5L 2s/Interval categories

< 5L 2s

The diatonic interval categories are used in most discussions of musical intervals, such as in diatonic music theory, the naming of just intervals, and the division of interval space into regions. This page provides an overview of diatonic intervals.

For the sake of space, infoboxes have only been provided for unisons, seconds, thirds, and fourths. The information on the octave complements (namely, octaves, sevenths, sixths, and fifths) may be generated by:

• Reversing the generator span, for example +7 generators to -7 generators.
• Taking 1200 minus the basic tuning to generate the complement's basic tuning.

Unisons and octaves

Perfect 0-mosstep

MOS	5L 2s
Other names	Perfect unison
Generator span	0 generators
Basic tuning	0c

Perfect unison and perfect octave

In nearly all systems, the perfect unison is represented by the JI ratio 1/1 and represents no change in pitch. It is a perfect 0-mosstep and is the interval from C to C in the same octave.

The perfect octave, its octave complement, is the most consonant musical interval, which is represented by the JI ratio 2/1. It is a perfect 7-mosstep in diatonic (hence the name octave when interval steps are counted inclusively) and is the interval from C to C an octave up.

Chromatic semitone and diminished octave

The chromatic semitone is the difference between a major and minor, perfect and augmented, or perfect and diminished interval in diatonic systems.

Augmented 0-mosstep

MOS	5L 2s
Other names	Chromatic semitone, augmented unison
Generator span	+7 generators
Basic tuning	100c

• In 12edo, it is represented by the interval of 100 cents.
• In 7-limit just intonation, the difference between supermajor and subminor intervals is 54/49, an interval of about 168 cents, though this is not usually used as a chromatic semitone.
• In 5-limit just intonation, it is represented by the interval 25/24, which is about 71 cents.
• In 3-limit just intonation, it is represented by the interval 2187/2048, which is about 114 cents.
  • In generalized diatonic systems, such as in meantone, the chromatic semitone is generated the same way as 2187/2048, by stacking 7 fifths.

The diminished octave is the octave complement of the chromatic semitone, and is the difference between a major interval and a minor interval an octave up.

The diminished octave and chromatic semitone are considered dissonances in conventional Western harmony, though this may be loosened by using a particularly wide tuning of the diatonic thirds, such as in extraclassical tonality

Seconds and sevenths

Diatonic semitone and major seventh

Minor 1-mosstep

MOS	5L 2s
Other names	Diatonic semitone, minor second
Generator span	-5 generators
Basic tuning	100c

The diatonic semitone, or minor second, is the difference between, for example, a major third and a perfect fourth. Two minor seconds occur in the C major diatonic scale, between E and F and between B and C.

• In 12edo, it is represented by the interval of 100 cents, and is enharmonic with the chromatic semitone (hence the name "semitone" for both intervals).
• In 7-limit just intonation, the difference between a supermajor second and a subminor third is 49/48, about 36 cents, and the difference between a supermajor third and a perfect fourth is 28/27, about 63 cents, though only the latter of these is commonly used as a minor second.
• In 5-limit just intonation, the minor second is represented by the interval 16/15, which is the difference between 5/4 and 4/3. In tunings such as 41 and 53edo, this is equated with the Pythagorean chromatic semitone.
• In 3-limit just intonation, the diatonic semitone is known as the limma of 256/243, which separates 81/64 and 4/3 and is an interval of about 90 cents.
  • In generalized diatonic systems, the diatonic semitone is constructed the same way, by stacking -5 fifths.

The major seventh is the octave complement of the minor second, and appears as the fourth note in maj7 chords in Western harmony.

The minor second and major seventh are conventionally considered dissonances.

Whole tone and minor seventh

Major 1-mosstep

MOS	5L 2s
Other names	Whole tone, major second
Generator span	+2 generators
Basic tuning	200c

The whole tone, or major second, is the larger variety of second (1-mosstep) found in the diatonic scale. 5 major seconds occur in the C major diatonic scale, between C, D, and E, and between F, G, A, and B.

• In 12edo, it is represented by the interval of 200 cents.
• In 7-limit just intonation, the supermajor second of 8/7 is found at about 231 cents.
• In 5-limit just intonation, two kinds of major seconds appear: the greater tone 9/8 of about 204 cents, and the lesser tone 10/9 of about 182 cents. This is because both kinds of major seconds are necessary to comprise a major third.
• In 3-limit just intonation, only the 9/8 major second exists.
  • The major second is generated the same way in diatonic systems, by stacking 2 fifths.

The minor seventh is the octave complement of the major second, and appears as the fourth note in dominant 7th chords in Western harmony. Additionally, the harmonic seventh is the interval 7/4, which appears in the harmonic series alongside the ratios of the major chord, and can be played alongside them to form a JI chord of 4:5:6:7.

The major second and minor seventh are conventionally considered dissonances, however they are somewhat on the line between dissonance and imperfect consonance, and can be used as a consonance in context.