e }} | |||
Badness: 0. | Badness (Sintel): 0.970 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 197: | Line 194: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.4134{{c}}, ~3/2 = 697.0933{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.3526{{c}} | ||
= | {{Optimal ET sequence|legend=0| 12fghh, 19egh, 31, 50e }} | ||
Badness (Sintel): 1.09 | |||
Badness: | |||
===== Huygens ===== | ===== Huygens ===== | ||
| Line 246: | Line 211: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 31 }} | |||
Badness: | Badness (Sintel): 1.02 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 261: | Line 226: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 31 }} | |||
Badness: | Badness (Sintel): 1.10 | ||
==== Grosstone ==== | ==== Grosstone ==== | ||
| Line 278: | Line 243: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 289: | Line 254: | ||
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | * 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | ||
{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }} | |||
Badness: | Badness (Sintel): 1.07 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 301: | Line 266: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}} | ||
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }} | |||
Badness: | Badness (Sintel): 1.06 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 316: | Line 281: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}} | ||
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }} | |||
Badness: | Badness (Sintel): 1.07 | ||
==== Meridetone ==== | ==== Meridetone ==== | ||
| Line 333: | Line 298: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/9 | ||
{{Optimal ET sequence|legend=0| 12f, 31f, 43 }} | |||
Badness: | Badness (Sintel): 1.09 | ||
===== Meridetonic ===== | ===== Meridetonic ===== | ||
| Line 352: | Line 317: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.9428{{c}}, ~3/2 = 697.4804{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5113{{c}} | ||
{{Optimal ET sequence|legend=0| 12fg, 31fg, 43 }} | |||
Badness: | Badness (Sintel): 1.41 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 367: | Line 332: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0089{{c}}, ~3/2 = 697.4864{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.4815{{c}} | ||
{{Optimal ET sequence|legend=0| 12fghh, 31fgh, 43 }} | |||
Badness: | Badness (Sintel): 1.54 | ||
===== | ===== Sauveuric ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 78/77, 81/80, 99/98, 120/119, 126/125 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 43 }} | |||
Badness: | Badness (Sintel): 1.22 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 43 }} | |||
Badness: | Badness (Sintel): 1.25 | ||
==== Hemimeantone ==== | |||
==== Hemimeantone ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 444: | Line 379: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}} | ||
{{Optimal ET sequence|legend=0| 19e, 43, 62 }} | |||
Badness: | Badness (Sintel): 1.30 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 459: | Line 394: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}} | ||
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }} | |||
Badness: | Badness (Sintel): 1.19 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 474: | Line 409: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}} | ||
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }} | |||
Badness: | Badness (Sintel): 1.15 | ||
==== Semimeantone ==== | ==== Semimeantone ==== | ||
| Line 491: | Line 426: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}} | ||
* | * CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }} | |||
Badness: | Badness (Sintel): 1.68 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 506: | Line 441: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }} | |||
Badness: | Badness (Sintel): 1.60 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 521: | Line 456: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }} | |||
Badness: | Badness (Sintel): 1.47 | ||
=== Meanpop === | === Meanpop === | ||
{{See also| Meantone vs meanpop }} | {{See also| Meantone vs meanpop }} | ||
Meanpop maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second. | Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 540: | Line 475: | ||
: mapping generator: ~2, ~3 | : mapping generator: ~2, ~3 | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }} | * 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }} | ||
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}] | : projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}] | ||
: | : unchanged-interval (eigenmonzo) basis: 2.5 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 558: | Line 491: | ||
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge. | Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge. | ||
{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }} | |||
Badness: 0. | Badness (Sintel): 0.712 | ||
; Music | ; Music | ||
| Line 572: | Line 505: | ||
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }} | Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/11 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 587: | Line 518: | ||
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | * 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | ||
{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }} | |||
Badness: 0. | Badness (Sintel): 0.863 | ||
===== Meanpoppic ===== | ===== Meanpoppic ===== | ||
| Line 599: | Line 530: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}} | ||
{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }} | |||
Badness: | Badness (Sintel): 1.02 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 614: | Line 545: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}} | ||
{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }} | |||
Badness: | Badness (Sintel): 1.08 | ||
===== Meanpoid ===== | ===== Meanpoid ===== | ||
| Line 629: | Line 560: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 31 }} | |||
Badness: | Badness (Sintel): 1.17 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 644: | Line 575: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 31 }} | |||
Badness: | Badness (Sintel): 1.25 | ||
==== Meanplop ==== | ==== Meanplop ==== | ||
| Line 657: | Line 588: | ||
