Meantone family: Difference between revisions
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The | {{Technical data page}} | ||
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 }} | ||
Subgroup: 2.3.5 | 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 | |||
[[Comma list]]: 81/80 | [[Comma list]]: 81/80 | ||
{{Mapping|legend=1| 1 0 -4 | 0 1 4 }} | |||
: mapping generators: ~2, ~3 | |||
[[ | [[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 = {{ | * [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma) | ||
: [[Eigenmonzo]] | : [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5) | * 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5) | ||
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] | * 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.) | ||
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }} | |||
[[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 {{ | * Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo. | ||
* Dominant adds [[64/63|{{ | * 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. | ||
* Sharptone adds [[28/27|{{ | * 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, 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 {{ | * Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period. | ||
* Mohajira adds {{ | * Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two. | ||
* Godzilla adds [[49/48|{{ | * 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|{{ | * Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth. | ||
* Liese adds {{ | * Liese adds {{monzo| -9 11 0 -3 }} with a ~[[10/7]] generator, three of which give the [[3/1|twelfth]]. | ||
* Squares adds {{ | * Squares adds {{monzo| -3 9 0 -4 }} with a ~[[9/7]] generator, four of which give the [[8/3|eleventh]]. | ||
* Jerome adds {{ | * Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five. | ||
==== | ==== Strong extensions ==== | ||
For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)<sup>3</sup> as [[7/5]], leading to septimal meantone, a very elegant extension to the 7-limit. | |||
For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant. | |||
Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in [[weak extension]]s. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason. | |||
==== Splitting the meantone fifth into two (243/242) ==== | |||
By tempering out [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)<sup>2</sup>]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp. There is also the consideration that tempering out [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups. | |||
==== Splitting the meantone fifth into three (1029/1024) ==== | |||
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 '''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]] 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 | |||
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]] | |||
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]] | |||
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]] | |||
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]] | |||
The rest are considered below. | |||
== Septimal meantone == | == Septimal meantone == | ||
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div> | <div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div> | ||
{{ | {{Main| Meantone #Septimal meantone}} | ||
{{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 (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 | ||
[[Comma list]]: 81/80, 126/125 | [[Comma list]]: 81/80, 126/125 | ||
{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }} | |||
[[ | [[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 = {{ | * [[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 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12) | * 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12) | ||
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] | * 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.) | ||
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] | * 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | ||
[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly. | [[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly. | ||
{{ | {{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.347 | ||
=== Undecimal meantone (huygens) === | |||
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}} | |||
{{See also| Huygens vs meanpop }} | |||
== | 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 138: | Line 126: | ||
Comma list: 81/80, 99/98, 126/125 | Comma list: 81/80, 99/98, 126/125 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }} | ||
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 }} | ||
: [{{ | : 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: | ||
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12) | * 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12) | ||
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] | * 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.) | ||
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 | {{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: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }} | ||
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| 8/13 0 0 1/26 0 -1/26 }} | ||
: | : 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 | {{Optimal ET sequence|legend=0| 12, 31, 43, 74 }} | ||
Badness: 0. | 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 | |||
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }} | |||
Badness (Sintel): 1.06 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }} | |||
Badness (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 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 81/80, 99/98, | Comma list: 66/65, 81/80, 99/98, 105/104 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{ | * 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 | |||
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 31 }} | |||
Badness (Sintel): 1.02 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }} | |||
Optimal tunings: | |||
* | * WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}} | ||
Optimal | {{Optimal ET sequence|legend=0| 12f, 31 }} | ||
Badness: | Badness (Sintel): 1.10 | ||
==== Meridetone ==== | ==== 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: 78/77, 81/80, 99/98, 126/125 | Comma list: 78/77, 81/80, 99/98, 126/125 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }} | ||
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 = {{ | * 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 | {{Optimal ET sequence|legend=0| 12f, 31f, 43 }} | ||
Badness: | Badness (Sintel): 1.09 | ||
==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 81/80, 99/98, 126/125 | Comma list: 78/77, 81/80, 99/98, 120/119, 126/125 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 43 }} | |||
Badness (Sintel): 1.22 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 43 }} | |||
Badness (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 | 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 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 81/80, | Comma list: 81/80, 99/98, 126/125, 169/168, 221/220 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}} | |||
{{Optimal ET sequence|legend=0| 19eg, 43, 62 }} | |||
Badness (Sintel): 1.19 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220 | |||
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }} | |||
==== | Optimal tunings: | ||
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}} | |||
{{Optimal ET sequence|legend=0| 19egh, 43, 62 }} | |||
Badness (Sintel): 1.15 | |||
==== Semimeantone ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
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: | |||
* | * WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}} | ||
* | * CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}} | ||
Optimal | {{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }} | ||
Badness (Sintel): 1.68 | |||
Subgroup: 2.3.5.7.11.13.17 | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 81/80, | Comma list: 81/80, 99/98, 126/125, 221/220, 289/288 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }} | ||
Optimal tunings: | |||
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}} | |||
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }} | |||
Badness (Sintel): 1.60 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220 | |||
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }} | |||
Optimal tunings: | |||
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}} | |||
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }} | |||
Badness (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 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 126/125, 385/384 | ||
Mapping: | 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 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 | |||
Comma list: 81/80, 105/104, 126/125, 144/143 | |||
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }} | |||
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 | {{Optimal ET sequence|legend=0| 19, 31, 50, 81 }} | ||
Badness: 0. | Badness (Sintel): 0.863 | ||
===== | ===== Meanpoppic ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 105/104, 126/125, 144/143, 273/272 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }} | ||
Badness: | Badness (Sintel): 1.02 | ||
===== 19-limit ===== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}} | |||
Optimal | {{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: | Comma list: 81/80, 105/104, 120/119, 126/125, 144/143 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 19, 31 }} | ||
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: | Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 19, 31 }} | ||
Badness: | Badness (Sintel): 1.25 | ||
==== | ==== Semimeanpop ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 126/125, 385/384, 847/845 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }} | ||
: mapping generators: ~55/39, ~3 | |||
Optimal | 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 | 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 | 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: | |||
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}} | |||
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=0| 7d, 12, 19, 31e }} | |||
Badness (Sintel): 0.708 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 45/44, 56/55, 78/77, 81/80 | |||
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}} | |||
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }} | |||
Badness (Sintel): 0.875 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119 | |||
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 19, 31e }} | |||
Badness (Sintel): 1.17 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119 | |||
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 19, 31e }} | |||
Badness (Sintel): 1.23 | |||
==== Vincenzo ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 45/44, 56/55, 65/64, 81/80 | |||
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }} | |||
== | 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 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80 | |||
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }} | |||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 19 }} | |||
Badness (Sintel): 1.30 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80 | |||
Mapping | Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 12, 19 }} | ||
Badness: | Badness (Sintel): 1.36 | ||
=== Bimeantone === | === Bimeantone === | ||
| Line 589: | Line 689: | ||
Comma list: 81/80, 126/125, 245/242 | Comma list: 81/80, 126/125, 245/242 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }} | ||
: mapping generators: ~63/44, ~3 | |||
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 | {{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }} | ||
Badness: | Badness (Sintel): 1.26 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 604: | Line 706: | ||
Comma list: 81/80, 105/104, 126/125, 245/242 | Comma list: 81/80, 105/104, 126/125, 245/242 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }} | ||
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 | {{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }} | ||
Badness: | Badness (Sintel): 1.19 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 619: | Line 721: | ||
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220 | Comma list: 81/80, 105/104, 126/125, 189/187, 221/220 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 12f, 38df, 50 }} | ||
Badness: | Badness (Sintel): 1.15 | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
| Line 634: | Line 736: | ||
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220 | Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220 | ||
Mapping: | 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: | |||
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}} | |||
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }} | |||
Badness (Sintel): 1.68 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 126/125, 144/143, 364/363 | |||
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}} | |||
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }} | |||
Badness (Sintel): 1.46 | |||
Badness: | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220 | |||
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}} | |||
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }} | |||
Badness (Sintel): 1.28 | |||
=== Migration === | |||
See [[Rastmic clan #Migration|Rastmic clan]]. | |||
== Flattone == | == Flattone == | ||
{{Main| Flattone }} | |||
