Hemimean family: Difference between revisions

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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  
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.
The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49).  


= Hemimean =
= Hemimean =
[[Comma]]: 3136/3125, hemimean
[[Comma list]]: 3136/3125 (hemimean)


7 and 9 limit minimax
[[Mapping]]: [{{val| 1 0 0 -3 }}, {{val| 0 1 0 0 }}, {{val| 0 0 2 5 }}]


[|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>]
Mapping generators: ~2, ~3, ~56/25


[[Eigenmonzo]]s: 2, 7/6, 4/3
Map to lattice: [{{val| 0 0 2 5 }}, {{val| 0 1 0 0 }}]


Lattice basis: 28/25 length 0.5055 3/2 length 1.5849
Lattice basis:
: 28/25 length = 0.5055, 3/2 length = 1.5849
: Angle (28/25, 3/2) = 90 degrees


Angle(28/25, 3/2) = 90 degrees
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s: 2, 7/6, 4/3


Map to lattice: [<0 0 2 5|, <0 1 0 0|]
{{Val list|legend=1| 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c }}


Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|]
[[Badness]]: 0.160 × 10<sup>-3</sup>


[[Generator]]s: 2, 3, 56/25
[[Complexity spectrum]]: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7
 
[[EDO|Edos]]: {{EDOs|68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c}}
 
Badness: 0.000160


[[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]]s: 441/440, 3136/3125
[[Comma list]]: 441/440, 3136/3125
 
11 limit minimax


[|1 0 0 0 0&gt;, |27/22 6/11 -5/22 -3/11 5/22&gt;,  
[[Mapping]]: [{{val| 1 0 0 -3 -9 }}, {{val| 0 1 0 0 2 }}, {{val| 0 0 2 5 8 }}]
|24/11 -4/11 -2/11 2/11 2/11&gt;,  
|27/11 -10/11 -5/11 5/11 5/11&gt;,
|24/11 -4/11 -13/11 2/11 13/11&gt;]


[[Eigenmonzo]]s: 2, 11/10, 9/7
Mapping generators: ~2, ~3, ~56/25


Lattice basis: 28/25 0.3829 16/15 1.1705
Map to lattice: [{{val| 0 -2 2 5 4 }}, {{val| 0 -1 0 0 -2 }}]


Angle(28/25, 16/15) = 93.2696
Lattice basis:
: 28/25 length = 0.3829, 16/15 length = 1.1705
: Angle (28/25, 16/15) = 93.2696


Map to lattice: [&lt;0 -2 2 5 4|, &lt;0 -1 0 0 -2|]
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 27/22 6/11 -5/22 -3/11 5/22 }}, {{monzo| 24/11 -4/11 -2/11 2/11 2/11 }}, {{monzo| 27/11 -10/11 -5/11 5/11 5/11 }}, {{monzo| 24/11 -4/11 -13/11 2/11 13/11 }}]
: [[Eigenmonzo]]s: 2, 11/10, 9/7


Map: [&lt;1 0 0 -3 -9|, &lt;0 1 0 0 2|, [0 0 2 5 8|]
{{Val list|legend=1| 31, 87, 118, 130, 217, 248, 378, 626, 961cd }}


[[Generator]]s: 2, 3, 56/25
[[Badness]]: 0.609 × 10<sup>-3</sup>


[[EDO|Edos]]: 12, 31, 87, 118, 130, 217, [[248edo|248]], [[378edo|378]], [[626edo|626]], [[961edo|961cd]]
[[Projection pair]]s: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7
 
Badness: 0.000609
 
Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7


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


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


Map: [&lt;1 0 0 -3 -9 15|, &lt;0 1 0 0 2 -2|, &lt;0 0 2 5 8 -7|]
Mapping: [{{val| 1 0 0 -3 -9 15 }}, {{val| 0 1 0 0 2 -2 }}, {{val| 0 0 2 5 8 -7 }}]


EDOs: 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e
{{Val list|legend=1| 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}


Badness: 0.00111
Badness: 1.11 × 10<sup>-3</sup>


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


Map: [&lt;1 0 0 -3 -9 -4|, &lt;0 1 0 0 2 -1|, &lt;0 0 2 5 8 8|]
Mapping: [{{val| 0 0 -3 -9 -4 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 0 0 2 5 8 8 }}]


