Diatonic notation

Diatonic notation or chain-of-fifths notation is the standard notation system in Western music. 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.

Basic concepts

The diatonic scale has a number of generalizations to tuning systems outside of 12edo. However, the one that has been chosen to be the "canonical" generalization for the purposes of notation is the MOS diatonic scale, LLsLLLs, for the following reasons:

• Most other diatonic scales are significantly more arbitrary or favor a certain style of composition at the expense of others (except in a few special cases)
• The MOS diatonic scale is closely connected to the chain of fifths and thus exists (or can be treated as if it exists) in any EDO that has one, with the chain of fifths itself being a useful structure compositionally
• Meantone, the main temperament used in the West outside 12edo, has a MOS diatonic scale
• The MOS diatonic scale is the only choice that perfectly preserves interval arithmetic.

The MOS diatonic scale is generated by a chain of 7 notes separated by fifths, with the chain of fifths being ... d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4 ..., and continuing with augmented and diminished intervals in their respective directions infinitely (in the case of a rank-2 tuning) or until the circle closes (in the case of an equal temperament).

This means that the accidentals # and b replace a note with the note 7 fifths in one direction or another on the chain of fifths, with the interval at 7 fifths being the augmented unison or chromatic semitone. (Note that in the general case, this is not equal to the diatonic semitone found as the small step of the diatonic scale. Additionally, sharp and flat do not always inflect by a single edostep; in fact, it is rare that they do so.) Additionally, the tunings of the intervals in a diatonic scale can vary significantly based on the tuning of the fifth used to create it. For example, a fifth tuned to 715 cents generates a major third of 460 cents, while a fifth tuned to 690 cents generates a major third of 360 cents, which is a difference of a whole semitone. Therefore, the primary compromise of the MOS diatonic scale is that the same label (say, "major third") can refer to wildly different intervals depending on the system. The tuning of a diatonic interval can be determined by stacking the corresponding fifth the necessary number of times (for example, the major third in 17edo is 424 cents, because the fifth of 706 cents stacks four times to 2824 cents, which results in 424 cents after octave-reducing) However, ultimately, this ends up being better than the alternatives in practice.

This idea of scale sensitivity to generator tuning is the fundamental idea behind monocot temperaments (such as meantone and archy), and more generally of regular temperament theory as a whole.

Ultimately, this means that diatonic notation 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 (unless harmonic antidiatonic notation is used), and 24edo cannot be notated with standard diatonic notation because it has two distinct chains of fifths.

Interval arithmetic

Interval arithmetic refers to the rules for determining a stacked interval's label. One way to calculate the result is to simply add together the positions of the intervals on a chain of fifths, for example:

major third (+4) + minor third (-3) = perfect fifth (+1)

Alternatively, one can examine the idea of having distinct diatonic and chromatic scales, corresponding to the 7edo and 12edo tunings of the diatonic intervals. In this case, a major third is two steps of the diatonic scale and four steps of the chromatic scale (because thirds are 2\7 and intervals enharmonically equivalent to the major third are 4\12 - note that on XenReference, "enharmonically equivalent" tends to specifically mean equivalent in the 12-form), meanwhile a minor third is two steps of the diatonic scale and three steps of the chromatic scale. Adding them results in an interval that is four steps of the diatonic scale and seven steps of the chromatic scale - in other words, a perfect fifth.

The mappings of the standard diatonic intervals are given below. Within a diatonic ordinal category, chromatic steps above the listed intervals represent increasing degrees of augmentation, and chromatic steps below the listed intervals represent increasing degrees of diminishment (so that the qualities go double-diminished - diminished - minor - major - augmented - double-augmented, or double-diminished - diminished - perfect - augmented - double-augmented), and beyond octave ordinals continue ninth, tenth, eleventh, etc.

Therefore, an interval mapped to 4 steps of the diatonic scale and 8 steps of the chromatic scale, although not listed, would be an augmented fifth, because it is one chromatic step above the perfect fifth and in the same ordinal category. (It would not be, in diatonic notation, a minor sixth, although it might be minor-sixth-sized; see Interval region to learn more).

Note that tritone is not its own interval category; a tritone is 6 chromatic steps but may be any number of diatonic steps (conventionally 3 (augmented fourth) or 4 (diminished fifth)).

Name	Abbreviation	Diatonic steps (\7)	Chromatic steps (\12)
Perfect unison	P1	0	0
Minor second	m2	1	1
Major second	M2	1	2
Minor third	m3	2	3
Major third	M3	2	4
Perfect fourth	P4	3	5
Perfect fifth	P5	4	7
Minor sixth	m6	5	8
Major sixth	M6	5	9
Minor seventh	m7	6	10
Major seventh	M7	6	11
Perfect octave	P8	7	12

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". Interval qualities go ..., sesquidiminished, diminished, semidiminished, perfect (or minor, neutral, major), semiaugmented, augmented, sesquiaugmented, ...

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 the Xenharmonic Reference 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, called harmonic notation. Other than that, however, the system remains fully coherent. A quick fix to make the antidiatonic notation intuitive, called melodic notation, 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).