In music theory, an interval is the difference (a ratio or logarithmic measure) in pitch between two notes and often refers to those two notes themselves (otherwise known as a dyad). An interval class is measured by the shortest distance possible between its two pitch classes.
Intervals may be labelled according their pitch ratios, as is commonly used in just intonation. Intervals may also be labelled according to their diatonic functionality, as is commonly done for tonal music, and according to the number of notes they span in a diatonic scale. The interval of a note from its tonic is its scale degree, thus the fifth degree of a scale is a fifth from its tonic. For atonal music, such as that written using the twelve tone technique or serialism, integer notation is often used, such as in musical set theory. Finally, it is also possible to label intervals using the logarithmic measure of cents, as is used to compare other intervals with those of twelve tone equal temperament.
Intervals may also be described as narrow and wide or small and large, consonant and dissonant or stable and unstable, weak and strong, simple and compound, vertical (or harmonic) and linear (or melodic), and, if linear as conjunct/steps or disjunct/skips. Simple intervals are those which lie within an octave and compound are those which are larger than a single octave. Thus a tenth is known as a compound third. Linear intervals are successive pitches while vertical intervals are simultaneous. Steps are linear intervals between consecutive scale degrees while skips are not.
Finally, the specification of any interval may be further refined, using the terms perfect, major, minor, augmented, and diminished. For example, in traditional Western tonal theory, a fourth is a fourth in virtue of the letter names of the notes involved (any sort of a C below and and any sort of an F above, say: C, D, E, F, so there are four letter-named pitches from C to F). But it may be further distinguished as a perfect fourth (C natural and F natural, or C sharp and F sharp, etc.), an augmented fourth (C natural and F sharp, C sharp and F double sharp, etc.), or a diminished fourth (C sharp and F natural, or C natural and F flat, etc.). Some intervals (unisons, octaves, fourths, and fifths) may be qualified only in the ways just illustrated; the others (seconds, sevenths, thirds, and sixths) may be qualified only as major, minor, diminished, or augmented. A major interval is always one semitone wider than the corresponding minor interval; an augmented interval is always one semitone wider than the corresponding perfect or major interval; and a diminished interval is always one semitone narrower than the corresponding perfect or minor interval. Note that there can also be double augmentation or double diminution of intervals (C flat and F sharp, for example, make a double-augmented fourth).
It is important to note that while intervals named by their harmonic functions, for instance, a major second, may be described by a ratio, cent, or integer, not every interval described by these more general terms may be described with the harmonic function name. For instance, all major seconds (in twelve tone equal temperament) are 200 cents, but not every interval of 200 cents is a major second. See: enharmonic.
Simple diatonic intervals
Below are listed the most commonly used harmonic function, ratio, integer, cents, and relative consonance or dissonance of common diatonic simple intervals. There are many other intervals and ratios, some of which follow.
Common simple intervals
- Unison: The ratio of 1:1 is a unison (specifically, a perfect unison; there are other variants), two notes at the same pitch. In integer notation it is a 0 and is also zero cents. It is the simplest and most consonant of intervals.
- Octave: The ratio of 2:1 is an octave (specifically, a perfect octave; there are other variants), two notes, the higher of which is double the pitch of the lower. (Such pitch ratios, we should note, depend ultimately on the physical vibrational frequency of the pitches involved.) It is 1200 cents and in integer notation it is a 0, like the unison. Octave equivalency describes the perception that octaves are the same note, that the same notes repeat throughout the pitch range. Thus C and C', C5 and C3, and C and any C any number of octaves above or below, are all the same note or pitch class. Thus the octave is slightly less then or just as consonant as the unison.
- Perfect fifth & perfect fourth: The ratio of 3:2 is a perfect fifth, two pitches, one note 1.5 times the pitch of another. In integer notation it is 7 and is 700 cents in equal temperament, which is a ratio two (1.955) cents flat of 3:2. The inverse of a perfect fifth is a perfect fourth. A perfect fourth is the ratio 4:3, 5 in integer notation, and 500 cents in equal temperament, which is two cents sharp of 4:3. The unison, octave, fifth, and fourth are considered "perfect intervals" and thus the most consonant, in that order.
- Major third & minor sixth: The ratio of 5:4 is a major third. In integer notation it is 4 and is 400 cents, which is 13.686 cents sharp of 5:4. Its inverse is a minor sixth, 8:5, which is 8 in integer notation and 800 cents in equal temperament, 13.686 cents flat of 8:5. The thirds and sixths are considered the most dynamic and interesting of the consonant intervals, and are thus the least consonant, in the following order: major third, major sixth, minor third, minor sixth.
