Parimo: Difference between revisions
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The '''parimo''' ([[ratio]]: 1771561/1771470, {{monzo|legend=1| -1 -11 -1 0 6 }}) is an [[Unnoticeable comma|unnoticeable]] [[11-limit]] [[comma]] with a value of approximately 0.09{{cent}}. It is the amount by which an [[octave-reduced]] stack of six [[11/9]] neutral thirds exceeds [[5/3]], as well as the amount by which a stack of three [[243/242|rastmas]] falls short of a [[81/80|syntonic comma]]. | |||
The '''parimo''' | |||
== Temperaments == | == Temperaments == | ||
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=== Parimic === | === Parimic === | ||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 1771561/1771470 | [[Comma list]]: 1771561/1771470 | ||
{{Mapping|legend=1| 1 0 5 0 1 | 0 1 1 0 2 | 0 0 -6 0 -1 | 0 0 0 1 0 }} | |||
: mapping generators: ~2, ~3, ~18/11, ~7 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073 | ||
{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e }} | {{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e }} | ||
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=== Tritomere === | === Tritomere === | ||
Subgroup: 2.3.5.11 | [[Subgroup]]: 2.3.5.11 | ||
[[Comma list]]: 1771561/1771470 | [[Comma list]]: 1771561/1771470 | ||
{{Mapping|legend=2| 1 0 5 1 | 0 1 1 2 | 0 0 -6 -1 }} | |||
: sval mapping generators: ~2, ~3, ~18/11 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073 | ||
{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231 }} | {{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231 }} | ||
[[Badness]]: 0.00712 × 10<sup>-3</sup> | [[Badness]]: 0.00712 × 10<sup>-3</sup> | ||
== Etymology == | |||
It is yet to be found out how this comma was named. However, as [[Petr Pařízek]] was the first to take note of it<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100716.html Yahoo! Tuning Group | ''The two smallest commas I have found so far'']</ref> and as it was tempered out in the [[mohaha]] temperament, it could be conjectured that the name was a contraction of ''Pařízek'' and ''mohaha''. | |||
== See also == | == See also == | ||
* [[Unnoticeable comma]] | * [[Unnoticeable comma]] | ||
== Notes == | |||
[[Category:Commas named after composers]] | |||
[[Category:Commas named after music theorists]] | |||
Latest revision as of 02:38, 4 November 2024
| Interval information |
The parimo (ratio: 1771561/1771470, monzo: [-1 -11 -1 0 6⟩) is an unnoticeable 11-limit comma with a value of approximately 0.09 ¢. It is the amount by which an octave-reduced stack of six 11/9 neutral thirds exceeds 5/3, as well as the amount by which a stack of three rastmas falls short of a syntonic comma.
Temperaments
Tempering it out leads to a form of parimic temperament, which can be said to be the undecimal counterpart of the metric temperament in that the syntonic comma is split into three equal parts in both systems. However, in parimic temperament, the parts represent different intervals – one part represents the rastma, and two represent the biyatisma. The corresponding 2.3.5.11 subgroup temperament is tritomere.
Parimic
Subgroup: 2.3.5.7.11
Comma list: 1771561/1771470
Mapping: [⟨1 0 5 0 1], ⟨0 1 1 0 2], ⟨0 0 -6 0 -1], ⟨0 0 0 1 0]]
- mapping generators: ~2, ~3, ~18/11, ~7
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073
Optimal ET sequence: 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e
Badness: 0.0279 × 10-6
Tritomere
Subgroup: 2.3.5.11
Comma list: 1771561/1771470
Sval mapping: [⟨1 0 5 1], ⟨0 1 1 2], ⟨0 0 -6 -1]]
- sval mapping generators: ~2, ~3, ~18/11
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073
Optimal ET sequence: 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231
Badness: 0.00712 × 10-3
Etymology
It is yet to be found out how this comma was named. However, as Petr Pařízek was the first to take note of it[1] and as it was tempered out in the mohaha temperament, it could be conjectured that the name was a contraction of Pařízek and mohaha.