5L 2s/Interval categories
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
Unisons and octaves are the distances between one note and a note with the same letter name. For example, you can take an octave from D to D an octave up, or a chromatic semitone from D to D♯.
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.
| MOS | 5L 2s |
| Other names | Chromatic semitone, augmented unison |
| Generator span | +7 generators |
| Basic tuning | 100 ¢ |
- In 12edo, it is represented by the interval of 100 ¢.
- In 7-limit just intonation, the difference between supermajor and subminor intervals is 54/49, an interval of about 168 ¢, 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 ¢.
- In 3-limit just intonation, it is represented by the interval 2187/2048, which is about 114 ¢.
- 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
Seconds are the steps of the diatonic scale, falling between one note and the next in the scale, for example D and E, or E and F. Larger intervals are considered skips, as they skip scale degrees.
Diatonic semitone and major seventh
| MOS | 5L 2s |
| Other names | Diatonic semitone, minor second |
| Generator span | −5 generators |
| Basic tuning | 100 ¢ |
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 ¢, and the difference between a supermajor third and a perfect fourth is 28/27, about 63 ¢, 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 ¢.
- 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
| MOS | 5L 2s |
| Other names | Whole tone, major second |
| Generator span | +2 generators |
| Basic tuning | 200 ¢ |
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 ¢.
- In 7-limit just intonation, the supermajor second of 8/7 is found at about 231 ¢.
- In 5-limit just intonation, two kinds of major seconds appear: the greater tone 9/8 of about 204 ¢, and the lesser tone 10/9 of about 182 ¢. 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, the octave complement of the supermajor second 8/7, 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 can be used to construct a sus2 chord, which has the property of being composed of consecutive generator steps.
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.
Thirds and sixths
Thirds are the backbone of diatonic harmony, providing scales, chords, and other harmonic elements a distinct "major" and "minor" flavor. Both varieties of third are considered imperfect consonances. Since they are both a good distance away from the unison fifth-wise, changing the tuning of the fifth can drastically alter the thirds, and thus, the flavor of the scale.
Minor third and major sixth
The minor third is the smaller variety of third, used to generate minor chords, scales, and keys. 4 minor thirds occur in the C major diatonic scale, on D, E, A, and B.
- In 12edo, it is represented by the interval of 300 ¢.
- In 13-limit just intonation, the inframinor third of 15/13 is found at about 247 ¢, and leads to the inframinor or arto chord 26:30:39.
- In 7-limit just intonation, the subminor third of 7/6 is found at about 267 ¢, and leads to the subminor or zo chord 6:7:9.
- In 5-limit just intonation, the minor third of 6/5 is found at about 316 ¢, and leads to the classical minor or gu chord 10:12:15.
- In 3-limit just intonation, the minor third of 32/27 is found at about 294 ¢, and leads to the Pythagorean minor or wa chord 54:64:81.
- In generalized diatonic systems, the minor third is generated the same way, leading to an interval between 240 and 343 ¢, by stacking −3 fifths.
The minor third is found in the diminished and minor chords in Western harmony.
The major sixth is its octave complement.
Major third and minor sixth
The major third is the larger variety of third, used to generate major chords, scales, and keys. 3 major thirds occur in the C major diatonic scale, on C, F, and G.
- In 12edo, it is represented by the interval of 400 ¢.
- In 13-limit just intonation, the ultramajor third of 13/10 is found at about 454 ¢, and leads to the ultramajor or tendo chord 10:13:15.
- In 7-limit just intonation, the supermajor third of 9/7 is found at about 436 ¢, and leads to the supermajor or ru chord 14:18:21.
- In 5-limit just intonation, the major third of 5/4 is found at about 386 ¢, and leads to the classical major or yo chord 4:5:6, which is also a harmonic series fragment that can be extended to 4:5:6:7 to produce the harmonic seventh chord.
- In 3-limit just intonation, the major third of 81/64 is found at about 408 ¢, and leads to the Pythagorean major or lawa chord 64:81:96.
- In generalized diatonic systems, the major third is generated the same way, leading to an interval between 343 and 480 ¢, by stacking 4 fifths.
The major third is found in the augmented and major chords in Western harmony.
The minor sixth is its octave complement.
Fourths and fifths
Fourths and fifths are the generators of the diatonic scale, and the simplest possible intervals other than the unison and octave.
Perfect fourth and perfect fifth
| MOS | 5L 2s |
| Other names | Perfect fourth |
| Generator span | −1 generators |
| Basic tuning | 500 ¢ |
The perfect fourth (and its octave complement, the perfect fifth) are the most basic diatonic intervals. A perfect fifth appears in most diatonic chords.
- In 12edo, the perfect fourth is represented by the interval of 500 ¢.
- In just intonation, the perfect fourth is usually represented, regardless of limit, by the interval 4/3 of about 498 ¢.
- In generalized diatonic systems, the perfect fourth is the dark generator, meaning it is generated by stacking −1 fifths, and can be used as a generator itself, as opposed to fifths.
The perfect fourth can be used to construct a sus4 chord, which has the same properties as the sus2 chord, being composed of consecutive generator steps. The perfect fourth and perfect fifth are considered perfect consonances.
Augmented fourth and diminished fifth
| MOS | 5L 2s |
| Other names | Augmented fourth |
| Generator span | 6 generators |
| Basic tuning | 600 ¢ |
The augmented fourth and diminished fifth, in contrast to the perfect counterparts, are harsh dissonances. They are collectively considered "tritones", as they are the size of approximately 3 whole tones. Only one of each occurs in the diatonic scale, an augmented fourth between F and B and a diminished fifth between B and F.
- In 12edo, they are enharmonic, represented by the interval of 600 ¢.
- In just intonation, an augmented fourth can be represented by the perfect fourth stacked with any chromatic semitone.
- In the 7-limit, this becomes 72/49, which is closer to a fifth than a fourth, at 667 ¢.
- A 7-limit interval that is closer to the expected size for a tritone is 7/5, at 583 ¢.
- In the 5-limit, this is 25/18, the classical augmented fourth of about 569 ¢.
- In the 3-limit, this is 729/512, the Pythagorean augmented fourth of about 612 ¢.
- In the 7-limit, this becomes 72/49, which is closer to a fifth than a fourth, at 667 ¢.
- In generalized diatonic systems, the augmented fourth is generated by stacking 6 fifths.
The diminished fifth appears as the fifth of a diminished chord.