mod, modulo

A mathematical operation which gives the remainder after division by a certain number, the modulus.

For example, in the regular 12-eq scale, two 'stacked 5ths' form a '9th'. To calculate which degree of the 12-eq scale this would be, modulo 12 is used, with a diatonic '5th' equal to the 7th degree of the 12-eq (i.e., chromatic) scale:

	'5th' + '5th'
=	7 + 7 mod 12
=	14 mod 12

14/12 = 1.&2/12, so the remainder, 2, is the answer.

So the '9th' would be a note which is the 2nd degree of the 12-eq scale, but an 'octave' higher than its 'original' position.

The English way of measuring the hours in a day works exactly like this: the 14th hour is 14 mod 12 = 2, so it is called 2 o'clock p.m.

For any 'octave'-equivalent equal-tempered system, the modulus is the number of equal degrees into which the 'octave' is divided.

When calculating cents in an 'octave'-equivalent system, the modulus is 1200. So two just '5ths' stacked on top of each other would give a just '9th' with an 'octave'-reduced cents value of:

	'5th' + '5th'
=	~702 + ~702 mod 1200
=	~1404 mod 1200

1404/1200 = 1.&~204/1200, so ~204 is the answer

which is the approximate cents value of a just 'major 2nd'.

[from Joe Monzo, JustMusic: A New Harmony]

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