Bug family
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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⟩.
Bug
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 its ploidacot is alpha-dicot. 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. Another notable tuning of bug is given by TE, CTE and POTE, all coinciding at 939.612 ¢ with pure octaves since prime 2 is not involved in the comma to begin with.
Subgroup: 2.3.5
Comma list: 27/25
Mapping: [⟨1 0 0], ⟨0 -2 -3]]
- mapping generators: ~2, ~5/3
- WE: ~2 = 1199.665 ¢, ~5/3 = 939.350 ¢
- error map: ⟨-0.335 -23.255 +31.736]
- CWE: ~2 = 1200.000 ¢, ~5/3 = 939.481 ¢
- error map: ⟨0.000 -22.993 +32.129]
Optimal ET sequence: 4, 5, 9, 14
Badness (Sintel): 0.769
Overview to extensions
Bug has an obvious 7-limit extension, beep, via the normal comma list {27/25, 36/35} which can also be obtained by adding 21/20. There is an alternative, mite, which adds 28/25 instead.
Temperaments discussed elsewhere include ugolino and codex. Considered below are beep and mite.
Beep
As bug divides 3/1 in half for 5/3~9/5, it only makes sense to also equate this interval with 7/4~12/7, joining the temperament with semaphore.
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.
Subgroup: 2.3.5.7
Comma list: 21/20, 27/25
Mapping: [⟨1 0 0 2], ⟨0 2 3 1]]
- WE: ~2 = 1204.399 ¢, ~5/3 = 940.039 ¢
- error map: ⟨+4.399 -21.877 +33.803 -19.988]
- CWE: ~2 = 1200.000 ¢, ~5/3 = 938.111 ¢
- error map: ⟨0.000 -25.734 +28.018 -30.715]
Optimal ET sequence: 4, 5, 9
Badness (Sintel): 0.472
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:
- WE: ~2 = 1205.296 ¢, ~5/3 = 939.817 ¢
- CWE: ~2 = 1200.000 ¢, ~5/3 = 936.415 ¢
Badness (Sintel): 0.749
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:
- WE: ~2 = 1205.291 ¢, ~5/3 = 940.192 ¢
- CWE: ~2 = 1200.000 ¢, ~5/3 = 936.788 ¢
Badness (Sintel): 0.874
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:
- WE: ~2 = 1205.575 ¢, ~5/3 = 938.493 ¢
- CWE: ~2 = 1200.000 ¢, ~5/3 = 935.485 ¢
Optimal ET sequence: 4e, 5e, 9
Badness (Sintel): 0.754
Mite
Subgroup: 2.3.5.7
Comma list: 27/25, 28/25
Mapping: [⟨1 0 0 -2], ⟨0 2 3 6]]
- WE: ~2 = 1187.604 ¢, ~5/3 = 949.379 ¢
- error map: ⟨-12.396 -3.196 +61.824 -47.759]
- CWE: ~2 = 1200.000 ¢, ~5/3 = 955.861 ¢
- error map: ⟨0.000 +9.766 +81.268 -33.663]
Optimal ET sequence: 1cdd, 4dd, 5
Badness (Sintel): 1.39