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[https://en.wikipedia.org/wiki/Septimal_meantone_temperament Wikipedia article]
[https://en.wikipedia.org/wiki/Septimal_meantone_temperament Wikipedia article]


The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7/4|7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7/5|7/5]], C-F#, the tritone. The [[Wedgies_and_Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo|31edo]] is a good tuning for it.
The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7/5]], C-F#, the tritone. The [[Wedgies_and_Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it.


[[Comma|Comma]]s: 81/80, 126/125
[[Comma]]s: 81/80, 126/125


7 and [[9-odd-limit|9-limit]] minimax
7 and [[9-odd-limit|9-limit]] minimax
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[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
[[Eigenmonzo]]s: 2, 5


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
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Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.
Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.


[[Map|Map]]: [<1 0 -4 -13|, <0 1 4 10|]
[[Map]]: [<1 0 -4 -13|, <0 1 4 10|]


[[generator|Generator]]s: 2, 3
[[Generator]]s: 2, 3


[[wedgie|Wedgie]]: <<1 4 10 4 13 12||
[[Wedgie]]: <<1 4 10 4 13 12||


EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[62edo|62]], [[74edo|74]], [[81edo|81]], [[93edo|93]], [[105edo|105]], [[143edo|143b]]
EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[74edo|74]], [[81edo|81]], [[105edo|105]], [[143edo|143b]]


[[Badness|Badness]]: 0.0137
[[Badness]]: 0.0137


==Bimeantone==
==Bimeantone==
11/8 is mapped to half octave minus the meantone diesis.
11/8 is mapped to half octave minus the meantone diesis.


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See also [[Meantone_vs_meanpop|Meantone vs meanpop]]
See also [[Meantone_vs_meanpop|Meantone vs meanpop]]


[[Comma|Comma]]s: 81/80, 126/125, 99/98
[[Comma]]s: 81/80, 126/125, 99/98


[[11-limit|11-limit]] minimax
[[11-odd-limit|11-limit]] minimax


[|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>,
[|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>]
|21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>]


[[Eigenmonzo|Eigenmonzo]]s: 2, 11/9
[[Eigenmonzo]]s: 2, 11/9


valid range: [696.774, 700.000] (31 to 12)
valid range: [696.774, 700.000] (31 to 12)
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[[Algebraic_generator|Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.
[[Algebraic_generator|Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.


[[Map|Map]]: [<1 0 -4 -13 -25|, <0 1 4 10 18|]
[[Map]]: [<1 0 -4 -13 -25|, <0 1 4 10 18|]


[[generator|Generator]]s: 2, 3
[[Generator]]s: 2, 3


EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[62edo|62]], [[74edo|74]], [[105edo|105]], [[198edo|198be]]
EDOs: [[12edo|12]], [[31edo|31]], [[43edo|43]], [[74edo|74]], [[105edo|105]], [[198edo|198be]]


[[Badness|Badness]]: 0.0170
[[Badness]]: 0.0170


[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]


===Tridecimal meantone===
===Tridecimal meantone===
[[Comma|Comma]]s: 66/65, 81/80, 99/98, 105/104
Commas: 66/65, 81/80, 99/98, 105/104


valid range: 697.674 (43)
valid range: 697.674 (43)
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Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]
Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]


EDOs: [[31edo|31]]
EDOs: 12f, 31, 43f


[[Badness|Badness]]: 0.0180
Badness: 0.0180


===Grosstone===
===Grosstone===
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Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|]
Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|]


EDOs: 43, 117df, 160bdf, 203bcdef
EDOs: 12f, 31f, 43, 117df, 160bdf, 203bcdef


Badness: 0.0264
Badness: 0.0264
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See also [[Meantone_vs_meanpop|Meantone vs meanpop]]
See also [[Meantone_vs_meanpop|Meantone vs meanpop]]


