Septimal ruthenia
Factorization | 2^{-263} × 3^{88} × 7^{44} |
Monzo | [-263 88 0 44⟩ |
Size in cents | 0.3799608¢ |
Names | septimal ruthenia, septiruthenia |
Color name | L^{19}z^{44}8, nela-lequadzo octave |
FJS name | [math]-[/math] |
Special properties | reduced, reduced harmonic |
Tenney height (log_{2} nd) | 526 |
Weil height (log_{2} max(n, d)) | 526.001 |
Wilson height (sopfr (nd)) | 1098 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.39811 bits |
Comma size | unnoticeable |
open this interval in xen-calc |
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 septiruthenic temperament, and in the 2.3.7 subgroup, rank-2 septiruthenian temperament (cf. ruthenium, the rank-2 2.3.5.7 temperament). You may find a list of good equal temperament supporting it below. See also 44th-octave temperaments.
Septiruthenic
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.