Archangelic comma

Interval information

Factorization	2301994 × 3-190537
Monzo	[301994 -190537⟩
Size in cents	0.000111619¢
Name	archangelic comma
FJS name	[math]\text{27219d18052}[/math]
Special properties	reduced, reduced subharmonic
Tenney height (log2 nd)	603988
Weil height (log2 max(n, d))	603988
Wilson height (sopfr(nd))	1175600
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.19982 bits
Comma size	unnoticeable
open this interval in xen-calc

The archangelic comma (monzo: [301994 -190537⟩) is the amount by which 190537 perfect fifths fall short of 111457 octaves, in other words 2¹¹¹⁴⁵⁷/(3/2)¹⁹⁰⁵³⁷. This difference is inaudible at roughly 0.0001 cents.

This comma was named by Aura in direct opposition to the satanic comma due to 190537edo actually surpassing 665edo in 3-2 telicity k-strength. However, it is opposite the satanic comma in other ways, such as the fact that tempering this comma results in a stretched fifth rather than a compressed fifth.

Temperament

Tempering out this comma leads to the archangelic family of temperaments. It is tempered out in 190537edo and its multiples (571611edo, 762148edo, 1714833edo, 1905370edo, and 4191814edo just to name a few), with 190537edo itself being a 28-strong 3-2 telic system due to tempering this comma out.