Septimal ruthenia: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
→Temperaments: -ian ending for 2.3.5.7, no ending for 2.3.7 as many other commas on this wiki have |
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== Temperaments == | == Temperaments == | ||
In the full 7-limit, tempering out this comma leads to the rank-3 '''septiruthenian temperament''' (cf. ruthenium, the rank-2 2.3.5.7 temperament). You may find a list of good equal temperament supporting it below. | In the full 7-limit, tempering out this comma leads to the rank-3 '''septiruthenian temperament''', and in the 2.3.7 subgroup, rank-2 '''septiruthen''' temperament (cf. ruthenium, the rank-2 2.3.5.7 temperament). You may find a list of good equal temperament supporting it below. | ||
=== Septiruthenian === | === Septiruthenian === | ||
Revision as of 14:30, 7 June 2023
| Interval information |
septiruthenia
reduced harmonic
The septimal ruthenia or septiruthenia (monzo: [-263 88 0 44⟩), known as the 44-septimal comma in earlier materials[1], is an unnoticeable 7-limit comma of about 0.38 cents which is defined as the difference between a stack of fourty-four 64/63's and the octave.
It's called "ruthenia" because of the 44th element, ruthenium, following the convention of naming certain fractional-octave temperaments after chemical elements. For edos N up to ~16000, it is tempered out if and only if 44 divides N. The relationship between 64/63 and one step of 44edo is highly precise, even up many thousands of edos.
Temperaments
In the full 7-limit, tempering out this comma leads to the rank-3 septiruthenian temperament, and in the 2.3.7 subgroup, rank-2 septiruthen temperament (cf. ruthenium, the rank-2 2.3.5.7 temperament). You may find a list of good equal temperament supporting it below.
Septiruthenian
Subgroup: 2.3.5.7
Comma list: [-263 88 0 44⟩
Mapping: [⟨44 0 0 263], ⟨0 1 0 -2], ⟨0 0 1 0]]
Mapping generators: ~64/63, ~3, ~5
Optimal ET sequence: 176, 308, 352, 484, 528, 836, 1848, 2684, 4532, 7744, 8580, 9592, 10428, 12276, 14960, 27236, 42196, 99352, …
Badness: 116 × 10-3
Another notable system tempering out the septiruthenia is 1012edo, which is a zeta peak edo.
Notes
- ↑ The first occurrence of the name seems to be in the page of unnoticeable comma, archived on October 26, 2016.