198edo
| ← 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 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 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.
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 | +18 | +26 | +14 | +3 | +31 | -32 | -9 | +33 | +12 | +7 | |
| 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 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
- Main article: Table of 198edo 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 it is distinct