#### Encyclopedia of Microtonal Music Theory

[Joe Monzo]

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
```
. . . . . . . . .

The standard epimoric approximation to the grad is the ratio 886:885 ( = ~1.955092302 cents), whose 2,3,5,59,443-monzo is [ 1, -1, -1, -1, 1 > . The tiny difference between them is:

```   -------- monzo --------
2    3   5  59  443

[  -1   -1  -1  -1   1 >   =  epimoric approximation, 886:885 ratio
-  [-19/12  1   0   0   0 >   =  actual grad, (2-(19/12)*3):1 ratio
----------------------------
[-31/12 -2  -1  -1   1 >   =  ~0.0000914362 cent ( = ~1/11000 cent)
```

Tuning treatises before c.1970 often defined the grad as the 886:885 ratio without emphasis on the fact of its being an approximation, particularly in the German literature of c.1850-1950.

. . . . . . . . .

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