Hemifamity 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 hemifamity family of rank-3 temperaments tempers out 5120/5103 (monzo: [10 -6 1 -1⟩), the hemifamity comma. These temperaments divide an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same chain of fifths inflected by the same comma to the opposite sides. In addition we may identify 10/7 with the augmented fourth (C–F#) and 50/49 with the Pythagorean comma.

Hemifamity can be further tempered to garibaldi, which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, hemifamity can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have 5/4 at the down major third (C–vE) and 7/4 at the down minor seventh (C–vBb).

Hemifamity

Subgroup: 2.3.5.7

Comma list: 5120/5103

Mapping: [⟨1 0 0 10], ⟨0 1 0 -6], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 2 -4], ⟨0 0 1 1]]

Lattice basis:

3/2 length = 0.5670, 10/9 length = 1.8063
Angle (3/2, 10/9) = 82.112 degrees

Optimal tunings:

  • WE: ~2 = 1199.7172 ¢, ~3/2 = 702.6636 ¢, ~5/4 = 386.7266 ¢
error map: ⟨-0.283 +0.426 -0.153 +0.222]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8166 ¢, ~5/4 = 386.5465 ¢
error map: ⟨0.000 +0.862 +0.233 +0.821]

Minimax tuning: c = 5120/5103

[[1 0 0 0⟩, [10/7 1/7 1/7 -1/7⟩, [0 0 1 0⟩, [10/7 -6/7 1/7 6/7⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7/3
[[1 0 0 0⟩, [5/4 1/4 1/8 -1/8⟩, [0 0 1 0⟩, [5/2 -3/2 1/4 3/4⟩]
unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness (Sintel): 0.675

Projection pairs: 7 5120/729

Music

Overview to extensions

11- and 13-limit extensions

Strong extensions of hemifamity are pele, laka, akea, and lono. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the 11/8 at the down diminished fifth (C–vGb); laka, up augmented third (C–^E#); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v3F#). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the 13/11 at the minor third (C–Eb), tempering out 352/351, 847/845, and 2080/2079.

Temperaments discussed elsewhere include:

Subgroup extensions

A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in #Subgroup extensions.

Pele

Main article: Pele
See also: Pentacircle clan

Pele tempers out 441/440 as well as 896/891 and may be described as the 41 & 46 & 58 temperament, finding the interval class of 11 at the down diminished fifth (C–vGb). It also extends parapyth. 145edo makes for an excellent tuning.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891

Mapping: [⟨1 0 0 10 17], ⟨0 1 0 -6 -10], ⟨0 0 1 1 1]]

Mapping to lattice: [⟨0 1 4 -2 -6], ⟨0 0 -1 -1 -1]]

Lattice basis:

3/2 length = 0.3812, 56/55 length = 1.5893
Angle(3/2, 56/55) = 90.4578 degrees

Optimal tunings:

  • WE: ~2 = 1199.5424 ¢, ~3/2 = 703.0109 ¢, ~5/4 = 387.6427 ¢
error map: ⟨-0.458 +0.598 +0.414 -1.995 +2.097]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2804 ¢, ~5/4 = 387.3911 ¢
error map: ⟨0.000 +1.325 +1.077 -1.117 +3.269]

Minimax tuning:

[[1 0 0 0 0⟩, [17/10 0 1/10 0 -1/10⟩, [17/5 -2 6/5 0 -1/5⟩, [16/5 -2 3/5 0 2/5⟩, [17/5 -2 1/5 0 4/5⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/9

Optimal ET sequence: 29, 41, 58, 87, 99e, 145, 186e

Badness (Sintel): 0.779

Projection pairs: 7 5120/729 11 655360/59049

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

Mapping: [⟨1 0 0 10 17 22], ⟨0 1 0 -6 -10 -13], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.4965 ¢, ~3/2 = 703.1192 ¢, ~5/4 = 388.0342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4225 ¢, ~5/4 = 387.7761 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.13/9

Optimal ET sequence: 29, 41, 46, 58, 87, 145, 232

Badness (Sintel): 0.658

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363

Mapping: [⟨1 0 0 10 17 22 8], ⟨0 1 0 -6 -10 -13 -1], ⟨0 0 1 1 1 1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.3960 ¢, ~3/2 = 703.0725 ¢, ~5/4 = 388.4246 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4518 ¢, ~5/4 = 388.4909 ¢