Mapping: {{mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }} | Mapping: {{mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1202.3237{{c}}, ~3/2 = 697.5502{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2135{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11 | ||
{{Optimal ET sequence|legend=0| 12e, 19, 31f }} | |||
Badness: | Badness (Sintel): 1.14 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 680: | Line 609: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.4737{{c}}, ~3/2 = 697.2690{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4129{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 19 }} | |||
Badness: | Badness (Sintel): 1.37 | ||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 695: | Line 624: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.8839{{c}}, ~3/2 = 697.0104{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4949{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 19 }} | |||
Badness: | Badness (Sintel): 1.43 | ||
=== Meanenneadecal === | |||
=== Meanenneadecal === | |||
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison. | Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison. | ||
| Line 740: | Line 639: | ||
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }} | Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}} | ||
Tuning ranges: | Tuning ranges: | ||
| Line 751: | Line 648: | ||
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377] | * 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377] | ||
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }} | |||
Badness: 0. | Badness (Sintel): 0.708 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 761: | Line 658: | ||
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }} | Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}} | ||
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }} | |||
Badness: 0. | Badness (Sintel): 0.875 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 780: | Line 675: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 19, 31e }} | |||
Badness: | Badness (Sintel): 1.17 | ||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 795: | Line 690: | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 19, 31e }} | |||
Badness: | Badness (Sintel): 1.23 | ||
==== | ==== Vincenzo ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 45/44, 56/55, 65/64, 81/80 | ||
Mapping: {{mapping| 1 0 -4 -13 -6 | Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 12, 19 }} | |||
Badness: | Badness (Sintel): 1.02 | ||
====== 19-limit | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80 | |||
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 19 }} | |||
Badness (Sintel): 1.30 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80 | ||
Mapping: {{mapping| 1 0 -4 -13 -6 | Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}} | ||
{{Optimal ET sequence|legend=0| 12, 19 }} | |||
Badness: | Badness (Sintel): 1.36 | ||
==== | ==== Meanundec ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 45/44, 56/55 | Comma list: 27/26, 40/39, 45/44, 56/55 | ||
Mapping: {{mapping| 1 0 -4 -13 -6 | Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1196.0359{{c}}, ~3/2 = 694.9504{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7474{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 12f, 19f }} | |||
Badness: | Badness (Sintel): 1.00 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 27/26, 34/33, 40/39, 45/44, 56/55 | ||
Mapping: {{mapping| 1 0 -4 -13 -6 | Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 | 0 1 4 10 6 3 7 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1196.8604{{c}}, ~3/2 = 695.7613{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1744{{c}} | ||
{{Optimal ET sequence|legend=0| 7dg, 12f }} | |||
Badness: | Badness (Sintel): 1.09 | ||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 39 | Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55 | ||
Mapping: {{mapping| 1 0 -4 -13 -6 10 | Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 -10 | 0 1 4 10 6 3 7 9 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1196.9296{{c}}, ~3/2 = 696.3321{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7122{{c}} | ||
{{Optimal ET sequence|legend=0| 7dgh, 12f }} | |||
Badness: | Badness (Sintel): 1.16 | ||
====== | === Meanundeci === | ||
Subgroup: 2.3.5.7.11 | Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C–F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C–A♭). | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 33/32, 55/54, 77/75 | ||
Mapping: {{mapping| 1 0 -4 -13 | Mapping: {{mapping| 1 0 -4 -13 5 | 0 1 4 10 -1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1205.7146{{c}}, ~3/2 = 697.9977{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.1805{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }} | |||
Badness: | Badness (Sintel): 1.04 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 33/32, 55/54, 65/64, 77/75 | ||
Mapping: {{mapping| 1 0 -4 -13 | Mapping: {{mapping| 1 0 -4 -13 5 10 | 0 1 4 10 -1 -4 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1205.5631{{c}}, ~3/2 = 697.9847{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.0144{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }} | |||
Badness: | Badness (Sintel): 1.09 | ||
=== | === Bimeantone === | ||
11/8 is mapped to half octave minus the [[128/125|meantone diesis]]. | |||
Subgroup: 2.3.5.7.11 | |||
Mapping: {{mapping| | Comma list: 81/80, 126/125, 245/242 | ||
Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }} | |||
: mapping generators: ~63/44, ~3 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}} | ||
* | * CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}} | ||
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }} | |||
Badness: | Badness (Sintel): 1.26 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 105/104, 126/125, 245/242 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}} | ||
* | * CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }} | |||
Badness: | Badness (Sintel): 1.19 | ||
==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 105/104, 126/125, 189/187, 221/220 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }} | |||
Badness: | Badness (Sintel): 1.15 | ||
==== | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}} | ||
