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Alphatricot family

Revision as of 18:34, 17 March 2025 by FloraC (talk | contribs) (Review, +CTE tunings and error maps)

The alphatricot family of temperaments tempers out the alphatricot comma (monzo: [39 -29 3⟩, ratio: 68 719 476 736 000 / 68 630 377 364 883).

Strong 7-limit extensions of this temperament include alphatrimot (53 & 70), alphatrident (53 & 229) and alphatrillium (53 & 441). Tempering out hemifamity comma (5120/5103) leads to alphatrimot, porwell comma (6144/6125) leads to alphatrident, and ragisma (4375/4374) leads to alphatrillium.

Alphatricot

Alphatricot is a microtemperament whose generator is the real cube root of the 3rd harmonic, 31/3, tuned between 63/44 and 13/9 and representing the acute augmented fourth of 59049/40960, that is, a Pythagorean augmented fourth plus a syntonic comma. Its ploidacot is alpha-tricot. It is a member of the schismic–Mercator equivalence continuum with n = 3, so unless 53edo is used as a tuning, the schisma is always observed.

The temperament was named by Paul Erlich in 2002 as tricot[1][2], but renamed in 2025 following the specifications of ploidacot.

Subgroup: 2.3.5

Comma list: [39 -29 3⟩

Mapping: [⟨1 0 -13], ⟨0 3 29]]

mapping generators: ~2, ~59049/40960

Wedgie: ⟨⟨ 3 29 39 ]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~59049/40960 = 634.0102
error map: ⟨0.0000 +0.0757 -0.0168]
  • POTE: ~2 = 1200.0000, ~59049/40960 = 634.0124
error map: ⟨0.0000 +0.0821 +0.0454]

Optimal ET sequence: 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc

Badness (Smith): 0.046093

2.3.5.13 subgroup

See also: No-fives subgroup temperaments § Threedic

Subgroup: 2.3.5.13

Comma list: 2197/2187, 41067/40960

Sval mapping: [⟨1 0 -13 0], ⟨0 3 29 7]]

Gencom mapping]: [⟨1 0 -13 0 0 0], ⟨0 3 29 0 0 7]]

gencom: [2 13/9; 2197/2187, 41067/40960]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/9 = 634.0179
  • POTE: ~2 = 1200.000, ~13/9 = 633.9970

Optimal ET sequence: 17c, 36c, 53

RMS error: 0.2342 cents

Scales

Alphatrillium

Alphatrillium, named by Xenllium in 2021 as trillium but renamed following the specifications of ploidacot, can be described as the 53 & 441 temperament, tempering out the ragisma aside from the alphatricot comma. 441edo is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest mos scale is the 123-tone one. For much simpler mappings of 7 at the cost of higher errors, you could try alphatrident and alphatrimot.

It can be extended to the 11-limit by tempering out 131072/130977, and to the 13-limit by tempering out 2080/2079, 4096/4095 and 4225/4224. The optimal tunings in the 11- and 13-limit lean towards 494edo; 935edo and especially 1429edo are recommendable tunings.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 1099511627776/1098337086315

Mapping: [⟨1 0 -13 53], ⟨0 3 29 -95]]

Wedgie: ⟨⟨ 3 29 -95 39 -159 -302 ]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~23625/16384 = 634.0121
error map: ⟨0.0000 +0.0813 +0.0372 +0.0247]
  • POTE: ~2 = 1200.0000, ~23625/16384 = 634.0118
error map: ⟨0.0000 +0.0804 +0.0283 +0.0537]

Optimal ET sequence: 53, …, 335, 388, 441, 935, 1376, 3193, 4569, 5945, 10514b

Badness (Smith): 0.030852

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 131072/130977, 759375/758912

Mapping: [⟨1 0 -13 53 -89], ⟨0 3 29 -95 175]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~3888/2695 = 634.0091
  • POTE: ~2 = 1200.0000, ~3888/2695 = 634.0094

