57/56

Interval information

Ratio	57/56
Factorization	2-3 × 3 × 7-1 × 19
Monzo	[-3 1 0 -1 0 0 0 1⟩
Size in cents	30.64211¢
Name	hendrix comma
Color name	19or1, noru 1sn, Noru comma
FJS name	[math]\displaystyle{ \text{P1}^{19}_{7} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	11.6402
Weil norm (log2 max(n, d))	11.6658
Wilson norm (sopfr(nd))	35
Comma size	medium
Open this interval in xen-calc

57/56 is a superparticular interval of about 30.6¢ which makes the difference between 4/3 (about 498¢) and 19/14 (about 528.7¢) in 19-limit just intonation; it is also the difference between 19/16 and 7/6. It is also called the hendrix comma, since by tempering it out, one can build an 8:10:14:19 hendrix chord, which has steps of 5/4-7/5-4/3. Patent vals which temper out the hendrix comma include those from 9, 10, 12, 16, 21, 22 and 26 edo.

57/56 is a comma of the 2.3.7.19 subgroup, and on that subgroup, patent vals tempering it out include those from 9, 12, 14, 17 and 26. The comma defines a rank three temperament on this subgroup, called hendrix. You can also temper it out in a rank-2 temperament in 2.3.19/7, called surprise.

Temperaments

Hendrix

This is the rank-3 temperament tempering out the hendrix comma, 57/56. This is simply the most sensible name for this temperament.

Subgroup: 2.3.7.19

Comma list: 57/56

Subgroup-val mapping: [⟨1 0 0 3], ⟨0 1 0 -1], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

• WE: ~2 = 1202.4345 ¢, ~3/2 = 697.4314 ¢, ~7/4 = 970.3487 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3981 ¢, ~7/4 = 972.1990 ¢

Optimal ET sequence: 5, 9, 12, 14, 17, 26, 31h

Badness (Sintel): 0.155

See also

• 28/19 – its fifth complement
• Medium comma
• List of superparticular intervals