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== Septimal meantone == | |||
Period: 1\1 | |||
Optimal ([[POTE]]) generator: ~3/2 = 696.495 | |||
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]] | |||
Scales: [[Meantone5]], [[Meantone7]], [[Meantone12]] | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div> | |||
<div class="mw-collapsible-content"> | |||
{| class="wikitable right-1 right-2 sortable" | |||
|+ | |||
|- | |||
! #Gens up | |||
! Cents <ref>octave-reduced</ref> | |||
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref> | |||
|- | |||
| 0 | |||
| 0.00 | |||
| 1/1 | |||
|- | |||
| 1 | |||
| 696.5 | |||
| 3/2 | |||
|- | |||
| 2 | |||
| 193.0 | |||
| 9/8, 10/9 | |||
|- | |||
| 3 | |||
| 889.5 | |||
| 5/3 | |||
|- | |||
| 4 | |||
| 386.0 | |||
| 5/4 | |||
|- | |||
| 5 | |||
| 1082.5 | |||
| 15/8, 28/15 | |||
|- | |||
| 6 | |||
| 579.0 | |||
| 7/5 | |||
|- | |||
| 7 | |||
| 75.5 | |||
| 21/20, 25/24, 28/27 | |||
|- | |||
| 8 | |||
| 772.0 | |||
| 14/9, 25/16 | |||
|- | |||
| 9 | |||
| 268.5 | |||
| 7/6 | |||
|- | |||
| 10 | |||
| 965.0 | |||
| 7/4 | |||
|- | |||
| 11 | |||
| 461.4 | |||
| 21/16 | |||
|} | |||
<references/></div></div> | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;"> | |||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
[[Mappings|Period-generator mapping]]: [<1 0 -4 -13|, <0 1 4 10|] | |||
[[Comma]]s: 81/80, 126/125 | |||
7 and [[9-odd-limit|9-limit]] minimax | |||
[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}] | |||
[[Eigenmonzo]]s: 2, 5 | |||
[[Tuning Ranges of Regular Temperaments|Valid range]]: [694.737, 700.000] (19 to 12) | |||
Nice range: [694.786, 701.955] | |||
Strict range: [694.786, 700.000] | |||
Mapping generator: ~3 | |||
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly. | |||
[[Wedgie]]: <<1 4 10 4 13 12|| | |||
[[Val]]s: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb | |||
[[Badness]]: 0.0137 | |||
</div></div> | |||
== Commas == | == Commas == | ||
Revision as of 07:19, 19 January 2021
Septimal meantone
Period: 1\1
Optimal (POTE) generator: ~3/2 = 696.495
EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50
Scales: Meantone5, Meantone7, Meantone12
Interval table (12-note MOS, 2.3.5.7 POTE tuning)
Technical data
Period-generator mapping: [<1 0 -4 -13|, <0 1 4 10|]
Commas: 81/80, 126/125
7 and 9-limit minimax
[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
Eigenmonzos: 2, 5
Valid range: [694.737, 700.000] (19 to 12)
Nice range: [694.786, 701.955]
Strict range: [694.786, 700.000]
Mapping generator: ~3
Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Wedgie: <<1 4 10 4 13 12||
Vals: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
Badness: 0.0137
Commas
41edo tempers out the following commas using its patent val, ⟨41 65 95 115 142 152 168 174 185 199 203].
| Prime limit |
Ratio[1] | Name(s) | Comment(s) |
|---|---|---|---|
| 3 | [65 -41⟩ | 41-comma | |
| 5 | [-5 -10 9⟩ | Shibboleth | |
| 5 | [-25 7 6⟩ | Ampersand | |
| 5 | 3125/3072 | Magic comma | |
| 5 | [5 -9 4⟩ | Tetracot comma | |
| 5 | [20 -17 3⟩ | Roda | |
| 5 | [-15 8 1⟩ | Schisma | |
| 7 | [0 -7 6 -1⟩ | Great BP diesis | |
| 7 | [-10 7 8 -7⟩ | Blackjackisma | |
| 7 | 875/864 | Keema | |
| 7 | 3125/3087 | Gariboh | |
| 7 | [10 -11 2 1⟩ | Tolerma | |
| 7 | [-15 3 2 2⟩ | Mirwomo comma | |
| 7 | 245/243 | Sensamagic | |
| 7 | 4000/3969 | Octagar | |
| 7 | [-15 0 -2 7⟩ | Quince | |
| 7 | 1029/1024 | Gamelisma | |
| 7 | 225/224 | Marvel comma | |
| 7 | [0 3 4 -5⟩ | Mirkwai | |
| 7 | [5 -7 -1 3⟩ | Hemimage | |
| 7 | 5120/5103 | Hemifamity | |
| 7 | [25 -14 0 -1⟩ | Garischisma | |
| 7 | 2401/2400 | Breedsma | |
| 11 | [15 0 1 0 -5⟩ | Thuja comma | |
| 11 | 245/242 | Cassacot | |
| 11 | 100/99 | Ptolemisma | |
| 11 | 1344/1331 | Hemimin | |
| 11 | 896/891 | Pentacircle | |
| 11 | [16 0 0 -2 -3⟩ | Orgonisma | |
| 11 | 243/242 | Rastma | |
| 11 | 385/384 | Keenanisma | |
| 11 | 441/440 | Werckisma | |
| 11 | 1375/1372 | Moctdel | |
| 11 | 540/539 | Swetisma | |
| 11 | 3025/3024 | Lehmerisma | |
| 11 | [-1 2 -4 5 -2⟩ | Odiheim | |
| 13 | 343/338 | ||
| 13 | 105/104 | Animist comma | |
| 13 | [12 -7 0 1 0 -1⟩ | Secorian | |
| 13 | 275/273 | Gassorma | |
| 13 | 144/143 | Grossma | |
| 13 | 196/195 | Mynucuma | |
| 13 | 640/637 | Huntma | |
| 13 | 1188/1183 | Kestrel comma | |
| 13 | 325/324 | Marveltwin | |
| 13 | 352/351 | Minthma | |
| 13 | 364/363 | Gentle comma | |
| 13 | 847/845 | Cuthbert | |
| 13 | 729/728 | Squbema | |
| 13 | 4096/4095 | Schismina | |
| 13 | [3 -2 0 -1 3 -2⟩ | Harmonisma | |
| 17 | 2187/2176 | Septendecimal schisma | |
| 17 | 256/255 | Septendecimal kleisma | |
| 17 | 715/714 | Septendecimal bridge comma | |
| 19 | 210/209 | Spleen comma | |
| 19 | 361/360 | Go comma | |
| 19 | 513/512 | Undevicesimal comma | |
| 19 | 1216/1215 | Eratosthenes' comma | |
| 23 | 736/729 | Vicesimotertial comma | |
| 29 | 145/144 | 29th-partial chroma |
- ↑ Ratios with more than 9 digits are presented in monzos