Breed family

The breed family of temperaments are rank-3 microtemperaments which temper out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is tempered at all, the optimal patent val 2749et will certainly do the trick.

Breed

Subgroup: 2.3.5.7

Comma list: 2401/2400

Mapping: [⟨1 1 1 2], ⟨0 2 1 1], ⟨0 0 2 1]]

mapping generators: ~2, ~49/40, ~10/7

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

Lattice basis:

49/40 length = 0.7858, 8/7 length = 1.1241

Angle (49/40, 8/7) = 107.367°

Minimax tuning:

• 7- and 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.3.5

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 350.9664, ~10/7 = 617.6720

Optimal ET sequence: 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d

Badness: 0.0153 × 10^-3

Projection pair: 3 = ~2401/800 to 2.5.7

Scales: breed11

Jove

Main article: Jove

Jove (formerly known as wonder) tempers out 243/242 and 441/440. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72, 130, 171 and 202 are good edos for jove.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440

Mapping: [⟨1 1 1 2 2], ⟨0 2 1 1 5], ⟨0 0 2 1 0]]

Minimax tuning:

• 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/5.11

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.5034, ~10/7 = 617.8275

Optimal ET sequence: 27e, 31, 41, 58, 72, 130, 202

Badness: 0.241 × 10^-3

Projection pairs: ~3 = ~242/81, ~5 = ~2200/441, ~7 = ~440/63, ~11 = ~644204/59049 to 2.7/5.11/9

Jovial

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440

Mapping: [⟨1 1 1 2 2 1], ⟨0 2 1 1 5 11], ⟨0 0 2 1 0 -1]]

Minimax tuning:

• 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.13/9
• 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/5.15/13

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.7179, ~10/7 = 617.8286

Optimal ET sequence: 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e

Badness: 0.624 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 2 1 1 5 11 9], ⟨0 0 2 1 0 -1 -3]]

Minimax tuning:

• 17-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.17/9

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.6947, ~10/7 = 617.5315

Optimal ET sequence: 27effg, 31fg, 41, 58, 72, 130, 171, 243

Badness: 0.741 × 10^-3

Heartlandia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 1452/1445

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 4 0 1 10 23 12], ⟨0 0 2 1 0 -1 -1]]

mapping generators: ~2, ~119/108, ~27/17

Optimal tuning (POTE): ~2 = 1\1, ~119/108 = 175.4177, ~27/17 = 793.9762

Optimal ET sequence: 14cf, 27effg, 41, 89, 130g

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 324/323, 364/363, 441/440

Mapping: [⟨1 1 1 2 2 1 3 3], ⟨0 4 0 1 10 23 12 4], ⟨0 0 2 1 0 -1 -1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.3862, ~19/12 = 793.9558

Optimal ET sequence: 14cf, 27effg, 41, 89, 130g

Jofur

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242

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

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.4477, ~10/7 = 618.8891

Optimal ET sequence: 27e, 31, 41, 58, 99ef, 157eff

Badness: 0.749 × 10^-3

Jovis

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440

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

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.3935, ~10/7 = 618.1036

Optimal ET sequence: 27e, 31, 45ef, 58, 72, 103, 130, 233, 363

Badness: 0.542 × 10^-3

Agni

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372

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

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

Lattice basis:

49/40 length = 0.756, 10/7 length = 0.819

Angle (49/40, 10/7) = 106.460 degrees

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [23/10 3/10 2/5 0 -2/5⟩, [12/5 2/5 1/5 0 -1/5⟩, [23/10 3/10 -3/5 0 3/5⟩]

eigenmonzo (unchanged-interval) basis: 2.3.11/5

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 350.7145, ~10/7 = 617.0044

Optimal ET sequence: 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee

Badness: 0.494 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 1375/1372

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

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 350.6803, ~10/7 = 617.1448

Optimal ET sequence: 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff

Badness: 0.923 × 10^-3

Zisa

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625

Mapping: [⟨1 1 1 2 -3], ⟨0 2 1 1 8], ⟨0 0 2 1 8]]

Optimal ET sequence: 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c

Badness: 0.640 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨1 1 1 2 -3 7], ⟨0 2 1 1 8 -6], ⟨0 0 2 1 8 -3]]

Optimal ET sequence: 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241

Badness: 0.830 × 10^-3

Lif

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977

Mapping: [⟨1 1 1 2 8], ⟨0 2 1 1 -12], ⟨0 0 2 1 -2]]

Optimal tuning (CTE): ~2 = 1\1, ~49/40 = 351.0959, ~10/7 = 617.6652

Optimal ET sequence: 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e

Badness: 0.793 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095

Mapping: [⟨1 1 1 2 8 7], ⟨0 2 1 1 -12 -6], ⟨0 0 2 1 -2 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~49/40 = 351.0960, ~10/7 = 617.6533

Optimal ET sequence: 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562

Badness: 0.579 × 10^-3

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400

Sval mapping: [⟨1 1 1 2 8 7 0], ⟨0 2 1 1 -12 -6 11], ⟨0 0 2 1 -2 -3 2]]

Optimal tuning (CTE): ~2 = 1\1, ~49/40 = 351.1007, ~10/7 = 617.6501

Optimal ET sequence: 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413

Badness: 0.499 × 10^-3

Baldur

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800

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

mapping generators: ~99/70, ~343/198, ~10/7

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [3/4 0 1/2 1/2 -1/2⟩, [0 0 1 0 0⟩, [23/16 0 5/8 1/8 -1/8⟩, [23/16 0 5/8 -7/8 7/8⟩]

eigenmonzo (unchanged-interval) basis: 2.5.11/7

Optimal ET sequence: 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de

Badness: 0.166 × 10^-3

Projection pairs: 2 9801/4900 3 117649/39204 7 9801/1400 11 913517247483640899/83082326424002500 to 5.7/2.99/4

Greenland

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715

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

Optimal ET sequence: 58, 72, 130, 198, 270, 940, 1210f

Badness: 0.433 × 10^-3

Complexity spectrum: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9

Projection pairs: 2 19600/9801 3 676/225 5 10400/2079 7 20384000/2910897 11 19208000000000000/1750211597736459 13 5026736/385875 to 10/7.200/99.26/15

Freya

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024

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

mapping generators: ~2, ~49/40, ~55/42

Minimax tuning:

• 11-odd-limit eigenmonzos (unchanged-intervals): 2, 14/11, 4/3

Optimal ET sequence: 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee

Badness: 0.170 × 10^-3

Projection pairs: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024 to 2.49/5.77/3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 3 3 2 4], ⟨0 2 3 2 1 -9], ⟨0 0 -4 -2 3 6]]

Optimal ET sequence: 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615

Badness: 0.855 × 10^-3

Projection pairs: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024, ~13 = ~1814078464000000000000000/139662717676432916098329 to 2.49/5.77/3

Eir

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024

Mapping: [⟨1 1 3 3 2 0], ⟨0 2 3 2 1 6], ⟨0 0 -4 -2 3 5]]

Optimal ET sequence: 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f

Badness: 0.581 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1225/1224, 2058/2057, 2080/2079, 2401/2400

Mapping: [⟨1 1 3 3 2 0 7], ⟨0 2 3 2 1 6 6], ⟨0 0 -4 -2 3 5 -12]]

Optimal ET sequence: 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815

Badness: 0.700 × 10^-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1225/1224, 1540/1539, 2080/2079, 3136/3135, 4200/4199

Mapping: [⟨1 1 3 3 2 0 7 6], ⟨0 2 3 2 1 6 6 -2], ⟨0 0 -4 -2 3 5 -12 -3]]

Mapping generators: ~2, ~49/40, ~55/42

Optimal ET sequence: 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815

Badness: 0.692 × 10^-3

Heimlaug

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024

Mapping: [⟨1 1 3 3 2 7], ⟨0 2 3 2 1 6], ⟨0 0 -4 -2 3 -13]]

Optimal ET sequence: 8bcef, 15bbccdeeff, 23bcf, 31, 72, 103, 167, 198, 270, 571, 643, 913f

Badness: 0.601 × 10^-3

17-limit

Equave 10/7 and 16-note scales with that period are of interest.

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 1716/1715

Mapping: [⟨1 1 3 3 2 7 7], ⟨0 2 3 2 1 6 6], ⟨0 0 -4 -2 3 -13 -12]]

mapping generators: ~2, ~49/40, ~17/13

Optimal ET sequence: 8bcefg, 15bbccdeeffggg, 23bcfg, 31, 64be, 72, 103, 167, 198g, 239, 270

Badness: 0.829 × 10^-3

Vili

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 391314/390625

Mapping: [⟨1 1 5 4 10], ⟨0 2 3 2 6], ⟨0 0 -6 -3 -14]]

Optimal ET sequence: 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583

Badness: 1.26 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4225/4224

Mapping: [⟨1 1 5 4 10 4], ⟨0 2 3 2 6 1], ⟨0 0 -6 -3 -14 -1]]

Optimal ET sequence: 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f

Badness: 0.738 × 10^-3

Frigg

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 644204/643125

Mapping: [⟨1 1 3 3 4], ⟨0 2 3 2 4], ⟨0 0 -10 -5 -11]]

Optimal ET sequence: 45e, 58, 103, 161, 212, 270, 643, 913, 1183e

Badness: 1.79 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 10648/10647

Mapping: [⟨1 1 3 3 4 5], ⟨0 2 3 2 4 3], ⟨0 0 -10 -5 -11 -14]]

Optimal ET sequence: 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef *

* optimal patent val: 1241

Badness: 0.934 × 10^-3

Ennealimmic

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374

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

mapping generators: ~27/25, ~5/3, ~11

Optimal ET sequence: 27, 45, 72, 171, 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd

Badness: 0.275 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12 0 -31], ⟨0 2 3 2 0 5], ⟨0 0 0 0 1 1]]

Optimal ET sequence: 27, 45f, 54cff, 72, 171, 198, 270, 639, 711, 981, 1692e

Badness: 0.755 × 10^-3