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| +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]]: | ||
| Line 34: | Line 37: | ||
{{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. | ||
| Line 63: | Line 66: | ||
==== Splitting the meantone fifth into three (1029/1024) ==== | ==== Splitting the meantone fifth into three (1029/1024) ==== | ||
By tempering [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] (S6/S7), the orwellisma. | By tempering out [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] (S6/S7), the orwellisma. | ||
This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called | This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called '''cynder'''. Though undecimal mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out [[176/175]] (S8/S10), which is (11/7)/(5/4)<sup>2</sup>, taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])<sup>2</sup> = [[36/25]] = ([[3/2]])/([[25/24]]). | ||
==== 31edo as splitting the fifth into two, three and nine ==== | ==== 31edo as splitting the fifth into two, three and nine ==== | ||
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering 225/224, which interestingly, though a rank 2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering 225/224). | [[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224). | ||
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. | ||
| Line 84: | Line 86: | ||
{{Wikipedia| Septimal meantone temperament }} | {{Wikipedia| Septimal meantone temperament }} | ||
In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth ( | In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). This mapping is rationalized by the fact that 81/80 factors as ([[126/125]])⋅([[225/224]]), and septimal meantone tempers out both of these commas as well as their difference, [[3136/3125]]. In fact it can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125, 225/224, and 3136/3125. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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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| +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|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 112: | Line 114: | ||
{{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 | 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𝄪). See [[chords of huygens]] for a list of dyadic chords in this temperament. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 125: | 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 143: | 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= | {{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }} | ||
Badness: 0. | Badness (Sintel): 0.563 | ||
; Music | ; Music | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] | ||
==== | ==== Grosstone ==== | ||
Grosstone, named for tempering out the [[grossma]], is the main extension of interest that extends undecimal meantone to the 13-limit. It maps 13/8 to the double-diminished seventh (C–B♭♭♭). Note also that 11/10 is a double-augmented unison; 12/11~13/12 is a double-diminished third; and 14/13 is a triple-augmented seventh octave reduced. Grosstone is flexible with its tunings; among the good tunings are [[31edo]], [[43edo]], and [[74edo]]. | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 99/98, 126/125, 144/143 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }} | ||
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: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| | * 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }} | ||
: eigenmonzo (unchanged-interval) | : eigenmonzo basis (unchanged-interval basis): 2.13/7 | ||
Tuning ranges: | |||
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43) | |||
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 31, 43, 74 }} | ||
Badness: | Badness (Sintel): 1.07 | ||
===== | ===== 17-limit ===== | ||
This extension maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth. | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 99/98, 120/119, 126/125, 144/143 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }} | ||
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= | {{Optimal ET sequence|legend=0| 12, 31, 43, 74g }} | ||
Badness: | Badness (Sintel): 1.06 | ||
===== 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, 96/95, 99/98, 120/119, 126/125, 144/143 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }} | ||
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= | {{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }} | ||
Badness: | Badness (Sintel): 1.07 | ||
==== | ==== Fokkertone ==== | ||
Fokkertone 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. 31edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning. | |||
This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025. | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 66/65, 81/80, 99/98, 105/104 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }} | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | |||
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: | ||
* 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 (Sintel): 0.746 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 243: | Line 238: | ||
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= | {{Optimal ET sequence|legend=0| 12f, 31 }} | ||
Badness: | Badness (Sintel): 1.02 | ||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 258: | Line 253: | ||
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= | {{Optimal ET sequence|legend=0| 12f, 31 }} | ||
Badness: | Badness (Sintel): 1.10 | ||
==== | ==== Meridetone ==== | ||
Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪). 43edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning. | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 81/80, 99/98, 126/125 | Comma list: 78/77, 81/80, 99/98, 126/125 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }} | ||
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| | * 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 (Sintel): 1.09 | |||
Badness: | |||
===== 17-limit ===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 81/80, 99/98, 120/119, 126/125 | Comma list: 78/77, 81/80, 99/98, 120/119, 126/125 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | 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= | {{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: 81/80, 96/95, 99/98, 120/119, 126/125 | Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 | 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= | {{Optimal ET sequence|legend=0| 12f, 43 }} | ||