Optimal ET sequence: 53, 388e, 441, 494, 935, 1429, 1923e

Badness (Smith): 0.046758

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078

Mapping: [⟨1 0 -13 53 -89 -28], ⟨0 3 29 -95 175 60]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~75/52 = 634.0091
  • POTE: ~2 = 1200.0000, ~75/52 = 634.0095

Optimal ET sequence: 53, 388e, 441, 494, 935, 1429, 1923e, 3352de

Badness (Smith): 0.019393

Pseudotrillium

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 4108797/4096000

Mapping: [⟨1 0 -13 53 -61], ⟨0 3 29 -95 122]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~231/160 = 634.0195
  • POTE: ~2 = 1200.0000, ~231/160 = 634.0190

Optimal ET sequence: 53, 335, 388

Badness (Smith): 0.111931

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374

Mapping: [⟨1 0 -13 53 -61 -28], ⟨0 3 29 -95 122 60]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~75/52 = 634.0185
  • POTE: ~2 = 1200.0000, ~75/52 = 634.0181

Optimal ET sequence: 53, 335, 388

Badness (Smith): 0.054837

Alphatrident

Alphatrident, also named by Xenllium in 2021 as trident but renamed following the specifications of ploidacot, can be described as the 53 & 229 temperament. It tempers out the garischisma, 33554432/33480783 ([25 -14 0 1⟩), and finds the harmonic 7 at -14 fifths or (-14) × 3 = -42 generator steps, so that the smallest mos scale that includes it is the 53-note one.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 14348907/14336000

Mapping: [⟨1 0 -13 25], ⟨0 3 29 -42]]

Wedgie: ⟨⟨ 3 29 -42 39 -75 -179 ]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~4096/2835 = 634.0484
error map: ⟨0.0000 +0.1901 +1.0893 +1.1421]
  • POTE: ~2 = 1200.0000, ~4096/2835 = 634.0480
error map: ⟨0.0000 +0.1890 +1.0784 +1.1579]

Optimal ET sequence: 53, 176, 229, 282, 511, 793cd

Badness (Smith): 0.101694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 6144/6125, 8019/8000

Mapping: [⟨1 0 -13 25 -33], ⟨0 3 29 -42 69]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~231/160 = 634.0630
  • POTE: ~2 = 1200.0000, ~231/160 = 634.0669

Optimal ET sequence: 53, 123, 176, 229

Badness (Smith): 0.074272

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125

Mapping: [⟨1 0 -13 25 -33 0], ⟨0 3 29 -42 69 7]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~13/9 = 634.0643
  • POTE: ~2 = 1200.0000, ~13/9 = 634.0652

Optimal ET sequence: 53, 123, 176, 229

Badness (Smith): 0.046593

Alphatrimot

Alphatrimot, named by Petr Pařízek in 2011[3] but renamed following the specifications of ploidacot, can be described as the 53 & 70 temperament. It finds prime 7 at only 11 generators up so that the generator is interpreted as a sharp ~81/56, but is more of a full 13-limit system in its own right. 123edo in the 123de val is a great tuning for it. Mos scales of 5, 7, 9, 11, 13, 15, 17, 19, 36 or 53 notes are available.

Subgroup: 2.3.5.7

Comma list: 2430/2401, 5120/5103

Mapping: [⟨1 0 -13 -3], ⟨0 3 29 11]]

Wedgie: ⟨⟨ 3 29 11 39 9 -56 ]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~81/56 = 633.9681
error map: ⟨0.0000 -0.0508 -1.2400 +4.8227]
  • POTE: ~2 = 1200.0000, ~81/56 = 634.0259
error map: ⟨0.0000 +0.1228 +0.4387 +5.4595]

Optimal ET sequence: 17c, 36c, 53, 229dd, 282dd

Badness (Smith): 0.100127

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 5120/5103

Mapping: [⟨1 0 -13 -3 -5], ⟨0 3 29 11 16]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~63/44 = 634.0214
  • POTE: ~2 = 1200.0000, ~63/44 = 634.0273

