Breed family: Difference between revisions

Revision as of 12:51, 11 February 2026

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 breed family of rank-3 temperaments are microtemperaments which temper out 2401/2400.

Breed

Breed has the same lattice structure as the 2.5.7 subgroup of JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7. 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.

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 unchanged-interval (eigenmonzo) basis: 2.3.5

Optimal tunings:

WE: ~2 = 1200.021 ¢, ~49/40 = 350.972 ¢, ~10/7 = 617.683 ¢
CWE: ~2 = 1200.000 ¢, ~49/40 = 350.974 ¢, ~10/7 = 617.681 ¢

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

Badness (Sintel): 0.067

Projection pair: 3 2401/800 to 2.5.7

Scales: breed11

Music

Bodacious Breed (archived 2010) by Gene Ward Smith – details | play – breed in 441edo tuning

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 unchanged-interval (eigenmonzo) basis: 2.7/5.11

Optimal tunings:

WE: ~2 = 1200.098 ¢, ~11/9 = 350.532 ¢, ~10/7 = 617.878 ¢
CWE: ~2 = 1200.000 ¢, ~11/9 = 350.537 ¢, ~10/7 = 617.871 ¢

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

Badness (Sintel): 0.290

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

Optimal tunings:

WE: ~2 = 1199.977 ¢, ~11/9 = 350.711 ¢, ~10/7 = 617.817 ¢
CWE: ~2 = 1200.000 ¢, ~11/9 = 350.713 ¢, ~10/7 = 617.818 ¢

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

Badness (Sintel): 0.583

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 unchanged-interval (eigenmonzo) basis: 2.5/3.17/9

Optimal tunings:

WE: ~2 = 1200.088 ¢, ~11/9 = 350.720 ¢, ~10/7 = 617.577 ¢
CWE: ~2 = 1200.000 ¢, ~11/9 = 350.714 ¢, ~10/7 = 617.561 ¢

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

Badness (Sintel): 0.704

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 tunings:

WE: ~2 = 1199.559 ¢, ~119/108 = 175.353 ¢, ~27/17 = 793.685 ¢
CWE: ~2 = 1200.000 ¢, ~119/108 = 175.371 ¢, ~27/17 = 793.756 ¢

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

Badness (Sintel): 2.843

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 tunings:

WE: ~2 = 1199.733 ¢, ~21/19 = 175.347 ¢, ~19/12 = 793.779 ¢
CWE: ~2 = 1200.000 ¢, ~21/19 = 175.358 ¢, ~19/12 = 793.819 ¢

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

Badness (Sintel): 2.294

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 tunings:

WE: ~2 = 1198.953 ¢, ~11/9 = 351.141 ¢, ~10/7 = 618.349 ¢
CWE: ~2 = 1200.000 ¢, ~11/9 = 351.254 ¢, ~10/7 = 618.593 ¢

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

Badness (Sintel): 0.701

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 tunings:

WE: ~2 = 1200.144 ¢, ~11/9 = 350.435 ¢, ~10/7 = 618.177 ¢
CWE: ~2 = 1200.000 ¢, ~11/9 = 350.440 ¢, ~10/7 = 618.176 ¢

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

Badness (Sintel): 0.507

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⟩]

unchanged-interval (eigenmonzo) basis: 2.3.11/5

Optimal tunings:

WE: ~2 = 1200.417 ¢, ~49/40 = 350.836 ¢, ~10/7 = 617.219 ¢
CWE: ~2 = 1200.000 ¢, ~49/40 = 350.837 ¢, ~10/7 = 616.991 ¢

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

Badness (Sintel): 0.593

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 tunings:

WE: ~2 = 1200.441 ¢, ~49/40 = 350.809 ¢, ~10/7 = 617.372 ¢
CWE: ~2 = 1200.000 ¢, ~49/40 = 350.805 ¢, ~10/7 = 617.157 ¢

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

Badness (Sintel): 0.863

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 (Smith): 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 (Smith): 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 = 1200.0000 ¢, ~49/40 = 351.0959 ¢, ~10/7 = 617.6652 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~49/40 = 351.0960 ¢, ~10/7 = 617.6533 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~49/40 = 351.1007 ¢, ~10/7 = 617.6501 ¢