Badness: | Badness (Sintel): 1.25 | ||
==== | ==== Hemimeantone ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 99/98, 126/125, 169/168 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }} | |||
: mapping generators: ~2, ~26/15 | |||
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 (Sintel): 1.30 | |||
===== 17-limit ===== | |||
===== | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 99/98, 126/125, 169/168, 221/220 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 - | Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }} | ||
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= | {{Optimal ET sequence|legend=0| 19eg, 43, 62 }} | ||
Badness: | Badness (Sintel): 1.19 | ||
===== 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, 99/98, 126/125, 153/152, 169/168, 221/220 | ||
Mapping: {{mapping| 1 0 -4 -13 -25 - | Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }} | ||
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= | {{Optimal ET sequence|legend=0| 19egh, 43, 62 }} | ||
Badness: | Badness (Sintel): 1.15 | ||
==== | ==== Semimeantone ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 99/98, 126/125, 847/845 | ||
Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }} | |||
: mapping generators: ~55/39, ~3 | |||
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= | {{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }} | ||
Badness: | Badness (Sintel): 1.68 | ||
===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 99/98, 126/125, 221/220, 289/288 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }} | ||
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= | {{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }} | ||
Badness: | Badness (Sintel): 1.60 | ||
===== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }} | ||
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= | {{Optimal ET sequence|legend=0| 12f, 50eff, 62 }} | ||
Badness: | Badness (Sintel): 1.47 | ||
=== | === Meanpop === | ||
{{See also| Huygens vs meanpop }} | |||
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 maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. 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 | |||
Comma list: 81/80, 126/125, 385/384 | |||
{{ | Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }} | ||
: mapping generator: ~2, ~3 | |||
== | Optimal tunings: | ||
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}} | |||
Minimax tuning: | |||
* 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 }}] | |||
: unchanged-interval (eigenmonzo) basis: 2.5 | |||
Tuning ranges: | |||
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31) | |||
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | |||
: | 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 | {{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }} | ||
Badness (Sintel): 0.712 | |||
; Music | |||
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}} | |||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] | |||
==== | ==== Tridecimal meanpop ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 81/80, | Comma list: 81/80, 105/104, 126/125, 144/143 | ||
Mapping: {{mapping| 1 0 -4 -13 - | 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: | ||
* 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: | |||
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31) | |||
* 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 }} | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Badness (Sintel): 0.863 | |||
===== Meanpoppic ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 81/80, | Comma list: 81/80, 105/104, 126/125, 144/143, 273/272 | ||
Mapping: {{mapping| 1 0 -4 -13 - | Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }} | ||
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= | {{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }} | ||
Badness: | Badness (Sintel): 1.02 | ||
==== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 81/80, | Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }} | ||
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= | {{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }} | ||
Badness: | Badness (Sintel): 1.08 | ||
===== | ===== Meanpoid ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 81/80, | Comma list: 81/80, 105/104, 120/119, 126/125, 144/143 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }} | ||
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= | {{Optimal ET sequence|legend=0| 19, 31 }} | ||
Badness: | Badness (Sintel): 1.17 | ||
===== 19-limit ===== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 81/80, | Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }} | ||
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 (Sintel): 1.25 | |||
==== Semimeanpop ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 126/125, 385/384, 847/845 | |||
Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }} | |||
: mapping generators: ~55/39, ~3 | |||
Optimal tunings: | |||
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}} | |||
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}} | |||
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }} | |||
Badness (Sintel): 1.78 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
: | Comma list: 81/80, 126/125, 221/220, 273/272, 289/288 | ||
{{ | Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }} | |||
Badness (Sintel): 1.45 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272 | |||
Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }} | |||
Optimal tunings: | |||
* | * WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }} | |||
Badness (Sintel): 1.28 | |||
=== | === Meanenneadecal === | ||
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal 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. | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 45/44, 56/55, 81/80 | |||
{{ | 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: | ||
* | * 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12) | ||
* | * 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377] | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }} | ||
Badness: 0. | Badness (Sintel): 0.708 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 45/44, 56/55, 78/77, 81/80 | ||
Mapping: {{mapping| 1 0 -4 -13 | 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= | {{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }} | ||
Badness: 0. | Badness (Sintel): 0.875 | ||
===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 45/44, 56/55, 78/77, 81/80, 120/119 | ||
Mapping: {{mapping| 1 0 -4 -13 | Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }} | ||
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= | {{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 | ||
Comma list: 77 | Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119 | ||
Mapping: {{mapping| 1 0 -4 -13 | Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }} | ||
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= | {{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: 65/64 | Comma list: 45/44, 56/55, 65/64, 81/80 | ||
Mapping: {{mapping| 1 0 -4 -13 | 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 (Sintel): 1.02 | |||
Badness: | |||
===== 17-limit ===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 52/51, 65/64 | Comma list: 45/44, 52/51, 56/55, 65/64, 81/80 | ||
Mapping: {{mapping| 1 0 -4 -13 | Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }} | ||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 12, 19 }} | ||
Badness: | 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: 39/38, 52/51, 65/64 | Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80 | ||
Mapping: {{mapping| 1 0 -4 -13 | 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 (Sintel): 1.36 | |||
=== Bimeantone === | |||
11/8 is mapped to half octave minus the [[128/125|meantone diesis]]. | |||
Subgroup: 2.3.5.7.11 | |||
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= | {{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= | {{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }} | ||
Badness: | Badness (Sintel): 1.19 | ||
=== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220 | |||
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }} | |||
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 (Sintel): 1.15 | |||
==== 19-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220 | |||
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }} | |||
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= | {{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= | {{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= | {{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }} | ||
Badness: | Badness (Sintel): 1.28 | ||
=== | === Migration === | ||
See [[Rastmic clan #Migration|Rastmic clan]]. | |||
== Flattone == | |||
{{Main| Flattone }} | |||
In flattone, 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. The fifth in flattone is typically flatter than that of [[19edo]]. Good tunings for flattone include [[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 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 | |||
Comma list: 45/44, 81/80, 385/384 | |||
{{ | Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }} | ||
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 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 45/44, 65/64, 78/77, 81/80 | |||
Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }} | |||
== | 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 | |||
=== Ptolemy === | |||
See [[Rastmic clan #Ptolemy|Rastmic clan]]. | |||
== 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 identifies 11/8 with the diminished fifth. Domination tempers out 77/75 and identifies 11/8 with the augmented third. Domineering identifies 11/8 with the augmented fourth, which is a very inaccurate mapping; it is however, notable for having the lowest badness among the extensions. 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 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= | {{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }} | ||
Badness: 0. | Badness (Sintel): 0.799 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 36/35, 56/55, 64/63, 66/65 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }} | ||
Optimal tunings: | 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 (Sintel): 0.996 | ||
Subgroup: 2.3.5.7.11.13 | |||
==== Dominion ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: | Comma list: 26/25, 36/35, 56/55, 64/63 | ||
Mapping: {{mapping| 1 0 -4 | 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= | {{Optimal ET sequence|legend=0| 5, 12, 17c }} | ||
Badness: | Badness (Sintel): 1.13 | ||
=== | === Domination === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 36/35, 64/63, 77/75 | ||
Mapping: {{mapping| 1 0 -4 - | Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }} | ||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 5e, 12e, 17c }} | ||
Badness: | Badness (Sintel): 1.21 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 26/25, 36/35, 64/63, 66/65 | ||
Mapping: {{mapping| 1 0 -4 - | Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }} | ||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 5e, 12e, 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 0 -4 | 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= | {{Optimal ET sequence|legend=0| 5e, 7, 12 }} | ||
Badness: 0. | Badness (Sintel): 0.727 | ||
=== | === Arnold === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 22/21, 33/32, 36/35 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }} | ||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 5, 7, 12e }} | ||
Badness: 0. | Badness (Sintel): 0.864 | ||
=== | === Neutrominant === | ||
See [[Rastmic clan #Neutrominant|Rastmic clan]]. | |||
== Flattertone == | |||
In flattertone, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). The fifth in flattertone is typically at least as flat as [[26edo]]. Here, 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. | |||
Flattertone was named by [[Flora Canou]] in 2024. | |||
[[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 | |||
Badness | |||
==== 11-limit ==== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 45/44, 81/80, 1375/1344 | ||
Mapping: {{mapping| 1 0 -4 - | 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= | {{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) | |||
== 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 }} | |||
[[Badness]] (Sintel): 0.629 | |||
=== Meanertone === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 21/20, 28/27, 33/32 | ||
Mapping: {{mapping| | 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= | {{Optimal ET sequence|legend=0| 5, 7d, 12de }} | ||
Badness: 0. | Badness (Sintel): 0.832 | ||
