Microtonal, just intonation, electronic music software Microtonal, just intonation, electronic music software

Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A number written as a superscript following another number, which indicates how many times the first number is to be multiplied by itself.

For example: 25 = 5 * 5, which can also be written 52. Here, 2 is the exponent.

Exponents were first written in their modern form in the early 1600s by Rene Descartes, who also published a treatise on music.

Exponents are important in my theory because they represent steps along each axis of a lattice-diagram, which thus shows information about the influence of each prime-factor in the tuning. My theories are based on the idea that the prime-series represents the ultimate basis to which our minds can reduce the mathematics of a tuning.

When a particular prime-series is known or given, the exponents can be written alone in a vector (specifically, in the form of a monzo), thus simplifying the notation of a tuning without sacrificing any mathematical precision (which is what happens when cents-values or other logarithmic measurements are used).

For example, if the series of prime-factors is given as {2,3,5}, the "25" cited above would be written [0 0 2], which indicates a point in 3-dimensional 2,3,5-prime-space which lies along the origin of both the 2- and the 3-axes, and 2 steps in the positive direction along the 5-axis.

The ASCII (i.e., plain text) convention for indicating exponents (in an email, for example) is to use the caret sign "^" before the exponent; for example, 32 would be typed as 3^2.

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[Joe Monzo]

A number or symbol indicating the power to which another number or symbol is to be raised. Examples: 32, xy, (81/80)(1/4), 2(a/b), which the exponents are 2, y, (1/4), and (a/b), respectively.

Exponents are important in my work in that they represent the powers of the prime-factors. In one form of my prime-factor notation, the primes themselves are omitted and only the exponents (including zeros) are used, as a vector. See Monzo, JustMusic Prime-factor notation for an explanation.

Graham Breed's matrix page is also concerned almost exclusively with exponents.

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