Hewuermera comma

Interval information

Ratio	589824/588245
Factorization	216 × 32 × 5-1 × 7-6
Monzo	[16 2 -1 -6⟩
Size in cents	4.640849¢
Name	Hewuermera comma
Color name	sr6g-3, satribiru-agu negative 3rd
FJS name	[math]\displaystyle{ \text{d}{-3}_{5,7,7,7,7,7,7} }[/math]
Special properties	reduced
Tenney height (log2 nd)	38.336
Weil height (log2 max(n, d))	38.3399
Wilson height (sopfr(nd))	85
Comma size	small
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589824/588245, the hewuermera comma, is a 7-limit small comma measuring about 4.6 cents. It is the amount by which a stack of six 8/7s exceed 20/9. It can also be expressed as the difference between three intervals of S7 = 49/48 and S4 = 16/15, or as (S6/S7 = 1728/1715)/(S7/S8 = 1029/1024).

Factorizations

As a generalized lopsided comma, 589824/588245 has the S-expression of (S46 = 2116/2115) × (S47 = 2209/2208)² × (S48 = 2304/2303)³. Equivalent factorizations include (S48 = 2304/2303) × (S(47/2) = 2209/2205) and (S46×S47×S48 = 736/735) × (S47×S48 = 1128/1127) × (S48 = 2304/2303).

The S-factorization indicates that 23 and 47 are particularly natural primes to incorporate into hewuermera temperaments (by tempering out S46, S47, and S48); in particular, three 8/7s reach a "1/2-comma meantone fifth" identified with 70/47~94/63.

Temperaments

Tempering it out alone in the 7-limit leads to the hewuermera rank-3 temperament, and in combination with another comma, to the hewuermera temperaments, the most notable of which are hemiwürschmidt and mothra. For highest accuracy, this comma demands that 8/7 be tuned slightly flat, as it is in EDOs such as 26, 68, and 99.

Etymology

The comma was named by Xenllium in 2021, from a combination of hemiwürschmidt and gamera, two 7-limit temperaments that share this tempered comma.