Xenharmonic Wiki

Hemimean family

Revision as of 08:28, 6 July 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings)
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:

  • WE: ~2 = 1199.8194 ¢, ~3/2 = 702.1353 ¢, ~28/25 = 193.7425 ¢
error map: ⟨-0.181 -0.000 +0.810 -0.474]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.1118 ¢, ~28/25 = 193.7167 ¢
error map: ⟨0.000 +0.157 +1.120 -0.243]

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

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:

  • WE: ~2 = 1200.0098 ¢, ~3/2 = 701.7170 ¢, ~28/25 = 193.5520 ¢
error map: ⟨+0.010 -0.228 +0.810 -1.046 +0.542]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7144 ¢, ~28/25 = 193.5518 ¢
error map: ⟨0.000 -0.241 +0.790 -1.067 +0.525]

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:

  • WE: ~2 = 1199.9154 ¢, ~3/2 = 701.7875 ¢, ~28/25 = 193.5853 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8346 ¢, ~28/25 = 193.5962 ¢

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

Badness (Sintel): 1.03

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:

  • WE: ~2 = 1200.000 ¢, ~3/2 = 702.5857 ¢, ~28/25 = 193.2930 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.6342 ¢, ~28/25 = 193.2932 ¢

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

Badness (Sintel): 1.18

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:

  • WE: ~2 = 1199.5790 ¢, ~3/2 = 701.8397 ¢, ~28/25 = 194.0111 ¢
error map: ⟨-0.421 -0.536 +0.867 +0.388 +0.849]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7235 ¢, ~28/25 = 193.9948 ¢
error map: ⟨0.000 -0.232 +1.676 +1.148 +1.894]

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): 1.05

Triglav

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125

Mapping: [⟨1 0 -2 -8 0], ⟨0 1 2 5 2], ⟨0 0 4 10 1]]

Mapping generators: ~2, ~3, ~11/9

Optimal tunings:

  • WE: ~2 = 1199.8764 ¢, ~3/2 = 702.4302 ¢, ~11/9 = 345.5856 ¢
error map: ⟨-0.124 +0.352 +0.889 -0.448 -1.119]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4067 ¢, ~11/9 = 345.6505 ¢
error map: ⟨0.000 +0.452 +1.102 -0.288 -0.854]

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 -8 0 5], ⟨0 1 2 5 2 -1], ⟨0 0 4 10 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.6554 ¢, ~3/2 = 702.8049 ¢, ~11/9 = 345.3412 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9366 ¢, ~11/9 = 345.4458 ¢

Optimal ET sequence: 24d, 31, 56, 80, 87, 111, 118, 167, 198

Badness (Sintel): 1.16

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:

  • WE: ~99/70 = 599.9102 ¢, ~3/2 = 702.1314 ¢, ~28/25 = 193.7431 ¢
error map: ⟨-0.180 -0.003 +0.813 -0.470 -0.008]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.1353 ¢, ~28/25 = 193.7122 ¢
error map: ⟨0.000 +0.180 +1.111 -0.265 +0.397]

Optimal ET sequence: 12, …, 38d, 50, 68, 80, 118, 130, 198, 248, 328, 446, 774c

Badness (Sintel): 1.79

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:

  • W: ~99/70 = 599.8791 ¢, ~3/2 = 702.1476 ¢, ~28/25 = 193.7877 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.1740 ¢, ~28/25 = 193.7869 ¢

Optimal ET sequence: 12, …, 50, 68, 80, 118, 130, 198, 328, 576cf, 774cf, 904cef

Badness (Sintel): 1.55

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:

  • WE: ~17/12 = 599.9481 ¢, ~3/2 = 702.2594 ¢, ~28/25 = 193.7763 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.2692 ¢, ~28/25 = 193.7762 ¢

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

Badness (Sintel): 1.74

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:

  • WE: ~17/12 = 599.9687 ¢, ~3/2 = 702.3501 ¢, ~19/17 = 193.7922 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.3549 ¢, ~19/17 = 193.7919 ¢

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

Badness (Sintel): 1.32

Subgroup extensions

Hemimean orion (2.3.5.7.17)

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 their difference 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:

  • WE: ~2 = 1199.7919 ¢, ~3/2 = 702.2561 ¢, ~28/25 = 193.7586 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3042 ¢, ~28/25 = 193.7365 ¢

Optimal ET sequence: 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328, 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:

  • WE: ~2 = 1199.8239 ¢, ~3/2 = 702.1623 ¢, ~19/17 = 193.7325 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2129 ¢, ~19/17 = 193.7161 ¢

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

Badness (Sintel): 0.578

Semiorion (2.3.5.7.17)

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:

  • WE: ~17/12 = 600.0551 ¢, ~3/2 = 702.1998 ¢, ~28/25 = 193.5929 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.2183 ¢, ~28/25 = 193.6044 ¢

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

Badness (Sintel): 1.69

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:

  • WE: ~17/12 = 600.0873 ¢, ~3/2 = 702.2450 ¢, ~19/17 = 193.5751 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 702.2791 ¢, ~19/17 = 193.5923 ¢

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

Badness (Sintel): 0.722

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