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }} | |||
Badness (Sintel): 1.08 | |||
=== Trimean === | |||
{{See also| No-sevens subgroup temperaments #Superpine }} | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 126/125, 1344/1331 | |||
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }} | |||
: mapping generators: ~2, ~11/10 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }} | |||
Badness: | Badness (Sintel): 1.68 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 126/125, 144/143, 364/363 | ||
Mapping: {{mapping| 1 0 - | Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }} | |||
Badness: | Badness (Sintel): 1.46 | ||
==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 126/125, 144/143, 189/187, 221/220 | ||
Mapping: {{mapping| 1 0 - | Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}} | ||
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }} | |||
Badness: | Badness (Sintel): 1.28 | ||
== | == Flattone == | ||
{{Main| Flattone }} | |||
In flattone tunings, the fifth is typically even flatter than that of [[19edo]]. Here, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]]. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 525/512 | |||
{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}} | |||
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}} | |||
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }} | |||
[[Minimax tuning]]: | |||
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }} | |||
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}] | |||
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5 | |||
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }} | |||
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}] | |||
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7 | |||
[[Tuning ranges]]: | |||
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19) | |||
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955] | |||
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] | |||
[[Algebraic generator]]: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4. | |||
Optimal | {{Optimal ET sequence|legend=1| 7, 19, 26, 45 }} | ||
[[Badness]] (Sintel): 0.976 | |||
=== 11-limit === | |||
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]]. | |||
Subgroup: 2.3.5.7.11 | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 45/44, 81/80, 385/384 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }} | ||
Optimal | Optimal tuning: | ||
* | * WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}} | ||
Tuning ranges: | |||
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19) | |||
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955] | |||
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }} | |||
Badness (Sintel): 1.12 | |||
Subgroup: 2.3.5.7.11 | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: | Comma list: 45/44, 65/64, 78/77, 81/80 | ||
Mapping: {{mapping| 1 0 -4 - | Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}} | ||
Tuning ranges: | |||
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19) | |||
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955] | |||
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }} | |||
Badness (Sintel): 0.920 | |||
== Flattertone == | |||
Flattertone tunings are typically at least as flat as [[26edo]]. Here, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). [[26edo]] and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 1875/1792 | |||
{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }} | |||
: mapping generators: ~2, ~3 | |||
== | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}} | |||
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}} | |||
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }} | |||
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }} | |||
[[Badness]] (Sintel): 2.43 | |||
==== 11-limit ==== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 45/44, 81/80, 1375/1344 | ||
: mapping | Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }} | |||
Badness: | Badness (Sintel): 1.53 | ||
; Music | |||
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024) | |||
== Dominant == | |||
{{Main| Dominant (temperament) }} | |||
{{See also| Archytas clan }} | |||
The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh (C–Bb). The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]]. | |||
Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension takes the tritone as 16/11, which it barely sounds like. The first alternative, domineering, takes the same step as 11/8, which it barely sounds like either. Domination tempers out 77/75 and identifies 11/8 with the augmented third; arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension [[neutrominant]], splitting the fifth as well as the chromatic semitone in two like in all [[rastmic clan|rastmic]] temperaments. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 36/35, 64/63 | |||
= | {{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }} | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}} | |||
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}} | |||
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }} | |||
[[Tuning ranges]]: | |||
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5) | |||
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587] | |||
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587] | |||
Optimal | {{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }} | ||
[[Badness]] (Sintel): 0.524 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 36/35, 56/55, 64/63 | |||
Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }} | |||
Tuning ranges: | |||
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17) | |||
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587] | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }} | |||
Badness: 0. | Badness (Sintel): 0.799 | ||
=== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 36/35, 56/55, 64/63, 66/65 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }} | ||
: | Optimal tunings: | ||
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}} | |||
Tuning ranges: | |||
* | * 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17) | ||
* | * 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587] | ||
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }} | |||
