Hemimean family

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The hemimean family of rank-3 temperaments tempers out 3136/3125, the hemimean comma.

The hemimean comma is the difference between the septimal semicomma (126/125) and the septimal kleisma (225/224). This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the syntonic comma (81/80) into two equal parts, each representing 126/125~225/224.

Other equivalences characteristic to hemimean are 128/125~50/49 and 49/45~(25/24)2.

Hemimean

Subgroup: 2.3.5.7

Comma list: 3136/3125

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~28/25 = 193.650 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.112 ¢, ~28/25 = 193.717 ¢

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c

Badness (Sintel): 0.706

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 10/9 and 9/8, 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. An interesting tuning not in the optimal ET sequence is 111edo. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.

The S-expression-based comma list for the 2.3.5.7.17.19 subgroup extension is {S16/S18, S17/S19, S18/S20(, (S16*S17)/(S19*S20) = S16/S18 * S17/S19 * S18/S20)}.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.196 ¢, ~28/25 = 193.655 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.304 ¢, ~28/25 = 193.737 ¢

Optimal ET sequence: 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg

Badness (Sintel): 0.884

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.132 ¢, ~19/17 = 193.647 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.213 ¢, ~19/17 = 193.716 ¢

Optimal ET sequence: 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446

Badness (Sintel): 0.578

Semiorion

Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The S-expression-based comma list for the 2.3.5.7.17.19 subgroup extension is {S17, S19, S16/S18(, S18/S20, 476/475 = S16/S20 * S17/S19)}.

Subgroup: 2.3.5.7.17

Comma list: 289/288, 3136/3125

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

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

Optimal tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.347 ¢, ~28/25 = 193.650 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.218 ¢, ~28/25 = 193.604 ¢

Optimal ET sequence: 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g

Badness (Sintel): 1.690

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 tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.509 ¢, ~19/17 = 193.669 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.279 ¢, ~19/17 = 193.592 ¢

Optimal ET sequence: 12, …, 50, 68, 106d, 118, 248g, 316g

Badness (Sintel): 0.722

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.720 ¢, ~28/25 = 193.554 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.714 ¢, ~28/25 = 193.552 ¢

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⟩]
Unchanged-interval (eigenmonzo) basis: 2.9/7.11/5

Optimal ET sequence: 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248

Badness (Sintel): 0.732

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.822 ¢, ~28/25 = 193.582 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.835 ¢, ~28/25 = 193.596 ¢

Optimal ET sequence: 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e

Badness (Sintel): 1.034

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.567 ¢, ~28/25 = 193.249 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.634 ¢, ~28/25 = 193.293 ¢

Optimal ET sequence: 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d

Badness (Sintel): 1.183

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 701.164 ¢, ~28/25 = 193.865 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 701.723 ¢, ~28/25 = 193.995 ¢

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⟩]
Unchanged-interval (eigenmonzo) basis: 2.3.11/5

Optimal ET sequence: 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee

Badness (Sintel): 1.045

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 tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.288 ¢, ~18/11 = 854.313 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.407 ¢, ~18/11 = 854.350 ¢

Optimal ET sequence: 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c

Badness (Sintel): 0.984

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024

Mapping: [⟨1 0 2 2 1 6], ⟨0 1 2 5 2 -1], ⟨0 0 -4 -10 -1 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 702.707 ¢, ~18/11 = 854.537 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.937 ¢, ~18/11 = 854.554 ¢

Optimal ET sequence: 24d, 31, 80, 87, 111, 118, 198

Badness (Sintel): 1.159

Semihemimean

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 9801/9800

Mapping: [⟨2 0 0 -6 -3], ⟨0 1 0 0 -2], ⟨0 0 2 5 7]]

Mapping generators: ~99/70, ~3, ~56/25

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~3/2 = 702.002 ¢, ~28/25 = 193.633 ¢
  • CWE: ~99/70 = 600.000 ¢, ~3/2 = 702.135 ¢, ~28/25 = 193.712 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.787

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3136/3125, 4459/4455

Mapping: [⟨2 0 0 -6 -3 15], ⟨0 1 0 0 -2 2], ⟨0 0 2 5 7 -6]]

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~3/2 = 701.838 ¢, ~28/25 = 193.671 ¢
  • CWE: ~99/70 = 600.000 ¢, ~3/2 = 702.174 ¢, ~28/25 = 193.787 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.550

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 561/560, 1001/1000, 1632/1625

Mapping: [⟨2 0 0 -6 -3 15 5], ⟨0 1 0 0 -2 2 1], ⟨0 0 2 5 7 -6 0]]

Optimal tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.108 ¢, ~28/25 = 193.723 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.269 ¢, ~28/25 = 193.776 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.743

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 361/360, 456/455, 476/475, 561/560

Mapping: [⟨2 0 0 -6 -3 15 5 3], ⟨0 1 0 0 -2 2 1 1], ⟨0 0 2 5 7 -6 0 1]]

Optimal tunings:

  • CTE: ~17/12 = 600.000 ¢, ~3/2 = 702.252 ¢, ~19/17 = 193.758 ¢
  • CWE: ~17/12 = 600.000 ¢, ~3/2 = 702.355 ¢, ~19/17 = 193.792 ¢

Optimal ET sequence: 12, 50, 68, 80, 118, 130, 198

Badness (Sintel): 1.318

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