Breed family: Difference between revisions
-music example (covered in jove) |
Cmloegcmluin (talk | contribs) adding parenthetical "(unchanged intervals)" to eigenmonzo lists |
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* 7- and [[9-odd-limit]] eigenmonzos: 2, 6/5, 5/4 | * 7- and [[9-odd-limit]] eigenmonzos (unchanged intervals): 2, 6/5, 5/4 | ||
{{Val list|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }} | {{Val list|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }} | ||
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[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[11-odd-limit]] eigenmonzos: 2, 7/5, 11/8 | * [[11-odd-limit]] eigenmonzos (unchanged intervals): 2, 7/5, 11/8 | ||
{{Val list|legend=1| 27e, 31, 41, 58, 72, 130, 202 }} | {{Val list|legend=1| 27e, 31, 41, 58, 72, 130, 202 }} | ||
| Line 55: | Line 55: | ||
Minimax tuning: | Minimax tuning: | ||
* 13-odd-limit eigenmonzos: 2, 10/9, 13/10 | * 13-odd-limit eigenmonzos (unchanged intervals): 2, 10/9, 13/10 | ||
* 15-odd-limit eigenmonzos: 2, 15/13, 7/5 | * 15-odd-limit eigenmonzos (unchanged intervals): 2, 15/13, 7/5 | ||
Mapping: [{{val| 1 1 1 2 2 1 }}, {{val| 0 2 1 1 5 11 }}, {{val| 0 0 2 1 0 -1 }}] | Mapping: [{{val| 1 1 1 2 2 1 }}, {{val| 0 2 1 1 5 11 }}, {{val| 0 0 2 1 0 -1 }}] | ||
| Line 76: | Line 76: | ||
Minimax tuning: | Minimax tuning: | ||
* 17-odd-limit eigenmonzos: 2, 18/17, 6/5 | * 17-odd-limit eigenmonzos (unchanged intervals): 2, 18/17, 6/5 | ||
Vals: {{Val list| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }} | Vals: {{Val list| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }} | ||
| Line 118: | Line 118: | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[11-odd-limit]] eigenmonzos: 2, 14/11, 4/3 | * [[11-odd-limit]] eigenmonzos (unchanged intervals): 2, 14/11, 4/3 | ||
{{Val list|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }} | {{Val list|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }} | ||
| Line 153: | Line 153: | ||
* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}] | ||
: [[Eigenmonzo]]s: 5/4, 14/11 | : [[Eigenmonzo]]s (unchanged intervals): 5/4, 14/11 | ||
{{Val list|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }} | {{Val list|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }} | ||
| Line 194: | Line 194: | ||
* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}] | ||
: [[Eigenmonzo]]s: 2, 4/3, 11/10 | : [[Eigenmonzo]]s (unchanged intervals): 2, 4/3, 11/10 | ||
{{Val list|legend=1| 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }} | {{Val list|legend=1| 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }} | ||
Revision as of 18:55, 23 June 2021
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
Map 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°
- 7- and 9-odd-limit eigenmonzos (unchanged intervals): 2, 6/5, 5/4
Badness: 0.0153 × 10-3
Projection pair: 3 = ~2401/800 to 2.5.7
Scales: breed11
Jove
Jove, formerly known as wonder, tempers out 243/242 and 441/440. Wonder has been deprecated as a name due to conflict with another temperament also given that name. 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]]
Mapping generators: ~2, ~11/9, ~10/7
- 11-odd-limit eigenmonzos (unchanged intervals): 2, 7/5, 11/8
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
Minimax tuning:
- 13-odd-limit eigenmonzos (unchanged intervals): 2, 10/9, 13/10
- 15-odd-limit eigenmonzos (unchanged intervals): 2, 15/13, 7/5
Mapping: [⟨1 1 1 2 2 1], ⟨0 2 1 1 5 11], ⟨0 0 2 1 0 -1]]
Mapping generators: ~2, ~11/9, ~10/7
Vals: Template:Val list
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]]
Mapping generators: ~2, ~11/9, ~10/7
Minimax tuning:
- 17-odd-limit eigenmonzos (unchanged intervals): 2, 18/17, 6/5
Vals: Template:Val list
Badness: 0.741 × 10-3
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]]
Mapping generators: ~2, ~11/9, ~10/7
Vals: Template:Val list
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]]
Mapping generators: ~2, ~11/9, ~10/7
Vals: Template:Val list
Badness: 0.542 × 10-3
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
- 11-odd-limit eigenmonzos (unchanged intervals): 2, 14/11, 4/3
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]]
Mapping generators: ~2, ~49/40, ~55/42
Vals: Template:Val list
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
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
- [[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⟩]
- Eigenmonzos (unchanged intervals): 5/4, 14/11
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]]
Vals: Template:Val list
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
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 generators: ~2, ~49/40, ~10/7
Map 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
- [[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⟩]
- Eigenmonzos (unchanged intervals): 2, 4/3, 11/10
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]]
Vals: Template:Val list
Badness: 0.923 × 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]]
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]]
Vals: Template:Val list
Badness: 0.738 × 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]]
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]]
Vals: Template:Val list
Badness: 0.830 × 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]]
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]]
Vals: Template:Val list *
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
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]]
Vals: Template:Val list
Badness: 0.755 × 10-3