# fifth-tone / 1/5-tone

[Joe Monzo]
###### 1. Approximate musical interval

Used in a general sense to refer to microtonal intervals approximately 1/5 the size of the diatonic whole-tone, measuring approximately 40 cents. Marchetto of Padua was the earliest theorist to say that the tone was divided into 5 parts, in his Lucidarium of 1318, and he did not explicitly claim that the division was into equal parts. Marchetto called his 1/5-tone a "diesis", and the terms are today more-or-less synonymous.

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###### 2. One degree of 30-edo

Calculated as the 30th root of 2 -- 30√2, or 2(1/30) -- an irrational proportion with the approximate ratio of 1 : 1.023373892, and an interval size of exactly 40 cents. It is the size of one degree, and thus the basic "step" size, in the 30-edo (or 30-tET) scale.

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###### 3. One degree of 31-edo

The term "fifth-tone" is also often used loosely to express one degree, and the basic "step" size, of 31-edo (or 31-tET), calculated as the 31st root of 2 -- 31√2, or 2(1/31) -- an irrational proportion with the approximate ratio 1 : 1.022611436 = ~38.70967742 or exactly 38 22/31 cents. The terms "fifth-tone scale" or "fifth-tone system" generally are meant to refer to 31-edo rather than 30-edo. Fokker called one degree of 31-edo a diesis.

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