198edo: Difference between revisions
Improve intro. Remove mentioning being the opv of semicanou since I'm planning on a redefinition of the 13-limit version. Add it in the categories instead cuz there's no harm, and remove those which are supported by 99et |
+RTT table |
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== Intervals == | == Intervals == | ||
{{main|Table of 198edo intervals}} | {{main|Table of 198edo intervals}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 3025/3024, 3136/3125, 4375/4374 | |||
| [{{val| 198 314 460 556 685 }}] | |||
| -0.344 | |||
| 0.291 | |||
| 4.80 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024 | |||
| [{{val| 198 314 460 556 685 733 }}] | |||
| -0.372 | |||
| 0.273 | |||
| 4.50 | |||
|} | |||
=== Rank-2 temperaments === | |||
Note: temperaments supported by 99et are not included. | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 2 | |||
| 38\198 | |||
| 230.30 | |||
| 8/7 | |||
| [[Hemigamera]] | |||
|- | |||
| 2 | |||
| 47\198<br>(52\198) | |||
| 315.15<br>(284.85) | |||
| 6/5<br>(33/28) | |||
| [[Semiparakleismic]] | |||
|- | |||
| 2 | |||
| 58\198<br>(41\198) | |||
| 351.52<br>(248.48) | |||
| 49/40<br>(15/13) | |||
| [[Semihemi]] | |||
|- | |||
| 2 | |||
| 56\198<br>(43\198) | |||
| 339.39<br>(260.61) | |||
| 243/200<br>(64/55) | |||
| [[Hemiamity]] | |||
|- | |||
| 2 | |||
| 74\198<br>(25\198) | |||
| 448.48<br>(151.51) | |||
| 35/27<br>(12/11) | |||
| [[Neusec]] | |||
|- | |||
| 3 | |||
| 41\198<br>(25\198) | |||
| 248.48<br>(151.51) | |||
| 15/13<br>(12/11) | |||
| [[Hemimist]] | |||
|- | |||
| 18 | |||
| 52\198<br>(3\198) | |||
| 315.15<br>(18.18) | |||
| 6/5<br>(99/98) | |||
| [[Hemiennealimmal]] | |||
|- | |||
| 22 | |||
| 82\198<br>(1\198) | |||
| 496.97<br>(6.06) | |||
| 4/3<br>(385/384) | |||
| [[Icosidillic]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
Revision as of 10:10, 18 July 2021
The 198 equal divisions of the octave (198edo), or the 198(-tone) equal temperament (198tet, 198et) when viewed from a regular temperament perspective, divides the octave into 198 parts of 6.061 cents each.
Theory
198edo is contorted in the 7-limit, with the same tuning as 99edo, but makes for a good 11- and 13-limit system. Like 99, it tempers out 2401/2400, 4375/4374, 3136/3125, 5120/5103 and 6144/6125 in the 7-limit; in the 11-limit it tempers 3025/3024, 9801/9800 and 14641/14580; and in the 13-limit 352/351, 676/675, 847/845, 1001/1000, 1716/1715 and 2080/2079.
It is the optimal patent val for the rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as hemimist, and namaka. It is distinctly consistent through the 15-odd-limit. It factors into 2 × 32 × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
Prime harmonics
Script error: No such module "primes_in_edo".
Intervals
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11 | 2401/2400, 3025/3024, 3136/3125, 4375/4374 | [⟨198 314 460 556 685]] | -0.344 | 0.291 | 4.80 |
| 2.3.5.7.11.13 | 352/351, 676/675, 847/845, 1716/1715, 3025/3024 | [⟨198 314 460 556 685 733]] | -0.372 | 0.273 | 4.50 |
Rank-2 temperaments
Note: temperaments supported by 99et are not included.
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 2 | 38\198 | 230.30 | 8/7 | Hemigamera |
| 2 | 47\198 (52\198) |
315.15 (284.85) |
6/5 (33/28) |
Semiparakleismic |
| 2 | 58\198 (41\198) |
351.52 (248.48) |
49/40 (15/13) |
Semihemi |
| 2 | 56\198 (43\198) |
339.39 (260.61) |
243/200 (64/55) |
Hemiamity |
| 2 | 74\198 (25\198) |
448.48 (151.51) |
35/27 (12/11) |
Neusec |
| 3 | 41\198 (25\198) |
248.48 (151.51) |
15/13 (12/11) |
Hemimist |
| 18 | 52\198 (3\198) |
315.15 (18.18) |
6/5 (99/98) |
Hemiennealimmal |
| 22 | 82\198 (1\198) |
496.97 (6.06) |
4/3 (385/384) |
Icosidillic |