Diatonic notation
Diatonic notation or chain-of-fifths notation is the standard notation system in Western music, which notates systems that are octave-periodic and generated by a (just or tempered) perfect fifth. Many tuning systems may be notated this way. EDOs can be notated using unmodified diatonic notation if they contain a single chain of fifths that generates a diatonic scale. For example, 23edo cannot be notated with standard diatonic notation because it does not have a diatonic scale, and 24edo cannot be notated with standard diatonic notation because it has two distinct chains of fifths.
Diatonic notation uses the 7 nominals (C, D, E, F, G, A, B) to form the C major scale, and the accidentals (#, b) to raise and lower by a chromatic semitone respectively. Note that a chromatic semitone is the difference between a diatonic major second and a diatonic semitone, and is in general not equal to a diatonic semitone, so, for example, C# and Db are in general not the same note.
Stacking intervals
The result of stacking two diatonic intervals can be determined by consulting this isomorphic table. Each step down is a major second, and each step to the right is a chromatic semitone. Add the two intervals as vectors from P1, and the result is the proper notation for the interval formed by stacking them.
| Degree | Quality | ||||||
|---|---|---|---|---|---|---|---|
| -3 | -2 | -1 | 0 | 1 | 2 | 3 | |
| 1sn | 4d1 | 3d1 | dd1 | d1 | P1 | A1 | AA1 |
| 2nd | 3d2 | dd2 | d2 | m2 | M2 | A2 | AA2 |
| 3rd | 3d3 | dd3 | d3 | m3 | M3 | A3 | AA3 |
| 4th | 3d4 | dd4 | d4 | P4 | A4 | AA4 | 3A4 |
| 5th | 3d5 | dd5 | d5 | P5 | A5 | AA5 | 3A5 |
| 6th | dd6 | d6 | m6 | M6 | A6 | AA6 | 3A6 |
| 7th | dd7 | d7 | m7 | M7 | A7 | AA7 | 3A7 |
| 8ve | dd8 | d8 | P8 | A8 | AA8 | 3A8 | 4A8 |
Extensions
Neutral diatonic notation
Neutral diatonic notation introduces two new accidentals: semisharp (+ or t) and semiflat (d) to raise and lower by half of a chromatic semitone respectively. This allows the notation of neutral intervals, specifically dicot intervals. #t is called "sesquisharp" and db is called "sesquiflat".
Ups and downs notation
Ups and downs notation allows edos to be easily notated by providing the up (^) and down (v) accidentals to raise and lower by a single edostep. For example, in 29edo, the 372c interval is often used as a classical major third, and can be notated C-vE (a "downmajor third") instead of notating it as a diminished fourth (C-Fb), which would otherwise be necessary. Note that ups and downs always go before notes and interval names. Neutral intervals are notated with the symbol ~ and called "mid" (e.g. neutral 3rd, neutral 4th = ~3, ~4).
Any sharp, flat, or arrow cancels any previous ones, meaning that an arrow by itself implies a natural sign.
The up and down arrows may also be used to notate steps smaller than a chromatic semitone in rank-2 systems. For example, the generator of porcupine, a submajor second of about 160c, may be notated C-vD.
When using ups and downs, Roman numerals should always be uppercase, alterations are enclosed in parentheses, and additions never are. Alterations always come last in the chord name. "a" and "d" are used for augmented and diminished.
Read the ups and downs page on the XenWiki for more information.
Modified ups and downs notation
A (non-standard) proposal for the form of ups and downs notation used on XenReference is a slight, but compatible given a modicum of intuition regarding the nature of intervals, alteration of Kite's version of ups and downs notation. The main change is that the basic spine is now neutral diatonic notation, meaning that the "mid" symbol and usage of "mid" is redundant.
Melodic antidiatonic notation
Antidiatonic systems, if notated with diatonic notation, would have sharps lowering the pitch and flats raising the pitch. Other than that, however, the system remains fully coherent. A quick fix to make the antidiatonic notation intuitive is to swap the meaning of sharp and flat, so that sharp always raises in pitch, and flat always lowers in pitch. So the interval that sounds like a major third would be called a major third, and it would be notated C-E# rather than C-Eb (note that in, say, 16edo, C-E is 300 cents).
