Clipper: Difference between revisions
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If c is a [[Comma|comma]], then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the [[Euler_genera|Euler genus]], the divisors of the integer N reduced to the octave, and Benedetti(c) is the [[Benedetti_height|Benedetti height]] of c = p/q, which is p*q if p/q is reduced to its lowest terms. Euler(Benedetti(c)) has exactly one interval of size c, which is removed when c is tempered out. Two [[Transversal|transversals]] of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c. | If c is a [[Comma|comma]], then clipper(c) is defined as Euler(Benedetti(c)), tempered by the codimension one temperament tempering out c. Here Euler(N) is the [[Euler_genera|Euler genus]], the divisors of the integer N reduced to the octave, and Benedetti(c) is the [[Benedetti_height|Benedetti height]] of c = p/q, which is p*q if p/q is reduced to its lowest terms. Euler(Benedetti(c)) has exactly one interval of size c, which is removed when c is tempered out. Two [[Transversal|transversals]] of clipper(c) are obtained by leaving out either one or the other of the pair of scale intervals separated by c. | ||
Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a [[Normal_lists#x-Normal interval lists|normal interval list]]. This group is characteristic of the comma, and is the group on which tempering by the comma takes place. For instance, Euler(Benedetti(225/224)) generates 2.3.5.7, the full 7-limit group, and tempering it out leads to a rank three temperament, marvel. However, Euler(Benedeti(3136/3125)) generates 2.5.7, and tempering it out generates a rank-two temperament of the 2.5.7 [[Just_intonation_subgroups|JI subgroup]], with mapping [<1 0 -3|, <0 2 5|] and an approximate 28/25 generator, which | Euler(Benedetti(c)) generates a JI group, which can be found by reducing it to a [[Normal_lists#x-Normal interval lists|normal interval list]]. This group is characteristic of the comma, and is the group on which tempering by the comma takes place. For instance, Euler(Benedetti(225/224)) generates 2.3.5.7, the full 7-limit group, and tempering it out leads to a rank three temperament, marvel. However, Euler(Benedeti(3136/3125)) generates 2.5.7, and tempering it out generates a rank-two temperament of the 2.5.7 [[Just_intonation_subgroups|JI subgroup]], with mapping [<1 0 -3|, <0 2 5|] and an approximate 28/25 generator, which is known as [[didacus]] temperament. | ||
= Scales = | == Scales == | ||
{{See also|Category:Clippers}} | |||
[[clipper1029|clipper(1029/1024)]], 7 notes, 2.3.7 | [[clipper1029|clipper(1029/1024)]], 7 notes, 2.3.7 | ||
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[[clipper245|clipper(245/243)]], 35 notes, 7-limit | [[clipper245|clipper(245/243)]], 35 notes, 7-limit | ||
= Links = | == Links == | ||
[http://tech.groups.yahoo.com/group/tuning-math/message/11429 http://tech.groups.yahoo.com/group/tuning-math/message/11429] | [http://tech.groups.yahoo.com/group/tuning-math/message/11429 http://tech.groups.yahoo.com/group/tuning-math/message/11429] | ||
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{{Navbox scale gallery}} | {{Navbox scale gallery}} | ||
[[Category:Euler-Fokker genera]] | [[Category:Euler-Fokker genera]] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Lists of scales]] | [[Category:Lists of scales]] | ||
[[Category:Todo:clarify]] | [[Category:Todo:clarify]] | ||