Septimal ruthenia

Interval information

Factorization	2-263 × 388 × 744
Monzo	[-263 88 0 44⟩
Size in cents	0.3799608¢
Names	septimal ruthenia, septiruthenia
Color name	L19z448, nela-lequadzo octave
FJS name	[math]\displaystyle{ - }[/math]
Special properties	reduced, reduced harmonic
Tenney height (log2 nd)	526
Weil height (log2 max(n, d))	526.001
Wilson height (sopfr(nd))	1098
Comma size	unnoticeable
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The septimal ruthenia or septiruthenia (monzo: [-263 88 0 44⟩), known as the 44-septimal comma in earlier materials, 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

Septiruthenian

Subgroup: 2.3.7

Comma list: [-263 88 44⟩

Mapping: [⟨44 0 263], ⟨0 1 -2]]

Mapping generators: ~64/63, ~3

Another notable system tempering out the septiruthenia is 1012edo, which is a zeta peak edo.

Trivia

Quite coincidentally, the color name for 64/63 is "Ru unison", and the 44th chemical element ruthenium, the namesake of this comma, has the symbol Ru.