Hemimean family: Difference between revisions

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Don't know why it doesn't start with order 1 edos ("order 1" means 225/224 and 126/125 are one step each)
m Cleanup; recategory
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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  
The hemimean comma, [[3136/3125]], is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with [[Spectrum_of_a_temperament|spectrum]] 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7.


[[Comma|Comma]]: 3136/3125, hemimean
The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with [[Spectrum of a temperament|spectrum]] 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7.
 
= Hemimean =
[[Comma]]: 3136/3125, hemimean


7 and 9 limit minimax
7 and 9 limit minimax
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[|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>]
[|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
[[Eigenmonzo]]s: 2, 7/6, 4/3


Lattice basis: 28/25 length 0.5055 3/2 length 1.5849
Lattice basis: 28/25 length 0.5055 3/2 length 1.5849
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Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|]  
Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|]  


[[generator|Generators]]: 2, 3, 56/25
[[Generator]]s: 2, 3, 56/25


[[EDO|Edos]]: {{EDOs|68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c}}
[[EDO|Edos]]: {{EDOs|68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c}}
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Badness: 0.000160
Badness: 0.000160


[[Projection_pair|Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7
[[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7


=Belobog=
= Belobog =
[[Comma|Commas]]: 3136/3125, 441/440
[[Comma]]s: 441/440, 3136/3125


11 limit minimax
11 limit minimax
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|24/11 -4/11 -13/11 2/11 13/11>]
|24/11 -4/11 -13/11 2/11 13/11>]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/10, 9/7
[[Eigenmonzo]]s: 2, 11/10, 9/7


Lattice basis: 28/25 0.3829 16/15 1.1705
Lattice basis: 28/25 0.3829 16/15 1.1705
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Map: [<1 0 0 -3 -9|, <0 1 0 0 2|, [0 0 2 5 8|]
Map: [<1 0 0 -3 -9|, <0 1 0 0 2|, [0 0 2 5 8|]


[[generator|Generators]]: 2, 3, 56/25
[[Generator]]s: 2, 3, 56/25


[[EDO|Edos]]: 12, 31, 87, 118, 130, 217, [[248edo|248]], [[378edo|378]], [[626edo|626]], [[961edo|961cd]]
[[EDO|Edos]]: 12, 31, 87, 118, 130, 217, [[248edo|248]], [[378edo|378]], [[626edo|626]], [[961edo|961cd]]
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Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7
Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7


Scales: [[belobog31|belobog31]]
Scales: [[belobog31]]


==13-limit==
== 13-limit ==
Commas: 441/440, 1001/1000, 3136/3125
Commas: 441/440, 1001/1000, 3136/3125


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Badness: 0.00111
Badness: 0.00111


==Bellowblog==
== Bellowblog ==
Commas: 196/195, 352/351, 625/624
Commas: 196/195, 352/351, 625/624


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Badness: 0.00126
Badness: 0.00126


=Siebog=
= Siebog =
[[Comma|Commas]]: 3136/3125, 540/539
[[Comma]]s: 540/539, 3136/3125


11 limit minimax
11 limit minimax
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|1 3/2 1/2 0 -1/2>, |8/5 3/5 -4/5 0 4/5>]
|1 3/2 1/2 0 -1/2>, |8/5 3/5 -4/5 0 4/5>]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/10, 4/3
[[Eigenmonzo]]s: 2, 11/10, 4/3


Map: [<1 0 0 -3 8|, <0 1 0 0 3|, <0 0 2 5 -8|]
Map: [<1 0 0 -3 8|, <0 1 0 0 3|, <0 0 2 5 -8|]
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Badness: 0.000870
Badness: 0.000870


=Triglav=
= Triglav =
Commas: 3136/3125, 3025/3024
Commas: 3025/3024, 3136/3125


Map: [<1 0 2 2 1|, <0 1 2 5 2|, <0 0 -4 -10 -1|]
Map: [<1 0 2 2 1|, <0 1 2 5 2|, <0 0 -4 -10 -1|]
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[[Category:Theory]]
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Planar temperament]]
[[Category:Hemimean]]
[[Category:Hemimean]]
[[Category:Rank 3]]

Revision as of 05:30, 6 January 2021

The hemimean family of temperaments are rank-3 temperaments tempering out 3136/3125.

The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with spectrum 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7.

Hemimean

Comma: 3136/3125, hemimean

7 and 9 limit minimax

[|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>]

Eigenmonzos: 2, 7/6, 4/3

Lattice basis: 28/25 length 0.5055 3/2 length 1.5849

Angle(28/25, 3/2) = 90 degrees

Map to lattice: [<0 0 2 5|, <0 1 0 0|]

Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|]

Generators: 2, 3, 56/25

Edos: 68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c

Badness: 0.000160

Projection pairs: 5 3136/625 7 68841472/9765625 to 2.3.25/7

Belobog

Commas: 441/440, 3136/3125

11 limit minimax

[|1 0 0 0 0>, |27/22 6/11 -5/22 -3/11 5/22>, |24/11 -4/11 -2/11 2/11 2/11>, |27/11 -10/11 -5/11 5/11 5/11>, |24/11 -4/11 -13/11 2/11 13/11>]

Eigenmonzos: 2, 11/10, 9/7

Lattice basis: 28/25 0.3829 16/15 1.1705

Angle(28/25, 16/15) = 93.2696

Map to lattice: [<0 -2 2 5 4|, <0 -1 0 0 -2|]

Map: [<1 0 0 -3 -9|, <0 1 0 0 2|, [0 0 2 5 8|]

Generators: 2, 3, 56/25

Edos: 12, 31, 87, 118, 130, 217, 248, 378, 626, 961cd

Badness: 0.000609

Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7

Scales: belobog31

13-limit

Commas: 441/440, 1001/1000, 3136/3125

Map: [<1 0 0 -3 -9 15|, <0 1 0 0 2 -2|, <0 0 2 5 8 -7|]

EDOs: 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e

Badness: 0.00111

Bellowblog

Commas: 196/195, 352/351, 625/624

Map: [<1 0 0 -3 -9 -4|, <0 1 0 0 2 -1|, <0 0 2 5 8 8|]

EDOs: 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df

Badness: 0.00126

Siebog

Commas: 540/539, 3136/3125

11 limit minimax

[|1 0 0 0 0>, |0 1 0 0 0>, |8/5 3/5 1/5 0 -1/5>, |1 3/2 1/2 0 -1/2>, |8/5 3/5 -4/5 0 4/5>]

Eigenmonzos: 2, 11/10, 4/3

Map: [<1 0 0 -3 8|, <0 1 0 0 3|, <0 0 2 5 -8|]

Generators: 2, 3, 768/343

Edos: 19, 31, 80, 99d, 111, 130, 241, 340ce, 371ce, 470cde, 711cde

Badness: 0.000870

Triglav

Commas: 3025/3024, 3136/3125

Map: [<1 0 2 2 1|, <0 1 2 5 2|, <0 0 -4 -10 -1|]

EDOs: 31, 80, 87, 111, 118, 198, 316, 545c, 861ce

Badness: 0.000819

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