Badness: 0. | Badness (Sintel): 0.996 | ||
==== | ==== Dominion ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 26/25, 36/35, 56/55, 64/63 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 12, 17c }} | |||
Badness: | Badness (Sintel): 1.13 | ||
=== | === Domineering === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 36/35, 45/44, 64/63 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 7, 12 }} | |||
Badness: 0. | Badness (Sintel): 0.727 | ||
== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 36/35, 45/44, 52/49, 64/63 | |||
Mapping: {{mapping| 1 0 -4 6 -6 10 | 0 1 4 -2 6 -4 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1198.1958{{c}}, ~3/2 = 694.7159{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.6809{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 12 }} | |||
Badness (Sintel): 1.12 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63 | |||
Mapping: {{mapping| 1 0 -4 6 -6 10 12 | 0 1 4 -2 6 -4 -5 }} | |||
Optimal tunings: | |||
* | * WE: ~2 = 1197.7959{{c}}, ~3/2 = 694.8362{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.0834{{c}} | ||
{{Optimal ET sequence|legend=0| 7, 12 }} | |||
Badness (Sintel): 1.25 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56 | |||
Mapping: {{mapping| 1 0 -4 6 -6 10 12 9 | 0 1 4 -2 6 -4 -5 -3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1197.6198{{c}}, ~3/2 = 694.8362{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2075{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }} | ||
Badness (Sintel): 1.24 | |||
==== Dominatrix ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 27/26, 36/35, 45/44, 64/63 | |||
Mapping: {{mapping| 1 0 -4 6 -6 -1 | 0 1 4 -2 6 3 }} | |||
Mapping: {{mapping| 1 0 -4 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1193.1574{{c}}, ~3/2 = 694.5610{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7268{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 7, 12f }} | |||
Badness (Sintel): 0.756 | |||
=== Domination === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 36/35, 64/63, 77/75 | |||
Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 5e, 12e, 17c }} | ||
: | Badness (Sintel): 1.21 | ||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 26/25, 36/35, 64/63, 66/65 | |||
Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}} | |||
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }} | |||
Badness (Sintel): 1.13 | |||
: | === Arnold === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 22/21, 33/32, 36/35 | |||
Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }} | |||
Optimal tunings: | |||
* | * WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}} | |||
{{Optimal ET sequence|legend=0| 5, 7, 12e }} | |||
{{ | |||
Badness (Sintel): 0.864 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 22/21, 27/26, 33/32, 36/35 | |||
Mapping: {{mapping| 1 0 -4 6 5 -1 | 0 1 4 -2 -1 3 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1197.8123{{c}}, ~3/2 = 695.4727{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5713{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 5, 7 }} | ||
Badness (Sintel): 0.963 | |||
==== 17-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49 | |||
Mapping: {{mapping| 1 0 -4 6 5 -1 12 | 0 1 4 -2 -1 3 -5 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1197.6327{{c}}, ~3/2 = 695.6030{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9316{{c}} | |||
{{Optimal ET sequence|legend=0| 5, 7 }} | |||
Badness (Sintel): 1.25 | |||
==== 19-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56 | |||
Mapping: {{mapping| 1 0 -4 6 5 -1 12 9 | 0 1 4 -2 -1 3 -5 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1197.5295{{c}}, ~3/2 = 695.6325{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0579{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }} | |||
Badness: | Badness (Sintel): 1.28 | ||
==== | == Sharptone == | ||
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val. | |||
However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone). | |||
The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 21/20, 28/27 | |||
{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}} | |||
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}} | |||
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }} | |||
==== | {{Optimal ET sequence|legend=1| 5, 7d, 12d }} | ||
Subgroup: 2.3.5.7.11 | |||
[[Badness]] (Sintel): 0.629 | |||
=== Meanertone === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 21/20, 28/27, 33/32 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 7d, 12de }} | |||
Badness (Sintel): 0.832 | |||
== Supermean == | |||
Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]]. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 81/80, 672/625 | ||
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }} | |||
Optimal | [[Optimal tuning]]s: | ||
* | * [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}} | ||
* | : [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}} | |||
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }} | |||
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }} | |||
Badness: | [[Badness]] (Sintel): 3.40 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 56/55, 81/80, 132/125 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}} | ||
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }} | |||
Badness: | Badness (Sintel): 2.09 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 26/25, 56/55, 66/65, 81/80 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}} | ||
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }} | |||
Badness: | Badness (Sintel): 1.67 | ||
== | == Mohajira == | ||
{{Main| Mohajira }} | |||
Mohajira can be viewed as derived from mohaha which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 6144/6125 | |||
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }} | |||
: mapping generators: ~2, ~128/105 | |||
==== | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}} | |||
: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}} | |||
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }} | |||
[[Minimax tuning]]: | |||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }} | |||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 6 0 -11/8 0 }} | |||
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.5 | |||
