User:Dave Keenan/sandbox

A fractional-3-limit notation is a kind of musical notation built on a chain-of-fifths notation, which is used for notating EDOs or ETs in a way that avoids favoring any mapping from JI, while preserving the notation of subset EDOs. Fractional-3-limit notations may be contrasted with two other kinds of chain-of-fifths notation for EDOs: JI-based notations, like the "good fifths" Sagittal notations, which assume specific JI mappings, and step-count notations, like Ups and Downs notations, which do not preserve the notation of subset EDOs. Fractional-3-limit notations assign symbols to fractions of some tempered 3-limit comma. In practice, this 3-limit comma is either the apotome (chromatic semitone) as represented by a sharp or flat, or the limma (diatonic semitone) as represented by the intervals B-C and E-F.

History

Stein-Zimmermann notation can be viewed as a very simple apotome-fraction notation, only capable of notating half-apotomes.

On 24-Sep-2016 in the Facebook Group: Microtonal Music and Tuning Theory, Cryptic Ruse introduced [combination of apotome-fraction and limma-fraction notations] to cover all EDOs up to 72.

When the EDO has fifths so wide that the limma becomes very small or negative (e.g. 32-EDO), an apotome-fraction notation must be used. When the EDO has fifths so narrow that the apotome becomes very small or negative (e.g. 33-EDO), limma-fraction notations must be used.

Although CRyptic later abandoned these notations, the principel was adopted by George Secor and Dave Keenan for the notation of EDOs with bad fifths, where a bad fidth is defined as one with an error of more tha 10 cents.