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

Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A musical interval with a size of approximately 116.7 cents.

The term honors the name of George Secor, original discoverer of the miracle family of temperaments. This interval (and other close approximations to it) is particularly important as the generator of the miracle tunings.

The minimax optimum value for the secor, based on its ability to generate temperaments which contain many close approximations of just intonation ratios, is (18/5)^(1/19) = ~116.7155940982074 cents. This satisfies the property that Secor mentions in his Xenharmonikôn article, that all the primary 11-limit ratios (aka 11-limit consonances) will be approximated with an error not greater than 3.32 cents.

The commonly-used EDOs which most closely approximate secor-generated tunings are 31-edo, 41-edo, and 72-edo. Their secors are:

EDO     ratio   ~cents

 31   2(3/31)  116.13 cents
 41   2(4/41)  117.07 cents
 72   2(7/72)  116.67 cents

Below is a graph showing the closest EDO approximations, those closest to "0" giving the best approximation. Above 72, the best approximations are given by 185-edo, 257-edo, and 329-edo.

The closest low-integer rational approximation of the secor is 46:43 (= ~116.7566416 cents), which is nearly identical to 2(18/185).


Secor, George. 1975.
A New Look at the Partch Monophonic Fabric
originally published in Xenharmonikôn 3.

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