Semaphoresmic family: Difference between revisions
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'''Semiphore family''' is a rank-3 temperament family of 2.3.5.7 that tempers out 49/48 and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-limit 49/48 temperament called "[[Chromatic pairs #Semaphore|semaphore]]," and the 2.3.5.7 49/48 and 81/80 temperament called "[[Meantone family #Godzilla|godzilla]]". | |||
Semiphore | = Semiphore = | ||
Comma list: 49/48 | |||
Mapping: [{{val| 1 0 2 2 }}, {{val| 0 2 0 1 }}, {{val| 0 0 1 0 }}] | |||
Mapping: Generators: 2, 7/4, 5 | |||
Angle(7/6, 5/4) = 90 degrees | Lattice basis: | ||
: 7/6 length = 0.7627, 5/4 length = 2.322 | |||
: Angle (7/6, 5/4) = 90 degrees | |||
{{Val list|legend=1| 4, 5, 9, 10, 15, 19 }} | |||
Badness: 0.000116 | Badness: 0.000116 | ||
=Selenium= | = Selenium = | ||
Comma list: 49/48, 56/55 | |||
Mapping: [{{val| 1 0 2 2 5 }}, {{val| 0 2 0 1 1 }}, {{val| 0 0 1 0 -1 }}] | |||
{{Val list|legend=1| 4, 5, 9, 10, 15, 19, 24, 34 }} | |||
Badness: 0.000665 | Badness: 0.000665 | ||
==13-limit== | == 13-limit == | ||
Comma list: 49/48, 56/55, 91/90 | |||
Mapping: [{{val| 1 0 2 2 5 -1 }}, {{val| 0 2 0 1 1 3 }}, {{val| 0 0 1 0 -1 1 }}] | |||
{{Val list|legend=1| 5, 9, 10, 15, 19, 24, 34 }} | |||
Badness: 0.000787 | Badness: 0.000787 | ||
=Negric= | = Negric = | ||
Comma list: 49/48, 225/224 | |||
Mapping: [{{val| 1 2 2 3 0 }}, {{val| 0 -4 3 -2 0 }}, {{val| 0 0 0 0 1 }}] | |||
{{Val list|legend=1| 9, 10, 19, 29, 38d, 67cde }} | |||
Badness: 0.001087 | Badness: 0.001087 | ||
==13-limit== | == 13-limit == | ||
Comma list: 49/48, 65/64, 91/90 | |||
Mapping: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}, {{val| 0 0 0 0 1 0 }}] | |||
{{Val list|legend=1| 9, 10, 19, 29, 38df, 67cdef }} | |||
Badness: 80.756 | Badness: 80.756 | ||
[[Category:Theory]] | |||
[[Category:Temperament family]] | |||
[[Category:Semiphore]] | |||
[[Category:Rank 3]] | |||
Revision as of 07:51, 19 February 2021
Semiphore family is a rank-3 temperament family of 2.3.5.7 that tempers out 49/48 and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-limit 49/48 temperament called "semaphore," and the 2.3.5.7 49/48 and 81/80 temperament called "godzilla".
Semiphore
Comma list: 49/48
Mapping: [⟨1 0 2 2], ⟨0 2 0 1], ⟨0 0 1 0]]
Mapping: Generators: 2, 7/4, 5
Lattice basis:
- 7/6 length = 0.7627, 5/4 length = 2.322
- Angle (7/6, 5/4) = 90 degrees
Badness: 0.000116
Selenium
Comma list: 49/48, 56/55
Mapping: [⟨1 0 2 2 5], ⟨0 2 0 1 1], ⟨0 0 1 0 -1]]
Badness: 0.000665
13-limit
Comma list: 49/48, 56/55, 91/90
Mapping: [⟨1 0 2 2 5 -1], ⟨0 2 0 1 1 3], ⟨0 0 1 0 -1 1]]
Badness: 0.000787
Negric
Comma list: 49/48, 225/224
Mapping: [⟨1 2 2 3 0], ⟨0 -4 3 -2 0], ⟨0 0 0 0 1]]
Badness: 0.001087
13-limit
Comma list: 49/48, 65/64, 91/90
Mapping: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3], ⟨0 0 0 0 1 0]]
Badness: 80.756