Diatonic
Diatonic is a kind of scale characterized by the division of the octave into 5 large steps (L) and 2 small steps (s) in an LLsLLLs pattern, or any rotation thereof. The most widespread diatonic is the MOS form, 5L 2s, where the large steps are equally sized and the small steps are also equally sized. When diatonic is used as an adjective on the wiki to describe something other than a scale, the MOS is what it usually refers to (for example, the diatonic major third is the major third in the MOS diatonic scale). However, other similar scales may be called diatonic as well.
MOS diatonic
The MOS diatonic scale is 5L 2s, tuned most simply in 12edo. It is the basic scale for a number of rank-2 temperaments, such as meantone, archy, and schismic. It can be formed by stacking seven perfect fifths and octave-reducing. MOS diatonic has seven modes, from classical music theory:
| Mode name | Pattern | 2nd | 3rd | 4th | 5th | 6th | 7th |
|---|---|---|---|---|---|---|---|
| Locrian | sLLsLLL | m | m | P | d | m | m |
| Phrygian | sLLLsLL | m | m | P | P | m | m |
| Aeolian (Minor) | LsLLsLL | M | m | P | P | m | m |
| Dorian | LsLLLsL | M | m | P | P | M | m |
| Mixolydian | LLsLLsL | M | M | P | P | M | m |
| Ionian (Major) | LLsLLLs | M | M | P | P | M | M |
| Lydian | LLLsLLs | M | M | A | P | M | M |
Intervals of MOS diatonic
Major second
| Interval information |
reduced,
reduced harmonic
The diatonic major second (M2), represented by the frequency ratio 9/8, is the larger of the two seconds (1-step intervals) in the MOS form of the diatonic scale. It is generated by stacking 2 fifths and octave-reducing. In Pythagorean tuning (and thus purely-tuned just intonation), it is approximately 203.9 cents in size, but as an interval in the abstract diatonic scale it may range from 171 to 240 cents, depending on the tuning.
It functions as the large step of diatonic, and along with the diatonic semitone (diatonic minor second) may be used to construct other diatonic intervals. For example, the diatonic major third is two major seconds stacked, and the diatonic minor third is a major second stacked with a minor second. The chromatic semitone is the difference between these two intervals.
As a harmonic interval, the diatonic major second is considered a dissonance in most contexts, due to its small size, but can in some contexts (such as arto and tendo theory) be considered a consonance or ambisonance. In 5edo, it is a consonant 8/7 interval much like the chromatic semitone.
The diatonic scale contains five major seconds. In the Ionian mode, major seconds are found on the 1st, 2nd, 4th, 5th, and 6th degrees of the scale; the other two degrees have minor seconds. The large number of major seconds compared to minor seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.
Major third
| Interval information |
reduced harmonic
The diatonic major third (M3), represented by the frequency ratio 81/64, is the larger of the two thirds (2-step intervals) in the MOS form of the diatonic scale. It is generated by stacking 4 fifths octave-reduced. In Pythagorean tuning (and thus purely-tuned just intonation), it is approximately 407.8 cents in size, but as an interval in the abstract diatonic scale it may range from 343 to 480 cents, depending on the tuning.
It can be constructed by stacking two diatonic major seconds, and as such may be called the ditone.
As a harmonic interval, the diatonic major third may be considered either a consonance or a dissonance depending on its tuning. Important tuning targets for the diatonic major third are 5/4 (Meantone temperament), 14/11 (Pentacircle temperament), 9/7 (Archytas temperament) and 13/10 (Oceanfront temperament).
The diatonic scale contains three major thirds. In the Ionian mode, major thirds are found on the first, fourth, and fifth degrees of the scale; the other four degrees have minor thirds. This roughly equal distribution leads to diatonic tonality being largely based on the distinction between major and minor thirds and triads.
Greek diatonic scales
Diatonic may be defined more broadly according to the Greek diatonic genus. In Greek music, a diatonic scale is constructed from identical tetrachords spanning a perfect fourth each, and separated by a Pythagorean major second, such that
a) the largest interval in the tetrachord is at most half of its total span
b) the middle interval in the tetrachord is not the smallest.
As such, the scale pattern for a generalized diatonic (in its Phrygian mode) may be sABCsAB, where C is an approximation of 9/8 and A and B are both intervals that are larger than s but no larger than half the tetrachord's span.
An example of a Greek diatonic is Ptolemy's intense diatonic, where C and A are equated to 9/8, B is 10/9, and s is 16/15. This results in the scale pattern sLMLsLM, which may more abstractly be called 'zarlino' or 'nicetone', and has fourteen modes rather than seven due to its chirality (for example, mosdiatonic Ionian LLsLLLs becomes two modes: LH-Ionian (MLsLMLs) and RH-Ionian (LMsLMLs).)
Another example of a Greek diatonic is the equable diatonic, which is a tuning where C is 9/8, B is 10/9, A is 11/10, and s is 12/11. This has the property of being easily expressible as a utonal sequence 1/(18:20:22:24:27:30:33:36).
The simplest rank-3 diatonic tunings are those found in 14edo (3-2-1-2-3-2-1) and 15edo (3-2-1-3-2-3-1). Note that in both cases, the largest interval in the tetrachord is exactly half of its width. This is similar to the edge-case found with the tuning of MOS diatonic in 5edo or 7edo.
Straddle-fifth diatonic scales
A straddle-fifth system has a sharp and a flat fifth, which combine to create a 9/8 wholetone. So, in those systems, a diatonic scale analogous to the MOS diatonic may be generated by alternating the two fifths. This creates an LLmLLLs pattern, notably meaning there are multiple varieties of small step, rather than multiple varieties of large step. Again, there are fourteen modes, because, for example, there are again two Ionian modes: LLsLLLm and LLmLLLs. Note that the perfect fourth is always based on one variety of fifth, and the perfect fifth is always the other.