[[Comma|Comma]]s: 81/80, 126/125, 385/384
[[Comma]]s: 81/80, 126/125, 385/384


[[11-limit|11-limit]] [[minimax|minimax]] 1/4 comma
[[11-odd-limit|11-limit]] minimax 1/4 comma


[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>,
[|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>]
|-3 0 5/2 0 0>, |11 0 -13/4 0 0>]


[[Eigenmonzo|Eigenmonzo]]s: 2, 5
[[Eigenmonzo]]s: 2, 5


valid range: [694.737, 696.774] (19 to 31)
valid range: [694.737, 696.774] (19 to 31)
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Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]
Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]


[[generator|Generator]]s: 2, 3
[[Generator]]s: 2, 3


EDOs: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[62edo|62]], [[81edo|81]]
EDOs: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]


[[Badness|Badness]]: 0.0215
[[Badness]]: 0.0215


[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]
[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]


===13-limit Meanpop===
===13-limit Meanpop===
[[Comma|Comma]]s: 81/80, 105/104, 144/143, 196/195
Commas: 81/80, 105/104, 126/125, 144/143


valid range: [694.737, 696.774] (19 to 31)
valid range: [694.737, 696.774] (19 to 31)
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Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|]
Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|]


EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]]
EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]


[[Badness|Badness]]: 0.0209
Badness: 0.0209


===Meanplop===
===Meanplop===
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==Meanenneadecal==
==Meanenneadecal==
[[Comma|Comma]]s: 45/44, 56/55, 81/80
Commas: 45/44, 56/55, 81/80


[[POTE_tuning|POTE generator]]: ~3/2 = 696.250
POTE generator: ~3/2 = 696.250


Mapping generator: ~3
Mapping generator: ~3
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Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]
Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]


EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]]
EDOs: [[7edo|7d]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50ee]]


[[Badness|Badness]]: 0.0214
Badness: 0.0214


===13-limit===
===13-limit===
[[Comma|Comma]]s: 45/44, 56/55, 78/77, 81/80
Commas: 45/44, 56/55, 78/77, 81/80


[[POTE_tuning|POTE generator]]: ~3/2 = 696.146
POTE generator: ~3/2 = 696.146


Mapping generator: ~3
Mapping generator: ~3
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Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]
Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]


EDOs: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]]
EDOs: 12f, 19, 31e, 50ee


[[Badness|Badness]]: 0.0212
Badness: 0.0212


===Vincenzo===
==Vincenzo==
Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22
Commas: 45/44, 56/55, 65/64, 81/80


POTE generator: ~3/2
POTE generator: ~3/2 = 695.060


Mapping generator: ~3
Mapping generator: ~3


Map: [<1 0 -4 -13 ... |, <0 1 4 10 6 -4 -5 -3 -6|]
Map: [<1 0 -4 -13 -6 10|, <0 1 4 10 6 -4|]
 
EDOs: 7d, 12, 19, 26d
 
Badness: 0.0248
 
===17-limit===
Commas: 45/44, 52/51, 56/55, 65/64, 81/80
 
POTE generator: ~3/2 = 695.858
 
Map: [<1 0 -4 -13 -6 10 12|, <0 1 4 10 6 -4 -5|]
 
EDOs: 7d, 12, 19
 
Badness: 0.0255
 
===19-limit===
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
 
POTE generator: ~3/2 = 696.131
 
Map: [<1 0 -4 -13 -6 10 12 9|, <0 1 4 10 6 -4 -5 -3|]
 
EDOs: 7d, 12, 19
 
Badness: 0.0223
 
===23-limit===
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
 
POTE generator: ~3/2 = 696.044
 
Map: [<1 0 -4 -13 -6 10 12 14|, <0 1 4 10 6 -4 -5 -3 -6|]


EDOs: 12
EDOs: 7d, 12, 19


Badness:
Badness: 0.0201


==Meanundeci==
==Meanundeci==
Retrieved from "https://en.xen.wiki/w/Meantone_family"