198edo: Difference between revisions
→Regular temperament properties: +no-17 19-limit and no-17 23-limit |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" Tags: Mobile edit Mobile web edit |
||
| (5 intermediate revisions by 4 users not shown) | |||
| Line 7: | Line 7: | ||
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 | 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. | ||
| Line 17: | Line 17: | ||
=== 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 === | ||
| Line 25: | Line 28: | ||
== 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 == | ||
| Line 97: | Line 105: | ||
| 13/12 | | 13/12 | ||
| [[Quasijerome]] | | [[Quasijerome]] | ||
|- | |||
| 1 | |||
| 29\198 | |||
| 175.76 | |||
| 72/65 | |||
| [[Quadrafifths]] | |||
|- | |- | ||
| 1 | | 1 | ||
| 65\198 | | 65\198 | ||
| 393. | | 393.94 | ||
| 49/39 | | 49/39 | ||
| [[Hitch]] | | [[Hitch]] | ||
| Line 118: | Line 132: | ||
| 2 | | 2 | ||
| 31\198 | | 31\198 | ||
| 187. | | 187.88 | ||
| 39/35 | | 39/35 | ||
| [[Semiwitch]] | | [[Semiwitch]] | ||
| Line 160: | Line 174: | ||
| 2 | | 2 | ||
| 74\198<br>(25\198) | | 74\198<br>(25\198) | ||
| 448.48<br>(151. | | 448.48<br>(151.52) | ||
| 35/27<br>(12/11) | | 35/27<br>(12/11) | ||
| [[Neusec]] | | [[Neusec]] | ||
| Line 172: | Line 186: | ||
| 3 | | 3 | ||
| 41\198<br>(25\198) | | 41\198<br>(25\198) | ||
| 248.48<br>(151. | | 248.48<br>(151.52) | ||
| 15/13<br>(12/11) | | 15/13<br>(12/11) | ||
| [[Hemimist]] | | [[Hemimist]] | ||
| Line 194: | Line 208: | ||
| [[Icosidillic]] | | [[Icosidillic]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <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]] | ||