Subgroup: 2.3.5.7 | 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 | [[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]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]]: ~3/2 = {{ | * [[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 = {{ | * [[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]]: | [[Tuning ranges]]: | ||
| Line 669: | Line 827: | ||
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955] | * 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955] | ||
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 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. | [[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]]: 0. | [[Badness]] (Sintel): 0.976 | ||
=== 11-limit === | |||
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]]. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 45/44, 81/80, 385/384 | Comma list: 45/44, 81/80, 385/384 | ||
Mapping: | 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: | Tuning ranges: | ||
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19) | * 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] | * 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 ==== | ==== 13-limit ==== | ||
| Line 705: | Line 860: | ||
Comma list: 45/44, 65/64, 78/77, 81/80 | Comma list: 45/44, 65/64, 78/77, 81/80 | ||
Mapping: | 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: | 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 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] | * 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 === | === 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 | [[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]]: | [[Tuning ranges]]: | ||
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5) | * [[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] | * 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587] | ||
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 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]]: 0. | [[Badness]] (Sintel): 0.524 | ||
=== 11-limit === | === 11-limit === | ||
| Line 774: | Line 911: | ||
Comma list: 36/35, 56/55, 64/63 | Comma list: 36/35, 56/55, 64/63 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }} | ||
Tuning ranges: | Tuning ranges: | ||
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17) | * 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] | * 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587] | ||
Optimal tunings: | |||
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }} | ||
Badness: 0. | Badness (Sintel): 0.799 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 792: | Line 930: | ||
Comma list: 36/35, 56/55, 64/63, 66/65 | Comma list: 36/35, 56/55, 64/63, 66/65 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}} | |||
Tuning ranges: | Tuning ranges: | ||
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17) | * 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] | * 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587] | ||
Optimal | {{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }} | ||
Badness: 0. | Badness (Sintel): 0.996 | ||
==== Dominion ==== | ==== Dominion ==== | ||
| Line 810: | Line 949: | ||
Comma list: 26/25, 36/35, 56/55, 64/63 | Comma list: 26/25, 36/35, 56/55, 64/63 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}} | |||
Optimal | {{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: 36/35, | Comma list: 36/35, 64/63, 77/75 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 5e, 12e, 17c }} | ||
Badness: | Badness (Sintel): 1.21 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 36/35, | Comma list: 26/25, 36/35, 64/63, 66/65 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}} | |||
Optimal | {{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: 36/35, 64/63 | Comma list: 36/35, 45/44, 64/63 | ||
Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }} | |||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}} | |||
{{Optimal ET sequence|legend=0| 5e, 7, 12 }} | |||
Badness (Sintel): 0.727 | |||
Badness: 0. | |||
=== Arnold === | === Arnold === | ||
| Line 914: | Line 1,009: | ||
Comma list: 22/21, 33/32, 36/35 | Comma list: 22/21, 33/32, 36/35 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}} | |||
Optimal | {{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: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}} | |||
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }} | |||
Badness (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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 21/20, 28/27 | [[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]]: 0. | [[Badness]] (Sintel): 0.629 | ||
=== Meanertone === | === Meanertone === | ||
| Line 1,018: | Line 1,091: | ||
Comma list: 21/20, 28/27, 33/32 | Comma list: 21/20, 28/27, 33/32 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}} | |||
Optimal | {{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 === | === 11-limit === | ||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 81/80, 176/175, 7056/6875 | |||
Mapping | {{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 | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: | [[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, 189/187, 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]]: 2.3.5.7.11.13.17.19 | |||
[[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 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: | [[Comma list]]: 81/80, 672/625 | ||
{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }} | |||
[[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 | |||
=== 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: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }} | ||
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: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }} | ||
Badness: | Badness (Sintel): 1.67 | ||
== Mohajira == | == Mohajira == | ||
{{Main| Mohajira }} | {{Main| Mohajira }} | ||
Mohajira can be viewed as derived from mohaha which maps the interval | 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, 6144/6125 | [[Comma list]]: 81/80, 6144/6125 | ||
{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }} | |||
: mapping generators: ~2, ~128/105 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}} | |||
[[ | : [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}} | |||
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{ | * [[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]]: | [[Tuning ranges]]: | ||
| Line 1,233: | Line 1,265: | ||
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978] | * 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978] | ||
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 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. | [[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]]: | [[Badness]] (Sintel): 1.41 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
| Line 1,249: | Line 1,279: | ||
Comma list: 81/80, 121/120, 176/175 | Comma list: 81/80, 121/120, 176/175 | ||
Mapping: | 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: | Minimax tuning: | ||
* | * 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }} | ||
: [{{Monzo| 1 0 0 0 0 }}, {{ | : 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 monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24) | ||
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978] | * 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978] | ||
Optimal | {{Optimal ET sequence|legend=0| 7, 24, 31 }} | ||
Badness: 0. | Badness (Sintel): 0.862 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
| Line 1,276: | Line 1,305: | ||
Comma list: 66/65, 81/80, 105/104, 121/120 | Comma list: 66/65, 81/80, 105/104, 121/120 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 7, 24, 31 }} | ||
Badness: 0. | Badness (Sintel): 0.966 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
| Line 1,293: | Line 1,322: | ||
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153 | Comma list: 66/65, 81/80, 105/104, 121/120, 154/153 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 7, 24, 31 }} | ||
Badness: | Badness (Sintel): 1.05 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
| Line 1,310: | Line 1,339: | ||
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152 | Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 7, 24, 31, 55 }} | ||
Badness: | Badness (Sintel): 1.05 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
== Mohamaq == | == Mohamaq == | ||
Subgroup: 2.3.5.7 | 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 | [[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 ET sequence|legend=1| 7d, 17c, 24 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.97 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
| Line 1,344: | Line 1,379: | ||
Comma list: 56/55, 77/75, 243/242 | Comma list: 56/55, 77/75, 243/242 | ||
Mapping: | Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 7d, 17c, 24 }} | ||
Badness: | Badness (Sintel): 1.20 | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
| Line 1,361: | Line 1,396: | ||
Comma list: 56/55, 66/65, 77/75, 243/242 | Comma list: 56/55, 66/65, 77/75, 243/242 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }} | ||
Badness: | Badness (Sintel): 1.19 | ||
Scales: [[mohaha7]], [[mohaha10]] | 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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, | [[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]]: | [[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges. | ||
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }} | |||
[[Badness]] (Sintel): 1.18 | |||
=== Liesel === | |||
=== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 56/55, 81/80, 540/539 | ||
Mapping | Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}} | |||
Optimal | {{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: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 19, 36 }} | |||
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: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 17c, 19e, 36e }} | ||
Badness: | 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: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }} | |||
Badness (Sintel): 1.11 | |||
=== Lisa === | |||
=== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 81/80, | Comma list: 45/44, 81/80, 343/330 | ||
Mapping: | Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}} | |||
{{Optimal ET sequence|legend=0| 17cee, 19 }} | |||
Badness (Sintel): 1.81 | |||
Badness: | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 81/80, | Comma list: 45/44, 81/80, 91/88, 147/143 | ||
Mapping: | 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 ET sequence|legend=0| 17cee, 19 }} | |||
Badness (Sintel): 1.49 | |||
== Superpine == | |||
{{See also| No-sevens subgroup temperaments #Superpine }} | |||
The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos ([[7L 8s]]) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 81/80, 1119744/1071875 | |||
{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }} | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}} | |||
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}} | |||
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }} | |||
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }} | |||
[[Badness]] (Sintel): 3.46 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 81/80, 176/175, 864/847 | ||
Mapping: | Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 7, 36, 43 }} | ||
Badness: | Badness (Sintel): 1.90 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | 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=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 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 81/80, 3125/3087 | ||
{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }} | |||
: mapping generators: ~56/45, ~3 | |||
Optimal | [[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 == | == Squares == | ||
{{Main| Squares }} | {{Main| Squares }} | ||
Squares splits the | 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, 2401/2400 | [[Comma list]]: 81/80, 2401/2400 | ||
{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }} | |||
: 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]]: ~9/7 = {{monzo| 1/2 0 -1/16 }} | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }} | ||
: [{{monzo| 1 0 0 0 | : [[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]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents. | ||
{{ | {{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.16 | ||
Scales: [[skwares8]], [[skwares11]], [[skwares14]] | Scales: [[skwares8]], [[skwares11]], [[skwares14]] | ||
| Line 1,637: | Line 1,637: | ||
Comma list: 81/80, 99/98, 121/120 | Comma list: 81/80, 99/98, 121/120 | ||
Mapping: | Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }} | ||
Badness: 0. | Badness (Sintel): 0.715 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 1,650: | Line 1,652: | ||
Comma list: 66/65, 81/80, 99/98, 121/120 | Comma list: 66/65, 81/80, 99/98, 121/120 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }} | ||
Badness: | Badness (Sintel): 1.05 | ||
==== Squad ==== | ==== Squad ==== | ||
| Line 1,663: | Line 1,667: | ||
Comma list: 78/77, 81/80, 91/90, 99/98 | Comma list: 78/77, 81/80, 91/90, 99/98 | ||
Mapping: | Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 14cf, 17c, 31f }} | ||
Badness: | Badness (Sintel): 1.11 | ||
==== Agora ==== | ==== Agora ==== | ||
| Line 1,676: | Line 1,682: | ||
Comma list: 81/80, 99/98, 105/104, 121/120 | Comma list: 81/80, 99/98, 105/104, 121/120 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }} | ||
Badness: | Badness (Sintel): 1.01 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 1,689: | Line 1,697: | ||
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119 | 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 tunings: | |||
