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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

When an EDO does not employ its patent val (i.e., uses one or more prime-mappings which are not the best, but rather the second- or third-best, etc.), the number specifying the cardinality of the EDO is supplemented by a letter or letters, which are the "warts". These letters correspond according to their order in the Roman alphabet with the series of prime-factors: thus, a = 2, b = 3, c = 5, d = 7, e = 11, f = 13, etc., with a single instance of a letter indicating the employment of the second-best mapping of that prime. In the case of a third-best mapping being employed, the letter is doubled, and so on for increasingly worse mappings.

Example: The patent val for 12-edo as a temperament of 11-limit JI is <12 19 28 34 42]. However, 11 is almost exactly midway between 41\12 and 42\12. If the val employed is instead <12 19 28 34 41], the tuning is indicated as the breed "12e". The wart in this case is "e" and indicates that the second-best mapping of prime-factor 11 is being used.

For an example of usage of 12e in a music-theory description, see Monzo, Searching For Schoenberg's Pantonality, especially the note at the beginning about how a misprint in the English translation led to my confusion over Schoenberg apparently employing both 12 and 12e simultaneously. In reality, according to the original 1911 German edition, Schoenberg consistently employed 12e in his diagram. In contrast, in his later paper Problems of Harmony, Schoenberg consistently employs the non-wart 12 patent-val.


[Paul Erlich, Facebook Xenharmonic Alliance group, 14 April 2023]

warts are the "p", "l", "hjln", etc. notations used to specify which breed the number refers to. See the xen wiki page on "val".

A Monzo is
(a) notationally a list of prime exponents, used in order to specify
(b) a frequency ratio (or a tempered version thereof, as an equivalent position or vector in the tempered version of the just lattice) representing a musical interval or pitch,
(c) but due to Gene's usage it refers to (b) in all cases, regardless of whether there is any explicit list of prime exponents.

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