198edo: Difference between revisions

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== Theory ==

198edo is ~~distinctly~~ [[consistent]] through the [[15-odd-limit]] with ~~harmonics~~ 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 [[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.

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]].

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]].

It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered chords]] including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.

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=== Prime harmonics ===

=== Subsets and supersets ===

198 factors into 2 × 32 × 11, ~~and~~ has ~~divisors~~ {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.

Since 198 factors into 2 × 32 × 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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| 2.3.5.7.11

| 2401/2400, 3025/3024, 3136/3125, 4375/4374

| [{{~~val~~| 198 314 460 556 685 }}]

| {{mapping| 198 314 460 556 685 }}

| -0.344

| 0.291

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| 2.3.5.7.11.13

| 352/351, 676/675, 847/845, 1716/1715, 3025/3024

| [{{~~val~~| 198 314 460 556 685 733 }}]

| {{mapping| 198 314 460 556 685 733 }}

| -0.372

| 0.273

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Icosidillic]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:99edo]]

[[Category:Major minthmic]]

[[Category:Namaka]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is distinctly [[consistent]] through the [[15-odd-limit]] with harmonics 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 [[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.
-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]].
+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]].
 It provides the [[optimal patent val]] for the 13-limit rank-5 temperament tempering out 352/351, plus other temperaments of lower rank also tempering it out, such as [[hemimist]] and [[namaka]]. Besides [[major minthmic chords]], it enables [[essentially tempered chords]] including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[island chords]] in the 15-odd-limit.
@@ Line 14: / Line 14: @@
 === Prime harmonics ===
-{{Harmonics in equal|198|columns=11}}
+{{Harmonics in equal|198}}
 === Subsets and supersets ===
-factors into 2 × 3<sup>2</sup> × 11, and has divisors {{EDOs| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}.
+Since 198 factors into 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.
@@ Line 37: / Line 37: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 3136/3125, 4375/4374
-| [{{val| 198 314 460 556 685 }}]
+| {{mapping| 198 314 460 556 685 }}
 | -0.344
 | 0.291
@@ Line 44: / Line 44: @@
 | 2.3.5.7.11.13
 | 352/351, 676/675, 847/845, 1716/1715, 3025/3024
-| [{{val| 198 314 460 556 685 733 }}]
+| {{mapping| 198 314 460 556 685 733 }}
 | -0.372
 | 0.273
@@ Line 57: / Line 57: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 176: / Line 176: @@
 | [[Icosidillic]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:99edo]]
 [[Category:Major minthmic]]
 [[Category:Namaka]]