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

Badness (Smith): 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⟩]

unchanged-interval (eigenmonzo) basis: 2.5.11/7

Optimal tunings:

WE: ~99/70 = 600.015 ¢, ~343/198 = 950.972 ¢, ~10/7 = 617.701 ¢
CWE: ~99/70 = 600.000 ¢, ~343/198 = 950.956 ¢, ~10/7 = 617.702 ¢

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

Badness (Sintel): 0.200

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 tunings:

WE: ~99/70 = 599.987 ¢, ~26/15 = 951.054 ¢, ~10/7 = 617.806 ¢
CWE: ~99/70 = 600.000 ¢, ~26/15 = 951.069 ¢, ~10/7 = 617.807 ¢

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

Badness (Sintel): 0.415

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

Midnatssol

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675

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

Optimal tunings:

WE: ~17/12 = 600.112 ¢, ~26/15 = 951.191 ¢, ~10/7 = 617.610 ¢
CWE: ~17/12 = 600.000 ¢, ~26/15 = 951.063 ¢, ~10/7 = 617.584 ¢

Optimal ET sequence: 58, 72, 130, 140, 198, 270g

Badness (Sintel): 0.720

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 (Smith): 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 (Smith): 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

VIxen named this extension after a healer goddess or valkyrie from the Norse mythology, as it is an extension of freya with the ibnsinma that evokes associations with medicine.

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 (Smith): 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 (Smith): 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 (Smith): 0.692 × 10^-3

Heimlaug

VIxen named this extension after a völva (seeress) from the Gull-Þóris saga of Icelanders. It is an extension of freya with the fairytale comma and the ainisma, both adding to the mystical theme. The one of prophecy is bolstered by that this extension has benediction as a subtemperament.

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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 0.934 × 10^-3

Ennealimmic

Not to be confused with Ennealimnic.

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 (Smith): 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 (Smith): 0.755 × 10^-3