- Minor third & major sixth: The ratio of 6:5 is a minor third. In integer notation it is 3, and is 300 cents in equal temperament, which is 15.641 cents flat of 6:5. Its inverse is a major sixth, 5:3, which is 9 in integer notation and 900 cents in equal temperament, which is 15.641 cents sharp of 5:3.
- Major second & minor seventh: The ratio of 9:8 is a major second. In integer notation it is 2 and is 200 cents, which is 3.91 cents flat of 9:8. Its inverse is a minor seventh, 16:9, which is 10 in integer notation and is 1,000 cents or 3.91 cents sharp of 16:9. It is the first dissonant interval and is commonly used between chord tones such as in the dominant seventh chord, which features the minor seventh between the fifth and second degress of a major scale. A non-equal tempered minor seventh is also one of the blue notes used in the blues and jazz. The major second is also know as a whole tone or whole step.
- Minor second and major seventh: Like the above intervals, many ratios are used for the minor second, but 16:15 is the most common. In integer notation it is 1 and is 100 cents, which is 11.731 cents flat of 16:15, but fairly close to 18:17. Its inverse is the major seventh, commonly 15:8, which is 11 in integer notation and 1,100 cents. The minor second and major seventh are the most dissonant intervals with the possible exception of the tritone, below. The minor second is also known as a semitone, half tone, or half step.
Augmented and diminished intervals
Along with all seconds and sevenths, all augmented and diminished intervals are considered dissonant. However, in twelve tone equal temperament, most intervals, when augmented or diminished, are enharmonically equivalent to another interval. For example, a diminished minor second is a unison and thus only the fourth and fifth are commonly altered.
- Tritone: The tritone, which may be a diminished fifth or augmented fourth, is 6 in integer notation and 600 cents. It can be approximated by the ratio 17:12, whose inverse is 24:17 and is 6 cents flatter than 17:12. (Ideally, the tritone should equal its own inverse.) It is called "tritone" because it spans three whole steps. It exactly, symmetrically, divides the octave in half and was considered the most dissonant interval, literally "the devil's interval" (diabolus in musica). It plays an important role in the dominant seventh chord.
As may be gathered from the above discussion, any interval may be subjected to inversion, by interchanging the positions of the upper and the lower pitches (though is less usual to speak of inverting unisons or octaves). For example, the fourth between a lower C and a higher F may be inverted to make a fifth, with a lower F and a higher C. Determining the exact nature of the inversion of any interval is easy. Here are two rules, applying to all simple (i.e., non-compound) intervals, and by straightforward extensions to compound intervals also:
1. The number of any interval and the number of its inversion always add up to nine (four + five = nine, in the example just given).
2. The inversion of a major interval is a minor interval (and vice versa); the inversion of a perfect interval is also perfect; the inversion of an augmented interval is a diminished interval (and vice versa); and the inversion of a double augmented interval is a double diminished interval (and vice versa).
A full example: E flat below and C natural above make a minor sixth. By the two rules just given, C natural below and E flat must make a major third.
Intervals are often abbreviated with a P for perfect, m for minor, M for major, d for diminished, A for augmented, followed by the diatonic interval number. The octave is P8, and a unison is usually referred to simply as "a unison" but can be labeled P1. The tritone, an augmented fourth or diminished fifth is often π or TT.
Thus a minor second is m2, a perfect fifth is P5, an diminished 3rd is d3, and an augmented fourth is A4.
The intervals in the chromatic scale are (in ascending melodic order): P1, m2, M2, m3, M3, P4, π, P5, m6, M6, m7, M7, P8.
Generations of intervals
The intervals can be divided into five "generations", which correspond to negative powers of two:
Zeroth generation (1+2−0): P1, P8.
First generation (1+2−1): P4, P5.
Second generation (1+2−2): M3, m3, M6, m6.
Third generation (1+2−3): M2, m7.
Fourth generation (1+2−4): m2, M7, π.
Each successive generation is more dissonant than the previous one.
Here is the derivation of each generation from the previous one: Start with the octave's ratio, 2:1. Multiply each of its two numbers by two, yielding 4:2. Then stick the missing number in the middle, which gives 4:3:2. This breaks up into a pair of ratios — 4:3 and 3:2. The minor one is 4:3 and the major one is 3:2. These are the perfect fourth and the perfect fifth, respectively, and they are the first generation.
Now take the perfect fifth's ratio, 3:2. Multiply each of its two numbers by two to obtain 6:4. Then stick the missing number in the middle, yielding 6:5:4. This breaks up into a pair of ratios — 6:5 and 5:4. The minor one is 6:5 and the major one is 5:4. These are the minor third and major third, respectively. Their inversions are 5:3 and 8:5, which are the major sixth and the minor sixth, respectively. So these are the second generation: M3, m3, M6, m6.
