Bug family: Difference between revisions

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Unify to 3 decimal places; +error maps; spell out 1\1 in decimal; note the badness metric being used
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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/3 = 939.6121
* [[CTE]] and [[POTE]]: ~2 = 1200.000, ~5/3 = 939.612
* [[POTE]]: ~2 = 1\1, ~5/3 = 939.612
: [[error map]]: {{val| 0.000 -22.731 +32.523 }}


{{Optimal ET sequence|legend=1| 4, 5, 9, 14 }}
{{Optimal ET sequence|legend=1| 4, 5, 9, 14 }}


[[Badness]]: 0.032801
[[Badness]] (Smith): 0.032801


== Beep ==
== Beep ==
{{Main| Beep }}
{{Main| Beep }}
{{See also| Semaphoresmic clan }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 36: Line 37:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/3 = 940.7060
* [[CTE]]: ~2 = 1200.000, ~5/3 = 940.706
* [[POTE]]: ~2 = 1\1, ~5/3 = 936.605
: [[error map]]: {{val| 0.000 -20.543 +35.804 -28.120 }}
* [[POTE]]: ~2 = 1200.000, ~5/3 = 936.605
: error map: {{val| 0.000 -28.744 +23.502 -32.221 }}


{{Optimal ET sequence|legend=1| 4, 5, 9 }}
{{Optimal ET sequence|legend=1| 4, 5, 9 }}


[[Badness]]: 0.018638
[[Badness]] (Smith): 0.018638


=== Pentoid ===
=== Pentoid ===
Line 51: Line 54:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/3 = 939.2367
* CTE: ~2 = 1200.000, ~5/3 = 939.237
* POTE: ~2 = 1\1, ~5/3 = 935.688
* POTE: ~2 = 1200.000, ~5/3 = 935.688


Optimal ET sequence: {{Optimal ET sequence| 4, 5, 9 }}
Optimal ET sequence: {{Optimal ET sequence| 4, 5, 9 }}


Badness: 0.022649
Badness (Smith): 0.022649


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/3 = 939.5890
* CTE: ~2 = 1200.000, ~5/3 = 939.589
* POTE: ~2 = 1\1, ~5/3 = 934.065
* POTE: ~2 = 1200.000, ~5/3 = 934.065


Optimal ET sequence: {{Optimal ET sequence| 4f, 5, 9 }}
Optimal ET sequence: {{Optimal ET sequence| 4f, 5, 9 }}


Badness: 0.021159
Badness (Smith): 0.021159


=== Pento ===
=== Pento ===
Line 81: Line 84:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~5/3 = 938.0777
* CTE: ~2 = 1200.000, ~5/3 = 938.078
* POTE: ~2 = 1\1, ~5/3 = 934.153
* POTE: ~2 = 1200.000, ~5/3 = 934.153


Optimal ET sequence: {{Optimal ET sequence| 4e, 5e, 9 }}
Optimal ET sequence: {{Optimal ET sequence| 4e, 5e, 9 }}


Badness: 0.022799
Badness (Smith): 0.022799


== Mite ==
== Mite ==
Line 98: Line 101:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~5/3 = 952.3649
* [[CTE]]: ~2 = 1200.000, ~5/3 = 952.365
* [[POTE]]: ~2 = 1\1, ~5/3 = 959.288
: [[error map]]: {{val| 0.000 +2.775 +70.781 -54.637 }}
* [[POTE]]: ~2 = 1200.000, ~5/3 = 959.288
: error map: {{val| 0.000 +16.622 +91.552 -13.095 }}


{{Optimal ET sequence|legend=1| 4dd, 5 }}
{{Optimal ET sequence|legend=1| 1cdd, 4dd, 5 }}


[[Badness]]: 0.054770
[[Badness]] (Smith): 0.054770


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Bug family]] <!-- Main article -->
[[Category:Bug family]] <!-- Main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 11:01, 25 December 2024

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 combining the two).

Temperaments discussed elsewhere include ugolino and codex.

Bug

Subgroup: 2.3.5

Comma list: 27/25

Mapping: [⟨1 0 0], ⟨0 -2 -3]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • CTE and POTE: ~2 = 1200.000, ~5/3 = 939.612
error map: ⟨0.000 -22.731 +32.523]

Optimal ET sequence: 4, 5, 9, 14

Badness (Smith): 0.032801

Beep

Main article: Beep
See also: Semaphoresmic clan

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 = 1200.000, ~5/3 = 940.706
error map: ⟨0.000 -20.543 +35.804 -28.120]
  • POTE: ~2 = 1200.000, ~5/3 = 936.605
error map: ⟨0.000 -28.744 +23.502 -32.221]

Optimal ET sequence: 4, 5, 9

Badness (Smith): 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 = 1200.000, ~5/3 = 939.237
  • POTE: ~2 = 1200.000, ~5/3 = 935.688

Optimal ET sequence: 4, 5, 9

Badness (Smith): 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 = 1200.000, ~5/3 = 939.589
  • POTE: ~2 = 1200.000, ~5/3 = 934.065

Optimal ET sequence: 4f, 5, 9

Badness (Smith): 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 = 1200.000, ~5/3 = 938.078
  • POTE: ~2 = 1200.000, ~5/3 = 934.153

Optimal ET sequence: 4e, 5e, 9

Badness (Smith): 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 = 1200.000, ~5/3 = 952.365
error map: ⟨0.000 +2.775 +70.781 -54.637]
  • POTE: ~2 = 1200.000, ~5/3 = 959.288
error map: ⟨0.000 +16.622 +91.552 -13.095]

Optimal ET sequence: 1cdd, 4dd, 5

Badness (Smith): 0.054770

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