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198edo: Difference between revisions

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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]].  
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.  
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.  
Notably, it is the last edo to map [[64/63]] and [[81/80]] to the same step consistently.  
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|198}}
{{Harmonics in equal|198}}
=== 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 ===
=== Subsets and supersets ===
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== Intervals ==
== Intervals ==
{{Main| Table of 198edo intervals }}
{{Main| Table of 198edo intervals }}
== Notation ==
=== Ups and downs notation ===
198edo can be notated using [[Kite's ups and downs notation|ups and downs]] with quarter-tone accidentals:
{{Ups and downs sharpness|198|true}}


== Regular temperament properties ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 3136/3125, 4375/4374
| 2401/2400, 3025/3024, 3136/3125, 4375/4374
| {{mapping| 198 314 460 556 685 }}
| {{Mapping| 198 314 460 556 685 }}
| −0.344
| −0.344
| 0.291
| 0.291
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| 352/351, 676/675, 847/845, 1716/1715, 3025/3024
| {{mapping| 198 314 460 556 685 733 }}
| {{Mapping| 198 314 460 556 685 733 }}
| −0.372
| −0.372
| 0.273
| 0.273
| 4.50
| 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]].  
* 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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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 13/12
| 13/12
| [[Quasijerome]]
| [[Quasijerome]]
|-
| 1
| 29\198
| 175.76
| 72/65
| [[Quadrafifths]]
|-
|-
| 1
| 1
| 65\198
| 65\198
| 393.93
| 393.94
| 49/39
| 49/39
| [[Hitch]]
| [[Hitch]]
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| 2
| 2
| 31\198
| 31\198
| 187.87
| 187.88
| 39/35
| 39/35
| [[Semiwitch]]
| [[Semiwitch]]
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|-
|-
| 2
| 2
| 52\198<br />(47\198)
| 52\198<br>(47\198)
| 315.15<br />(284.85)
| 315.15<br>(284.85)
| 6/5<br />(33/28)
| 6/5<br>(33/28)
| [[Semiparakleismic]]
| [[Semiparakleismic]]
|-
|-
| 2
| 2
| 58\198<br />(41\198)
| 58\198<br>(41\198)
| 351.52<br />(248.48)
| 351.52<br>(248.48)
| 49/40<br />(15/13)
| 49/40<br>(15/13)
| [[Semihemi]]
| [[Semihemi]]
|-
|-
| 2
| 2
| 67\198<br />(32\198)
| 67\198<br>(32\198)
| 406.06<br />(193.94)
| 406.06<br>(193.94)
| 495/392<br />(28/25)
| 495/392<br>(28/25)
| [[Semihemiwürschmidt]]
| [[Semihemiwürschmidt]]
|-
|-
| 2
| 2
| 74\198<br />(25\198)
| 74\198<br>(25\198)
| 448.48<br />(151.51)
| 448.48<br>(151.52)
| 35/27<br />(12/11)
| 35/27<br>(12/11)
| [[Neusec]]
| [[Neusec]]
|-
|-
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|-
|-
| 3
| 3
| 41\198<br />(25\198)
| 41\198<br>(25\198)
| 248.48<br />(151.51)
| 248.48<br>(151.52)
| 15/13<br />(12/11)
| 15/13<br>(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
| 6
| 6
| 82\198<br />(16\198)
| 82\198<br>(16\198)
| 496.97<br />(96.97)
| 496.97<br>(96.97)
| 4/3<br />(200/189)
| 4/3<br>(200/189)
| [[Semimist]]
| [[Semimist]]
|-
|-
| 18
| 18
| 52\198<br />(3\198)
| 52\198<br>(3\198)
| 315.15<br />(18.18)
| 315.15<br>(18.18)
| 6/5<br />(99/98)
| 6/5<br>(99/98)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 22
| 22
| 82\198<br />(1\198)
| 82\198<br>(1\198)
| 496.97<br />(6.06)
| 496.97<br>(6.06)
| 4/3<br />(385/384)
| 4/3<br>(385/384)
| [[Icosidillic]]
| [[Icosidillic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:99edo]]
[[Category:99edo]]
[[Category:Major minthmic]]
[[Category:Major minthmic]]
[[Category:Namaka]]
[[Category:Namaka]]
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