198edo: Difference between revisions
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== Theory == | == Theory == | ||
198edo | 198edo is [[enfactoring|enfactored]] in the [[7-limit]], with the same tuning as [[99edo]], but makes for a good [[11-limit|11-]] and [[13-limit]] system. It is [[consistency|distinctly consistent]] through the [[15-odd-limit]], and demonstrates a sharp tendency, with [[harmonic]]s 3 through 13 all tuned sharp. | ||
Like 99, it [[tempering out|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]]. | Like 99, it [[tempering out|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]]. | ||
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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. | ||
Extending it beyond the 13-limit can be tricky, as the approximated [[17/1|harmonic 17]] is almost 1/3-edostep flat of just, which does not blend well with the sharp tendency from the lower harmonics. The 198g val in turn gives you an alternative that is more than 2/3-edostep sharp. However, if we skip prime 17 altogether and treat 198edo as a no-17 [[23-limit]] system, it is almost consistent to the no-17 [[23-odd-limit]] with the sole exception of [[19/15]] and its [[octave complement]]. It tempers out [[361/360]] and [[456/455]] in the [[19-limit]], and [[484/483]] and [[576/575]] in the [[23-limit]]. Finally, the harmonics [[29/1|29]] and [[31/1|31]] are quite accurate, though the [[25/1|25]] and [[27/1|27]] are sharp enough to have incurred more inconsistencies. | |||
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]]. | 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]]. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 198 factors into {{ | Since 198 factors into primes as {{nowrap| 2 × 3<sup>2</sup> × 11 }}, 198edo has subset edos {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}. | ||
A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s. | A step of 198edo is exactly 50 [[purdal]]s or 62 [[prima]]s. | ||
Revision as of 14:31, 23 March 2025
| ← 197edo | 198edo | 199edo → |
198 equal divisions of the octave (abbreviated 198edo or 198ed2), also called 198-tone equal temperament (198tet) or 198 equal temperament (198et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 198 equal parts of about 6.06 ¢ each. Each step represents a frequency ratio of 21/198, or the 198th root of 2.
Theory
198edo is enfactored in the 7-limit, with the same tuning as 99edo, but makes for a good 11- and 13-limit system. It is distinctly consistent through the 15-odd-limit, and demonstrates a sharp tendency, with harmonics 3 through 13 all tuned sharp.
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 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as hemimist and namaka. Besides major minthmic chords, it enables essentially tempered chords including cuthbert chords, sinbadmic chords, and petrmic chords in the 13-odd-limit, in addition to island chords in the 15-odd-limit.
Notably, it is the last edo to map 64/63 and 81/80 to the same step consistently.
Extending it beyond the 13-limit can be tricky, as the approximated harmonic 17 is almost 1/3-edostep flat of just, which does not blend well with the sharp tendency from the lower harmonics. The 198g val in turn gives you an alternative that is more than 2/3-edostep sharp. However, if we skip prime 17 altogether and treat 198edo as a no-17 23-limit system, it is almost consistent to the no-17 23-odd-limit with the sole exception of 19/15 and its octave complement. It tempers out 361/360 and 456/455 in the 19-limit, and 484/483 and 576/575 in the 23-limit. Finally, the harmonics 29 and 31 are quite accurate, though the 25 and 27 are sharp enough to have incurred more inconsistencies.
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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.08 | +1.57 | +0.87 | +0.20 | +1.90 | -1.93 | -0.54 | +2.03 | +0.73 | +0.42 |
| Relative (%) | +0.0 | +17.7 | +25.8 | +14.4 | +3.3 | +31.3 | -31.8 | -9.0 | +33.5 | +12.0 | +6.9 | |
| Steps (reduced) |
198 (0) |
314 (116) |
460 (64) |
556 (160) |
685 (91) |
733 (139) |
809 (17) |
841 (49) |
896 (104) |
962 (170) |
981 (189) | |
Subsets and supersets
Since 198 factors into primes as 2 × 32 × 11, 198edo has subset edos 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
A step of 198edo is exactly 50 purdals or 62 primas.
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 |
- 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past 190 and followed by 224.
Rank-2 temperaments
Note: temperaments supported by 99et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 7\198 | 42.42 | 40/39 | Humorous |
| 1 | 19\198 | 115.15 | 77/72 | Semigamera |
| 1 | 23\198 | 139.39 | 13/12 | Quasijerome |
| 1 | 65\198 | 393.93 | 49/39 | Hitch |
| 1 | 83\198 | 503.03 | 147/110 | Quadrawürschmidt |
| 2 | 14\198 | 84.85 | 21/20 | Floral |
| 2 | 31\198 | 187.87 | 39/35 | Semiwitch |
| 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 | 5\198 | 30.30 | 55/54 | Hemichromat |
| 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct