198edo: Difference between revisions
→Theory: discuss the extensions beyond the 13-limit |
→Regular temperament properties: +no-17 19-limit and no-17 23-limit |
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| Line 32: | Line 32: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 40: | Line 40: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3025/3024, 3136/3125, 4375/4374 | | 2401/2400, 3025/3024, 3136/3125, 4375/4374 | ||
| {{ | | {{Mapping| 198 314 460 556 685 }} | ||
| −0.344 | | −0.344 | ||
| 0.291 | | 0.291 | ||
| Line 47: | Line 47: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024 | | 352/351, 676/675, 847/845, 1716/1715, 3025/3024 | ||
| {{ | | {{Mapping| 198 314 460 556 685 733 }} | ||
| −0.372 | | −0.372 | ||
| 0.273 | | 0.273 | ||
| 4.50 | | 4.50 | ||
|- | |||
| 2.3.5.7.11.13.19 | |||
| 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330 | |||
| {{Mapping| 198 314 460 556 685 733 841 }} | |||
| −0.301 | |||
| 0.307 | |||
| 5.07 | |||
|- | |||
| 2.3.5.7.11.13.19.23 | |||
| 352/351, 361/360, 456/455, 484/483, 576/575, 676/675, 847/845 | |||
| {{Mapping| 198 314 460 556 685 733 841 896 }} | |||
| −0.319 | |||
| 0.291 | |||
| 4.81 | |||
|} | |} | ||
* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]]. | * 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]]. | ||
| Line 60: | Line 74: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 127: | Line 141: | ||
|- | |- | ||
| 2 | | 2 | ||
| 52\198<br | | 52\198<br>(47\198) | ||
| 315.15<br | | 315.15<br>(284.85) | ||
| 6/5<br | | 6/5<br>(33/28) | ||
| [[Semiparakleismic]] | | [[Semiparakleismic]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 58\198<br | | 58\198<br>(41\198) | ||
| 351.52<br | | 351.52<br>(248.48) | ||
| 49/40<br | | 49/40<br>(15/13) | ||
| [[Semihemi]] | | [[Semihemi]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 67\198<br | | 67\198<br>(32\198) | ||
| 406.06<br | | 406.06<br>(193.94) | ||
| 495/392<br | | 495/392<br>(28/25) | ||
| [[Semihemiwürschmidt]] | | [[Semihemiwürschmidt]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 74\198<br | | 74\198<br>(25\198) | ||
| 448.48<br | | 448.48<br>(151.51) | ||
| 35/27<br | | 35/27<br>(12/11) | ||
| [[Neusec]] | | [[Neusec]] | ||
|- | |- | ||
| Line 157: | Line 171: | ||
|- | |- | ||
| 3 | | 3 | ||
| 41\198<br | | 41\198<br>(25\198) | ||
| 248.48<br | | 248.48<br>(151.51) | ||
| 15/13<br | | 15/13<br>(12/11) | ||
| [[Hemimist]] | | [[Hemimist]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 82\198<br | | 82\198<br>(16\198) | ||
| 496.97<br | | 496.97<br>(96.97) | ||
| 4/3<br | | 4/3<br>(200/189) | ||
| [[Semimist]] | | [[Semimist]] | ||
|- | |- | ||
| 18 | | 18 | ||
| 52\198<br | | 52\198<br>(3\198) | ||
| 315.15<br | | 315.15<br>(18.18) | ||
| 6/5<br | | 6/5<br>(99/98) | ||
| [[Hemiennealimmal]] | | [[Hemiennealimmal]] | ||
|- | |- | ||
| 22 | | 22 | ||
| 82\198<br | | 82\198<br>(1\198) | ||
| 496.97<br | | 496.97<br>(6.06) | ||
| 4/3<br | | 4/3<br>(385/384) | ||
| [[Icosidillic]] | | [[Icosidillic]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:99edo]] | [[Category:99edo]] | ||
[[Category:Major minthmic]] | [[Category:Major minthmic]] | ||
[[Category:Namaka]] | [[Category:Namaka]] | ||
Revision as of 14:49, 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 |
| 2.3.5.7.11.13.19 | 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330 | [⟨198 314 460 556 685 733 841]] | −0.301 | 0.307 | 5.07 |
| 2.3.5.7.11.13.19.23 | 352/351, 361/360, 456/455, 484/483, 576/575, 676/675, 847/845 | [⟨198 314 460 556 685 733 841 896]] | −0.319 | 0.291 | 4.81 |
- 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