[[Tuning ranges]]: | |||
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24) | |||
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978] | |||
* | * 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978] | ||
* | |||
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly. | |||
{{Optimal ET sequence|legend=1| 7, 24, 31 }} | |||
[[Badness]] (Sintel): 1.41 | |||
Scales: [[mohaha7]], [[mohaha10]] | |||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 121/120, 176/175 | ||
Mapping: {{mapping| 1 0 | Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}} | ||
Minimax tuning: | |||
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }} | |||
: projection map: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}] | |||
: unchanged-interval (eigenmonzo) basis: 2.5 | |||
Tuning ranges: | |||
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24) | |||
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978] | |||
= | {{Optimal ET sequence|legend=0| 7, 24, 31 }} | ||
Badness (Sintel): 0.862 | |||
Scales: [[mohaha7]], [[mohaha10]] | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 66/65, 81/80, 105/104, 121/120 | |||
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 24, 31 }} | |||
Badness (Sintel): 0.966 | |||
Scales: [[mohaha7]], [[mohaha10]] | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153 | |||
Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 24, 31 }} | |||
Badness (Sintel): 1.05 | |||
Scales: [[mohaha7]], [[mohaha10]] | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152 | |||
Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }} | |||
Mapping: {{mapping| 1 0 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}} | ||
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }} | |||
Badness: | Badness (Sintel): 1.05 | ||
Scales: [[mohaha7]], [[mohaha10]] | |||
== Mohamaq == | |||
Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as {{nowrap| 17c & 24 }}; its ploidacot is dicot, the same as mohajira. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 81/80, 392/375 | ||
{{Mapping|legend=1| 1 0 - | {{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }} | ||
: mapping generators: ~2, ~25/21 | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}} | ||
* [[ | : [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}} | |||
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 7d, 17c, 24 }} | ||
[[Badness]] (Sintel): 1.97 | |||
[[ | Scales: [[mohaha7]], [[mohaha10]] | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 56/55, 77/75, 243/242 | ||
Mapping: {{mapping| 1 0 - | Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}} | ||
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }} | |||
Badness: | Badness (Sintel): 1.20 | ||
Scales: [[mohaha7]], [[mohaha10]] | |||
[[ | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 56/55, 66/65, 77/75, 243/242 | |||
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}} | |||
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }} | |||
Badness (Sintel): 1.19 | |||
Scales: [[mohaha7]], [[mohaha10]] | |||
== Liese == | |||
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span> | |||
Liese splits the [[3/1|perfect twelfth]] into three generators of ~10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. It may be described as {{nowrap| 17c & 19 }}; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 686/675 | |||
= | {{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }} | ||
: mapping generators: ~2, ~10/7 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}} | |||
: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}} | |||
: error map: {{val| 0.000 -4.263 +4.454 -10.622 }} | |||
[[Minimax tuning]]: | |||
* | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }} | ||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }} | |||
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | |||
[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges. | |||
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }} | |||
[[Badness]] (Sintel): 1.18 | |||
=== Liesel === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 56/55, 81/80, 540/539 | |||
Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 19, 36 }} | |||
Badness (Sintel): 1.35 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 56/55, 78/77, 81/80, 91/90 | |||
Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 19, 36 }} | ||
Badness (Sintel): 1.13 | |||
=== Elisa === | |||
=== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 77/75, 81/80, 99/98 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 17c, 19e, 36e }} | ||
Badness (Sintel): 1.37 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 66/65, 77/75, 81/80, 99/98 | |||
Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}} | |||
= | {{Optimal ET sequence|legend=0| 17c, 19e, 36e }} | ||
Badness (Sintel): 1.11 | |||
=== Lisa === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 45/44, 81/80, 343/330 | |||
Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}} | |||
{{Optimal ET sequence|legend=0| 17cee, 19 }} | |||
Badness (Sintel): 1.81 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 45/44, 81/80, 91/88, 147/143 | |||
Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }} | |||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}} | |||
{{Optimal ET sequence|legend=0| 17cee, 19 }} | |||
Badness (Sintel): 1.49 | |||
== | == Superpine == | ||
{{See also| No-sevens subgroup temperaments #Superpine }} | |||
The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos ([[7L 8s]]) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 1119744/1071875 | |||
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}} | |||
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}} | |||
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }} | |||
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }} | |||
[[Badness]] (Sintel): 3.46 | |||
[[ | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 176/175, 864/847 | |||
: mapping | Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 7, 36, 43 }} | ||
Badness (Sintel): 1.90 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 78/77, 81/80, 144/143, 176/175 | |||
Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 7, 36, 43 }} | ||
Badness (Sintel): 1.52 | |||
== Lithium == | |||
Lithium is named after the 3rd element for having a 3rd-octave period (and also for lithium's molar mass of 6.9 g/mol since 69edo supports it). Its ploidacot is triploid monocot. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 3125/3087 | |||
{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }} | |||
: mapping generators: ~56/45, ~3 | |||
Optimal tuning | [[Optimal tuning]]s: | ||
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}}) | |||
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }} | |||
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}}) | |||
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }} | |||
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }} | |||
Badness: | [[Badness]] (Sintel): 1.75 | ||
== Squares == | |||
{{Main| Squares }} | |||
Squares splits the [[6/1|6th harmonic]] into four subminor sixths of [[11/7]]~[[14/9]] (or splits a [[8/3|perfect eleventh]] into four supermajor thirds of [[9/7]]~[[14/11]]), and uses it for a generator. It may be described as {{nowrap| 14c & 17c }}; its ploidacot is beta-tetracot. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8-, 11-, and 14-note mos scales available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, | [[Comma list]]: 81/80, 2401/2400 | ||
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }} | |||
: mapping generators: ~2, ~14/9 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}} | |||
: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}} | |||
: error map: {{val| 0.000 -5.331 +0.183 -1.422 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~ | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }} | ||
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2 | : [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }} | ||
: [[eigenmonzo basis| | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Algebraic generator]]: | [[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents. | ||
{{Optimal ET sequence|legend=1| 17c, | {{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.16 | ||
=== | Scales: [[skwares8]], [[skwares11]], [[skwares14]] | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 99/98, 121/120 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}} | |||
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }} | |||
Badness: 0. | Badness (Sintel): 0.715 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 66/65, 81/80, 99/98, 121/120 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}} | |||
{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }} | |||
Badness: | Badness (Sintel): 1.05 | ||
=== | ==== Squad ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 77 | Comma list: 78/77, 81/80, 91/90, 99/98 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 14cf, 17c, 31f }} | ||
Badness (Sintel): 1.11 | |||
==== Agora ==== | |||
==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 99/98, 105/104, 121/120 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}} | |||
{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }} | |||
Badness: | Badness (Sintel): 1.01 | ||
=== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 99/98, 105/104, 120/119, 121/119 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}} | |||
{{Optimal ET sequence|legend=0| 14cf, 31 }} | |||
Badness: | Badness (Sintel): 1.15 | ||
==== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}} | |||
{{Optimal ET sequence|legend=0| 14cf, 31 }} | |||
Badness (Sintel): 1.15 | |||
=== Cuboctahedra === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 385/384, 1375/1372 | |||
Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}} | |||
{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }} | |||
Badness (Sintel): 1.88 | |||
== Jerome == | |||
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, | [[Comma list]]: 81/80, 17280/16807 | ||
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }} | |||
: mapping generators: ~2, ~54/49 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}} | |||
: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}} | |||
: error map: {{val| 0.000 -5.191 +0.741 +6.643 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 17c, 26, 43 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.75 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 81/80, 99/98, 864/847 | ||
Mapping: {{mapping| 1 2 | Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 26, 43 }} | |||
Badness: | Badness (Sintel): 1.58 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 78/77, 81/80, 144/143 | Comma list: 78/77, 81/80, 99/98, 144/143 | ||
Mapping: {{mapping| 1 2 | Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 26, 43 }} | |||
Badness (Sintel): 1.21 | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187 | |||
== | Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}} | |||
{{Optimal ET sequence|legend=0| 17cg, 26, 43 }} | |||
Badness (Sintel): 1.06 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
: | Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143 | ||
Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}} | |||
{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }} | |||
Badness (Sintel): 1.11 | |||
== Meantritone == | |||
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, | [[Comma list]]: 81/80, 16875/16807 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~10/7 | ||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1201.3832{{c}}, ~10/7 = 619.9478{{c}} | |||
: [[error map]]: {{val| +1.383 -3.599 +1.576 +0.499 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.3176{{c}} | |||
: error map: {{val| 0.000 -5.367 +0.038 -1.791 }} | |||
{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }} | |||
[[ | [[Badness]] (Sintel): 2.08 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, 99/98, | Comma list: 81/80, 99/98, 2541/2500 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 29cde, 31 }} | ||
Badness (Sintel): 1.42 | |||
== Injera == | |||
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as {{nowrap| 12 & 26 }}; its ploidacot is diploid monocot. It tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera. | |||
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name] | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 50/49, 81/80 | |||
{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }} | |||
