Hemimean family: Difference between revisions
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The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). | 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 == | ||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 3136/3125 (hemimean) | [[Comma list]]: 3136/3125 (hemimean) | ||
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* 7- and [[9-odd-limit]] | * 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 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}] | ||
: [[Eigenmonzo]] | : [[Eigenmonzo basis]]: 2.3.7 | ||
{{Val list|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c }} | {{Val list|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c }} | ||
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[[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 == | ||
[[Subgroup]]: 2.3.5.7.11 | |||
Subgroup: 2.3.5.7.11 | |||
[[Comma list]]: 441/440, 3136/3125 | [[Comma list]]: 441/440, 3136/3125 | ||
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* [[11-odd-limit]] | * [[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 }}] | : [{{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]] | : [[Eigenmonzo basis]]: 2.9/7.11/5 | ||
{{Val list|legend=1| 12, 19e, 31, 87, 99e, 118, 130, 217, 248, 378, 626, 961cd }} | {{Val list|legend=1| 12, 19e, 31, 87, 99e, 118, 130, 217, 248, 378, 626, 961cd }} | ||
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Scales: [[belobog31]] | Scales: [[belobog31]] | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Mapping: [{{val| 1 0 0 -3 -9 15 }}, {{val| 0 1 0 0 2 -2 }}, {{val| 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 }}] | ||
{{Val list | Optimal GPV sequence: {{Val list| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }} | ||
Badness: 1.11 × 10<sup>-3</sup> | Badness: 1.11 × 10<sup>-3</sup> | ||
=== Bellowblog | === Bellowblog === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Mapping: [{{val| 0 0 -3 -9 -4 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 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 }}] | ||
{{Val list | Optimal GPV sequence: {{Val list| 12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df }} | ||
Badness: 1.26 × 10<sup>-3</sup> | Badness: 1.26 × 10<sup>-3</sup> | ||
== Siebog | == Siebog == | ||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 540/539, 3136/3125 | |||
[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}] | |||
Mapping generators: ~2, ~3, ~768/343 | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[11-odd-limit]] | * [[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 }}] | : [{{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]] | : [[Eigenmonzo basis]]: 2.3.11/10 | ||
{{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }} | {{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }} | ||
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[[Badness]]: 0.870 × 10<sup>-3</sup> | [[Badness]]: 0.870 × 10<sup>-3</sup> | ||
== Triglav | == Triglav == | ||
[[Subgroup]]: 2.3.5.7.11 | |||
Subgroup: 2.3.5.7.11 | |||
[[Comma list]]: 3025/3024, 3136/3125 | [[Comma list]]: 3025/3024, 3136/3125 | ||
[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}] | [[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}] | ||
Mapping generators: ~2, ~3, ~18/11 | |||
{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce }} | {{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce }} | ||
Revision as of 11:04, 17 December 2022
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
Subgroup: 2.3.5.7
Comma list: 3136/3125 (hemimean)
Mapping: [⟨1 0 0 -3], ⟨0 1 0 0], ⟨0 0 2 5]]
Mapping generators: ~2, ~3, ~56/25
Mapping 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
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
- Eigenmonzo basis: 2.3.7
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
Subgroup: 2.3.5.7.11
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
Mapping 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
- [[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⟩]
- Eigenmonzo basis: 2.9/7.11/5
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
Subgroup: 2.3.5.7.11.13
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]]
Optimal GPV sequence: Template:Val list
Badness: 1.11 × 10-3
Bellowblog
Subgroup: 2.3.5.7.11.13
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]]
Optimal GPV sequence: Template:Val list
Badness: 1.26 × 10-3
Siebog
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125
Mapping: [⟨1 0 0 -3 8], ⟨0 1 0 0 3], ⟨0 0 2 5 -8]]
Mapping generators: ~2, ~3, ~768/343
- [[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⟩]
- Eigenmonzo basis: 2.3.11/10
Badness: 0.870 × 10-3
Triglav
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 3136/3125
Mapping: [⟨1 0 2 2 1], ⟨0 1 2 5 2], ⟨0 0 -4 -10 -1]]
Mapping generators: ~2, ~3, ~18/11
Badness: 0.819 × 10-3