Tetracot comma

Interval information

Ratio	20000/19683
Factorization	25 × 3-9 × 54
Monzo	[5 -9 4⟩
Size in cents	27.65985¢
Names	tetracot comma, minimal diesis
Color name	sy41, Saquadyo comma
FJS name	[math]\text{A1}^{5,5,5,5}[/math]
Special properties	reduced
Tenney height (log2 nd)	28.5524
Weil height (log2 max(n, d))	28.5754
Wilson height (sopfr(nd))	57
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.37587 bits
Comma size	small
open this interval in xen-calc

The tetracot comma (monzo: [5 -9 4⟩, ratio: 20000/19683), or minimal diesis, is a small 5-limit comma of 27.660 cents which is the amount by which four minor whole tones exceed a fifth; that is, (10/9)⁴/(3/2). It can also be expressed as the octave-reduced difference between four just major thirds and nine just perfect fifths, which directly follows from its monzo. Another characterization is the interval between four syntonic commas and a Pythagorean apotome, making the corresponding temperament a member of the syntonic–chromatic equivalence continuum. Since an apotome minus two syntonic commas is a classical chroma, the comma can be expressed as the difference between an apotome and a stack of two classical chromas, or between a classical chroma and a stack of two syntonic commas.

Temperaments

Tempering out this comma leads to the tetracot family of temperaments.