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 [[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 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 [[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.

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 chord]]s including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] 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.

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]].

=== Prime harmonics ===

=== Octave stretch ===

198edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[512ed6]] or [[710ed12]]. This improves the approximated harmonics 3, 5, 7, 13, and 23; the 11 may become less accurate depending on the specific tuning. The 19 also gets worse on compression, so the compression should be very mild if the target is the no-17 23-limit.

=== 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 primes as {{nowrap| 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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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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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

| −0.344

| 0.291

| 4.80

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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

| −0.372

| 0.273

| 4.50

|-

| 2.3.5.7.11.13.19

| 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330

| {{Mapping| 198 314 460 556 685 733 841 }}

| −0.301

| 0.307

| 5.07

|-

| 2.3.5.7.11.13.19.23

| 352/351, 361/360, 456/455, 484/483, 576/575, 676/675, 847/845

| {{Mapping| 198 314 460 556 685 733 841 896 }}

| −0.319

| 0.291

| 4.81

|}

* 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]].

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

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | 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/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:99edo]]

[[Category:~~198edo]]~~

[[Category:Major minthmic]]

~~[[Category:Equal divisions of the octave|###]] ~~

~~[[Category:Minthmic~~]]

[[Category:Namaka]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|198}}
+{{ED intro}}
 == 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 [[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 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 [[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.
+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 chord]]s including [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] 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.
+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]].
 === Prime harmonics ===
-{{Harmonics in equal|198|columns=11}}
+{{Harmonics in equal|198}}
+=== Octave stretch ===
+edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[512ed6]] or [[710ed12]]. This improves the approximated harmonics 3, 5, 7, 13, and 23; the 11 may become less accurate depending on the specific tuning. The 19 also gets worse on compression, so the compression should be very mild if the target is the no-17 23-limit.
 === 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 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.
@@ Line 26: / Line 31: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 37: / Line 43: @@
 | 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
+| −0.344
 | 0.291
 | 4.80
@@ Line 44: / Line 50: @@
 | 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
+| −0.372
 | 0.273
 | 4.50
+|-
+| 2.3.5.7.11.13.19
+| 352/351, 361/360, 456/455, 676/675, 847/845, 1331/1330
+| {{Mapping| 198 314 460 556 685 733 841 }}
+| −0.301
+| 0.307
+| 5.07
+|-
+| 2.3.5.7.11.13.19.23
+| 352/351, 361/360, 456/455, 484/483, 576/575, 676/675, 847/845
+| {{Mapping| 198 314 460 556 685 733 841 896 }}
+| −0.319
+| 0.291
+| 4.81
 |}
 * 198et has a lower absolute error in the 13-limit than any previous equal temperaments, past [[190edo|190]] and followed by [[224edo|224]].
@@ Line 55: / Line 75: @@
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | 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 197: @@
 | [[Icosidillic]]
 |}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:99edo]]
-[[Category:198edo]]
+[[Category:Major minthmic]]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Minthmic]]
 [[Category:Namaka]]