: mapping generators: ~7/5, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~7/5 = 600.6662{{c}}, ~3/2 = 695.1463{{c}} (~21/20 = 94.4801{{c}}) | |||
: [[error map]]: {{val| +1.332 -5.476 -5.729 +12.425 }} | |||
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.7712{{c}} (~21/20 = 94.7712{{c}}) | |||
: error map: {{val| 0.000 -7.184 -7.229 +10.259 }} | |||
[[Tuning ranges]]: | |||
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12) | |||
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955] | |||
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | |||
{{Optimal ET sequence|legend=1| 12, 26, 38 }} | |||
[[Badness]] (Sintel): 0.788 | |||
; Music | |||
* [https://web.archive.org/web/20201127013520/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 ''Two Pairs of Socks''] by [[Igliashon Jones]] – in [[26edo]] tuning | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 45/44, 50/49, 81/80 | |||
Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}}) | |||
==== | Tuning ranges: | ||
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12) | |||
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | |||
{{Optimal ET sequence|legend=0| 12, 26 }} | |||
Badness (Sintel): 0.764 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 45/44, 50/49, 78/77, 81/80 | ||
Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 600.9982{{c}}, ~3/2 = 693.8249{{c}} (~21/20 = 92.8267{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.0992{{c}} (~21/20 = 93.0992{{c}}) | |||
Tuning ranges: | |||
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26) | |||
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | |||
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }} | |||
Badness: 0. | Badness (Sintel): 0.891 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 45/44, 50/49, 78/77, 81/80, 85/84 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~7/5 = 601.1757{{c}}, ~3/2 = 693.8441{{c}} (~21/20 = 92.6684{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.8879{{c}} (~21/20 = 92.8879{{c}}) | |||
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }} | |||
Badness: 0. | Badness (Sintel): 0.935 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~7/5 = 601.4245{{c}}, ~3/2 = 693.9426{{c}} (~21/20 = 92.5181{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.7606{{c}} (~21/20 = 92.7606{{c}}) | |||
{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }} | |||
Badness: 0. | Badness (Sintel): 0.920 | ||
=== | ==== Enjera ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 27/26, 40/39, 45/44, 50/49 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }} | ||
{{ | Optimal tunings: | ||
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}}) | |||
Optimal | {{Optimal ET sequence|legend=0| 10cdeef, 12f }} | ||
Badness (Sintel): 1.10 | |||
=== Injerous === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 33/32, 50/49, 55/54 | |||
Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}}) | |||
{{ | {{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }} | ||
: | Badness (Sintel): 1.28 | ||
=== Lahoh === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 50/49, 56/55, 81/77 | |||
{{ | Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }} | ||
Optimal tunings: | |||
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}}) | |||
= | {{Optimal ET sequence|legend=0| 10cd, 12 }} | ||
Badness (Sintel): 1.42 | |||
Mapping: {{mapping| 1 | === Teff === | ||
{{Main| Teff }} | |||
Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot. | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 50/49, 81/80, 864/847 | |||
Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }} | |||
: mapping generators: ~7/5, ~16/11 | |||
Optimal | Optimal tunings: | ||
* WE: ~7/5 = 600.2802{{c}}, ~16/11 = 647.7720{{c}} (~33/32 = 47.4918{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5224{{c}} (~33/32 = 47.5224{{c}}) | |||
{{Optimal ET sequence|legend=0| 24d, 26, 50d }} | |||
Badness: | Badness (Sintel): 2.34 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 78/77, 81/80 | Comma list: 50/49, 78/77, 81/80, 144/143 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~7/5 = 600.3037{{c}}, ~16/11 = 647.7954{{c}} (~33/32 = 47.4917{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5256{{c}} (~33/32 = 47.5256{{c}}) | |||
{{Optimal ET sequence|legend=0| 24d, 26, 50d }} | |||
Badness: | Badness (Sintel): 1.65 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 78/77, 81/80, | Comma list: 50/49, 78/77, 81/80, 85/84, 144/143 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~7/5 = 600.5123{{c}}, ~16/11 = 647.8970{{c}} (~34/33 = 47.3846{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4314{{c}} (~34/33 = 47.4314{{c}}) | |||
{{Optimal ET sequence|legend=0| 24d, 26 }} | |||
Badness: | Badness (Sintel): 1.50 | ||
=== 19-limit === | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 78/77, 81/80, | Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~7/5 = 600.6308{{c}}, ~16/11 = 648.0424{{c}} (~34/33 = 47.4116{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4715{{c}} (~34/33 = 47.4715{{c}}) | |||
{{Optimal ET sequence|legend=0| 24d, 26 }} | |||
Badness: | Badness (Sintel): 1.41 | ||
== | == Pombe == | ||
Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, | [[Comma list]]: 81/80, 300125/294912 | ||
{{Mapping|legend=1| 1 4 | {{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }} | ||
: mapping generators: ~735/512, ~35/24 | |||
[[Optimal tuning]] ([[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~735/512 = 601.0652{{c}}, ~35/24 = 648.9295{{c}} (~36/35 = 47.8642{{c}}) | |||
: [[error map]]: {{val| +2.130 -3.031 +0.861 -1.756 }} | |||
* [[CWE]]: ~735/512 = 600.0000{{c}}, ~35/24 = 647.8628{{c}} (~36/35 = 47.8628{{c}}) | |||
: error map: {{val| 0.000 -6.229 -3.411 -8.140 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.94 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 81/80, 245/242, 385/384 | ||
Mapping: {{mapping| 1 4 | Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.7890{{c}}, ~16/11 = 648.7592{{c}} (~36/35 = 47.9701{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.9516{{c}} (~36/35 = 47.9516{{c}}) | |||
{{Optimal ET sequence|legend=0| 24, 26, 50 }} | |||
Badness: | Badness (Sintel): 1.72 | ||
== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 105/104, 144/143, 245/242 | |||
Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }} | |||
Optimal tunings: | |||
* WE: ~99/70 = 600.6971{{c}}, ~16/11 = 648.6029{{c}} (~36/35 = 47.9058{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}}) | |||
{{ | {{Optimal ET sequence|legend=0| 24, 26, 50 }} | ||
: | Badness (Sintel): 1.28 | ||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272 | |||
Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }} | |||
{{ | Optimal tunings: | ||
* WE: ~17/12 = 600.7610{{c}}, ~16/11 = 648.6638{{c}} (~36/35 = 47.9028{{c}}) | |||
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}}) | |||
{{Optimal ET sequence|legend=0| 24, 26, 50 }} | |||
Badness (Sintel): 1.08 | |||
=== | === 19-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209 | ||
Mapping: {{mapping| 2 0 | Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }} | ||
{{ | Optimal tunings: | ||
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}}) | |||
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}}) | |||
Optimal | {{Optimal ET sequence|legend=0| 24, 26, 50 }} | ||
Badness (Sintel): 1.01 | |||
== Orphic == | |||
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 5898240/5764801 | |||
{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }} | |||
: mapping generators: ~2401/1728, ~343/288 | |||
Optimal tuning ( | [[Optimal tuning]]s: | ||
* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}}) | |||
: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }} | |||
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}}) | |||
: error map: {{val| 0.000 -5.041 +1.342 +3.860 }} | |||
{{Optimal ET sequence|legend=1| 26, 48c, 74 }} | |||
[[Badness]] (Sintel): 6.55 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 99/98, 73728/73205 | |||
Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }} | |||
Optimal tunings: | |||
* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}}) | |||
* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}}) | |||
Optimal | {{Optimal ET sequence|legend=0| 26, 48c, 74 }} | ||
Badness (Sintel): 3.36 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 99/98, 144/143, 2200/2197 | |||
Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }} | |||
Optimal tunings: | |||
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}}) | |||
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}}) | |||
Optimal | {{Optimal ET sequence|legend=0| 26, 48c, 74 }} | ||
Badness (Sintel): 2.21 | |||
== Cloudtone == | |||
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 81/80, 16807/16384 | ||
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }} | |||
: mapping generators: ~8/7, ~3 | |||
Optimal | [[Optimal tuning]]s: | ||
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}}) | |||
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }} | |||
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}}) | |||
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }} | |||
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }} | |||
=== | [[Badness]] (Sintel): 2.59 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 385/384, 2401/2376 | ||
Mapping: {{mapping| | Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}}) | |||
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}}) | |||
{{Optimal ET sequence|legend=0| 5, 45, 50 }} | |||
Badness: | Badness (Sintel): 2.33 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 105/104, 144/143, 2401/2376 | ||
Mapping: {{mapping| | Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}}) | |||
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}}) | |||
{{Optimal ET sequence|legend=0| 5, 45f, 50 }} | |||
Badness: | Badness (Sintel): 2.02 | ||
=== | == Subgroup extensions == | ||
=== Stützel (2.3.5.19) === | |||
[[Subgroup]]: 2.3.5.19 | |||
[[Comma list]]: 81/80, 96/95 | |||
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }} | |||
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }} | |||
: mapping generators: ~2, ~3 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}} | |||
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}} | |||
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }} | |||
Optimal | {{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }} | ||
[[Badness]] (Sintel): 0.324 | |||
=== Hypnotone === | |||
Hypnotone is no-sevens [[#Flattone|flattone]]. | |||
[[Subgroup]]: 2.3.5.11 | |||
Subgroup: 2.3.5 | |||
Comma list: | [[Comma list]]: 45/44, 81/80 | ||
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }} | |||
{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}} | |||
: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}} | |||
: error map: {{val| 0.000 -8.047 -10.680 +12.133 }} | |||
= | {{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }} | ||
[[Badness]] (Sintel): 0.326 | |||
==== 2.3.5.11.13 subgroup ==== | |||
Subgroup: 2.3.5.11.13 | |||
Comma list: 45/44, 65/64, 81/80 | |||
Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }} | |||
Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }} | |||
Badness (Sintel): 0.561 | |||
=== Dequarter === | |||
[[Subgroup]]: 2.3.5.11 | |||
[[Comma list]]: 33/32, 55/54 | |||
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }} | |||
{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }} | |||
{{Mapping|legend= | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}} | |||
[[Optimal tuning]] | : [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }} | ||
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}} | |||
{{ | : error map: {{val| 0.000 -8.834 -13.831 -44.439 }} | ||
[[ | |||
{{ | |||
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }} | {{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.451 | ||
==== Dreamtone ==== | ==== Dreamtone ==== | ||
| Line 2,605: | Line 2,479: | ||
Comma list: 33/32, 55/54, 975/968 | Comma list: 33/32, 55/54, 975/968 | ||
Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }} | |||
: | Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }} | ||
Badness (Sintel): 1.40 | |||
== References == | |||
<references/> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Meantone family| ]] <!-- main article --> | [[Category:Meantone family| ]] <!-- main article --> | ||
[[Category:Meantone| ]] <!-- key article --> | [[Category:Meantone| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||