Diatonic notation
Diatonic notation or chain-of-fifths notation is the generalization of the standard notation system for 12edo in Western music that aligns with its historical usage to notate meantone or Pythagorean temperaments with greater than 12 notes. 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. (As a shortcut in mental arithmetic, you may use only the difference between the fifth and 12edo's fifth - 6*4 = 24, therefore a 706c fifth generates a 424c major third; similarly, 4*4=16, therefore a 696c fifth (700-4) generates a 384c major third (400-16).)
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
Diamond-MOS features
These features are listed as part of Diamond-MOS notation on the Xenharmonic Wiki but are best categorized as microtonal additions to standard diatonic notation.
Transpositions may be written explicitly, i.e. "1\11 higher than written", at the top of the score or at a key change. This avoids having to write in difficult-to-notate microtonal keys.
The EDO which a score is played in may be written explicitly, i.e. "53edo". If an edo has multiple diatonic scales, or if an arbitrary diatonic scale is warranted, it may be specified by using the generator tuning, e.g. "54edo, diatonic fifth = 31\54", or "Non-edo mos tuning, diatonic fifth = 696.2144 cents".
Key signatures may feature non-MOS alterations, placed after a vertical dotted line labelled "ALT" below it.
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
Main article: Ups and downs notation
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).