== | == Mildtone == | ||
Mildtone tempers out [[16128/15625]] and finds the interval class of 7 at 22 generators up, as a triple-augmented fifth (C–G#x). [[55edo]] and [[67edo]] are among the possible tunings. | |||
Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 16128/15625 | |||
{{ | {{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }} | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}} | |||
: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}} | |||
: error map: {{val| 0.000 -3.415 +7.845 -0.952 }} | |||
= | {{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }} | ||
[[Badness]] (Sintel): 2.67 | |||
=== 11-limit === | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 81/80, 176/175, 7058/6875 | |||
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}} | |||
== | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}} | |||
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }} | |||
[[Badness]] (Sintel): 2.15 | |||
=== 13-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13 | |||
[[Comma list]]: 81/80, 176/175, 196/195, 832/825 | |||
= | {{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}} | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}} | |||
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }} | |||
: | [[Badness]] (Sintel): 2.04 | ||
=== 17-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | |||
[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825 | |||
= | {{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}} | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}} | |||
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }} | |||
[[Badness]] (Sintel): 1.98 | |||
=== 19-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13.19 | |||
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825 | |||
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}} | |||
Optimal | {{Optimal ET sequence|legend=1| 12f, 55f, 67 }} | ||
[[Badness]] (Sintel): 1.95 | |||
{{Todo|unify precision|review}} | |||
== | == 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]]: 81/80, | [[Comma list]]: 81/80, 672/625 | ||
{{ | {{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }} | ||
[[Optimal tuning]]s: | [[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]] (Sintel): 3.40 | |||
[[Badness]]: | |||
=== 11-limit === | === 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 | 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 (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 (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. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, | [[Comma list]]: 81/80, 6144/6125 | ||
{{Mapping|legend=1| 1 0 | {{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~128/105 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}} | ||
* [[CWE]]: ~2 = | : [[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 | |||
Comma list: 81/80, 121/120, 176/175 | |||
{{ | Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }} | ||
== | 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 }} | ||
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 (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 | |||
[[Comma list]]: 81/80, 392/375 | |||
= | {{Mapping|legend=1| 1 1 0 -1 | 0 2 8 13 }} | ||
: mapping generators: ~2, ~25/21 | |||
[[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 | {{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 | 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= | {{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| | {{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }} | ||
Badness: | [[Badness]] (Sintel): 1.18 | ||
=== | === Liesel === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 56/55, 81/80, 540/539 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 17c, 19, 36 }} | ||
Badness: | Badness (Sintel): 1.35 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 56/55, 78/77, 81/80, 91/90 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }} | ||
Optimal | 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: | Badness (Sintel): 1.13 | ||
=== | === Elisa === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 77/75, 81/80, 99/98 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 19e, 36e }} | ||
Badness: | Badness (Sintel): 1.37 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 66/65, 77/75, 81/80, 99/98 | ||
Mapping: {{mapping| 1 0 -4 | Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 17c, 19e, 36e }} | ||
Badness: | Badness (Sintel): 1.11 | ||
=== | === Lisa === | ||
Subgroup: 2.3.5.7.11 | 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 | {{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 tunings: | |||
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}} | |||
Optimal | {{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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 81/80, 1119744/1071875 | ||
{{Mapping|legend=1| 1 | {{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 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 176/175, 864/847 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }} | ||
Optimal | 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: | 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 ET sequence|legend= | {{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]]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]] (Sintel): 1.75 | |||
== Squares == | |||
{{Main| Squares }} | |||
== | |||
{{Main| | |||
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, ~ | : mapping generators: ~2, ~14/9 | ||
{{ | [[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 | 3/2 0 9/16 0 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents. | |||
[[Algebraic generator]]: | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.16 | ||
Scales: [[ | Scales: [[skwares8]], [[skwares11]], [[skwares14]] | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, 121/120 | Comma list: 81/80, 99/98, 121/120 | ||
Mapping: | Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }} | ||
Optimal | 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 (Sintel): 0.715 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 66/65, 81/80, 99/98, 121/120 | |||
Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }} | |||
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 (Sintel): 1.05 | |||
==== Squad ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 78/77, 81/80, 91/90, 99/98 | |||
{{ | 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 ET sequence|legend=0| 14cf, 17c, 31f }} | |||
