Clippers

If c is a 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 genus, the divisors of the integer N reduced to the octave, and Benedetti(c) is the 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 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 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 JI subgroup, with mapping [<1 0 -3|, <0 2 5|] and an approximate 28/25 generator, which might be called 7-limit roulette temperament.

Scales

clipper(1029/1024), 7 notes, 2.3.7

clipper(81/80), 9 notes, 5-limit

clipper(3125/3072), 11 notes, 5-limit

clipper(121/120), 11 notes, 2.3.5.11

clipper(176/175), 11 notes, 2.5.7.11

clipper(65536/65219), 11 notes, 2.7.11

clipper(144/143), 11 notes, 2.3.11.13

clipper(169/168), 11 notes, 2.3.7.13

clipper(640/637), 11 notes, 2.5.7.13

clipper(2048/2025), 14 notes, 5-limit

clipper(385/384), 15 notes, 11-limit

clipper(105/104), 15 notes, 2.3.5.7.13

clipper(225/224), 17 notes, 7-limit

clipper(32805/32768), 17 notes, 5-limit

clipper(3136/3125), 17 notes, 2.5.7

clipper(99/98), 17 notes, 2.3.7.11

clipper(100/99), 17 notes, 2.3.5.11

clipper(243/242), 17 notes, 2.3.11

clipper(245/242), 17 notes, 2.5.7.11

clipper(896/891), 19 notes, 2.3.7.11

clipper(625/624), 19 notes, 2.3.5.13

clipper(126/125), 23 notes, 7-limit

clipper(6144/6125), 23 notes, 7-limit

clipper(65625/65536), 23 notes, 7-limit

clipper(5120/5103), 27 notes, 7-limit

clipper(4000/3993), 31 notes, 2.3.5.11

clipper(245/243), 35 notes, 7-limit

Links

http://tech.groups.yahoo.com/group/tuning-math/message/11429

http://tech.groups.yahoo.com/group/tuning-math/message/11432

http://tech.groups.yahoo.com/group/tuning-math/message/11439

http://tech.groups.yahoo.com/group/tuning-math/message/11441