405/392: Difference between revisions
Note that it's the sum of 81/80 and 50/49 |
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'''405/392''', the '''greenwoodma''', is a [[medium comma|medium]] [[7-limit]] [[comma]]. It is the sum of the [[81/80|syntonic comma (81/80)]] and the [[50/49|jubilisma (50/49)]]. | '''405/392''', the '''greenwoodma''', is a [[medium comma|medium]] [[7-limit]] [[comma]]. It is the difference between [[8/5]] and a stack of two [[9/7]]'s. In terms of commas, it is the sum of the [[81/80|syntonic comma (81/80)]] and the [[50/49|jubilisma (50/49)]]. | ||
== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma leads to the '''greenwoodmic temperament'''. See [[Greenwoodmic temperaments]] for a collection of rank-2 temperaments where it is tempered out. | [[Tempering out]] this comma leads to the rank-3 '''greenwoodmic temperament'''. This temperament splits a [[~]][[5/2]] in two, each representing ~[[14/9]]. See [[Greenwoodmic temperaments]] for a collection of rank-2 temperaments where it is tempered out. | ||
=== Greenwoodmic === | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 405/392 | |||
{{Mapping|legend=1| 1 0 1 -1 | 0 1 0 2 | 0 0 2 1 }} | |||
: mapping generators: ~2, ~3, ~14/9 | |||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~2 = 1200.000, ~3/2 = 694.595, ~14/9 = 791.182 | |||
: [[error map]]: {{val| 0.000 -7.360 -3.949 +11.546 }} | |||
* [[CWE]]: ~2 = 1200.000, ~3/2 = 693.344, ~14/9 = 790.072 | |||
: [[error map]]: {{val| 0.000 -8.611 -6.169 +7.935 }} | |||
{{Optimal ET sequence|legend=1| 9, 12, 26, 38, 73bc }} | |||
[[Badness]] (Smith): 0.412 × 10<sup>-3</sup> | |||
== Notation == | == Notation == | ||
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[[Category:Greenwoodmic]] | [[Category:Greenwoodmic]] | ||
[[Category:Commas with unknown etymology]] | [[Category:Commas with unknown etymology]] | ||
Latest revision as of 11:58, 6 January 2025
| Interval information |
405/392, the greenwoodma, is a medium 7-limit comma. It is the difference between 8/5 and a stack of two 9/7's. In terms of commas, it is the sum of the syntonic comma (81/80) and the jubilisma (50/49).
Temperaments
Tempering out this comma leads to the rank-3 greenwoodmic temperament. This temperament splits a ~5/2 in two, each representing ~14/9. See Greenwoodmic temperaments for a collection of rank-2 temperaments where it is tempered out.
Greenwoodmic
Subgroup: 2.3.5.7
Comma list: 405/392
Mapping: [⟨1 0 1 -1], ⟨0 1 0 2], ⟨0 0 2 1]]
- mapping generators: ~2, ~3, ~14/9
- CTE: ~2 = 1200.000, ~3/2 = 694.595, ~14/9 = 791.182
- error map: ⟨0.000 -7.360 -3.949 +11.546]
- CWE: ~2 = 1200.000, ~3/2 = 693.344, ~14/9 = 790.072
- error map: ⟨0.000 -8.611 -6.169 +7.935]
Optimal ET sequence: 9, 12, 26, 38, 73bc
Badness (Smith): 0.412 × 10-3
Notation
Sagittal notation
In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal and is called the 49/5 medium diesis, or 49/5M for short, because the simplest interval it notates is 49/5 = 7×7/5 (equiv. 49/40), as for example in C-E . The upward version is called 5/49M or 49/5M up and is represented by .