Mystery comma: Difference between revisions
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{{monzo|46 -29}}, the '''29-comma''' of 43.305 cents, is the difference between 46 octaves and 29 fifths, in other words 2<sup>46</sup>/3<sup>29</sup>. For EDOs up to 400, the 29-comma is tempered out if and only if 29 divides it, for example [[29edo]], [[58edo]] or [[87edo]]. | {{monzo|46 -29}}, the '''29-comma''' of 43.305 cents, is the difference between 46 octaves and 29 fifths, in other words 2<sup>46</sup>/3<sup>29</sup>. For EDOs up to 400, the 29-comma is tempered out if and only if 29 divides it, for example [[29edo]], [[58edo]] or [[87edo]]. | ||
== Temperaments == | |||
Tempering out the 29-comma splits the octave into 29 equal parts and maps the harmonic 3 to 17\29. It leads to a number of regular temperaments including [[Hemifamity temperaments|mystery]]. | |||
=== 29-commatic (58&87) === | |||
Subgroup: 2.3.5 | |||
[[Comma list]]: {{monzo|46 -29}} | |||
[[Mapping]]: [{{val| 29 46 0 }}, {{val| 0 0 1 }}] | |||
Mapping generators: ~531441/524288, ~5 | |||
[[POTE generator]]: ~5/4 = 387.408 | |||
{{Val list|legend=1| 29, 58, 87, 232, 319 }} | |||
[[Badness]]: 1.020556 | |||
=== 7-limit 58&87 === | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 5120/5103, 50421/50000 | |||
[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}] | |||
[[POTE generator]]: ~5/4 = 388.646 | |||
{{Val list|legend=1| 29, 58, 87, 145 }} | |||
[[Badness]]: 0.103734 | |||
=== 11-limit 58&87 (Mystery) === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 441/440, 896/891, 3388/3375 | |||
Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}] | |||
POTE generator: ~5/4 = 388.460 | |||
Vals: {{Val list| 29, 58, 87, 145 }} | |||
Badness: 0.034291 | |||
== See also == | == See also == | ||
Revision as of 03:58, 22 April 2021
| Interval information |
29-3-comma
reduced subharmonic
[46 -29⟩, the 29-comma of 43.305 cents, is the difference between 46 octaves and 29 fifths, in other words 246/329. For EDOs up to 400, the 29-comma is tempered out if and only if 29 divides it, for example 29edo, 58edo or 87edo.
Temperaments
Tempering out the 29-comma splits the octave into 29 equal parts and maps the harmonic 3 to 17\29. It leads to a number of regular temperaments including mystery.
29-commatic (58&87)
Subgroup: 2.3.5
Comma list: [46 -29⟩
Mapping: [⟨29 46 0], ⟨0 0 1]]
Mapping generators: ~531441/524288, ~5
POTE generator: ~5/4 = 387.408
Badness: 1.020556
7-limit 58&87
Subgroup: 2.3.5.7
Comma list: 5120/5103, 50421/50000
Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]
POTE generator: ~5/4 = 388.646
Badness: 0.103734
11-limit 58&87 (Mystery)
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 3388/3375
Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]
POTE generator: ~5/4 = 388.460
Vals: Template:Val list
Badness: 0.034291