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==== Splitting the meantone fifth into three (1029/1024) ====
==== Splitting the meantone fifth into three (1029/1024) ====
By tempering [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] (S6/S7), the orwellisma.
By tempering out [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])<sup>3</sup> is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)<sup>3</sup> = [[1728/1715]] (S6/S7), the orwellisma.


This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though 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]]).
This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called [[cynder]], though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out [[176/175]] (S8/S10), which is (11/7)/(5/4)<sup>2</sup>, taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])<sup>2</sup> = [[36/25]] = ([[3/2]])/([[25/24]]).


==== 31edo as splitting the fifth into two, three and nine ====
==== 31edo as splitting the fifth into two, three and nine ====
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering 225/224, which interestingly, though a rank 2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering 225/224).
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).


Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
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{{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♯). Septimal meantone tempers out the common 7-limit commas [[126/125]], [[225/224]], and [[3136/3125]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.  
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♯). Septimal meantone tempers out the common 7-limit commas [[126/125]], [[225/224]], and [[3136/3125]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{See also| Meantone vs meanpop }}
{{See also| Meantone vs meanpop }}


Undecimal meantone maps the [[11/8]] to the double augmented third (C-E𝄪), and tridecimal meantone maps the [[13/8]] to the double augmented fifth (C-G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is the double augmented unison; 12/11 is a double diminished third; and 14/13 is a minor second.  
Undecimal meantone maps the [[11/8]] to the double-augmented third (C–E𝄪), and tridecimal meantone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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===== Meantonic =====
===== Meantonic =====
Dubbed ''meantonic'' here, this extension maps the 17/16 to the octave-reduced triple augmented seventh (C-B𝄪♯), and 19/16 to the quadruple augmented unison (C-C𝄪𝄪). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.  
Dubbed ''meantonic'' here, this extension maps the 17/16 to the octave-reduced triple-augmented seventh (C–B𝄪♯), and 19/16 to the quadruple-augmented unison (C–C𝄪𝄪). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.  


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
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===== Meantoid =====
===== Meantoid =====
Dubbed ''meantoid'' here, this extension maps 17/16~19/18 to the augmented unison (C-C♯) and 19/16 to the augmented second (C-D♯). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverted, so genuine septendecimal and undevicesimal harmony cannot be expected.  
Dubbed ''meantoid'' here, this extension maps 17/16~19/18 to the augmented unison (C–C♯) and 19/16 to the augmented second (C–D♯). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverted, so genuine septendecimal and undevicesimal harmony cannot be expected.  


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
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===== Huygens =====
===== Huygens =====
Dubbed ''huygens'' here, this extension is perhaps the most practical, as it 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.  
Dubbed ''huygens'' here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.  


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
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==== Grosstone ====
==== Grosstone ====
Grosstone maps 13/8 to the double diminished seventh (C-B♭♭♭).  
Grosstone maps 13/8 to the double-diminished seventh (C–B♭♭♭).  


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
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==== Meridetone ====
==== Meridetone ====
Meridetone maps the 13/8 to the quadruple augmented fourth (C-F𝄪𝄪).  
Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪).  


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
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{{See also| Meantone vs meanpop }}
{{See also| Meantone vs meanpop }}


Meanpop maps the 11/8 to the double diminished fifth (C-G𝄫), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the double diminished third; 12/11~13/12, double augmented unison; and 14/13, minor second.  
Meanpop maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
: eigenmonzo (unchanged-interval) basis: 2.5
: eigenmonzo (unchanged-interval) basis: 2.5


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Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
: Eigenmonzo (unchanged-interval) basis: 2.11
: eigenmonzo (unchanged-interval) basis: 2.11


Optimal ET sequence: {{Optimal ET sequence| 12e, 19, 31f }}
Optimal ET sequence: {{Optimal ET sequence| 12e, 19, 31f }}
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=== Meanenneadecal ===
=== Meanenneadecal ===
Meanenneadecal maps the 11/8 to the augmented fourth (C-F♯), and tridecimal meanenneadecal still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the major second; 12/11~14/13, minor second; and 13/12, double augmented unison.  
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal still maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Meanundeci ===
=== Meanundeci ===
Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C-F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C-A♭).  
Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C–F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C–A♭).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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{{Main| Flattone }}
{{Main| Flattone }}


In flattone tunings, the fifth is typically even flatter than that of [[19edo]]. Here, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C-B𝄫), [[7/6]] is a diminished third (C-E𝄫), and [[7/5]] is a doubly-diminished fifth (C-G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].
In flattone tunings, the fifth is typically even flatter than that of [[19edo]]. Here, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are [[45edo]], [[64edo]], and [[71edo]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [{{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 }}]
: [[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|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [{{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 }}]
: [[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|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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== Flattertone ==
== Flattertone ==
Flattertone tunings are typically at least as flat as [[26edo]]. Here, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C-Ax). [[26edo]] and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.
Flattertone tunings are typically at least as flat as [[26edo]]. Here, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). [[26edo]] and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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==== 11-limit ====
==== 11-limit ====
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 6 0 -11/8 0 }}]
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 6 0 -11/8 0 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5


[[Tuning ranges]]:
[[Tuning ranges]]:
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Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: [{{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 }}]
: 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 }}]
: Eigenmonzo (unchanged-interval) basis: 2.5
: eigenmonzo (unchanged-interval) basis: 2.5


Tuning ranges:
Tuning ranges:
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<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[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
Line 1,806: Line 1,805:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 2/3 0 11/12 0 }}]
: [[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|Eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
[[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.
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== Lithium ==
== Lithium ==
Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.
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. 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
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{{Main| Squares }}
{{Main| Squares }}


Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[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
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[[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 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 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|Eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) 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.
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Mapping: {{mapping| 1 3 8 6 7 | 0 -4 -16 -9 -10 }}
Mapping: {{mapping| 1 3 8 6 7 | 0 -4 -16 -9 -10 }}


{{Multival|legend=1| 4 16 9 10 16 3 2 -24 -32 -3 }}
Wedgie: {{multival| 4 16 9 10 16 3 2 -24 -32 -3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 425.957
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 425.957
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== Meantritone ==
== Meantritone ==
The ''meantritone'' temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus [[15/11]]-wide super-fourth) and five of them equals ~[[16/3]] (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus [[15/11]]-wide super-fourth) and five of them equals ~[[16/3]] (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== 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|38EDO]], which is two parallel [[19edo]]s, is an excellent tuning for 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]], 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]
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; Music
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]
* [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 ===
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{{Main| Teff }}
{{Main| Teff }}


Teff (found by [[Mason Green]]) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.
Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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== Cloudtone ==
== Cloudtone ==
The ''cloudtone'' temperament (5&amp;50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
The cloudtone temperament (5 &amp; 50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. 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
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== Subgroup extensions ==
== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
=== Stützel (2.3.5.19) ===
Subgroup: 2.3.5.19
[[Subgroup]]: 2.3.5.19


[[Comma list]]: 81/80, 96/95
[[Comma list]]: 81/80, 96/95


{{Mapping|legend=2| 1 2 4 3 | 0 -1 -4 3 }}
{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}
 
: sval mapping generators: ~2, ~3


{{Mapping|legend=3| 1 2 4 0 0 0 0 3 | 0 -1 -4 0 0 0 0 3 }}
{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}


: [[gencom]]: [2 4/3; 81/80 96/95]
: [[gencom]]: [2 3; 81/80 96/95]


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 697.867
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 697.867
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