Optimal ET sequence: 17c, 36ce, 53

Badness (Smith): 0.056134

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -13 -3 -5 0], ⟨0 3 29 11 16 7]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~13/9 = 634.0275
  • POTE: ~2 = 1200.0000, ~13/9 = 634.0115

Optimal ET sequence: 17c, 36ce, 53

Badness (Smith): 0.032102

Tritricot

Subgroup: 2.3.5.7

Comma list: 250047/250000, 11785390260224/11767897353375

Mapping: [⟨3 6 19 30], ⟨0 -3 -29 -52]]

Wedgie: ⟨⟨ 9 87 156 117 222 118 ]]

Optimal tuning (POTE): ~63/50 = 400.0000, ~100352/91125 = 165.9837

Optimal ET sequence: 159, 282, 441, 2487, 2928, 3369

Badness (Smith): 0.086081

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 166698/166375, 200704/200475

Mapping: [⟨3 6 19 30 22], ⟨0 -3 -29 -52 -28]]

Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9835

Optimal ET sequence: 159, 282, 441

Badness (Smith): 0.074002

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875

Mapping: [⟨3 6 19 30 22 36], ⟨0 -3 -29 -52 -28 -60]]

Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9842

Optimal ET sequence: 159, 282, 441

Badness (Smith): 0.035641

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619

Mapping: [⟨3 6 19 30 22 36 16], ⟨0 -3 -29 -52 -28 -60 -9]]

Optimal tuning (POTE): ~34/27 = 400.0000, ~11/10 = 165.9805

Optimal ET sequence: 159, 282, 441

Badness (Smith): 0.025972

Noletaland

Noletaland is described as 282 & 1323, and it combines the smallest consistent edo in the 29-odd-limit with the smallest uniquely consistent. It reaches 4/3 in nine generators (noleta-…) and tempers out the landscape comma (…-land). Noletaland reaches 13/11 in 2 generators, and 29/19 in 5. Then there is 44/25 in 4, and 152/115 in also 4.

Subgroup: 2.3.5.7.11

Comma list: 250047/250000, 56723625/56689952, 78675968/78594219

Mapping: [⟨3 6 19 30 35], ⟨0 -9 -87 -156 -178]]

mappin generators: ~63/50, ~1936/1875

Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3290

Optimal ET sequence: 282, 759de, 1041, 1323, 4251e

Badness (Smith): 0.158

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 10648/10647, 43904/43875, 85750/85683, 250047/250000

Mapping: [⟨3 6 19 30 35 36], ⟨0 -9 -87 -156 -178 -180]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3294

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0725

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2058/2057, 4914/4913, 8624/8619, 12376/12375, 250047/250000

Mapping: [⟨3 6 19 30 35 36 29], ⟨0 -9 -87 -156 -178 -180 -121]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0380

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2058/2057, 2926/2925, 3136/3135, 4200/4199, 4914/4913, 250047/250000

Mapping: [⟨3 6 19 30 35 36 29 18], ⟨0 -9 -87 -156 -178 -180 -121 -38]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0269

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 2058/2057, 2926/2925, 3136/3135, 3381/3380, 3520/3519, 4914/4913, 18515/18513

Mapping: [⟨3 6 19 30 35 36 29 18 31], ⟨0 -9 -87 -156 -178 -180 -121 -38 -126]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0194

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 2058/2057, 2755/2754, 2926/2925, 3136/3135, 3381/3380, 3451/3450, 3520/3519, 4914/4913

Mapping: [⟨3 6 19 30 35 36 29 18 31 19], ⟨0 -9 -87 -156 -178 -180 -121 -38 -126 -32]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0168

Notes

  1. ↑ Yahoo! Tuning Group | Paul's new names
  2. ↑ Yahoo! Tuning Group | Ultimate 5-limit comma list
  3. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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