198edo: Difference between revisions

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== Theory ==

198edo ~~is [[consistency|distinctly consistent]] through the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned sharp. It~~ 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.

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]].

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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]].

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=== Subsets and supersets ===

Since 198 factors into {{~~factorisation~~|~~198~~}}, 198edo has subset edos {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.

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.

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 == Theory ==
-edo is [[consistency|distinctly consistent]] through the [[15-odd-limit]] with [[harmonic]]s of 3 through 13 all tuned sharp. It 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.
+edo 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]].
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 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]].
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 === Subsets and supersets ===
-Since 198 factors into {{factorisation|198}}, 198edo has subset edos {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.
+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.