198edo: Difference between revisions
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Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently. | Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently. | ||
The 198b val | The 198b val [[support]]s a [[septimal meantone]] close to the [[CTE tuning]], although [[229edo]] is even closer, and besides, the 198be val supports an undecimal meantone almost identical to the [[POTE tuning]]. | ||
198 factors into 2 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}. | 198 factors into 2 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}. | ||
Revision as of 18:40, 25 January 2022
| ← 197edo | 198edo | 199edo → |
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 about 6.06 cents each.
Theory
198edo is distinctly consistent through the 15-odd-limit with harmonics of 3 through 13 all tuned sharp. It is enfactored 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, 3136/3125, 4375/4374, 5120/5103, 6144/6125 and 10976/10935 in the 7-limit. In the 11-limit, 3025/3024, 3388/3375, 9801/9800, 14641/14580, and 16384/16335; in the 13-limit, 352/351, 676/675, 847/845, 1001/1000, 1716/1715, 2080/2079, 2200/2197 and 6656/6655.
It provides 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. Besides minthmic chords, it enables the cuthbert triad, the island chords, the sinbadmic chords, and the petrmic triad.
Notably, it is the last edo to map 64/63 and 81/80 to the same step consistently.
The 198b val supports a septimal meantone close to the CTE tuning, although 229edo is even closer, and besides, the 198be val supports an undecimal meantone almost identical to the POTE tuning.
198 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 |
|---|---|---|---|---|
| 1 | 7\198 | 42.42 | 40/39 | Humorous |
| 1 | 23\198 | 139.39 | 13/12 | Quasijerome |
| 1 | 83\198 | 503.03 | 147/110 | Quadrawürschmidt |
| 2 | 14\198 | 84.85 | 21/20 | Floral |
| 2 | 38\198 | 230.30 | 8/7 | Hemigamera |
| 2 | 40\198 | 242.42 | 121/105 | Semiseptiquarter |
| 2 | 43\198 | 260.61 | 64/55 | Hemiamity |
| 2 | 52\198 (47\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 | 67\198 (32\198) |
406.06 (193.94) |
495/392 (28/25) |
Semihemiwürschmidt |
| 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 |
| 6 | 82\198 (16\198) |
496.97 (96.97) |
4/3 (200/189) |
Semimist |
| 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 |