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 14cf, 31 }} | ||
Badness: | Badness (Sintel): 1.15 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 1,702: | Line 1,712: | ||
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119 | Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 14cf, 31 }} | ||
Badness: | Badness (Sintel): 1.15 | ||
=== Cuboctahedra === | === Cuboctahedra === | ||
| Line 1,715: | Line 1,727: | ||
Comma list: 81/80, 385/384, 1375/1372 | Comma list: 81/80, 385/384, 1375/1372 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }} | ||
Badness: | Badness (Sintel): 1.88 | ||
== Jerome == | == Jerome == | ||
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the | Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, 17280/16807 | [[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]]: | [[Badness]] (Sintel): 2.75 | ||
=== 11-limit === | === 11-limit === | ||
| Line 1,747: | Line 1,763: | ||
Comma list: 81/80, 99/98, 864/847 | Comma list: 81/80, 99/98, 864/847 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 17c, 26, 43 }} | ||
Badness: | Badness (Sintel): 1.58 | ||
=== 13-limit === | === 13-limit === | ||
| Line 1,762: | Line 1,778: | ||
Comma list: 78/77, 81/80, 99/98, 144/143 | 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 | {{Optimal ET sequence|legend=0| 17c, 26, 43 }} | ||
Badness: | Badness (Sintel): 1.21 | ||
=== 17-limit === | === 17-limit === | ||
| Line 1,777: | Line 1,793: | ||
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187 | Comma list: 78/77, 81/80, 99/98, 144/143, 189/187 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 17cg, 26, 43 }} | ||
Badness: | Badness (Sintel): 1.06 | ||
=== 19-limit === | === 19-limit === | ||
| Line 1,792: | Line 1,808: | ||
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143 | Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143 | ||
Mapping: | 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 | |||
Badness: 0. | {{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 == | ||
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 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 | 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] | [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name] | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 50/49, 81/80 | [[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]]: | [[Tuning ranges]]: | ||
| Line 1,823: | Line 1,877: | ||
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955] | * 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955] | ||
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | * 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | ||
{{ | {{Optimal ET sequence|legend=1| 12, 26, 38 }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.788 | ||
; Music | ; Music | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] | * [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 === | === 11-limit === | ||
| Line 1,838: | Line 1,890: | ||
Comma list: 45/44, 50/49, 81/80 | Comma list: 45/44, 50/49, 81/80 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }} | ||
Optimal tunings: | |||
* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}}) | |||
Tuning ranges: | Tuning ranges: | ||
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12) | * 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] | * 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | ||
Optimal | {{Optimal ET sequence|legend=0| 12, 26 }} | ||
Badness: 0. | Badness (Sintel): 0.764 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 1,858: | Line 1,909: | ||
Comma list: 45/44, 50/49, 78/77, 81/80 | Comma list: 45/44, 50/49, 78/77, 81/80 | ||
Mapping: | 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: | Tuning ranges: | ||
* 13 | * 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26) | ||
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | * 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955] | ||
Optimal | {{Optimal ET sequence|legend=0| 12f, 14cf, 26 }} | ||
Badness: 0. | Badness (Sintel): 0.891 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 1,878: | Line 1,928: | ||
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84 | Comma list: 45/44, 50/49, 78/77, 81/80, 85/84 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }} | ||
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 | {{Optimal ET sequence|legend=0| 12f, 14cf, 26 }} | ||
Badness: 0. | Badness (Sintel): 0.935 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 1,891: | Line 1,943: | ||
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84 | Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }} | ||
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 | {{Optimal ET sequence|legend=0| 12f, 14cf, 26 }} | ||
Badness: 0. | Badness (Sintel): 0.920 | ||
==== Enjera ==== | ==== Enjera ==== | ||
| Line 1,904: | Line 1,958: | ||
Comma list: 27/26, 40/39, 45/44, 50/49 | Comma list: 27/26, 40/39, 45/44, 50/49 | ||
Mapping: | Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }} | ||
Optimal tunings: | |||
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}}) | |||
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}}) | |||
{{Optimal ET sequence|legend=0| 10cdeef, 12f }} | |||
Badness (Sintel): 1.10 | |||
Badness: | |||
=== Injerous === | === Injerous === | ||
| Line 1,919: | Line 1,973: | ||
Comma list: 33/32, 50/49, 55/54 | Comma list: 33/32, 50/49, 55/54 | ||
Mapping: | 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 | |||
Badness: | |||
=== Lahoh === | === Lahoh === | ||
| Line 1,934: | Line 1,988: | ||
Comma list: 50/49, 56/55, 81/77 | Comma list: 50/49, 56/55, 81/77 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 10cd, 12 }} | ||
Badness: | Badness (Sintel): 1.42 | ||
=== Teff === | === Teff === | ||
{{Main| Teff }} | {{Main| Teff }} | ||
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 | ||
| Line 1,953: | Line 2,007: | ||
Comma list: 50/49, 81/80, 864/847 | Comma list: 50/49, 81/80, 864/847 | ||
Mapping: | Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }} | ||
: mapping generators: ~7/5, ~16/11 | |||
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 | {{Optimal ET sequence|legend=0| 24d, 26, 50d }} | ||
Badness: | Badness (Sintel): 2.34 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 1,968: | Line 2,024: | ||
Comma list: 50/49, 78/77, 81/80, 144/143 | 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 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 | {{Optimal ET sequence|legend=0| 24d, 26, 50d }} | ||
Badness: | Badness (Sintel): 1.65 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 1,981: | Line 2,039: | ||
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143 | 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 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 | {{Optimal ET sequence|legend=0| 24d, 26 }} | ||
Badness: | Badness (Sintel): 1.50 | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
| Line 1,994: | Line 2,054: | ||
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143 | 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 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 | {{Optimal ET sequence|legend=0| 24d, 26 }} | ||
Badness: | Badness (Sintel): 1.41 | ||
== Pombe == | == Pombe == | ||
Pombe (named after the African millet beer) is a variant of [[#Teff]] by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. 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. | Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, 300125/294912 | [[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]]: | [[Badness]] (Sintel): 2.94 | ||
=== 11-limit === | === 11-limit === | ||
| Line 2,026: | Line 2,090: | ||
Comma list: 81/80, 245/242, 385/384 | Comma list: 81/80, 245/242, 385/384 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 24, 26, 50 }} | ||
Badness: | Badness (Sintel): 1.72 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,039: | Line 2,105: | ||
Comma list: 81/80, 105/104, 144/143, 245/242 | Comma list: 81/80, 105/104, 144/143, 245/242 | ||
Mapping: | 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 | {{Optimal ET sequence|legend=0| 24, 26, 50 }} | ||
Badness: | Badness (Sintel): 1.28 | ||
=== 17-limit === | === 17-limit === | ||
| Line 2,052: | Line 2,120: | ||
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272 | 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 | {{Optimal ET sequence|legend=0| 24, 26, 50 }} | ||
Badness: | Badness (Sintel): 1.08 | ||
=== 19-limit === | === 19-limit === | ||
| Line 2,065: | Line 2,135: | ||
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209 | Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209 | ||
Mapping: | Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }} | ||
Optimal tunings: | |||
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}}) | |||
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}}) | |||
Optimal | {{Optimal ET sequence|legend=0| 24, 26, 50 }} | ||
Badness: | Badness (Sintel): 1.01 | ||
== Orphic == | == Orphic == | ||
Subgroup: 2.3.5.7 | 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 | [[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]]s: | ||
* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}}) | |||
: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }} | |||
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}}) | |||
: error map: {{val| 0.000 -5.041 +1.342 +3.860 }} | |||
{{Optimal ET sequence|legend=1| 26, 48c, 74 }} | |||
[[Badness]] (Sintel): 6.55 | |||
[[Badness]]: | |||
=== 11-limit === | === 11-limit === | ||
| Line 2,095: | Line 2,171: | ||
Comma list: 81/80, 99/98, 73728/73205 | Comma list: 81/80, 99/98, 73728/73205 | ||
Mapping: | Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }} | ||
Optimal tunings: | |||
* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}}) | |||
* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}}) | |||
{{Optimal ET sequence|legend=0| 26, 48c, 74 }} | |||
Badness (Sintel): 3.36 | |||
Badness: | |||
=== 13-limit === | === 13-limit === | ||
| Line 2,110: | Line 2,186: | ||
Comma list: 81/80, 99/98, 144/143, 2200/2197 | 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 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=0| 26, 48c, 74 }} | |||
Badness (Sintel): 2.21 | |||
Badness: | |||
== Cloudtone == | == 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, 16807/16384 | [[Comma list]]: 81/80, 16807/16384 | ||
{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }} | |||
: mapping generators: ~8/7, ~3 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}}) | |||
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }} | |||
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}}) | |||
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }} | |||
{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }} | |||
[[Badness]] (Sintel): 2.59 | |||
[[Badness]]: | |||
=== 11-limit === | === 11-limit === | ||
| Line 2,144: | Line 2,222: | ||
Comma list: 81/80, 385/384, 2401/2376 | Comma list: 81/80, 385/384, 2401/2376 | ||
Mapping: | Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }} | ||
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 | {{Optimal ET sequence|legend=0| 5, 45, 50 }} | ||
Badness: | Badness (Sintel): 2.33 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,159: | Line 2,237: | ||
Comma list: 81/80, 105/104, 144/143, 2401/2376 | Comma list: 81/80, 105/104, 144/143, 2401/2376 | ||
Mapping: | 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 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 tunings: | |||
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }} | |||
Badness (Sintel): 1.40 | |||
== References == | |||
<references/> | |||
[[Category:Temperament families]] | |||
[[Category:Temperament | |||
[[Category:Meantone family| ]] <!-- main article --> | [[Category:Meantone family| ]] <!-- main article --> | ||
[[Category:Meantone| | [[Category:Meantone| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
Latest revision as of 15:18, 23 June 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The meantone family is the family of rank-2 temperaments that temper out the syntonic comma, 81/80, and thus can all be seen as extensions of meantone.
Meantone
Meantone is characterized by an octave period, a fifth generator, and the relationship that four fifths go to make up a 5th harmonic.
Subgroup: 2.3.5
Comma list: 81/80
Mapping: [⟨1 0 -4], ⟨0 1 4]]
- mapping generators: ~2, ~3
- WE: ~2 = 1201.3906 ¢, ~3/2 = 697.0455 ¢
- error map: ⟨+1.391 -3.519 +1.868]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6512 ¢
- error map: ⟨0.000 -5.304 +0.291]
- 5-odd-limit: ~3/2 = [0 0 1/4⟩ (1/4-comma)
- 5-odd-limit diamond monotone: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
- 5-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b
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.
- Flattertone adds [-24 17 0 -1⟩, finding the ~7/4 at the double-augmented sixth, for a tuning between 33edo and 26edo.
- Flattone adds [-17 9 0 1⟩, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
- Septimal meantone adds [-13 10 0 -1⟩, finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
- Dominant adds [6 -2 0 -1⟩, finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
- Sharptone adds [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.
- Injera adds [-7 8 0 -2⟩ with a half-octave period.
- Mohajira adds [-23 11 0 2⟩ and splits the fifth in two.
- Godzilla adds [-4 -1 0 2⟩ with an ~8/7 generator, two of which give the fourth.
- Mothra adds [-10 1 0 3⟩ with an ~8/7 generator, three of which give the fifth.
- Liese adds [-9 11 0 -3⟩ with a ~10/7 generator, three of which give the twelfth.
- Squares adds [-3 9 0 -4⟩ with a ~9/7 generator, four of which give the eleventh.
- Jerome adds [3 7 0 -5⟩ and slices the fifth in five.