Now take the major third's ratio, 5:4. Multiply each of its two numbers by two, which yields 10:8. Then stick the missing number in the middle, giving 10:9:8. This breaks up into a pair of ratios — 10:9 and 9:8. The minor one is 10:9 and the major one is 9:8. Both of these are whole-tones, i.e. major seconds. The inversion of 9:8 is 16:9, a minor seventh. So these are the third generation: M2, m7.
Now take the whole-tone's ratio, 9:8. Multiply each of its two numbers by two, giving 18:16. Then stick the missing number in the middle to obtain 18:17:16. This breaks up into a pair of ratios — 18:17 and 17:16. The minor one is 18:17 and the major one is 17:16. Both of these are semitones, i.e. minor seconds. The inversion of 18:17 is 17:9, a major seventh. The last interval is the tritone. The tritone is ideally equal to the square root of two, which is irrational, but can be approximated by adding a semitone to a perfect fourth:
which is the inversion of 17:12. 24:17 has the same denominator as 18:17, and 17:12 has the same numerator as 17:16. So these are the intervals of the fourth generation: m2, M7, π.
Ordered and unordered pitch and pitch class intervals
In atonal or musical set theory there are numerous types of intervals, the first being ordered pitch interval, the distance between two pitches upward or downward. For instance, the interval from C to G upward is 7, but the interval from G to C downward is −7. Using integer notation and modulo 12, ordered pitch interval, ip, may be defined, for any two pitches x and y, as:
the other way.
One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, similar to the interval of tonal theory. This may be defined as:
The interval between pitch classes may be measured with ordered and unordered pitch class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. Thus the ordered pitch class interval, i<x, y>, may be defined as:
mod 12, of course.
For unordered pitch class interval see interval class.
Interval cycles, "unfold a single recurrent interval in a series that closes with a return to the initial pitch class", and are notated by George Perle using the letter "C", for cycle, with an interval class integer to distinguish the interval. Thus the diminished seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0-11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the harmonic organization of post-diatonic music and can easily be identified by naming the cycle." (Perle, 1990)
- Perle, George (1990). The Listening Composer, p. 21. California: University of California Press. ISBN 0520069919.
Interval strength and root
David Cope suggests the concept of interval strength, in which an interval's strength is determined by its approximation to a lower, stronger, or higher, weaker, position in the harmonic series. See also: Lipps-Meyer law.
Hindemith and David Cope both suggest the concept of interval roots. To determine an interval's root, one locates its nearest approximation in the harmonic series. The root of a perfect fourth, then, is its top note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered intervals are the roots, as are the tops of all even numbered intervals. The root of a collection of intervals or a chord is thus determined by the interval root of its strongest interval.
As to its usefulness, Cope provides the example of the final tonic chord of some popular music being traditionally analyzable as a "submediant six-five chord" (added sixth chords by popular terminology), or a first inversion seventh chord (possibly the dominant of the mediant V/iii). According the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C-G), is the bottom C, the tonic.
- Cope, David (1997). Techniques of the Contemporary Composer, p.40-41. New York, New York: Schirmer Books. ISBN 0028647378.
There are also a number of intervals not found in the chromatic scale or labeled with a diatonic function which have names of their own. Many of these intervals describe small discrepancies between notes tuned according to the tuning systems used. Most of the following intervals may be described as microtones.
- A Pythagorean comma is the difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequency ratio 531441:524288, and is equal to 23.46 cents
- A syntonic comma is the difference between four justly tuned perfect fifths and two octaves plus a major third. It is expressed by the ratio 81:80, and is equal to 21.51 cents
- A Septimal comma is 64/63, and is the difference between the Pythagorean or 3-limit "7th" and the "harmonic 7th".
- Diesis is generally used to mean the difference between three justly tuned major thirds and one octave. It is expressed by the ratio 128:125, and is equal to 41.06 cents. However, it has been used to mean other small intervals: see diesis for details
- A schisma (also skhisma) is the difference between five octaves and eight justly tuned fifths plus one justly tuned major third. It is expressed by the ratio 32805:32768, and is equal to 1.95 cents. It is also the difference between the Pythagorean and syntonic commas.
- A schismic major third is a schisma different than a just major third, eight fifths down and five octaves up, Fb in C.
- A quarter tone is half the width of a semitone, which is half the width of a whole tone.
- A kleisma is six major thirds up, five fifths down and one octave up, or, more commonly, 225:224.
- A limma is the ratio 256:243, which is the semitone in Pythagorean tuning.
- A ditone is the pythagorean ratio 81:64, two 9:8 tones.
- Additionally, some cultures around the world have their own names for intervals found in their music. See: sargam, Bali
- See also specific and generic interval.
For the mathematical use of the word "interval", see interval (mathematics).