198edo: Difference between revisions

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== Theory ==

198edo 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~~]] it tempers~~ [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; ~~and~~ in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]]~~. It is distinctly [[consistent]] through the [[15-odd-limit]] with harmonics of 3 through 13 all tuned sharp, and is the last edo to map [[64/63]] and [[81/80]] to the same step consistently~~.

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-limit|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]].

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 == Theory ==
-edo 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]] it tempers [[3025/3024]], [[3388/3375]], [[9801/9800]], [[14641/14580]], and [[16384/16335]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[2200/2197]] and [[6656/6655]]. It is distinctly [[consistent]] through the [[15-odd-limit]] with harmonics of 3 through 13 all tuned sharp, and is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.
+edo 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-limit|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]].