Strong extensions
For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)3 as 7/5, leading to septimal meantone, a very elegant extension to the 7-limit.
For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant.
Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in weak extensions. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into n parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.
Splitting the meantone fifth into two (243/242)
By tempering out 243/242 we equate the distance from 9/8 to 10/9 (= S9) with the distance between 11/10 to 12/11 (= S11), leading to mohaha which is in some sense thus a trivial tuning of rastmic (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone rastmic temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full 11-limit by finding 7/4 as the semi-diminished seventh, leading to mohajira, which inflates 64/63 to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened (5/4)2 with 11/7, which is also natural as meantone tempering usually has 5/4 slightly sharp. There is also the consideration that tempering out 121/120 leads to similarly high damage in the 11-limit as tempering 81/80 in the 5-limit, because both erase key distinctions of their respective JI subgroups.
Splitting the meantone fifth into three (1029/1024)
By tempering out 1029/1024 we equate the distance from 7/6 to 8/7 (= S7) with the distance from 8/7 to 9/8 (= S8), so that (8/7)3 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)3 = 1728/1715 (S6/S7), the orwellisma.
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)2, 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)2 = 36/25 = (3/2)/(25/24).
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 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
- Plutus (+15/14) → Very low accuracy temperaments
- Godzilla (+49/48) → Semaphoresmic clan
- Mothra (+1029/1024) → Gamelismic clan
- Mohaha (+121/120) → Rastmic clan
The rest are considered below.
Septimal meantone
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
Comma list: 81/80, 126/125
Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]
- WE: ~2 = 1201.2358 ¢, ~3/2 = 697.2122 ¢
- error map: ⟨+1.236 -3.507 +2.535 -0.412]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6562 ¢
- error map: ⟨0.000 -5.299 +0.311 -2.264]
- 7- and 9-odd-limit: ~3/2 = [0 0 1/4⟩ (1/4-comma)
- projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.5
- 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
- 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, 503.4257 cents. The recurrence converges quickly.
Optimal ET sequence: 12, 19, 31, 81, 112b, 143b
Badness (Sintel): 0.347
Undecimal meantone (huygens)
- "Huygens" redirects here. For the Dutch mathematician, physicist and astronomer, see Wikipedia: Christiaan Huygens.
Undecimal meantone[1] a.k.a. huygens[2][3] 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
Comma list: 81/80, 99/98, 126/125
Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]
Optimal tunings:
- WE: ~2 = 1200.7636 ¢, ~3/2 = 697.4122 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.0315 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
- projection map: [[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
- unchanged-interval (eigenmonzo) basis: 2.11/9
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.
Optimal ET sequence: 12, 19e, 31, 105, 136b
Badness (Sintel): 0.563
- Music
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
Comma list: 81/80, 99/98, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29], ⟨0 1 4 10 18 -16]]
Optimal tunings:
- WE: ~2 = 1199.9389 ¢, ~3/2 = 697.2282 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.2627 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [8/13 0 0 1/26 0 -1/26⟩
- 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: 12, 31, 43, 74
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
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29 12], ⟨0 1 4 10 18 -16 -5]]
Optimal tunings:
- WE: ~2 = 1199.5811 ¢, ~3/2 = 697.0918 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3303 ¢
Optimal ET sequence: 12, 31, 43, 74g
Badness (Sintel): 1.06
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29 12 9], ⟨0 1 4 10 18 -16 -5 -3]]
Optimal tunings:
- WE: ~2 = 1199.2931 ¢, ~3/2 = 696.9690 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3736 ¢
Optimal ET sequence: 12, 31, 43, 74gh
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: [⟨1 0 -4 -13 -25 -20], ⟨0 1 4 10 18 15]]
Optimal tunings:
- WE: ~2 = 1200.8149 ¢, ~3/2 = 697.1155 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.7085 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
- unchanged-interval (eigenmonzo) basis: 2.11/9
Optimal ET sequence: 12f, 19e, 31
Badness (Sintel): 0.746
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
Mapping: [⟨1 0 -4 -13 -25 -20 12], ⟨0 1 4 10 18 15 -5]]
Optimal tunings:
- WE: ~2 = 1199.5548 ¢, ~3/2 = 696.7449 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.9823 ¢
Badness (Sintel): 1.02
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
Mapping: [⟨1 0 -4 -13 -25 -20 12 9], ⟨0 1 4 10 18 15 -5 -3]]
Optimal tunings:
- WE: ~2 = 1199.0408 ¢, ~3/2 = 696.5824 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.1061 ¢
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
Comma list: 78/77, 81/80, 99/98, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39], ⟨0 1 4 10 18 27]]
Optimal tunings:
- WE: ~2 = 1199.9122 ¢, ~3/2 = 697.4779 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.5241 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [14/25 -2/25 0 0 0 1/25⟩
- unchanged-interval (eigenmonzo) basis: 2.13/9
Optimal ET sequence: 12f, 31f, 43
Badness (Sintel): 1.09
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39 12], ⟨0 1 4 10 18 27 -5]]
Optimal tunings:
- WE: ~2 = 1199.3793 ¢, ~3/2 = 697.2833 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.6222 ¢
Badness (Sintel): 1.22
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39 12 9], ⟨0 1 4 10 18 27 -5 -3]]
Optimal tunings:
- WE: ~2 = 1199.0260 ¢, ~3/2 = 697.1486 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.6887 ¢
Badness (Sintel): 1.25
Hemimeantone
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 169/168
Mapping: [⟨1 0 -4 -13 -25 -5], ⟨0 2 8 20 36 11]]
- mapping generators: ~2, ~26/15
Optimal tunings:
- WE: ~2 = 1201.0387 ¢, ~26/15 = 949.2863 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 948.5065 ¢
Optimal ET sequence: 19e, 43, 62
Badness (Sintel): 1.30
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
Mapping: [⟨1 0 -4 -13 -25 -5 -22], ⟨0 2 8 20 36 11 33]]
Optimal tunings:
- WE: ~2 = 1201.0270 ¢, ~26/15 = 949.2892 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 948.5169 ¢
Optimal ET sequence: 19eg, 43, 62
Badness (Sintel): 1.19
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
Mapping: [⟨1 0 -4 -13 -25 -5 -22 -25], ⟨0 2 8 20 36 11 33 37]]
Optimal tunings:
- WE: ~2 = 1201.0339 ¢, ~19/11 = 949.2902 ¢
- CWE: ~2 = 1200.0000 ¢, ~19/11 = 948.5111 ¢
Optimal ET sequence: 19egh, 43, 62
Badness (Sintel): 1.15
Semimeantone
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 847/845
Mapping: [⟨2 0 -8 -26 -50 -59], ⟨0 1 4 10 18 21]]
- mapping generators: ~55/39, ~3
Optimal tunings:
- WE: ~55/39 = 600.3606 ¢, ~3/2 = 697.4241 ¢
- CWE: ~55/39 = 600.0000 ¢, ~3/2 = 697.0545 ¢
Optimal ET sequence: 12f, …, 50eff, 62, 136b
Badness (Sintel): 1.68
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
Mapping: [⟨2 0 -8 -26 -50 -59 5], ⟨0 1 4 10 18 21 1]]
Optimal tunings:
- WE: ~17/12 = 600.5426 ¢, ~3/2 = 697.5571 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.9858 ¢
Optimal ET sequence: 12f, 50eff, 62, 136bg
Badness (Sintel): 1.60
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
Mapping: [⟨2 0 -8 -26 -50 -59 5 -1], ⟨0 1 4 10 18 21 1 3]]
Optimal tunings:
- WE: ~17/12 = 600.5959 ¢, ~3/2 = 697.5985 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.9638 ¢
Optimal ET sequence: 12f, 50eff, 62
Badness (Sintel): 1.47
Meanpop
Meanpop[1][3] 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: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]
- mapping generator: ~2, ~3
Optimal tunings:
- WE: ~2 = 1201.3464 ¢, ~3/2 = 697.2159 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4509 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [0 0 1/4⟩
- projection map: [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [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 3x3 + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Optimal ET sequence: 12e, 19, 31, 81, 112b
Badness (Sintel): 0.712
- Music
- Scott Joplin's "The Entertainer" tuned into meanpop[dead link]
- Twinkle canon – 50 edo by Claudi Meneghin
Tridecimal meanpop
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 24 -20], ⟨0 1 4 10 -13 15]]
Optimal tunings:
- WE: ~2 = 1201.0765 ¢, ~3/2 = 696.8361 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2347 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [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: 19, 31, 50, 81
Badness (Sintel): 0.863
Meanpoppic
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
Mapping: [⟨1 0 -4 -13 24 -20 -37], ⟨0 1 4 10 -13 15 26]]
Optimal tunings:
- WE: ~2 = 1201.0727 ¢, ~3/2 = 696.8168 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2195 ¢
Optimal ET sequence: 19g, 31, 50, 81, 131bd
Badness (Sintel): 1.02
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
Mapping: [⟨1 0 -4 -13 24 -20 -37 -40], ⟨0 1 4 10 -13 15 26 28]]
Optimal tunings:
- WE: ~2 = 1201.0719 ¢, ~3/2 = 696.8101 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2137 ¢
Optimal ET sequence: 19gh, 31, 50, 81
Badness (Sintel): 1.08
Meanpoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 24 -20 12], ⟨0 1 4 10 -13 15 -5]]
