Bug family: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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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, | 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, approximately one step of [[9edo]]. The monzo for 27/25 is {{monzo| 0 3 -2 }}, the dual of which is {{multival| 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 | 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 [[Ragismic microtemperaments #Ennealimmal|ennealimmal]] tempering of a given 7-limit interval ''x'', that is enough to know what JI ratio ''x'' is. | ||
Temperaments discussed elsewhere include [[Very low accuracy temperaments #Ugolino|ugolino]]. | Temperaments discussed elsewhere include [[Very low accuracy temperaments #Ugolino|ugolino]]. | ||
== Bug == | == Bug == | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 27/25 | [[Comma list]]: 27/25 | ||
{{Mapping|legend=1| 1 0 0 | 0 -2 -3 }} | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/3 = 939.612 | ||
{{Optimal ET sequence|legend=1| 4, 5, 9, 14, 23c }} | {{Optimal ET sequence|legend=1| 4, 5, 9, 14, 23c }} | ||
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== Beep == | == Beep == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 21/20, 27/25 | [[Comma list]]: 21/20, 27/25 | ||
{{Mapping|legend=1| 1 0 0 2 | 0 2 3 1 }} | |||
{{Multival|legend=1| 2 3 1 0 -4 -6 }} | {{Multival|legend=1| 2 3 1 0 -4 -6 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/3 = 936.605 | ||
{{Optimal ET sequence|legend=1| 4, 5, 9, 23cd, 32bcd, 41bcdd }} | {{Optimal ET sequence|legend=1| 4, 5, 9, 23cd, 32bcd, 41bcdd }} | ||
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Comma list: 21/20, 27/25, 33/32 | Comma list: 21/20, 27/25, 33/32 | ||
Mapping: | Mapping: {{mapping| 1 0 0 2 5 | 0 2 3 1 -2 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~5/3 = 935.688 | ||
{{Optimal ET sequence|legend=1| 4, 5, 9, 32bcde, 41bcdde, 50bbcdde }} | {{Optimal ET sequence|legend=1| 4, 5, 9, 32bcde, 41bcdde, 50bbcdde }} | ||
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Comma list: 21/20, 26/25, 27/25, 33/32 | Comma list: 21/20, 26/25, 27/25, 33/32 | ||
Mapping: | Mapping: {{mapping| 1 0 0 2 5 -1 | 0 2 3 1 -2 6 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~5/3 = 934.065 | ||
{{Optimal ET sequence|legend=1| 4f, 5, 9, 32bcde, 41bcddef, 50bbcddeff }} | {{Optimal ET sequence|legend=1| 4f, 5, 9, 32bcde, 41bcddef, 50bbcddeff }} | ||
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Comma list: 21/20, 27/25, 45/44 | Comma list: 21/20, 27/25, 45/44 | ||
Mapping: | Mapping: {{mapping| 1 0 0 2 -2 | 0 2 3 1 7 }} | ||
POTE | Optimal tuning (POTE): ~2 = 1\1, ~5/3 = 934.153 | ||
{{Optimal ET sequence|legend=1| 4e, 5e, 9 }} | {{Optimal ET sequence|legend=1| 4e, 5e, 9 }} | ||
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== Mite == | == Mite == | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 27/25, 28/25 | [[Comma list]]: 27/25, 28/25 | ||
{{Mapping|legend=1| 1 0 0 -2 | 0 2 3 6 }} | |||
{{Multival|legend=1| 2 3 6 0 4 6 }} | {{Multival|legend=1| 2 3 6 0 4 6 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/3 = 959.288 | ||
{{Optimal ET sequence|legend=1| 4dd, 5 }} | {{Optimal ET sequence|legend=1| 4dd, 5 }} | ||
Revision as of 12:43, 6 August 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, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~5/3 = 959.288
Badness: 0.054770