Breed family: Difference between revisions

Revision as of 12:51, 11 February 2026

Contents

Breed

Jove

Jovial

17-limit

Heartlandia

19-limit

Jofur

Jovis

Agni

13-limit

Zisa

13-limit

Lif

13-limit

2.3.5.7.11.13.19 subgroup

Baldur

Greenland

Midnatssol

Freya

13-limit

Eir

17-limit

19-limit

Heimlaug

17-limit

Vili

13-limit

Frigg

13-limit

Ennealimmic

13-limit

Navigation menu

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The '''breed family''' of [[temperament]]s are [[Rank-3 temperament|rank-3]] [[microtemperament]]s which [[Tempering out|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.
+The '''breed family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperament]]s are [[microtemperament]]s which [[tempering out|temper out]] [[2401/2400]].
 == Breed ==
-Breed has the same lattice structure as 2.5.7 JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7.
+Breed has the same lattice structure as the [[2.5.7 subgroup]] of JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7. 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]] [[2749edo|2749et]] will certainly do the trick.
 [[Subgroup]]: 2.3.5.7
@@ Line 10: / Line 10: @@
 {{Mapping|legend=1| 1 1 1 2 | 0 2 1 1 | 0 0 2 1 }}
 : Mapping generators: ~2, ~49/40, ~10/7
@@ Line 20: / Line 19: @@
 [[Minimax tuning]]:
-* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.021¢, ~49/40 = 350.972¢, ~10/7 = 617.683¢
+* [[WE]]: ~2 = 1200.021{{c}}, ~49/40 = 350.972{{c}}, ~10/7 = 617.683{{c}}
-* [[CWE]]: ~2 = 1200.000¢, ~49/40 = 350.974¢, ~10/7 = 617.681¢
+* [[CWE]]: ~2 = 1200.000{{c}}, ~49/40 = 350.974{{c}}, ~10/7 = 617.681{{c}}
 {{Optimal ET sequence|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }}
@@ Line 30: / Line 29: @@
 [[Badness]] (Sintel): 0.067
-[[Projection pair]]: ~3 = ~2401/800 to 2.5.7
+[[Projection pair]]: <code>3 2401/800</code> to 2.5.7
 Scales: [[breed11]]
@@ Line 52: / Line 51: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.098¢, ~11/9 = 350.532¢, ~10/7 = 617.878¢
+* [[WE]]: ~2 = 1200.098{{c}}, ~11/9 = 350.532{{c}}, ~10/7 = 617.878{{c}}
-* [[CWE]]: ~2 = 1200.000¢, ~11/9 = 350.537¢, ~10/7 = 617.871¢
+* [[CWE]]: ~2 = 1200.000{{c}}, ~11/9 = 350.537{{c}}, ~10/7 = 617.871{{c}}
 {{Optimal ET sequence|legend=1| 27e, 31, 41, 58, 72, 130, 202 }}
@@ Line 59: / Line 58: @@
 [[Badness]] (Sintel): 0.290
-Projection pairs: ~3 = ~242/81, ~5 = ~2200/441, ~7 = ~440/63, ~11 = ~644204/59049 to 2.7/5.11/9
+[[Projection pair]]s: <code>3 242/81, 5 2200/441, 7 440/63, 11 644204/59049</code> to 2.7/5.11/9
 === Jovial ===
@@ Line 73: / Line 72: @@
 Optimal tunings:
-* WE: ~2 = 1199.977¢, ~11/9 = 350.711¢, ~10/7 = 617.817¢
+* WE: ~2 = 1199.977{{c}}, ~11/9 = 350.711{{c}}, ~10/7 = 617.817{{c}}
-* CWE: ~2 = 1200.000¢, ~11/9 = 350.713¢, ~10/7 = 617.818¢
+* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.713{{c}}, ~10/7 = 617.818{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}
+{{Optimal ET sequence|legend=0| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}
 Badness (Sintel): 0.583
@@ Line 91: / Line 90: @@
 Optimal tunings:
-* WE: ~2 = 1200.088¢, ~11/9 = 350.720¢, ~10/7 = 617.577¢
+* WE: ~2 = 1200.088{{c}}, ~11/9 = 350.720{{c}}, ~10/7 = 617.577{{c}}
-* CWE: ~2 = 1200.000¢, ~11/9 = 350.714¢, ~10/7 = 617.561¢
+* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.714{{c}}, ~10/7 = 617.561{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}
+{{Optimal ET sequence|legend=0| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}
 Badness (Sintel): 0.704
@@ Line 108: / Line 107: @@
 Optimal tunings:
-* WE: ~2 = 1199.559¢, ~119/108 = 175.353¢, ~27/17 = 793.685¢
+* WE: ~2 = 1199.559{{c}}, ~119/108 = 175.353{{c}}, ~27/17 = 793.685{{c}}
-* CWE: ~2 = 1200.000¢, ~119/108 = 175.371¢, ~27/17 = 793.756¢
+* CWE: ~2 = 1200.000{{c}}, ~119/108 = 175.371{{c}}, ~27/17 = 793.756{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27effg, 41, 89, 130g }}
+{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 89, 130g }}
 Badness (Sintel): 2.843
@@ Line 123: / Line 122: @@
 Optimal tunings:
-* WE: ~2 = 1199.733¢, ~21/19 = 175.347¢, ~19/12 = 793.779¢
+* WE: ~2 = 1199.733{{c}}, ~21/19 = 175.347{{c}}, ~19/12 = 793.779{{c}}
-* CWE: ~2 = 1200.000¢, ~21/19 = 175.358¢, ~19/12 = 793.819¢
+* CWE: ~2 = 1200.000{{c}}, ~21/19 = 175.358{{c}}, ~19/12 = 793.819{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27effg, 41, 89, 130g }}
+{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 89, 130g }}
 Badness (Sintel): 2.294
@@ Line 138: / Line 137: @@
 Optimal tunings:
-* WE: ~2 = 1198.953¢, ~11/9 = 351.141¢, ~10/7 = 618.349¢
+* WE: ~2 = 1198.953{{c}}, ~11/9 = 351.141{{c}}, ~10/7 = 618.349{{c}}
-* CWE: ~2 = 1200.000¢, ~11/9 = 351.254¢, ~10/7 = 618.593¢
+* CWE: ~2 = 1200.000{{c}}, ~11/9 = 351.254{{c}}, ~10/7 = 618.593{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27e, 31, 41, 58, 99ef, 157eff }}
+{{Optimal ET sequence|legend=0| 27e, 31, 41, 58, 99ef, 157eff }}
 Badness (Sintel): 0.701
@@ Line 153: / Line 152: @@
 Optimal tunings:
-* WE: ~2 = 1200.144¢, ~11/9 = 350.435¢, ~10/7 = 618.177¢
+* WE: ~2 = 1200.144{{c}}, ~11/9 = 350.435{{c}}, ~10/7 = 618.177{{c}}
-* CWE: ~2 = 1200.000¢, ~11/9 = 350.440¢, ~10/7 = 618.176¢
+* CWE: ~2 = 1200.000{{c}}, ~11/9 = 350.440{{c}}, ~10/7 = 618.176{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
+{{Optimal ET sequence|legend=0| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
 Badness (Sintel): 0.507
@@ Line 179: / Line 178: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.417¢, ~49/40 = 350.836¢, ~10/7 = 617.219¢
+* [[WE]]: ~2 = 1200.417{{c}}, ~49/40 = 350.836{{c}}, ~10/7 = 617.219{{c}}
-* [[CWE]]: ~2 = 1200.000¢, ~49/40 = 350.837¢, ~10/7 = 616.991¢
+* [[CWE]]: ~2 = 1200.000{{c}}, ~49/40 = 350.837{{c}}, ~10/7 = 616.991{{c}}
 {{Optimal ET sequence|legend=1| 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }}
@@ Line 194: / Line 193: @@
 Optimal tunings:
-* WE: ~2 = 1200.441¢, ~49/40 = 350.809¢, ~10/7 = 617.372¢
+* WE: ~2 = 1200.441{{c}}, ~49/40 = 350.809{{c}}, ~10/7 = 617.372{{c}}
-* CWE: ~2 = 1200.000¢, ~49/40 = 350.805¢, ~10/7 = 617.157¢
+* CWE: ~2 = 1200.000{{c}}, ~49/40 = 350.805{{c}}, ~10/7 = 617.157{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}
+{{Optimal ET sequence|legend=0| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}
 Badness (Sintel): 0.863
@@ Line 210: / Line 209: @@
 {{Optimal ET sequence|legend=1| 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c }}
-[[Badness]]: 0.640 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.640 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 219: / Line 218: @@
 Mapping: {{mapping| 1 1 1 2 -3 7 | 0 2 1 1 8 -6 | 0 0 2 1 8 -3 }}
-{{Optimal ET sequence|legend=1| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}
+{{Optimal ET sequence|legend=0| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}
-Badness: 0.830 × 10<sup>-3</sup>
+Badness (Smith): 0.830 × 10<sup>-3</sup>
 == Lif ==
@@ Line 230: / Line 229: @@
 {{Mapping|legend=1| 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 tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~49/40 = 351.0959{{c}}, ~10/7 = 617.6652{{c}}
 {{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e }}
-[[Badness]]: 0.793 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.793 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 243: / Line 242: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.0000{{c}}, ~49/40 = 351.0960{{c}}, ~10/7 = 617.6533{{c}}
-{{Optimal ET sequence|legend=1| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}
+{{Optimal ET sequence|legend=0| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}
-Badness: 0.579 × 10<sup>-3</sup>
+Badness (Smith): 0.579 × 10<sup>-3</sup>