Optimal tunings:
- WE: ~2 = 1200.2768 ¢, ~3/2 = 696.5683 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4114 ¢
Badness (Sintel): 1.17
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
Mapping: [⟨1 0 -4 -13 24 -20 12 9], ⟨0 1 4 10 -13 15 -5 -3]]
Optimal tunings:
- WE: ~2 = 1199.7905 ¢, ~3/2 = 696.3779 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4973 ¢
Badness (Sintel): 1.25
Semimeanpop
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 126/125, 385/384, 847/845
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 ¢, ~3/2 = 697.2151 ¢
- CWE: ~55/39 = 600.0000 ¢, ~3/2 = 696.4341 ¢
Optimal ET sequence: 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: [⟨2 0 -8 -26 48 39 5], ⟨0 1 4 10 -13 -10 1]]
Optimal tunings:
- WE: ~17/12 = 600.7232 ¢, ~3/2 = 697.2820 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.4411 ¢
Optimal ET sequence: 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: [⟨2 0 -8 -26 48 39 5 -1], ⟨0 1 4 10 -13 -10 1 3]]
Optimal tunings:
- WE: ~17/12 = 600.7527 ¢, ~3/2 = 697.3244 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 696.4525 ¢
Optimal ET sequence: 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: [⟨1 0 -4 -13 -6], ⟨0 1 4 10 6]]
Optimal tunings:
- WE: ~2 = 1199.6946 ¢, ~3/2 = 696.0729 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.2083 ¢
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: 7d, 12, 19, 31e
Badness (Sintel): 0.708
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 56/55, 78/77, 81/80
Mapping: [⟨1 0 -4 -13 -6 -20], ⟨0 1 4 10 6 15]]
Optimal tunings:
- WE: ~2 = 1199.7931 ¢, ~3/2 = 696.0258 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1241 ¢
Optimal ET sequence: 7df, 12f, 19, 31e
Badness (Sintel): 0.875
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
Mapping: [⟨1 0 -4 -13 -6 -20 12], ⟨0 1 4 10 6 15 -5]]
Optimal tunings:
- WE: ~2 = 1198.6665 ¢, ~3/2 = 695.8010 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.4998 ¢
Optimal ET sequence: 12f, 19, 31e
Badness (Sintel): 1.17
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Mapping: [⟨1 0 -4 -13 -6 -20 12 9], ⟨0 1 4 10 6 15 -5 -3]]
Optimal tunings:
- WE: ~2 = 1198.2880 ¢, ~3/2 = 695.7123 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6370 ¢
Optimal ET sequence: 12f, 19, 31e
Badness (Sintel): 1.23
Vincenzo
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10], ⟨0 1 4 10 6 -4]]
Optimal tunings:
- WE: ~2 = 1202.1684 ¢, ~3/2 = 696.3160 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 695.2045 ¢
Optimal ET sequence: 7d, 12, 19
Badness (Sintel): 1.02
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12], ⟨0 1 4 10 6 -4 -5]]
Optimal tunings:
- WE: ~2 = 1200.5137 ¢, ~3/2 = 696.1561 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 695.8771 ¢
Badness (Sintel): 1.30
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9], ⟨0 1 4 10 6 -4 -5 -3]]
Optimal tunings:
- WE: ~2 = 1199.8261 ¢, ~3/2 = 696.0298 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1262 ¢
Badness (Sintel): 1.36
Bimeantone
11/8 is mapped to half octave minus the meantone diesis.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 245/242
Mapping: [⟨2 0 -8 -26 -31], ⟨0 1 4 10 12]]
- mapping generators: ~63/44, ~3
Optimal tunings:
- WE: ~63/44 = 600.7492 ¢, ~3/2 = 696.8853 ¢
- CWE: ~63/44 = 600.0000 ¢, ~3/2 = 696.1908 ¢
Optimal ET sequence: 12, 26de, 38d, 50
Badness (Sintel): 1.26
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125, 245/242
Mapping: [⟨2 0 -8 -26 -31 -40], ⟨0 1 4 10 12 15]]
Optimal tunings:
- WE: ~55/39 = 600.8309 ¢, ~3/2 = 696.8000 ¢
- CWE: ~55/39 = 600.0000 ¢, ~3/2 = 696.0066 ¢
Optimal ET sequence: 12f, 26deff, 38df, 50
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: [⟨2 0 -8 -26 -31 -40 5], ⟨0 1 4 10 12 15 1]]
Optimal tunings:
- WE: ~17/12 = 600.9234 ¢, ~3/2 = 696.8536 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 695.9317 ¢
Optimal ET sequence: 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: [⟨2 0 -8 -26 -31 -40 5 -1], ⟨0 1 4 10 12 15 1 3]]
Optimal tunings:
- WE: ~17/12 = 600.9845 ¢, ~3/2 = 696.8939 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 695.8947 ¢
Optimal ET sequence: 12f, 26deff, 38df, 50
Badness (Sintel): 1.08
Trimean
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 1344/1331
Mapping: [⟨1 2 4 7 5], ⟨0 -3 -12 -30 -11]]
- mapping generators: ~2, ~11/10
Optimal tunings:
- WE: ~2 = 1200.7155 ¢, ~11/10 = 167.9055 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.7749 ¢
Optimal ET sequence: 7d, 36d, 43, 50, 93
Badness (Sintel): 1.68
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 126/125, 144/143, 364/363
Mapping: [⟨1 2 4 7 5 3], ⟨0 -3 -12 -30 -11 5]]
Optimal tunings:
- WE: ~2 = 1200.6104 ¢, ~11/10 = 167.8749 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.7728 ¢
Optimal ET sequence: 7d, 43, 50, 93
Badness (Sintel): 1.46
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
Mapping: [⟨1 2 4 7 5 3 8], ⟨0 -3 -12 -30 -11 5 -28]]
Optimal tunings:
- WE: ~2 = 1200.6144 ¢, ~11/10 = 167.8716 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.7682 ¢
Optimal ET sequence: 7dg, 43, 50, 93
Badness (Sintel): 1.28
Migration
See Rastmic clan.
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: [⟨1 0 -4 17], ⟨0 1 4 -9]]
- WE: ~2 = 1203.6308 ¢, ~3/2 = 695.8782 ¢
- error map: ⟨+3.631 -2.446 -2.801 -2.684]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.7334 ¢
- error map: ⟨0.000 -8.222 -11.380 -12.426]
- 7-odd-limit: ~3/2 = [8/13 0 1/13 -1/13⟩
- projection map: [[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
- unchanged-interval (eigenmonzo) basis: 2.7/5
- 9-odd-limit: ~3/2 = [6/11 2/11 0 -1/11⟩
- projection map: [[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/7
- 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 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Optimal ET sequence: 7, 19, 26, 45
Badness (Sintel): 0.976
11-limit
This can also be considered a no-sevens temperament: hypnotone.
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 385/384
Mapping: [⟨1 0 -4 17 -6], ⟨0 1 4 -9 6]]
Optimal tuning:
- WE: ~2 = 1202.3247 ¢, ~3/2 = 694.4688 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.1467 ¢
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: 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: [⟨1 0 -4 17 -6 10], ⟨0 1 4 -9 6 -4]]
Optimal tunings:
- WE: ~2 = 1202.5156 ¢, ~3/2 = 694.5107 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.0538 ¢
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: 7, 19, 26, 45f, 71bcf, 116bcdef
Badness (Sintel): 0.920
Ptolemy
See Rastmic clan.
Dominant
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 temperaments.
Subgroup: 2.3.5.7
Comma list: 36/35, 64/63
Mapping: [⟨1 0 -4 6], ⟨0 1 4 -2]]
- WE: ~2 = 1195.3384 ¢, ~3/2 = 698.8478 ¢
- error map: ⟨-4.662 -7.769 +9.077 +14.832]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1125 ¢
- error map: ⟨0.000 -0.842 +18.136 +28.949]
- 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: 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: [⟨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:
- WE: ~2 = 1194.0169 ¢, ~3/2 = 699.7473 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2672 ¢
Optimal ET sequence: 5, 12, 17c, 29cde
Badness (Sintel): 0.799
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 56/55, 64/63, 66/65
Mapping: [⟨1 0 -4 6 13 18], ⟨0 1 4 -2 -6 -9]]
Optimal tunings:
- WE: ~2 = 1193.8055 ¢, ~3/2 = 700.0042 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8254 ¢
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: 12f, 17c, 29cdef
Badness (Sintel): 0.996
Dominion
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 64/63
Mapping: [⟨1 0 -4 6 13 -9], ⟨0 1 4 -2 -6 8]]
Optimal tunings:
- WE: ~2 = 1195.0293 ¢, ~3/2 = 701.9847 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7698 ¢
Optimal ET sequence: 5, 12, 17c
Badness (Sintel): 1.13
Domination
Subgroup: 2.3.5.7.11
Comma list: 36/35, 64/63, 77/75
Mapping: [⟨1 0 -4 6 -14], ⟨0 1 4 -2 11]]
Optimal tunings:
- WE: ~2 = 1194.8645 ¢, ~3/2 = 701.9872 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5945 ¢
Optimal ET sequence: 5e, 12e, 17c
Badness (Sintel): 1.21
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 64/63, 66/65
Mapping: [⟨1 0 -4 6 -14 -9], ⟨0 1 4 -2 11 8]]
Optimal tunings:
- WE: ~2 = 1195.1324 ¢, ~3/2 = 702.6343 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.0791 ¢
Optimal ET sequence: 5e, 12e, 17c
Badness (Sintel): 1.13
Domineering
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 64/63
Mapping: [⟨1 0 -4 6 -6], ⟨0 1 4 -2 6]]
Optimal tunings:
- WE: ~2 = 1194.7102 ¢, ~3/2 = 695.6962 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.1765 ¢
Optimal ET sequence: 5e, 7, 12
Badness (Sintel): 0.727
Arnold
Subgroup: 2.3.5.7.11
Comma list: 22/21, 33/32, 36/35
Mapping: [⟨1 0 -4 6 5], ⟨0 1 4 -2 -1]]
Optimal tunings:
- WE: ~2 = 1199.8507 ¢, ~3/2 = 698.4045 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.4822 ¢
Optimal ET sequence: 5, 7, 12e
Badness (Sintel): 0.864
Neutrominant
See 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 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: [⟨1 0 -4 -24], ⟨0 1 4 17]]
- mapping generators: ~2, ~3
- WE: ~2 = 1204.4511 ¢, ~3/2 = 694.3258 ¢
- error map: ⟨+4.451 -3.178 -9.011 +3.554]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 692.0479 ¢
- error map: ⟨0.000 -9.907 -18.122 -4.012]
Optimal ET sequence: 7d, 19d, 26, 59bcd, 85bccd
Badness (Sintel): 2.43
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 1375/1344
Mapping: [⟨1 0 -4 -24 -6], ⟨0 1 4 17 6]]
Optimal tunings:
- WE: ~2 = 1203.4653 ¢, ~3/2 = 693.8144 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 692.0422 ¢
Optimal ET sequence: 7d, 19d, 26
Badness (Sintel): 1.53
- Music
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[3].
