198edo

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← 197edo 198edo 199edo →
Prime factorization 2 × 32 × 11
Step size 6.06061 ¢ 
Fifth 116\198 (703.03 ¢) (→ 58\99)
Semitones (A1:m2) 20:14 (121.2 ¢ : 84.85 ¢)
Consistency limit 15
Distinct consistency limit 15

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.

Table of rank-2 temperaments by generator
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