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Breedsmic temperaments: Difference between revisions

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: mapping generators: ~2, ~49/40
: mapping generators: ~2, ~49/40
{{Multival|legend=1| 2 25 13 35 15 -40 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


[[Algebraic generator]]: (2 + sqrt(2))/2
[[Algebraic generator]]: (2 + sqrt(2))/2
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: Mapping generators: ~2, ~256/245
: Mapping generators: ~2, ~256/245
{{Multival|legend=1| 22 -5 3 -59 -57 21 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
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In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.


Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning.
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: Mapping generators: ~2, ~875/512
: Mapping generators: ~2, ~875/512
{{Multival|legend=1| 38 -3 8 -93 -94 27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
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: Mapping generators: ~2, ~189/160
: Mapping generators: ~2, ~189/160
{{Multival|legend=1| 6 41 22 51 18 -64 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
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: Mapping generators: ~2, ~2187/1372
: Mapping generators: ~2, ~2187/1372
{{Multival|legend=1| 14 59 33 61 13 -8 9 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
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: Mapping generators: ~2, ~42/25
: Mapping generators: ~2, ~42/25
{{Multival|legend=1| 34 29 23 -33 -59 -28 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
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== Unthirds ==
== Unthirds ==
The generator for unthirds temperament is undecimal major third, 14/11.
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
 
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: Mapping generators: ~2, ~6125/3888
: Mapping generators: ~2, ~6125/3888
{{Multival|legend=1| 42 47 34 -23 -64 -53 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
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: mapping generators: ~2, ~49/40
: mapping generators: ~2, ~49/40
{{Multival|legend=1| 2 -57 -28 -95 -50 95 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
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: Mapping generators: ~2, ~28/15
: Mapping generators: ~2, ~28/15
{{Multival|legend=1| 26 -37 -12 -119 -92 76 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
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Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281


Optimal ET sequence: {{Optimal ET sequence|l 10, 151, 161, 171, 332, 503ef, 835eeff }}
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}


Badness: 0.027322
Badness: 0.027322
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: Mapping generators: ~2, ~1296/875
: Mapping generators: ~2, ~1296/875
{{Multival|legend=1| 52 56 41 -32 -81 -62 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
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: Mapping generators: ~2, ~57344/46875
: Mapping generators: ~2, ~57344/46875
{{Multival|legend=1| 60 -8 11 -152 -151 48 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
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: Mapping generators: ~2, ~2800/2187
: Mapping generators: ~2, ~2800/2187
{{Multival|legend=1| 32 86 51 62 -9 -123 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
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: Mapping generators: ~2, ~8/7
: Mapping generators: ~2, ~8/7
{{Multival|legend=1| 18 -7 1 -53 -49 22 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
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: Mapping generators: ~2, ~125/96
: Mapping generators: ~2, ~125/96
{{Multival|legend=1| 46 15 19 -83 -99 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
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: Mapping generators: ~2, ~10/9
: Mapping generators: ~2, ~10/9
{{Multival|legend=1| 22 43 27 17 -19 -58 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
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: Mapping generators: ~2, ~250/189
: Mapping generators: ~2, ~250/189
{{Multival|legend=1| 28 36 25 -8 -39 -43 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
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: Mappping generators: ~2, ~10/7
: Mappping generators: ~2, ~10/7
{{Multival|legend=1| 30 13 14 -49 -62 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
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== Lockerbie ==
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''  
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}


: mapping generators: ~2, ~3828125/2985984
: Mapping generators: ~2, ~3828125/2985984


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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Badness (Smith): 0.0160
Badness (Smith): 0.0160
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
Badness (Smith): 0.0210
=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 549755813888/533935546875
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
: mapping generators: ~2, ~7/4
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
[[Badness]] (Sintel): 4.40


== Surmarvelpyth ==
== Surmarvelpyth ==
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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