6656/6655: Difference between revisions
Invert the clause of 1789edo vs good edos; +natural 17-limit extension |
→Temperaments: formatting |
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| Line 17: | Line 17: | ||
[[Mapping]]: <br> | [[Mapping]]: <br> | ||
[ | {| class="right-all" | ||
|- | |||
| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || ], | |||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || ]] | |||
|} | |||
: mapping generators: ~2, ~3, ~5, ~7, ~11 | : mapping generators: ~2, ~3, ~5, ~7, ~11 | ||
| Line 33: | Line 40: | ||
Mapping: <br> | Mapping: <br> | ||
[ | {| class="right-all" | ||
|- | |||
| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || 6 || ], | |||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 2 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 2 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || -2 || ]] | |||
|} | |||
Optimal ET sequence: {{Optimal ET sequence| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }} | Optimal ET sequence: {{Optimal ET sequence| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }} | ||
Revision as of 12:25, 19 July 2023
| Interval information |
reduced
6656/6655, the jacobin comma, apparently named by Gene Ward Smith in 2014, is a 13-limit (also 2.5.11.13 subgroup) superparticular interval of about 0.26 ¢. It is the difference between a stack of three 11/8 superfourths and one 13/10 naiadic plus an octave. In terms of commas, it is the difference between 364/363 and 385/384, between 2080/2079 and 3025/3024 as well as between 4096/4095 and 10648/10647. In the 17-limit, it factors neatly into (12376/12375)(14400/14399).
Temperaments
By tempering it out, the jacobin temperament is defined. Interestingly, 1789edo is an edo that supports the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament.
The 17-limit factorization shows us a natural path of extension, also given below.
Jacobin
Subgroup: 2.3.5.7.11.13
Comma list: 6656/6655
| [⟨ | 1 | 0 | 0 | 0 | 0 | -9 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 3 | ]] |
- mapping generators: ~2, ~3, ~5, ~7, ~11
Optimal ET sequence: 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee
Septendecimal jacobin
Subgroup: 2.3.5.7.11.13.17
Comma list: 6656/6655, 12376/12375
Mapping:
| [⟨ | 1 | 0 | 0 | 0 | 0 | -9 | 6 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 2 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 1 | 2 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | -1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 3 | -2 | ]] |
Optimal ET sequence: 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg