198edo: Difference between revisions

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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 6.061 [[cent]]s each.

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 6.061 [[cent]]s each.

== Theory ==

~~198EDO~~ is ~~contorted (or~~ [[enfactored]]) in the [[7-limit]], with the same tuning as [[99edo~~|99EDO~~]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] ~~and~~ [[14641/14580]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] ~~and~~ [[2080/2079]].

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]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] 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]] and [[6656/6655]].

It is 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 [[~~Misty family #Hemimist|~~hemimist]], and [[~~Hemifamity family #Namaka|~~namaka]]. It is distinctly [[consistent]] through the [[15-odd-limit]]. It factors into 2 × 3<sup>2</sup> × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.

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]]. It is distinctly [[consistent]] through the [[15-odd-limit]].

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 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.

=== Prime harmonics ===

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=== Rank-2 temperaments ===

Note: temperaments supported by ~~99EDO~~ are not included.

Note: temperaments supported by 99edo are not included.

{| class="wikitable center-all left-5"

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-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 6.061 [[cent]]s each.
+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 6.061 [[cent]]s each.
 == Theory ==
-EDO is contorted (or [[enfactored]]) in the [[7-limit]], with the same tuning as [[99edo|99EDO]], but makes for a good 11- and 13-limit system. Like 99, it tempers out [[2401/2400]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] in the 7-limit; in the [[11-limit]] it tempers [[3025/3024]], [[9801/9800]] and [[14641/14580]]; and in the [[13-limit]] [[352/351]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
+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]], [[4375/4374]], [[3136/3125]], [[5120/5103]] and [[6144/6125]] 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]] and [[6656/6655]].
-It is 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 [[Misty family #Hemimist|hemimist]], and [[Hemifamity family #Namaka|namaka]]. It is distinctly [[consistent]] through the [[15-odd-limit]]. It factors into 2 × 3<sup>2</sup> × 11, and has divisors 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.
+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]]. It is distinctly [[consistent]] through the [[15-odd-limit]].
+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]].
+factors into 2 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.
 === Prime harmonics ===
@@ Line 39: / Line 43: @@
 === Rank-2 temperaments ===
-Note: temperaments supported by 99EDO are not included.
+Note: temperaments supported by 99edo are not included.
 {| class="wikitable center-all left-5"