198edo: Difference between revisions
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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]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|198|columns=11}} | {{Harmonics in equal|198|columns=11}} | ||
=== Miscellany === | |||
198 factors into 2 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}. | |||
== Intervals == | == Intervals == | ||
| Line 23: | Line 24: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 46: | Line 47: | ||
| 4.50 | | 4.50 | ||
|} | |} | ||
* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]]. | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
| Line 52: | Line 54: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
Revision as of 10:03, 26 November 2022
| ← 197edo | 198edo | 199edo → |
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 essentially tempered chords including cuthbert triad, sinbadmic chords, and petrmic triad 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 (%) | +0.0 | +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) | |
Miscellany
198 factors into 2 × 32 × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
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 (Reduced) |
Cents (Reduced) |
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 | 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 |