Bug family: Difference between revisions
m Removing from Category:Regular temperament theory using Cat-a-lot |
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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| Line 14: | Line 14: | ||
[[POTE generator]]: ~5/3 = 939.612 | [[POTE generator]]: ~5/3 = 939.612 | ||
{{ | {{Optimal ET sequence|legend=1| 4, 5, 9, 14, 23c }} | ||
[[Badness]]: 0.032801 | [[Badness]]: 0.032801 | ||
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[[POTE generator]]: ~5/3 = 936.605 | [[POTE generator]]: ~5/3 = 936.605 | ||
{{ | {{Optimal ET sequence|legend=1| 4, 5, 9, 23cd, 32bcd, 41bcdd }} | ||
[[Badness]]: 0.018638 | [[Badness]]: 0.018638 | ||
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POTE generator: ~5/3 = 935.688 | POTE generator: ~5/3 = 935.688 | ||
Optimal | {{Optimal ET sequence|legend=1| 4, 5, 9, 32bcde, 41bcdde, 50bbcdde }} | ||
Badness: 0.022649 | Badness: 0.022649 | ||
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POTE generator: ~5/3 = 934.065 | POTE generator: ~5/3 = 934.065 | ||
Optimal | {{Optimal ET sequence|legend=1| 4f, 5, 9, 32bcde, 41bcddef, 50bbcddeff }} | ||
Badness: 0.021159 | Badness: 0.021159 | ||
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POTE generator: ~5/3 = 934.153 | POTE generator: ~5/3 = 934.153 | ||
Optimal | {{Optimal ET sequence|legend=1| 4e, 5e, 9 }} | ||
Badness: 0.022799 | Badness: 0.022799 | ||
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[[POTE generator]]: ~5/3 = 959.288 | [[POTE generator]]: ~5/3 = 959.288 | ||
{{ | {{Optimal ET sequence|legend=1| 4dd, 5 }} | ||
[[Badness]]: 0.054770 | [[Badness]]: 0.054770 | ||
Revision as of 16:29, 7 May 2023
The 5-limit parent of the bug family is bug, a temperament of sorts (that is, an exotemperament) which tempers out 27/25, the large limma, or approximately one step of 9edo. The monzo for 27/25 is [0 3 -2⟩, the dual of which is ⟨⟨ 2 3 0 ]]. The generator for bug is 5/3, two of which give the 3, and three of which give the 5. Bug may be described as the 4&5 temperament, and 14edo is a good bug tuning, though wide latitude in these matters is possible. 4, 5, or 9 note MOS are a place to start with it.
Bug has a 7-limit extension, beep, via the normal comma list [27/25, 36/35] which can also be obtained by adding 21/20. Beep has the curious property that if we know both the beep tempering and the ennealimmal tempering of a given 7-limit interval x, that is enough to know what JI ratio x is.
Temperaments discussed elsewhere include ugolino.
Bug
Subgroup: 2.3.5
Comma list: 27/25
Mapping: [⟨1 0 0], ⟨0 -2 -3]]
POTE generator: ~5/3 = 939.612
Optimal ET sequence: 4, 5, 9, 14, 23c
Badness: 0.032801
Beep
Subgroup: 2.3.5.7
Comma list: 21/20, 27/25
Mapping: [⟨1 0 0 2], ⟨0 2 3 1]]
Wedgie: ⟨⟨ 2 3 1 0 -4 -6 ]]
POTE generator: ~5/3 = 936.605
Optimal ET sequence: 4, 5, 9, 23cd, 32bcd, 41bcdd
Badness: 0.018638
Pentoid
Subgroup: 2.3.5.7.11
Comma list: 21/20, 27/25, 33/32
Mapping: [⟨1 0 0 2 5], ⟨0 2 3 1 -2]]
POTE generator: ~5/3 = 935.688
Optimal ET sequence: 4, 5, 9, 32bcde, 41bcdde, 50bbcdde
Badness: 0.022649
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 21/20, 26/25, 27/25, 33/32
Mapping: [⟨1 0 0 2 5 -1], ⟨0 2 3 1 -2 6]]
POTE generator: ~5/3 = 934.065
Optimal ET sequence: 4f, 5, 9, 32bcde, 41bcddef, 50bbcddeff
Badness: 0.021159
Pento
Subgroup: 2.3.5.7.11
Comma list: 21/20, 27/25, 45/44
Mapping: [⟨1 0 0 2 -2], ⟨0 2 3 1 7]]
POTE generator: ~5/3 = 934.153
Optimal ET sequence: 4e, 5e, 9
Badness: 0.022799
Mite
Subgroup: 2.3.5.7
Comma list: 27/25, 28/25
Mapping: [⟨1 0 0 -2], ⟨0 2 3 6]]
Wedgie: ⟨⟨ 2 3 6 0 4 6 ]]
POTE generator: ~5/3 = 959.288
Badness: 0.054770