Whole tone

A whole tone is an interval that is the size of a major second that functions as the large step of a diatonic scale.

Whole tones range roughly from 160 to 240 cents, and are preferably closer to 200 cents. Whole tones may be defined to be between ~165 and ~250 cents (1\3edP4 and 1\2edP4), based on the definition of a diatonic tetrachord.

Sometimes, the term whole tone refers specifically to the interval 9/8 (the Pythagorean major second), or to the intervals 9/8 and 10/9 together (the two sizes of whole tone in the Zarlino diatonic scale). These two sizes of whole tone are separated by a syntonic comma, which is tempered out in meantone temperament. Therefore, meantone has a single whole tone in between 9/8 and 10/9 in size, hence the name. In 12edo, the whole tone divides the octave into six, and as a result whole-tone-generated scales are generally 6-form - thus, from a scale structure perspective, the whole tones of 22edo (218c) and 26edo (185c) may be used as upper and lower bounds.

Name

Whole tone comes from diatonic scale theory, which comprises the diatonic scale of five whole tones and two semitones; the step ratio of diatonic is either exactly or roughly 2:1 in most historical tunings, thus a semitone is roughly half of a whole tone. This is not to be confused with the sense of tone to refer to a note.

Qualities

While the whole tone is itself a quality of second (the major second), the various commatic inflections are considered by some theorists to represent subqualities of whole tone; 9/8 is the "major tone" and 10/9 the "minor tone".

As generators of temperaments

The most prominent whole-tone-generated temperament is Didacus, which is a no-threes subgroup temperament generated by a whole tone of about 194 cents that finds 5/4 at 2 steps and 7/4 at 5 steps; this is the most obvious temperament interpretation of the 12edo whole-tone scale. Didacus is a 6 cluster temperament.

Another notable whole tone temperament is Machine, where a ~215c whole tone, interpreted as a very flat 8/7 or a sharp 9/8, stacks three times to 16/11. A more accurate but more complex interpretation is Contraption, which instead finds 7/4 at 10 steps as a 2.7.11 subgroup temperament. 11edo supports both temperaments, and in fact Contraption is an 11-form temperament. In either case, the generator may be interpreted as 17/15 and 6 steps (octave-reduced) as a flat 17/16, so that they extend to 2.15.7.11.17.

Additional temperaments associated with whole tone generators include Tetracot (which stacks a flat 10/9 four times to reach 3/2), and Slendric (which stacks a just or slightly sharpened 8/7 three times to reach 3/2). The edges of the tetrachordal definition mentioned above represent Porcupine (equalizing 9:10:11:12) and Semaphore (equalizing 6:7:8).

In just intonation

TODO: complete section

3-limit

Overwhelmingly, "whole tone" in just intonation refers to 9/8. However, as a subcategory of second (specifically, major seconds), a couple more intervals meet the criteria.

5-limit

The minor tone is 10/9. 10/9 and 9/8 stack to 5/4, and as previously mentioned in meantone temperament they are equated. An equal temperament that is distinctly consistent to the 9-odd-limit thus cannot support meantone.

7-limit

28/25

28/25 sits between 9/8 and 10/9 in size, and is the generator for the didacus temperament along with being the approximate size of a meantone-tempered whole tone.

8/7

8/7 is a very large major second, generally out of the bounds of whole tones but approximated as such in flat tunings of archy (7edo, 12edo, 17edo, and 22edo). It is the octave complement of the harmonic seventh, which is also the seventh harmonic (octave-reduced). It may be considered the "major tone".

11-limit

11/10

11/10, conversely, is on the flat side, but is equated with 10/9 in temperaments like porcupine and tetracot (widening the syntonic comma) and further with 9/8 (tempering it out entirely) in flattone.

25/22

25/22 is an 11-limit major second that functions as a sharp neogothic major second. It is the interval separating 11/10 and 5/4.

Higher limit

17/15

17/15 however is perhaps the more proper neogothic major second, being the mediant of 8/7 and 9/8 much as 14/11 is the mediant of 9/7 and 5/4.