Subgroup: 2.3.5.7
Comma list: 21/20, 28/27
Mapping: [⟨1 0 -4 -2], ⟨0 1 4 3]]
- WE: ~2 = 1204.2961 ¢, ~3/2 = 702.6463 ¢
- error map: ⟨+4.296 +4.987 +24.271 -56.591]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4928 ¢
- error map: ⟨0.000 -0.462 +19.657 -64.347]
Optimal ET sequence: 5, 7d, 12d
Badness (Sintel): 0.629
Meanertone
Subgroup: 2.3.5.7.11
Comma list: 21/20, 28/27, 33/32
Mapping: [⟨1 0 -4 -2 5], ⟨0 1 4 3 -1]]
Optimal tunings:
- WE: ~2 = 1208.5304 ¢, ~3/2 = 701.5669 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.1117 ¢
Optimal ET sequence: 5, 7d, 12de
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 Lucius Chiaraviglio in 2024.
Subgroup: 2.3.5.7
Comma list: 81/80, 16128/15625
Mapping: [⟨1 0 -4 -32], ⟨0 1 4 22]]
- WE: ~2 = 1199.7304 ¢, ~3/2 = 698.3953 ¢
- error map: ⟨-0.270 -3.829 +7.267 -1.434]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.5397 ¢
- error map: ⟨0.000 -3.415 +7.845 -0.952]
Optimal ET sequence: 12, 43d, 55, 67
Badness (Sintel): 2.67
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 176/175, 7056/6875
Mapping: [⟨1 0 -4 -32], ⟨0 1 4 22 30]]
Optimal ET sequence: 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: [⟨1 0 -4 -32 -44], ⟨0 1 4 22 30]]
Optimal ET sequence: 12f, 55f, 67
Badness (Sintel): 2.04
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 176/175, 189/187, 196/195, 832/825
Mapping: [⟨1 0 -4 -32 -44 12], ⟨0 1 4 22 30 -5]]
Optimal ET sequence: 12f, 55f, 67
Badness (Sintel): 1.98
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825
Mapping: [⟨1 0 -4 -32 -44 12 9], ⟨0 1 4 22 30 -5 -3]]
Optimal ET sequence: 12f, 55f, 67
Badness (Sintel): 1.95
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
Comma list: 81/80, 672/625
Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]
- WE: ~2 = 1195.4372 ¢, ~3/2 = 702.2086 ¢
- error map: ⟨-4.563 -4.309 +22.521 -8.319]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5375 ¢
- error map: ⟨0.000 +2.583 +31.836 -0.763]
Optimal ET sequence: 5d, 12d, 17c
Badness (Sintel): 3.40
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 132/125
Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]
Optimal tunings:
- WE: ~2 = 1195.7270 ¢, ~3/2 = 702.5848 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7471 ¢
Optimal ET sequence: 5de, 12de, 17c
Badness (Sintel): 2.09
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 56/55, 66/65, 81/80
Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]
Optimal tunings:
- WE: ~2 = 1196.3958 ¢, ~3/2 = 702.9766 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7940 ¢
Optimal ET sequence: 5de, 12de, 17c, 29c
Badness (Sintel): 1.67
Mohajira
Mohajira can be viewed as derived from mohaha which maps the interval half a 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 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
Comma list: 81/80, 6144/6125
Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]
- mapping generators: ~2, ~128/105
- WE: ~2 = 1200.8160 ¢, ~128/105 = 348.6518 ¢
- error map: ⟨+0.816 -3.835 +2.901 +0.900]
- CWE: ~2 = 1200.0000 ¢, ~128/105 = 348.4194 ¢
- error map: ⟨0.000 -5.116 +1.041 -1.439]
- 7- and 9-odd-limit: ~128/105 = [0 0 1/8⟩
- projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
- Unchanged-interval (eigenmonzo) basis: 2.5
- 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 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Optimal ET sequence: 7, 24, 31
Badness (Sintel): 1.41
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 176/175
Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]
Optimal tunings:
- WE: ~2 = 1201.1562 ¢, ~11/9 = 348.8124 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.4910 ¢
Minimax tuning:
- 11-odd-limit: ~11/9 = [0 0 1/8⟩
- projection map: [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [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: 7, 24, 31
Badness (Sintel): 0.862
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 105/104, 121/120
Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]
Optimal tunings:
- WE: ~2 = 1200.4256 ¢, ~11/9 = 348.6819 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.5622 ¢
Optimal ET sequence: 7, 24, 31
Badness (Sintel): 0.966
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]
Optimal tunings:
- WE: ~2 = 1200.0382 ¢, ~11/9 = 348.7471 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.7360 ¢
Optimal ET sequence: 7, 24, 31
Badness (Sintel): 1.05
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: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]
Optimal tunings:
- WE: ~2 = 1199.7469 ¢, ~11/9 = 348.7367 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.8117 ¢
Optimal ET sequence: 7, 24, 31, 55
Badness (Sintel): 1.05
Mohamaq
Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as 17c & 24; its ploidacot is dicot, the same as mohajira.
Subgroup: 2.3.5.7
Comma list: 81/80, 392/375
Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]
- mapping generators: ~2, ~25/21
- WE: ~2 = 1199.0661 ¢, ~25/21 = 350.3127 ¢
- error map: ⟨-0.934 -2.264 +16.188 -13.827]
- CWE: ~2 = 1200.0000 ¢, ~25/21 = 350.4856 ¢
- error map: ⟨0.000 -0.984 +17.571 -12.513]
Optimal ET sequence: 7d, 17c, 24
Badness (Sintel): 1.97
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 243/242
Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]
Optimal tunings:
- WE: ~2 = 1199.1924 ¢, ~11/9 = 350.3286 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4821 ¢
Optimal ET sequence: 7d, 17c, 24
Badness (Sintel): 1.20
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 66/65, 77/75, 243/242
Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]
Optimal tunings:
- WE: ~2 = 1198.5986 ¢, ~11/9 = 350.3353 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.6459 ¢
Optimal ET sequence: 7d, 17c, 24, 41c
Badness (Sintel): 1.19
Liese
Liese splits the 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 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: [⟨1 0 -4 -3], ⟨0 3 12 11]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 1201.5548 ¢, ~10/7 = 633.2251 ¢
- error map: ⟨+1.555 -2.280 +6.168 -8.015]
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 632.5640 ¢
- error map: ⟨0.000 -4.263 +4.454 -10.622]
- 7- and 9-odd-limit: ~10/7 = [1/3 0 1/12⟩
- projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.5
Algebraic generator: Radix, the real root of x5 - 2x4 + 2x3 - 2x2 + 2x - 2, also a root of x6 - x5 - 2. The recurrence converges.
Optimal ET sequence: 17c, 19, 55, 74d
Badness (Sintel): 1.18
Liesel
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 540/539
Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]
Optimal tunings:
- WE: ~2 = 1198.8507 ¢, ~10/7 = 632.4668 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 632.9963 ¢
Optimal ET sequence: 17c, 19, 36
Badness (Sintel): 1.35
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 78/77, 81/80, 91/90
Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]
Optimal tunings:
- WE: ~2 = 1199.4968 ¢, ~10/7 = 632.7766 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.0082 ¢
Optimal ET sequence: 17c, 19, 36
Badness (Sintel): 1.13
Elisa
Subgroup: 2.3.5.7.11
Comma list: 77/75, 81/80, 99/98
Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 16]]
Optimal tunings:
- WE: ~2 = 1201.0489 ¢, ~10/7 = 633.6147 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.1644 ¢
Optimal ET sequence: 17c, 19e, 36e
Badness (Sintel): 1.37
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 77/75, 81/80, 99/98
Mapping: [⟨1 0 -4 -3 -5 0], ⟨0 3 12 11 16 7]]
Optimal tunings:
- WE: ~2 = 1201.4815 ¢, ~10/7 = 633.7720 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.1281 ¢
Optimal ET sequence: 17c, 19e, 36e
Badness (Sintel): 1.11
Lisa
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 343/330
Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 18]]
Optimal tunings:
- WE: ~2 = 1202.6773 ¢, ~10/7 = 632.7783 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 631.6175 ¢
Optimal ET sequence: 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: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 18 7]]
Optimal tunings:
- WE: ~2 = 1203.6086 ¢, ~10/7 = 633.1193 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 631.5346 ¢
Optimal ET sequence: 17cee, 19
Badness (Sintel): 1.49
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 36 & 43; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent 6/5 – harmonics other than 3 all require the 15-tone mos (7L 8s) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as 11/10 as in porcupine, which makes 11/8 high-complexity like the other harmonics, but in the 13-limit 5 generators up closely approximates 13/8. 43edo is a good tuning especially for the higher-limit extensions.
Subgroup: 2.3.5.7
Comma list: 81/80, 1119744/1071875
Mapping: [⟨1 2 4 1], ⟨0 -3 -12 13]]
- WE: ~2 = 1199.3652 ¢, ~35/32 = 167.1615 ¢
- error map: ⟨-0.635 -4.709 +5.209 +3.639]
- CWE: ~2 = 1200.0000 ¢, ~35/32 = 167.2561 ¢
- error map: ⟨0.000 -3.723 +6.613 +5.503]
Optimal ET sequence: 7, 36, 43, 79c
Badness (Sintel): 3.46
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 176/175, 864/847
Mapping: [⟨1 2 4 1 5], ⟨0 -3 -12 13 -11]]
Optimal tunings:
- WE: ~2 = 1199.0522 ¢, ~11/10 = 167.1904 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.3382 ¢
Optimal ET sequence: 7, 36, 43
Badness (Sintel): 1.90
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 144/143, 176/175
Mapping: [⟨1 2 4 1 5 3], ⟨0 -3 -12 13 -11 5]]
Optimal tunings:
- WE: ~2 = 1199.4286 ¢, ~11/10 = 167.3105 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 167.3958 ¢
Optimal ET sequence: 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: [⟨3 0 -12 -20], ⟨0 1 4 6]]
- mapping generators: ~56/45, ~3
- WE: ~56/45 = 400.6744 ¢, ~3/2 = 695.8474 ¢ {~15/14 = 105.5015 ¢)
- error map: ⟨+2.023 -4.084 -2.924 +4.910]
- CWE: ~56/45 = 400.0000 ¢, ~3/2 = 695.1413 ¢ {~15/14 = 104.8587 ¢)
- error map: ⟨0.000 -6.814 -5.748 +2.022]
Optimal ET sequence: 12, 33cd, 45, 57
Badness (Sintel): 1.75
Squares
Squares splits the 6th harmonic into four subminor sixths of 11/7~14/9 (or splits a perfect eleventh into four supermajor thirds of 9/7~14/11), and uses it for a generator. It may be described as 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
Comma list: 81/80, 2401/2400
Mapping: [⟨1 -1 -8 -3], ⟨0 4 16 9]]
- mapping generators: ~2, ~14/9
- WE: ~2 = 1201.2488 ¢, ~14/9 = 774.8640 ¢
- error map: ⟨+1.249 -3.748 +1.520 +1.204]
- CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.1560 ¢
- error map: ⟨0.000 -5.331 +0.183 -1.422]
- 7- and 9-odd-limit: ~9/7 = [1/2 0 -1/16⟩
- projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.5
Algebraic generator: Sceptre2, the positive root of 9x2 + x - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Optimal ET sequence: 14c, 17c, 31, 169b, 200b
Badness (Sintel): 1.16
Scales: skwares8, skwares11, skwares14
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 121/120
Mapping: [⟨1 -1 -8 -3 -3], ⟨0 4 16 9 10]]
Optimal tunings:
- WE: ~2 = 1201.6657 ¢, ~11/7 = 775.1171 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.1754 ¢
Optimal ET sequence: 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: [⟨1 -1 -8 -3 -3 5], ⟨0 4 16 9 10 -2]]
Optimal tunings:
- WE: ~2 = 1199.8419 ¢, ~11/7 = 774.3484 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4422 ¢
Optimal ET sequence: 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: [⟨1 -1 -8 -3 -3 -6], ⟨0 4 16 9 10 15]]
Optimal tunings:
- WE: ~2 = 1202.0312 ¢, ~11/7 = 775.5589 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/7 = 774.4140 ¢
Optimal ET sequence: 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: [⟨1 -1 -8 -3 -3 -15], ⟨0 4 16 9 10 29]]
Optimal tunings:
- WE: ~2 = 1202.3228 ¢, ~11/7 = 775.2214 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8617 ¢
Optimal ET sequence: 14cf, 31, 45ef, 76e
Badness (Sintel): 1.01
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
Mapping: [⟨1 -1 -8 -3 -3 -15 -3], ⟨0 4 16 9 10 29 11]]
Optimal tunings:
- WE: ~2 = 1201.4340 ¢, ~11/7 = 774.7375 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8955 ¢
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: [⟨1 -1 -8 -3 -3 -15 -3 -8], ⟨0 4 16 9 10 29 11 19]]
Optimal tunings:
- WE: ~2 = 1201.2461 ¢, ~11/7 = 774.5783 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/7 = 773.8479 ¢
Badness (Sintel): 1.15
Cuboctahedra
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 1375/1372
Mapping: [⟨1 -1 -8 -3 17], ⟨0 4 16 9 -21]]
Optimal tunings:
- WE: ~2 = 1201.4436 ¢, ~14/9 = 774.9386 ¢
- CWE: ~2 = 1200.0000 ¢, ~14/9 = 774.0243 ¢
Optimal ET sequence: 31, 107b, 138b, 169be, 200be
Badness (Sintel): 1.88
Jerome
Jerome is related to Hieronymus' tuning; the Hieronymus generator is 51/20, or 139.316 cents. It may be described as 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: [⟨1 1 0 2], ⟨0 5 20 7]]
- mapping generators: ~2, ~54/49
- WE: ~2 = 1200.1640 ¢, ~54/49 = 139.3624 ¢
- error map: ⟨+0.164 -4.979 +0.934 +7.039]
- CWE: ~2 = 1200.0000 ¢, ~54/49 = 139.3528 ¢
- error map: ⟨0.000 -5.191 +0.741 +6.643]
Optimal ET sequence: 17c, 26, 43
Badness (Sintel): 2.75
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 864/847
Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]
Optimal tunings:
- WE: ~2 = 1201.4436 ¢, ~12/11 = 139.3714 ¢
- CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.4038 ¢
Optimal ET sequence: 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: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]
Optimal tunings:
- WE: ~2 = 1199.8860 ¢, ~13/12 = 139.3737 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3817 ¢
Optimal ET sequence: 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: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]
Optimal tunings:
- WE: ~2 = 1199.8346 ¢, ~13/12 = 139.3431 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3544 ¢
Optimal ET sequence: 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: [⟨1 1 0 2 3 3 2 1], ⟨0 5 20 7 4 6 18 28]]
Optimal tunings:
- WE: ~2 = 1199.8891 ¢, ~13/12 = 139.3001 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.3080 ¢
Optimal ET sequence: 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: [⟨1 -1 -8 -7], ⟨0 5 20 19]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 1201.3832 ¢, ~10/7 = 619.9478 ¢
- error map: ⟨+1.383 -3.599 +1.576 +0.499]
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.3176 ¢
- error map: ⟨0.000 -5.367 +0.038 -1.791]
Optimal ET sequence: 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: [⟨1 -1 -8 -7 -11], ⟨0 5 20 19 28]]
Optimal tunings:
- WE: ~2 = 1201.2054 ¢, ~10/7 = 619.9752 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.4223 ¢
Optimal ET sequence: 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 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 19edos, is an excellent tuning for injera.
Subgroup: 2.3.5.7
Comma list: 50/49, 81/80
Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]
- mapping generators: ~7/5, ~3
- WE: ~7/5 = 600.6662 ¢, ~3/2 = 695.1463 ¢ (~21/20 = 94.4801 ¢)
- error map: ⟨+1.332 -5.476 -5.729 +12.425]
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 694.7712 ¢ (~21/20 = 94.7712 ¢)
- error map: ⟨0.000 -7.184 -7.229 +10.259]
- 7- and 9-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
- 7-odd-limit diamond tradeoff: ~3/2 = [688.957, 701.955]
- 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
Optimal ET sequence: 12, 26, 38
Badness (Sintel): 0.788
- Music
- Two Pairs of Socks by Igliashon Jones – in 26edo tuning
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 50/49, 81/80
Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]
Optimal tunings:
- WE: ~7/5 = 600.9350 ¢, ~3/2 = 693.9198 ¢ (~21/20 = 92.9848 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.3539 ¢ (~21/20 = 93.3539 ¢)
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]
Badness (Sintel): 0.764
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 50/49, 78/77, 81/80
Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]
Optimal tunings:
- WE: ~7/5 = 600.9982 ¢, ~3/2 = 693.8249 ¢ (~21/20 = 92.8267 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.0992 ¢ (~21/20 = 93.0992 ¢)
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: 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: [⟨2 0 -8 -7 -12 -21 5], ⟨0 1 4 4 6 9 1]]
Optimal tunings:
- WE: ~7/5 = 601.1757 ¢, ~3/2 = 693.8441 ¢ (~21/20 = 92.6684 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 692.8879 ¢ (~21/20 = 92.8879 ¢)
Optimal ET sequence: 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: [⟨2 0 -8 -7 -12 -21 5 -1], ⟨0 1 4 4 6 9 1 3]]
Optimal tunings:
- WE: ~7/5 = 601.4245 ¢, ~3/2 = 693.9426 ¢ (~21/20 = 92.5181 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 692.7606 ¢ (~21/20 = 92.7606 ¢)
Optimal ET sequence: 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: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]
Optimal tunings:
- WE: ~7/5 = 599.1863 ¢, ~3/2 = 693.1791 ¢ (~21/20 = 93.9929 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 693.6809 ¢ (~21/20 = 93.6809 ¢)
Optimal ET sequence: 10cdeef, 12f
Badness (Sintel): 1.10
Injerous
Subgroup: 2.3.5.7.11
Comma list: 33/32, 50/49, 55/54
Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]
Optimal tunings:
- WE: ~7/5 = 603.1682 ¢, ~3/2 = 694.1945 ¢ (~21/20 = 91.0264 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 691.6107 ¢ (~21/20 = 91.6107 ¢)
Optimal ET sequence: 12e, 14c, 26e, 40cee
Badness (Sintel): 1.28
Lahoh
Subgroup: 2.3.5.7.11
Comma list: 50/49, 56/55, 81/77
Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]
Optimal tunings:
- WE: ~7/5 = 597.3179 ¢, ~3/2 = 695.8759 ¢ (~21/20 = 98.5581 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~3/2 = 697.8757 ¢ (~21/20 = 97.8757 ¢)
Badness (Sintel): 1.42
Teff
Teff, found and named by Mason Green, is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.
