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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[http://tech.groups.yahoo.com/group/tuning-math/message/11441 http://tech.groups.yahoo.com/group/tuning-math/message/11441] | [http://tech.groups.yahoo.com/group/tuning-math/message/11441 http://tech.groups.yahoo.com/group/tuning-math/message/11441] | ||
{{ | {{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]] | ||
Latest revision as of 01:46, 8 February 2026
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 is known as didacus 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
| View • Talk • EditScale galleries | |
|---|---|
| JI scales | 12-tone JI • Combination product set • Constant structure • Harry Partch-related • Maximal harmony epimorphic • MOS transversal • Non-octave JI • Wakalix • Z-polygon transversal • Other JI Full list: Category:Just intonation scales |
| Tempered scales | 11-tone MOS • 12-tone tempered • Chromatic pair • Clipper • Double mode • Essentially tempered • Fantasy detemper • Marvel woo • Meantone • Min ambiguity • MOS cradle • Negri-9 • Neutral third • Non-octave tempered • Scalesmith systematic • Ternary • Other tempered Full list: Category:Tempered scales |
| Scales in EDOs | in 10edo • 11 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 33 • 34 • 35 • 36 • 37 • 38 • 40 • 41 • 42 • 43 • 46 • 49 • 53 • 72 • 80 |
All other scale gallery pages are included in Category:Lists of scales | |