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 | ||
| Line 32: | Line 32: | ||
[[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 | ||
| Line 61: | Line 61: | ||
Scales: [[belobog31]] | Scales: [[belobog31]] | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 73: | Line 73: | ||
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 | ||
| Line 85: | Line 85: | ||
Badness: 1.26 × 10<sup>-3</sup> | Badness: 1.26 × 10<sup>-3</sup> | ||
= Siebog = | == Siebog == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 104: | Line 104: | ||
[[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 | ||
Revision as of 21:32, 1 June 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
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
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
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
- Eigenmonzos: 2, 7/6, 4/3
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
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
- [[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
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]]
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]]
Badness: 1.26 × 10-3
Siebog
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125
- [[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
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]]
Badness: 0.819 × 10-3