Badness (Sintel): 1.11 | |||
==== Agora ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 99/98, 105/104, 121/120 | |||
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }} | |||
{{ | 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 (Sintel): 1.01 | |||
=== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 99/98, 105/104, 120/119, 121/119 | ||
Mapping: | 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= | {{Optimal ET sequence|legend=0| 14cf, 31 }} | ||
Badness: | Badness (Sintel): 1.15 | ||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119 | |||
Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }} | |||
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 | |||
[[Comma list]]: 81/80, 17280/16807 | |||
{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }} | |||
: mapping generators: ~2, ~54/49 | |||
[[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| 17c, 26, 43 }} | |||
[[Badness]] (Sintel): 2.75 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 99/98, 864/847 | |||
Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }} | |||
== | 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 (Sintel): 1.58 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 78/77, 81/80, 99/98, 144/143 | |||
: mapping | 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 | |||
[[Comma list]]: 81/80, 16875/16807 | |||
{{ | {{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }} | ||
: 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 === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 99/98, 2541/2500 | |||
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 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 | {{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 | Subgroup: 2.3.5.7.11 | ||
Comma list: 45/44, 81/80 | Comma list: 45/44, 50/49, 81/80 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }} | ||
Optimal | 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 | |||
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-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 (Sintel): 0.891 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84 | |||
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 (Sintel): 0.935 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84 | |||
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 (Sintel): 0.920 | |||
==== Enjera ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 27/26, 40/39, 45/44, 50/49 | |||
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 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 | |||
=== 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 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | 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= | {{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: | Comma list: 50/49, 78/77, 81/80, 144/143 | ||
Mapping: | 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= | {{Optimal ET sequence|legend=0| 24d, 26, 50d }} | ||
Badness: | Badness (Sintel): 1.65 | ||
==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | 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: {{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= | {{Optimal ET sequence|legend=0| 24d, 26 }} | ||
Badness: | Badness (Sintel): 1.50 | ||
==== | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 81/80, | Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143 | ||
Mapping: | 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= | {{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 | |||
[[Comma list]]: 81/80, 300125/294912 | |||
{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }} | |||
: mapping generators: ~735/512, ~35/24 | |||
[[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| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }} | ||
[[Badness]] (Sintel): 2.94 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 245/242, 385/384 | |||
{{ | Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }} | ||
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 (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: {{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: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209 | |||
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 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| | {{Optimal ET sequence|legend=1| 26, 48c, 74 }} | ||
Badness: | [[Badness]] (Sintel): 6.55 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 99/98, 73728/73205 | ||
Mapping: | Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }} | ||
Optimal | 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 ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 48c, 74 }} | ||
Badness: | Badness (Sintel): 3.36 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 99/98, 144/143, 2200/2197 | ||
Mapping: | Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }} | ||
Optimal | 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 ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 48c, 74 }} | ||
Badness: | Badness (Sintel): 2.21 | ||
== | == Cloudtone == | ||
The | 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]]: 81/80, | [[Comma list]]: 81/80, 16807/16384 | ||
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }} | |||
: mapping generators: ~8/7, ~3 | |||
[[Optimal tuning]] | [[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| | {{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.59 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 81/80, 385/384, 2401/2376 | ||
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= | {{Optimal ET sequence|legend=0| 5, 45, 50 }} | ||
Badness: | Badness (Sintel): 2.33 | ||
== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 105/104, 144/143, 2401/2376 | |||
Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }} | |||
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 (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 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 | |||
[[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 generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}} | |||
: [[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 | {{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }} | ||
[[Badness]] (Sintel): 0.451 | |||
==== Dreamtone ==== | |||
Subgroup: 2.3.5.11.13 | |||
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 | 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]] | ||