Bird's eye view of temperaments by accuracy: Difference between revisions
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Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]]. | Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]]. | ||
Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). | Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). This structure is practically identical to that of [[cassaschismic]], only that the minicomma is not an independent generator and is instead found in the deep in the diploid chain of fifths. | ||
It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure | It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure. It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]. | ||
It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]]. | It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]]. | ||