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. | |||
= | = Octagar = | ||
[[ | [[Comma]] c = 4000/3969 | ||
7-limit minimax: 3 and 7 1/6c sharp, 5 just | 7-limit minimax: 3 and 7 1/6c sharp, 5 just | ||
| Line 9: | Line 9: | ||
|0 0 1 0>, |5/6 -2/3 1/2 2/3>] | |0 0 1 0>, |5/6 -2/3 1/2 2/3>] | ||
[[Eigenmonzo | [[Eigenmonzo]]s: 2, 7/6, 5/4 | ||
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp | 9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp | ||
| Line 16: | Line 16: | ||
|0 0 1 0>, |5/4 -1 3/4 1/2>] | |0 0 1 0>, |5/4 -1 3/4 1/2>] | ||
[[Eigenmonzo | [[Eigenmonzo]]s: 2, 5/4, 9/7 | ||
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605 | Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605 | ||
| Line 28: | Line 28: | ||
Badness: 0.000216 | Badness: 0.000216 | ||
[[ | [[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 | ||
| Line 39: | Line 39: | ||
23: 12800000/12252303, 107163/102400 | 23: 12800000/12252303, 107163/102400 | ||
==Nakika== | == Nakika == | ||
Commas: 100/99, 245/242 | Commas: 100/99, 245/242 | ||
Associated linear temperament: [[ | Associated linear temperament: [[Tetracot family|octacot]] | ||
Lattice basis: 11/7 0.798 22/21 0.906 | Lattice basis: 11/7 0.798 22/21 0.906 | ||
| Line 58: | Line 58: | ||
Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7 | Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7 | ||
Scales: [[ | Scales: [[nakika12]] | ||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Octagar]] | [[Category:Octagar]] | ||
[[Category:Rank 3]] | |||
Revision as of 06:00, 6 January 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