EDOs: 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df
{{Val list|legend=1| 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df }}


Badness: 0.00126
Badness: 1.26 × 10<sup>-3</sup>


= Siebog =
= Siebog =
[[Comma]]s: 540/539, 3136/3125
[[Comma list]]: 540/539, 3136/3125
 
11 limit minimax
 
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |8/5 3/5 1/5 0 -1/5&gt;,
|1 3/2 1/2 0 -1/2&gt;, |8/5 3/5 -4/5 0 4/5&gt;]


[[Eigenmonzo]]s: 2, 11/10, 4/3
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 8/5 3/5 1/5 0 -1/5 }}, {{monzo| 1 3/2 1/2 0 -1/2 }}, {{monzo| 8/5 3/5 -4/5 0 4/5 }}]
: [[Eigenmonzo]]s: 2, 11/10, 4/3


Map: [&lt;1 0 0 -3 8|, &lt;0 1 0 0 3|, &lt;0 0 2 5 -8|]
[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]


[[generator|Generators]]: 2, 3, 768/343
Mapping generators: ~2, ~3, ~768/343


[[EDO|Edos]]: 19, 31, 80, 99d, 111, [[130edo|130]], [[241edo|241]], 340ce, 371ce, 470cde, 711cde
{{Val list|legend=1| 19, 31, 80, 99, 111, 130, 241, 340ce, 371ce, 470cde, 711cde }}


Badness: 0.000870
[[Badness]]: 0.870 × 10<sup>-3</sup>


= Triglav =
= Triglav =
Commas: 3025/3024, 3136/3125
[[Comma list]]: 3025/3024, 3136/3125


Map: [&lt;1 0 2 2 1|, &lt;0 1 2 5 2|, &lt;0 0 -4 -10 -1|]
[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}]


EDOs: 31, 80, 87, 111, 118, 198, 316, 545c, 861ce
{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce


Badness: 0.000819
[[Badness]]: 0.819 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Theory]]

Revision as of 18:43, 5 April 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).

Hemimean

Comma list: 3136/3125 (hemimean)

Mapping: [⟨1 0 0 -3], ⟨0 1 0 0], ⟨0 0 2 5]]

Mapping generators: ~2, ~3, ~56/25

Map to lattice: [⟨0 0 2 5], ⟨0 1 0 0]]

Lattice basis:

28/25 length = 0.5055, 3/2 length = 1.5849
Angle (28/25, 3/2) = 90 degrees

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Eigenmonzos: 2, 7/6, 4/3

Template:Val list

Badness: 0.160 × 10-3

Complexity spectrum: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7

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

Belobog

Comma list: 441/440, 3136/3125

Mapping: [⟨1 0 0 -3 -9], ⟨0 1 0 0 2], ⟨0 0 2 5 8]]

Mapping generators: ~2, ~3, ~56/25

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

Lattice basis:

28/25 length = 0.3829, 16/15 length = 1.1705
Angle (28/25, 16/15) = 93.2696

Minimax tuning:

[[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

Template:Val list

Badness: 0.609 × 10-3

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

Scales: belobog31

13-limit

Comma list: 441/440, 1001/1000, 3136/3125

Mapping: [⟨1 0 0 -3 -9 15], ⟨0 1 0 0 2 -2], ⟨0 0 2 5 8 -7]]

Template:Val list

Badness: 1.11 × 10-3

Bellowblog

Comma list: 196/195, 352/351, 625/624

Mapping: [⟨0 0 -3 -9 -4], ⟨0 1 0 0 2 -1], ⟨0 0 2 5 8 8]]

Template:Val list

Badness: 1.26 × 10-3

Siebog

Comma list: 540/539, 3136/3125

Minimax tuning:

[[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

Mapping: [⟨1 0 0 -3 8], ⟨0 1 0 0 3], ⟨0 0 2 5 -8]]

Mapping generators: ~2, ~3, ~768/343

Template:Val list

Badness: 0.870 × 10-3

Triglav

Comma list: 3025/3024, 3136/3125

Mapping: [⟨1 0 2 2 1], ⟨0 1 2 5 2], ⟨0 0 -4 -10 -1]]

{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce

Badness: 0.819 × 10-3

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