Bug family

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.

Bug divides 3/1 in half so it may be viewed as a less accurate version of semaphore (with beep temperament combining the two).

Temperaments discussed elsewhere include ugolino.

Bug

Subgroup: 2.3.5

Comma list: 27/25

Mapping: [⟨1 0 0], ⟨0 -2 -3]]

mapping generators: ~2, ~5/3

Optimal tunings:

• CTE: ~2 = 1\1, ~5/3 = 939.6121
• POTE: ~2 = 1\1, ~5/3 = 939.612

Optimal ET sequence: 4, 5, 9, 14

Badness: 0.032801

Beep

Main article: 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 tunings:

• CTE: ~2 = 1\1, ~5/3 = 940.7060
• POTE: ~2 = 1\1, ~5/3 = 936.605

Optimal ET sequence: 4, 5, 9

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 tunings:

• CTE: ~2 = 1\1, ~5/3 = 939.2367
• POTE: ~2 = 1\1, ~5/3 = 935.688

Optimal ET sequence: 4, 5, 9

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 tunings:

• CTE: ~2 = 1\1, ~5/3 = 939.5890
• POTE: ~2 = 1\1, ~5/3 = 934.065

Optimal ET sequence: 4f, 5, 9

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 tunings:

• CTE: ~2 = 1\1, ~5/3 = 938.0777
• 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 tunings:

• CTE: ~2 = 1\1, ~5/3 = 952.3649
• POTE: ~2 = 1\1, ~5/3 = 959.288

Optimal ET sequence: 4dd, 5

Badness: 0.054770