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Encyclopedia of Microtonal Music Theory

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mod / modulo

[Joe Monzo]

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

For example, in the regular 12-edo tuning, two 'stacked 5ths' form a '9th'. To calculate which degree of the 12-edo scale this would be, modulo 12 is used, with a diatonic '5th' equal to the 7th degree of the 12-edo (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-edo scale, but an octave higher than its "normalized" position.

The English way of measuring the hours in a day works exactly like this (with the same modulo of 12): 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 (i.e., the cardinality of the EDO).

For example, to calculate 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'.

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