Ups and downs notation

Ups and downs notation refers broadly to a number of notations devised for notating equal temperaments or similar systems with up and down arrows (^ and v) to indicate alterations smaller than a semitone. Most commonly, it refers to Kite's ups and downs notation, which notates any EDO (up to a certain extent) by letting the arrows stand for a single edostep and combining them with standard diatonic notation. Stein-Zimmermann-Gould notation, the main system covered on this page, is a system built on a similar philosophy, with further refinements.

Note: This page currently covers a slightly modified version of Kite's ups and downs notation altered to include neutral accidentals. It is up to the designers of Stein-Zimmermann-Gould to modify it to fit their notation.

Basic premise

Standard diatonic notation covers the chain of fifths. If present, neutral diatonic notation covers half-chroma alterations. (This is sufficient for covering systems such as 17edo and 31edo.)

Ups (^) and downs (v) raise and lower the pitch of a note by 1 EDOstep respectively, in at the very least all edos up to and including 67edo. Above that point, it may be useful in some systems to allow an up or down accidental to stand for more than 1 EDOstep. For instance, in 94edo, it is best to let the arrows stand for the garibaldi comma. In those systems, lifts (/) and drops (\) may be used to stand for a single EDOstep. Ups/downs and lifts/drops are always placed before the note/degree/interval name.

Interval arithmetic is preserved, with ups and downs added in. For instance, an upmajor third on an upminor third is a double-up ("dup") perfect fifth. Note that the same terminology is used for the up and down accidentals as for the qualities they define, so for example an "upflat three" degree makes an "upminor third" interval with the unison. This becomes potentially ambiguous when the quality of the interval is not specified, but in most EDOs this is not a concern.

Note that any arrow or Pythagorean accidental cancels any previous ones.

Chord names

Alterations are enclosed in parentheses, additions never are. Alterations always come last in the chord name. In edos with 7edo's fifth, every interval is perfect, with no major or minor; this serves to resolve the ambiguity introduced by using up/down for both qualities and accidentals. Do not use lowercase roman numerals for chords; a "v" symbol for down might be confused with the "v" symbol in Roman numerals.

Ups and downs apply to chord qualities as well as to intervals, so that "Cv" is a downmajor chord and "C^m" is an upminor chord. Ups and downs may appear in bass figures as well. Note that in general, ups and downs also apply to the sixth, seventh, and eleventh if placed before the normal chord quality, so that "C^m7" is a minor seventh chord with both the third and the seventh raised by a step. If a distance other than a perfect fifth is warranted, the added seventh is placed after a comma punctuation mark (,), like with added intervals in general.