Optimal ET sequence: 29, 41, 46, 58, 87, 99ef, 145

Badness (Sintel): 0.884

Laka

Main article: Laka

Laka can be described as the 41 & 53 & 58 temperament, tempering out 540/539, and finds the interval class of 11 at the up augmented third (C–^E#). Gene Ward Smith considered it a 17-limit temperament, assigning the vanishing of 442/441 (41g & 53 & 58) as the main extension, but 41 & 53g & 58 also makes for a competitive extension.[1] Indeed, laka makes most sense as a 2.3.5.7.11.13.19-subgroup temperament, skipping prime 17, as the 19 is accurate and easily available in a 24-tone scale. 152edo makes for an excellent tuning, using the 152f val for prime 13.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5120/5103

Mapping: [⟨1 0 0 10 -18], ⟨0 1 0 -6 15], ⟨0 0 1 1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.6201 ¢, ~3/2 = 702.4416 ¢, ~5/4 = 386.6781 ¢
error map: ⟨-0.380 +0.107 -0.395 +0.924 +0.527]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6175 ¢, ~5/4 = 386.4170 ¢
error map: ⟨0.000 +0.663 +0.103 +1.886 +1.528]

Minimax tuning

[[1 0 0 0 0⟩, [4/3 0 2/21 -1/21 1/21⟩, [0 0 1 0 0⟩, [2 0 3/7 2/7 -2/7⟩, [2 0 3/7 -5/7 5/7⟩]
unchanged-interval (eigenmonzo) basis: 2.5.11/7

Optimal ET sequence: 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee

Badness (Sintel): 0.992

Projection pairs: 7 5120/729 11 14348907/1310720

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728

Mapping: [⟨1 0 0 10 -18 -13], ⟨0 1 0 -6 15 12], ⟨0 0 1 1 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.4742 ¢, ~3/2 = 702.3385 ¢, ~5/4 = 387.0965 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5780 ¢, ~5/4 = 386.7718 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [13/8 -1/2 1/8 0 0 1/8⟩, [13/4 -3 5/4 0 0 1/4⟩, [7/2 0 1/2 0 0 -1/2⟩, [25/8 -9/2 5/8 0 0 13/8⟩, [13/4 -3 1/4 0 0 5/4⟩]
unchanged-interval (eigenmonzo) basis: 2.11.13/7

Optimal ET sequence: 41, 53, 58, 94, 111, 152f, 415dff *

* optimal patent val: 205

Badness (Sintel): 0.769

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 352/351, 400/399, 456/455, 495/494

Mapping: [⟨1 0 0 10 -18 -13 -6], ⟨0 1 0 -6 15 12 5], ⟨0 0 1 1 -1 -1 1]]

Optimal tunings:

  • WE: ~2 = 1199.4881 ¢, ~3/2 = 702.3224 ¢, ~5/4 = 386.8881 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5613 ¢, ~5/4 = 386.6230 ¢

Optimal ET sequence: 41, 53, 58h, 94, 111, 152f, 415dffhh *

* optimal patent val: 205

Badness (Sintel): 0.647

Akea

Lattice for 13-limit akea.
Ditto, but rearranged to basis {~2, ~3, ~81/80}.

Akea tempers out 385/384 and may be described as the 41 & 46 & 53 temperament, finding the interval class of 11 at the double-up fourth (C–^^F). 140edo, 181edo and especially 321edo can be used as tunings. Note that 94edo is a notable tuning not appearing on the optimal ET sequence.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187

Mapping: [⟨1 0 0 10 -3], ⟨0 1 0 -6 7], ⟨0 0 1 1 -2]]

Optimal tunings:

  • WE: ~2 = 1200.1396 ¢, ~3/2 = 702.9241 ¢, ~5/4 = 385.1817 ¢
error map: ⟨+0.140 +1.109 -0.853 -0.351 -1.213]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8511 ¢, ~5/4 = 385.1712 ¢
error map: ⟨0.000 +0.896 -1.143 -0.761 -1.703]

Minimax tuning:

[[1 0 0 0 0⟩, [5/3 0 1/6 -1/6 0⟩, [26/9 0 13/18 -7/18 -1/3⟩, [26/9 0 -5/18 11/18 -1/3⟩, [26/9 0 -5/18 -7/18 2/3⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 53, 87, 140, 181, 321

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384

Mapping: [⟨1 0 0 10 -3 2], ⟨0 1 0 -6 7 4], ⟨0 0 1 1 -2 -2]]

Lattice basis:

3/2 length = 0.5354, 27/20 length = 1.0463
Angle (3/2, 27/20) = 80.5628 degrees

Mapping to lattice: [⟨0 1 3 -3 1 -2], ⟨0 0 -1 -1 2 2]]

Optimal tunings:

  • WE: ~2 = 1200.0943 ¢, ~3/2 = 702.9377 ¢, ~5/4 = 385.4278 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8853 ¢, ~5/4 = 385.4002 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [5/3 0 1/6 -1/6 0 0⟩, [26/9 0 13/18 -7/18 -1/3 0⟩, [26/9 0 -5/18 11/18 -1/3 0⟩, [26/9 0 -5/18 -7/18 2/3 0⟩, [26/9 0 -7/9 1/9 2/3 0⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 46, 53, 87, 140, 321, 461e

Badness (Sintel): 0.769

Scales: akea46_13

Lono

Lono tempers out 176/175 and may be described as the 46 & 53 & 58 temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v3F#). It notably also tempers out 8019/8000, thus setting 11/10, 10/9, 9/8, and 8/7 a comma apart from each other. 111edo is a great tuning for it. 157edo is a viable alternative, which is almost as good.

