Hemimean family: Difference between revisions

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Line 17: Line 17:
: 28/25 length = 0.5055, 3/2 length = 1.5849
: 28/25 length = 0.5055, 3/2 length = 1.5849
: Angle (28/25, 3/2) = 90 degrees
: Angle (28/25, 3/2) = 90 degrees
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 23: Line 25:
: [[Eigenmonzo basis]]: 2.3.7
: [[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, 675c }}


[[Badness]]: 0.160 × 10<sup>-3</sup>
[[Badness]]: 0.160 × 10<sup>-3</sup>
Line 103: Line 105:
: 28/25 length = 0.3829, 16/15 length = 1.1705
: 28/25 length = 0.3829, 16/15 length = 1.1705
: Angle (28/25, 16/15) = 93.2696
: Angle (28/25, 16/15) = 93.2696
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 109: Line 113:
: [[Eigenmonzo basis]]: 2.9/7.11/5
: [[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, 68e, 87, 99e, 118, 130, 217, 248 }}


[[Badness]]: 0.609 × 10<sup>-3</sup>
[[Badness]]: 0.609 × 10<sup>-3</sup>
Line 123: Line 127:


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 }}]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816


Optimal GPV sequence: {{Val list| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
Optimal GPV sequence: {{Val list| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
Line 133: Line 139:
Comma list: 196/195, 352/351, 625/624
Comma list: 196/195, 352/351, 625/624


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


Optimal GPV sequence: {{Val list| 12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df }}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493
 
Optimal GPV sequence: {{Val list| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


Badness: 1.26 × 10<sup>-3</sup>
Badness: 1.26 × 10<sup>-3</sup>
Line 146: Line 154:
[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]
[[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
Mapping generators: ~2, ~3, ~56/25
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 153: Line 163:
: [[Eigenmonzo basis]]: 2.3.11/10
: [[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, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}


[[Badness]]: 0.870 × 10<sup>-3</sup>
[[Badness]]: 0.870 × 10<sup>-3</sup>
Line 166: Line 176:
Mapping generators: ~2, ~3, ~18/11
Mapping generators: ~2, ~3, ~18/11


{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132
 
{{Val list|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}


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

Revision as of 12:04, 18 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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Eigenmonzo basis: 2.3.7

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

Hemimean orion

As the second generator of hemimean, 28/25, is close to 19/17, and as the latter is the mediant of 5/4, it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out (28/25)/(19/17) = 476/475, or equivalently stated, the semiparticular (5/4)/(19/17)2 = 1445/1444. Notice 3136/3125 = (476/475)(2128/2125) and that 2128/2125 = (1216/1215)(1701/1700), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

Sval mapping: [⟨1 0 0 -3 -5], ⟨0 1 0 0 5], ⟨0 0 2 5 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548

Optimal GPV sequence: Template:Val list

Badness: 0.573

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 476/475, 1216/1215, 1445/1444

Sval mapping: [⟨1 0 0 -3 -5 -6], ⟨0 1 0 0 5 5], ⟨0 0 2 5 1 2]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647

Optimal GPV sequence: Template:Val list

Badness: 0.456

Semiorion

Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two.

Subgroup: 2.3.5.7.17

Comma list: 289/288, 3136/3125

Sval mapping: [⟨2 0 0 -6 5], ⟨0 1 0 0 1], ⟨0 0 2 5 0]]

Sval mapping generators: ~17/12, ~3, ~56/25

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499

Optimal GPV sequence: Template:Val list

Badness: 1.095

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 289/288, 361/360, 476/475

Mapping: [⟨2 0 0 -6 5 3], ⟨0 1 0 0 1 1], ⟨0 0 2 5 0 1]]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669

Optimal GPV sequence: Template:Val list

Badness: 0.569

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545

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⟩]
Eigenmonzo basis: 2.9/7.11/5

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

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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493

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, ~56/25

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645

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⟩]
Eigenmonzo basis: 2.3.11/10

Template:Val list

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132

Template:Val list

Badness: 0.819 × 10-3

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