41-comma: Difference between revisions
Propose "counterpyth comma" as a name |
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{{Infobox Interval | {{Infobox Interval | ||
| Icon = | | Icon = | ||
| Ratio = | | Ratio = 36 893 488 147 419 103 232 / <br>36 472 996 377 170 786 403 | ||
| Monzo = 65 -41 | | Monzo = 65 -41 | ||
| Cents = 19.84496 | | Cents = 19.84496 | ||
| Name = 41-comma, <br> | | Name = 41-comma, <br>counterpyth comma | ||
| Color name = | | Color name = | ||
| FJS name = | | FJS name = | ||
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}} | }} | ||
{{monzo|65 -41}}, the '''41-comma''', is an interval of 19.845 [[cent]]s. It is the amount by which 41 fifths ([[3/2]]) fall short of 24 [[octave]]s, in other words 2<sup>24</sup>/(3/2)<sup>41</sup> | {{monzo|65 -41}}, the '''41-comma''' or '''counterpyth comma''', is an interval of 19.845 [[cent]]s. It is the amount by which 41 fifths ([[3/2]]) fall short of 24 [[octave]]s, in other words 2<sup>24</sup>/(3/2)<sup>41</sup>. | ||
== Temperaments == | == Temperaments == | ||
Tempering out | Tempering out this comma splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. It leads to a number of regular temperaments in the [[counterpyth family]]. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]]. | ||
=== | === Counterpyth (41&205) === | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
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[[POTE generator]]: ~5/4 = 386.668 | [[POTE generator]]: ~5/4 = 386.668 | ||
{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779 }} | {{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }} | ||
[[Badness]]: 0.934310 | [[Badness]]: 0.934310 | ||
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== See also == | == See also == | ||
* [[ | * [[Small comma]] | ||
[[Category:3-limit]] | [[Category:3-limit]] | ||
[[Category:Small comma]] | [[Category:Small comma]] | ||
[[Category:Counterpyth]] | |||
Revision as of 13:16, 23 April 2021
| Interval information |
36 472 996 377 170 786 403
counterpyth comma
reduced subharmonic
[65 -41⟩, the 41-comma or counterpyth comma, is an interval of 19.845 cents. It is the amount by which 41 fifths (3/2) fall short of 24 octaves, in other words 224/(3/2)41.
Temperaments
Tempering out this comma splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. It leads to a number of regular temperaments in the counterpyth family. For equal divisions N up to 1230, the comma is tempered out if and only if 41 divides N. Examples are 41edo, 164edo, 205edo, 246edo, 328edo and 369edo.
Counterpyth (41&205)
Subgroup: 2.3.5
Comma list: [65 -41⟩
Mapping: [⟨41 65 0], ⟨0 0 1]]
Mapping generators: ~531441/524288, ~5
POTE generator: ~5/4 = 386.668
Badness: 0.934310
7-limit 41&123
Subgroup: 2.3.5.7
Comma list: 1029/1024, 537824/531441
Mapping: [⟨41 65 0 115], ⟨0 0 1 0]]
POTE generator: ~5/4 = 385.731
Badness: 0.161056
7-limit 41&205
Subgroup: 2.3.5.7
Comma list: 5120/5103, 2500000/2470629
Mapping: [⟨41 65 0 20], ⟨0 0 1 1]]
POTE generator: ~5/4 = 385.667
Badness: 0.142344
7-limit 41&369
Subgroup: 2.3.5.7
Comma list: 2401/2400, 52613349376/52301766015
Mapping: [⟨41 65 95 115], ⟨0 0 2 1]]
POTE generator: ~8/7 = 231.045
Badness: 0.134559