Octagar family: Difference between revisions
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The '''octagar family''' of temperaments are [[planar temperament]]s tempering out [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds. | The '''octagar family''' of temperaments are [[planar temperament]]s tempering out [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds. | ||
= Octagar = | == Octagar == | ||
[[Comma]] c = 4000/3969 | [[Comma]] c = 4000/3969 | ||
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[[Projection pair]]s: 5 3969/800 7 27783/4000 to 2.3.7/5 | [[Projection pair]]s: 5 3969/800 7 27783/4000 to 2.3.7/5 | ||
=== [[Hobbits|Hobbit bases]] === | ==== [[Hobbits|Hobbit bases]] ==== | ||
{2, 3, 7/5} subgroup | {2, 3, 7/5} subgroup | ||
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23: 12800000/12252303, 107163/102400 | 23: 12800000/12252303, 107163/102400 | ||
== Nakika == | === Nakika === | ||
Commas: 100/99, 245/242 | Commas: 100/99, 245/242 | ||
Revision as of 21:43, 1 June 2021
The octagar family of temperaments are planar temperaments tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.
Octagar
Comma c = 4000/3969
7-limit minimax: 3 and 7 1/6c sharp, 5 just
[|1 0 0 0>, |5/6 1/3 1/2 -1/3>, |0 0 1 0>, |5/6 -2/3 1/2 2/3>]
Eigenmonzos: 2, 7/6, 5/4
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
[|1 0 0 0>, |5/8 1/2 3/8 -1/4>, |0 0 1 0>, |5/4 -1 3/4 1/2>]
Eigenmonzos: 2, 5/4, 9/7
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
Angle(63/50, 21/20) = 97.743 cents
Map to lattice: [<0 -1 -2 -1|, <0 -1 0 2|]
EDOs: 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230
Badness: 0.000216
Projection pairs: 5 3969/800 7 27783/4000 to 2.3.7/5
Hobbit bases
{2, 3, 7/5} subgroup
12: 50/49, 256000/250047
15: 256000/250047, 1029/1000
23: 12800000/12252303, 107163/102400
Nakika
Commas: 100/99, 245/242
Associated linear temperament: octacot
Lattice basis: 11/7 0.798 22/21 0.906
Angle(11/7, 22/21) = 97.747 degrees
Map to lattice: [<0 1 2 1 2|, <0 -1 0 2 2|]
Map: [<1 0 1 4 4|, <0 1 0 -2 -2|, <0 0 2 3 4|]
EDOs: 12, 15, 26, 29, 41
Badness: 0.000539
Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7
Scales: nakika12