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.

One early adopter of this is Kite's ups and downs (KUD) 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. More commonly used is the Stein–Zimmermann–Gould (SZG) notation, which is a system built on a similar philosophy with more conventional-looking symbols, and is the main system covered on this page.

Basic premise

Standard diatonic notation covers the chain of fifths. If present, neutral diatonic notation covers half-chroma alterations. In SZG, its use alternates between edos of odd and even sharpnesses i.e. the number of steps the chromatic semitone is mapped to. For sharp-2 systems such as 17edo, 24edo and 31edo, its use is sufficient to notate the full edo and no arrows are needed.

Ups (^) and downs (v) raise and lower the pitch of a note by a small amount, most commonly one edostep in at the very least all edos up to and including 67edo. Above that point, it may be useful in some systems to let an up or down accidental stand for more than one 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.

Kite's chord names

Kite has specified a naming standard for chords in his notation.

In this standard, 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.