17-comma

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Interval information
Ratio 134217728/129140163
Factorization 227 × 3-17
Monzo [27 -17
Size in cents 66.764985¢
Names 17-comma,
gothic comma,
Pythagorean gothma
Color name ssw3, sasawa 3rd,
Wa-17 comma
FJS name [math]\text{dd3}[/math]
Special properties reduced
Tenney height (log2 nd) 53.9444
Weil height (log2 max(n, d)) 54
Wilson height (sopfr (nd)) 105
Harmonic entropy
(Shannon, [math]\sqrt{nd}[/math])
~4.36401 bits
Comma size medium
open this interval in xen-calc

The 17-comma, otherwise known as the gothic comma or Pythagorean gothma (monzo[27 -17, ratio: 134217728/129140163), is an interval of 66.765 ¢. It is the amount by which seventeen perfect fifths fall short of ten octaves, or 210/(3/2)17.

Temperaments

Tempering out this comma in the 5-limit leads to the gothic temperament. For edos up to 150 the comma is tempered out if and only if 17 divides it, for example 17edo, 34edo or 68edo.

Terminology

The names were given by Flora Canou in 2023. The first term gothic comma draws two inspirations:

  1. the neogothic tuning available in 17edo, and
  2. the link of TAMNAMS Extension name and comma names by the analogy pair gothic : pythagotonic :: mystery : pythamystonic.

The second term Pythagorean gothma is part of a series of step names in the Pythagorean diatonic scale, where gothma refers to the double diminished 2-step, i.e. |3s - L|.

See also