grad

A term used by Andreas Werckmeister to designate the small interval describing 1/12th part of the Pythagorean comma.

The grad is calculated as the 12th root of the ratio 531441:524288, or ((2^-19)*(3¹²))^(1/12), thus with a ratio itself of approximately 1:1.001129891. It is an irrational number. The width of this grad interval is ~1.955000865 (pretty close to 1 & 21/22) cents. The grad is thus precisely the amount by which the 12-EDO "5th" is narrower than the Pythagorean "5th" with the ratio 3:2, and therefore is the unit of measurement used in tuning that temperament.

This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into ~613.81047 equal parts. Thus the grad represents one degree in ~613.81047-EDO "non-octave" tuning, or its audibly identical "octave"-based relative 614-EDO.

There are just over 51 grads (a more exact figure is ~51.1508725, about 51 & 1/7 or almost exactly 51 & 8/53) in a Semitone.

The formula for calculating the grad-value of any ratio is:

grads = log₁₀(ratio) / log₁₀[ 2^(-19/12) * 3 ]


Note that the grad is nearly the same size as the skhisma, the difference between them being only ~0.001280077 (= ~1/781) cent:

      2^x    3^y  5^z	
	
   [ -19/12   1    0 ]   grad
-  [ -15      8    1 ]   skhisma
----------------------
   [ 161/12  -7   -1 ]   difference between grad and skhisma


See also Manuel Op de Coul's Logarithmic Interval Measures.

[from Joe Monzo, JustMusic: A New Harmony]

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