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103

Mapping: [⟨1 0 0 10 6], ⟨0 1 0 -6 -6], ⟨0 0 1 1 3]]

Optimal tunings:

  • WE: ~2 = 1199.3368 ¢, ~3/2 = 702.5643 ¢, ~5/4 = 389.5319 ¢
error map: ⟨-0.663 -0.054 +1.892 +1.341 -2.088]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9356 ¢, ~5/4 = 389.4076 ¢
error map: ⟨0.000 +0.981 +3.094 +2.968 -0.708]

Optimal ET sequence: 46, 53, 58, 99, 111, 268cd

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845

Mapping: [⟨1 0 0 10 6 11], ⟨0 1 0 -6 -6 -9], ⟨0 0 1 1 3 3]]

Optimal tunings:

  • WE: ~2 = 1199.3329 ¢, ~3/2 = 702.5519 ¢, ~5/4 = 389.5508 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.9205 ¢, ~5/4 = 389.4341 ¢

Optimal ET sequence: 46, 53, 58, 99, 104c, 111, 268cd

Badness (Sintel): 0.850

Kapo

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5120/5103

Mapping: [⟨1 0 0 10 7], ⟨0 1 1 -5 -2], ⟨0 0 2 2 -1]]

mapping generators: ~2, ~3, ~128/99

Optimal tunings:

  • WE: ~2 = 1199.7125 ¢, ~3/2 = 702.6631 ¢, ~128/99 = 441.8973 ¢
error map: ⟨-0.287 +0.421 -0.143 +0.216 +0.021]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8413 ¢, ~128/99 = 441.9493 ¢
error map: ⟨0.000 +0.886 +0.426 +0.866 +1.050]

Minimax tuning:

[[1 0 0 0 0⟩, [8/5 2/5 0 -1/15 -2/15⟩, [14/5 6/5 0 7/15 -16/15⟩, [16/5 -6/5 0 13/15 -4/15⟩, [16/5 -6/5 0 -2/15 11/15⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7.11/9

Optimal ET sequence: 41, 65d, 87, 111, 152, 239, 391, 980bcde, 1132bcdde, 1371bbcddee

Badness (Sintel): 1.19

Namaka

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 5120/5103

Mapping: [⟨1 0 0 10 -6], ⟨0 2 0 -12 9], ⟨0 0 1 1 1]]

mapping generators: ~2, ~400/231, ~5

Optimal tunings:

  • WE: ~2 = 1199.7179 ¢, ~400/231 = 951.2909 ¢, ~5/4 = 387.4982 ¢
error map: ⟨-0.282 +0.627 +0.620 -0.203 -1.074]
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.5081 ¢, ~5/4 = 387.3182 ¢
error map: ⟨0.000 +1.061 +1.004 +0.395 -0.426]

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845

Mapping: [⟨1 0 0 10 -6 -1], ⟨0 2 0 -12 9 3], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.7072 ¢, ~26/15 = 951.2767 ¢, ~5/4 = 387.4314 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.5016 ¢, ~5/4 = 387.2360 ¢

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198, 536f

Badness (Sintel): 0.731

Subgroup extensions

Counterpyth (2.3.5.7.19)

Main article: Counterpyth

Developed analogous to parapyth, counterpyth is an extension of hemifamity with an even milder fifth, as it finds 19/15 at the major third (C–E) and 19/10 at the major seventh (C–B). Notice the factorization 5120/5103 = (400/399)⋅(1216/1215). Other important ratios are 21/19 at the diminished third (C–Ebb) and 19/14 at the augmented third (C–E#).

It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths.

Subgroup: 2.3.5.7.19

Comma list: 400/399, 1216/1215

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

Optimal tunings:

  • WE: ~2 = 1199.6953 ¢, ~3/2 = 702.5169 ¢, ~5/4 = 386.2648 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6771 ¢, ~5/4 = 386.0544 ¢

Optimal ET sequence: 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh

Badness (Sintel): 0.347

References

  1. ↑ Yahoo! Tuning Group | Laka 17-limit minimax planar temperament
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