229edo: Difference between revisions
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== Theory == | == Theory == | ||
While not highly accurate for its size, 229edo is the point where a few important temperaments meet, and is [[consistency|distinctly consistent]] in the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 39 -29 3 }} ([[tricot comma]]) in the 5-limit; [[2401/2400]], [[3136/3125]], [[6144/6125]], and [[14348907/14336000]] in the 7-limit; [[3025/3024]], [[3388/3375]], [[8019/8000]], [[14641/14580]] and 15488/15435 in the 11-limit, and | While not highly accurate for its size, 229edo is the point where a few important temperaments meet, and is [[consistency|distinctly consistent]] in the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 39 -29 3 }} ([[tricot comma]]) in the 5-limit; [[2401/2400]], [[3136/3125]], [[6144/6125]], and [[14348907/14336000]] in the 7-limit; [[3025/3024]], [[3388/3375]], [[8019/8000]], [[14641/14580]] and 15488/15435 in the 11-limit, notably [[support|supporting]] [[hemiwürschmidt]], [[newt]], and [[trident]]. | ||
It extends less well to the 13-limit. Using the [[patent val]] {{val| 229 363 532 643 792 '''847''' }}, it tempers out [[351/350]], [[1573/1568]], [[2080/2079]], and [[4096/4095]]. Using the alternative 229f val {{val| 229 363 532 643 792 '''848''' }}, it tempers out [[352/351]], [[729/728]], [[1001/1000]], and [[1716/1715]]. | |||
Higher [[harmonic]]s like [[17/1|17]], [[19/1|19]], and [[23/1|23]] are well approximated, so it shows great potential in the no-13 23-limit. It tempers out [[561/560]], [[1701/1700]] in the 17-limit; [[476/475]], [[1216/1215]], [[1540/1539]], and [[1729/1728]] in the 19-limit; and [[576/575]] in the 23-limit. | |||
The 229b [[val]] supports a [[septimal meantone]] close to the [[CTE tuning]]. | The 229b [[val]] supports a [[septimal meantone]] close to the [[CTE tuning]]. | ||
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| 99.56 | | 99.56 | ||
| 18/17 | | 18/17 | ||
| [[Quintagar]] / [[quinsandra]] / [[quinsandric]] | | [[Quintagar]] / [[quinsandra]] (229) / [[quinsandric]] (229) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 105: | Line 109: | ||
| 351.09 | | 351.09 | ||
| 49/40 | | 49/40 | ||
| [[Newt]] | | [[Newt]] (229) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 129: | Line 133: | ||
| 565.94 | | 565.94 | ||
| 18/13 | | 18/13 | ||
| [[Trident]] | | [[Trident]] (229) | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||