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 | ||