Subgroup: 2.3.5.7.11
Comma list: 50/49, 81/80, 864/847
Mapping: [⟨2 1 -4 -3 8], ⟨0 2 8 8 -1]]
- mapping generators: ~7/5, ~16/11
Optimal tunings:
- WE: ~7/5 = 600.2802 ¢, ~16/11 = 647.7720 ¢ (~33/32 = 47.4918 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.5224 ¢ (~33/32 = 47.5224 ¢)
Optimal ET sequence: 24d, 26, 50d
Badness (Sintel): 2.34
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 78/77, 81/80, 144/143
Mapping: [⟨2 1 -4 -3 8 2], ⟨0 2 8 8 -1 5]]
Optimal tunings:
- WE: ~7/5 = 600.3037 ¢, ~16/11 = 647.7954 ¢ (~33/32 = 47.4917 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.5256 ¢ (~33/32 = 47.5256 ¢)
Optimal ET sequence: 24d, 26, 50d
Badness (Sintel): 1.65
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
Mapping: [⟨2 1 -4 -3 8 2 6], ⟨0 2 8 8 -1 5 2]]
Optimal tunings:
- WE: ~7/5 = 600.5123 ¢, ~16/11 = 647.8970 ¢ (~34/33 = 47.3846 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.4314 ¢ (~34/33 = 47.4314 ¢)
Badness (Sintel): 1.50
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
Mapping: [⟨2 1 -4 -3 8 2 6 2], ⟨0 2 8 8 -1 5 2 6]]
Optimal tunings:
- WE: ~7/5 = 600.6308 ¢, ~16/11 = 648.0424 ¢ (~34/33 = 47.4116 ¢)
- CWE: ~7/5 = 600.0000 ¢, ~16/11 = 647.4715 ¢ (~34/33 = 47.4715 ¢)
Badness (Sintel): 1.41
Pombe
Pombe (named after the African millet beer) is a variant of #Teff by 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: [⟨2 1 -4 11], ⟨0 2 8 -5]]
- mapping generators: ~735/512, ~35/24
- WE: ~735/512 = 601.0652 ¢, ~35/24 = 648.9295 ¢ (~36/35 = 47.8642 ¢)
- error map: ⟨+2.130 -3.031 +0.861 -1.756]
- CWE: ~735/512 = 600.0000 ¢, ~35/24 = 647.8628 ¢ (~36/35 = 47.8628 ¢)
- error map: ⟨0.000 -6.229 -3.411 -8.140]
Optimal ET sequence: 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: [⟨2 1 -4 11 8], ⟨0 2 8 -5 -1]]
Optimal tunings:
- WE: ~99/70 = 600.7890 ¢, ~16/11 = 648.7592 ¢ (~36/35 = 47.9701 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~16/11 = 647.9516 ¢ (~36/35 = 47.9516 ¢)
Optimal ET sequence: 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: [⟨2 1 -4 11 8 2], ⟨0 2 8 -5 -1 5]]
Optimal tunings:
- WE: ~99/70 = 600.6971 ¢, ~16/11 = 648.6029 ¢ (~36/35 = 47.9058 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~16/11 = 647.8990 ¢ (~36/35 = 47.8990 ¢)
Optimal ET sequence: 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: [⟨2 1 -4 11 8 2 6], ⟨0 2 8 -5 -1 5 2]]
Optimal tunings:
- WE: ~17/12 = 600.7610 ¢, ~16/11 = 648.6638 ¢ (~36/35 = 47.9028 ¢)
- CWE: ~17/12 = 600.0000 ¢, ~16/11 = 647.8990 ¢ (~36/35 = 47.8990 ¢)
Optimal ET sequence: 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: [⟨2 1 -4 11 8 2 6 2], ⟨0 2 8 -5 -1 5 2 6]]
Optimal tunings:
- WE: ~17/12 = 600.8048 ¢, ~16/11 = 648.7494 ¢ (~36/35 = 47.9446 ¢)
- CWE: ~17/12 = 600.0000 ¢, ~16/11 = 647.9425 ¢ (~36/35 = 47.9425 ¢)
Optimal ET sequence: 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: [⟨2 1 -4 4], ⟨0 4 16 3]]
- mapping generators: ~2401/1728, ~343/288
- WE: ~2401/1728 = 600.1767 ¢, ~343/288 = 324.3015 ¢ (~7/6 = 275.8751 ¢)
- error map: ⟨+0.353 -4.572 +1.804 +4.785]
- CWE: ~2401/1728 = 600.0000 ¢, ~343/288 = 324.2285 ¢ (~7/6 = 275.7715 ¢)
- error map: ⟨0.000 -5.041 +1.342 +3.860]
Optimal ET sequence: 26, 48c, 74
Badness (Sintel): 6.55
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 73728/73205
Mapping: [⟨2 1 -4 4 8], ⟨0 4 16 3 -2]]
Optimal tunings:
- WE: ~363/256 = 600.1011 ¢, ~77/64 = 324.2923 ¢ (~7/6 = 275.8088 ¢)
- CWE: ~363/256 = 600.0000 ¢, ~77/64 = 324.2463 ¢ (~7/6 = 275.7537 ¢)
Optimal ET sequence: 26, 48c, 74
Badness (Sintel): 3.36
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 144/143, 2200/2197
Mapping: [⟨2 1 -4 4 8 2], ⟨0 4 16 3 -2 10]]
Optimal tunings:
- WE: ~55/39 = 600.0540 ¢, ~77/64 = 324.2551 ¢ (~7/6 = 275.7989 ¢)
- CWE: ~55/39 = 600.0000 ¢, ~77/64 = 324.2307 ¢ (~7/6 = 275.7693 ¢)
Optimal ET sequence: 26, 48c, 74
Badness (Sintel): 2.21
Cloudtone
The cloudtone temperament tempers out the cloudy comma, 16807/16384 and the syntonic comma, 81/80 in the 7-limit. It may be described as 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
Comma list: 81/80, 16807/16384
Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]
- mapping generators: ~8/7, ~3
- WE: ~8/7 = 240.4267 ¢, ~3/2 = 696.9566 ¢ (~49/48 = 24.3235 ¢)
- error map: ⟨+2.133 -2.865 +1.513 -2.852]
- CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.1637 ¢ (~49/48 = 23.8373 ¢)
- error map: ⟨0.000 -5.791 -1.659 -8.826]
Optimal ET sequence: 5, 40c, 45, 50
Badness (Sintel): 2.59
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 2401/2376
Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]
Optimal tunings:
- WE: ~8/7 = 240.2740 ¢, ~3/2 = 697.3317 ¢ (~56/55 = 23.4904 ¢)
- CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.6269 ¢ (~56/55 = 23.3731 ¢)
Optimal ET sequence: 5, 45, 50
Badness (Sintel): 2.33
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 144/143, 2401/2376
Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]
Optimal tunings:
- WE: ~8/7 = 240.2435 ¢, ~3/2 = 696.8686 ¢ (~91/90 = 23.8618 ¢)
- CWE: ~8/7 = 240.0000 ¢, ~3/2 = 696.2653 ¢ (~91/90 = 23.7347 ¢)
Optimal ET sequence: 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
Subgroup-val mapping: [⟨1 0 -4 9], ⟨0 1 4 -3]]
Gencom mapping: [⟨1 0 -4 0 0 0 0 9], ⟨0 1 4 0 0 0 0 -3]]
- mapping generators: ~2, ~3
- WE: ~2 = 1199.5513 ¢, ~3/2 = 697.6058 ¢
- error map: ⟨-0.448 -4.798 +4.110 +6.977]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.8222 ¢
- error map: ⟨0.000 -4.133 +4.975 +9.020]
Optimal ET sequence: 5, 7, 12, 31, 43, 98h
Badness (Sintel): 0.324
Hypnotone
Hypnotone is no-sevens flattone.
Subgroup: 2.3.5.11
Comma list: 45/44, 81/80
Subgroup-val mapping: [⟨1 0 -4 -6], ⟨0 1 4 6]]
Gencom mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6]]
- mapping generators: ~2, ~3
- WE: ~2 = 1202.0621 ¢, ~3/2 = 694.5448 ¢
- error map: ⟨+2.062 -5.348 -8.135 +15.951]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.9085 ¢
- error map: ⟨0.000 -8.047 -10.680 +12.133]
Optimal ET sequence: 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: [⟨1 0 -4 -6 10], ⟨0 1 4 6 -4]]
Gencom mapping: [⟨1 0 -4 0 -6 10], ⟨0 1 4 0 6 -4]]
Optimal tunings:
- WE: ~2 = 1202.6916 ¢, ~3/2 = 694.4181 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.0870 ¢
Optimal ET sequence: 7, 12, 19, 26, 45f
Badness (Sintel): 0.561
Dequarter
Subgroup: 2.3.5.11
Comma list: 33/32, 55/54
Subgroup-val mapping: [⟨1 0 -4 5], ⟨0 1 4 -1]]
Gencom mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1]]
- mapping generators: ~2, ~3
- WE: ~2 = 1206.5832 ¢, ~3/2 = 695.8763 ¢
- error map: ⟨+6.583 +0.504 -2.809 -20.862]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 693.1206 ¢
- error map: ⟨0.000 -8.834 -13.831 -44.439]
Optimal ET sequence: 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: [⟨1 0 -4 5 21], ⟨0 1 4 -1 -11]]
Gencom mapping: [⟨1 0 -4 0 5 21], ⟨0 1 4 0 -1 -11]]
Optimal tunings:
- WE: ~2 = 1207.8248 ¢, ~3/2 = 694.7806 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 690.1826 ¢
Optimal ET sequence: 7, 19eff, 26eff, 33ceeff, 40ceeff
Badness (Sintel): 1.40