Alpharabian comma

Interval information

Ratio	131769/131072
Factorization	2-17 × 32 × 114
Monzo	[-17 2 0 0 4⟩
Size in cents	9.181771¢
Name	Alpharabian comma
Color name	L1o4-2, Laquadlo comma
FJS name	[math]\displaystyle{ \text{M}{-2}^{11,11,11,11} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	34.0077
Weil norm (log2 max(n, d))	34.0153
Wilson norm (sopfr(nd))	84
Comma size	small
Open this interval in xen-calc

The Alpharabian comma (monzo: [-17 2 0 0 4⟩, ratio: 131769/131072) is a small 11-limit (also 2.3.11 subgroup) comma measuring about 9.2 ¢. It is the amount by which a stack of four 33/32 quartertones exceeds a 9/8 whole tone, and the amount by which a stack of two 128/121 diatonic semitones falls short of a 9/8 whole tone.

The term Alpharabian comes from Alpharabius – another name for Al-Farabi – and was chosen due to the fact that 33/32, also known as the the Al-Farabi quartertone, is the primary parachroma of the 11-limit, a fact which lends itself to the idea of just 2.3.11 tuning being called Alpharabian tuning in the same way that just 3-limit tuning is called Pythagorean tuning. Of note is that the Alpharabian comma and the Pythagorean comma are similar in that both commas represent the difference between two of their respective p-limit's primary diatonic semitones and a 9/8 whole tone.

Temperaments

Tempering out the Alpharabian comma in the 11-limit results in the rank-4 alphaxenic temperament (→ Catalog of rank-4 temperaments #Alphaxenic (131769/131072)), or in the 2.3.11 subgroup, the rank-2 alphaxenean temperament (→ No-fives subgroup temperaments #Alphaxenean).

See also

• Alphaxenic rank-3 clan