198edo: Difference between revisions
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The '''198 equal divisions of the octave''' (''' | 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 [[cent]]s each. | ||
== Theory == | == Theory == | ||
198EDO is contorted in the [[7-limit]], with the same tuning as [[99edo|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 [[Misty family #Hemimist|hemimist]], and [[Hemifamity family #Namaka|namaka]]. It is distinctly [[consistent]] through the [[15-odd-limit]]. It factors into 2 × 3<sup>2</sup> × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99. | 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 [[Misty family #Hemimist|hemimist]], and [[Hemifamity family #Namaka|namaka]]. It is distinctly [[consistent]] through the [[15-odd-limit]]. It factors into 2 × 3<sup>2</sup> × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99. | ||
| Line 39: | Line 39: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Note: temperaments supported by | Note: temperaments supported by 99EDO are not included. | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
| Line 48: | Line 48: | ||
! Associated<br>ratio | ! Associated<br>ratio | ||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 23\198 | |||
| 139.39 | |||
| 13/12 | |||
| [[Quasijerome]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 74: | Line 80: | ||
|- | |- | ||
| 2 | | 2 | ||
| | | 52\198<br>(47\198) | ||
| 315.15<br>(284.85) | | 315.15<br>(284.85) | ||
| 6/5<br>(33/28) | | 6/5<br>(33/28) | ||
Revision as of 13:32, 13 August 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 99EDO are not included.
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 23\198 | 139.39 | 13/12 | Quasijerome |
| 1 | 83\198 | 503.03 | 147/110 | Quadrawürschmidt |
| 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 |
| 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 |