 === 2.3.5.7.11.13.19 subgroup ===
@@ Line 256: / Line 255: @@
 Sval mapping: {{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 tuning (CTE): ~2 = 1200.0000{{c}}, ~49/40 = 351.1007{{c}}, ~10/7 = 617.6501{{c}}
-{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}
+{{Optimal ET sequence|legend=0| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}
-Badness: 0.499 × 10<sup>-3</sup>
+Badness (Smith): 0.499 × 10<sup>-3</sup>
 == Baldur ==
@@ Line 268: / Line 267: @@
 {{Mapping|legend=1| 2 0 1 3 7 | 0 2 1 1 -2 | 0 0 2 1 3 }}
 : Mapping generators: ~99/70, ~343/198, ~10/7
@@ Line 277: / Line 275: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~99/70 = 600.015¢, ~343/198 = 950.972¢, ~10/7 = 617.701¢
+* [[WE]]: ~99/70 = 600.015{{c}}, ~343/198 = 950.972{{c}}, ~10/7 = 617.701{{c}}
-* [[CWE]]: ~99/70 = 600.000¢, ~343/198 = 950.956¢, ~10/7 = 617.702¢
+* [[CWE]]: ~99/70 = 600.000{{c}}, ~343/198 = 950.956{{c}}, ~10/7 = 617.702{{c}}
 {{Optimal ET sequence|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }}
@@ Line 294: / Line 292: @@
 Optimal tunings:
-* WE: ~99/70 = 599.987¢, ~26/15 = 951.054¢, ~10/7 = 617.806¢
+* WE: ~99/70 = 599.987{{c}}, ~26/15 = 951.054{{c}}, ~10/7 = 617.806{{c}}
-* CWE: ~99/70 = 600.000¢, ~26/15 = 951.069¢, ~10/7 = 617.807¢
+* CWE: ~99/70 = 600.000{{c}}, ~26/15 = 951.069{{c}}, ~10/7 = 617.807{{c}}
-{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 270, 940, 1210f }}
+{{Optimal ET sequence|legend=0| 58, 72, 130, 198, 270, 940, 1210f }}
 Badness (Sintel): 0.415
@@ Line 303: / Line 301: @@
 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
+Projection pairs: <code>2 19600/9801, 3 676/225, 5 10400/2079, 7 20384000/2910897, 11 19208000000000000/1750211597736459, 13 5026736/385875</code> to 10/7.200/99.26/15
 ==== Midnatssol ====
@@ Line 313: / Line 311: @@
 Optimal tunings:
-* WE: ~17/12 = 600.112¢, ~26/15 = 951.191¢, ~10/7 = 617.610¢
+* WE: ~17/12 = 600.112{{c}}, ~26/15 = 951.191{{c}}, ~10/7 = 617.610{{c}}
-* CWE: ~17/12 = 600.000¢, ~26/15 = 951.063¢, ~10/7 = 617.584¢
+* CWE: ~17/12 = 600.000{{c}}, ~26/15 = 951.063{{c}}, ~10/7 = 617.584{{c}}
-{{Optimal ET sequence|legend=1| 58, 72, 130, 140, 198, 270g }}
+{{Optimal ET sequence|legend=0| 58, 72, 130, 140, 198, 270g }}
 Badness (Sintel): 0.720
@@ Line 326: / Line 324: @@
 {{Mapping|legend=1| 1 1 3 3 2 | 0 2 3 2 1 | 0 0 -4 -2 3 }}
 : Mapping generators: ~2, ~49/40, ~55/42
@@ Line 334: / Line 331: @@
 {{Optimal ET sequence|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
-[[Badness]]: 0.170 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.170 × 10<sup>-3</sup>
 [[Projection pair]]s: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024 to 2.49/5.77/3
@@ Line 345: / Line 342: @@
 Mapping: {{mapping| 1 1 3 3 2 4 | 0 2 3 2 1 -9 | 0 0 -4 -2 3 6 }}
-{{Optimal ET sequence|legend=1| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}
+{{Optimal ET sequence|legend=0| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}
-Badness: 0.855 × 10<sup>-3</sup>
+Badness (Smith): 0.855 × 10<sup>-3</sup>
-Projection pairs: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024, ~13 = ~1814078464000000000000000/139662717676432916098329 to 2.49/5.77/3
+Projection pairs: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329</code> to 2.49/5.77/3
 === Eir ===
@@ Line 360: / Line 357: @@
 Mapping: {{mapping| 1 1 3 3 2 0 | 0 2 3 2 1 6 | 0 0 -4 -2 3 5 }}
-{{Optimal ET sequence|legend=1| 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}
+{{Optimal ET sequence|legend=0| 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}
-Badness: 0.581 × 10<sup>-3</sup>
+Badness (Smith): 0.581 × 10<sup>-3</sup>
 ==== 17-limit ====
@@ Line 371: / Line 368: @@
 Mapping: {{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|legend=1| 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815 }}
