2.3.5.13 subgroup: Difference between revisions
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[[Tridecaschismic]], which adds prime 13 to schismic via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and especially 53edo. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively. | [[Tridecaschismic]], which adds prime 13 to schismic via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and especially 53edo. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively. | ||
Cata and tridecaschismic unite | Cata and tridecaschismic unite in 53edo. | ||
Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping. | Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping. | ||