+{{Optimal ET sequence|legend=0| 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815 }}
-Badness: 0.700 × 10<sup>-3</sup>
+Badness (Smith): 0.700 × 10<sup>-3</sup>
 ==== 19-limit ====
@@ Line 381: / Line 378: @@
 Mapping: {{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
-Mapping generators: ~2, ~49/40, ~55/42
+{{Optimal ET sequence|legend=0| 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815 }}
-{{Optimal ET sequence|legend=1| 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815 }}
+Badness (Smith): 0.692 × 10<sup>-3</sup>
-Badness: 0.692 × 10<sup>-3</sup>
 === Heimlaug ===
@@ Line 397: / Line 393: @@
 Mapping: {{mapping| 1 1 3 3 2 7 | 0 2 3 2 1 6 | 0 0 -4 -2 3 -13 }}
-{{Optimal ET sequence|legend=1| 8bcef, 15bbccdeeff, 23bcf, 31, 72, 103, 167, 198, 270, 571, 643, 913f }}
+{{Optimal ET sequence|legend=0| 8bcef, 15bbccdeeff, 23bcf, 31, 72, 103, 167, 198, 270, 571, 643, 913f }}
-Badness: 0.601 × 10<sup>-3</sup>
+Badness (Smith): 0.601 × 10<sup>-3</sup>
 ==== 17-limit ====
@@ Line 409: / Line 405: @@
 Mapping: {{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|legend=1| 8bcefg, 15bbccdeeffggg, 23bcfg, 31, 64be, 72, 103, 167, 198g, 239, 270 }}
+{{Optimal ET sequence|legend=0| 8bcefg, 15bbccdeeffggg, 23bcfg, 31, 64be, 72, 103, 167, 198g, 239, 270 }}
-Badness: 0.829 × 10<sup>-3</sup>
+Badness (Smith): 0.829 × 10<sup>-3</sup>
 == Vili ==
@@ Line 425: / Line 420: @@
 {{Optimal ET sequence|legend=1| 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583 }}
-[[Badness]]: 1.26 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.26 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 434: / Line 429: @@
 Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 3 2 6 1 | 0 0 -6 -3 -14 -1 }}
-{{Optimal ET sequence|legend=1| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}
+{{Optimal ET sequence|legend=0| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}
-Badness: 0.738 × 10<sup>-3</sup>
+Badness (Smith): 0.738 × 10<sup>-3</sup>
 == Frigg ==
@@ Line 447: / Line 442: @@
 {{Optimal ET sequence|legend=1| 45e, 58, 103, 161, 212, 270, 643, 913, 1183e }}
-[[Badness]]: 1.79 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.79 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 456: / Line 451: @@
 Mapping: {{mapping| 1 1 3 3 4 5 | 0 2 3 2 4 3 | 0 0 -10 -5 -11 -14 }}
-{{Optimal ET sequence|legend=1| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} <nowiki>*</nowiki>
+{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} <nowiki>*</nowiki>
 <nowiki>*</nowiki> [[optimal patent val]]: [[1241edo|1241]]
-Badness: 0.934 × 10<sup>-3</sup>
+Badness (Smith): 0.934 × 10<sup>-3</sup>
 == Ennealimmic ==
 {{Distinguish|Ennealimnic}}
 [[Subgroup]]: 2.3.5.7.11
@@ Line 469: / Line 464: @@
 {{Mapping|legend=1| 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|legend=1| 27, 45, 72, 171, 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd }}
-[[Badness]]: 0.275 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.275 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 483: / Line 477: @@
 Mapping: {{mapping| 9 1 1 12 0 -31 | 0 2 3 2 0 5 | 0 0 0 0 1 1 }}
-{{Optimal ET sequence|legend=1| 27, 45f, 54cff, 72, 171, 198, 270, 639, 711, 981, 1692e }}
+{{Optimal ET sequence|legend=0| 27, 45f, 54cff, 72, 171, 198, 270, 639, 711, 981, 1692e }}
-Badness: 0.755 × 10<sup>-3</sup>
+Badness (Smith): 0.755 × 10<sup>-3</sup>
 [[Category:Temperament families]]

Breed family: Difference between revisions

Revision as of 12:51, 11 February 2026

Breed

Jove

Jovial

17-limit

Heartlandia

19-limit

Jofur

Jovis

Agni

13-limit

Zisa

13-limit

Lif

13-limit

2.3.5.7.11.13.19 subgroup

Baldur

Greenland

Midnatssol

Freya

13-limit

Eir

17-limit

19-limit

Heimlaug

17-limit

Vili

13-limit

Frigg

13-limit

Ennealimmic

13-limit

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