In music theory
, an interval
is the difference 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 distance between its two pitch classes, the shortest distance between its two pitches transposed to be close as possible.
Intervals may be labelled according their pitch ratios, as is commonly used in just intonation. Intervals may also be labelled according to their harmonic functions, 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 centss, 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, simple and compound, and as steps or 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. Finally, intervals may be labelled with or modified by the addition of perfect, major, minor, augmented, and diminished before the number of notes apart (for instance, augmented fourth). Perfect intervals are never major or minor and major and minor intervals are never perfect. Major and minor intervals are one semitone above, or below, their minor and major counterparts, respectively (see minor second below). Augmented and diminished intervals are raised or lowered a step and any interval may be augmented or diminished and may even be double augmented or diminished.
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 other intervals and ratios, some of which follow.
Common simple intervals
- Unison: The ratio of 1:1 is a unison, two notes playing 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, two notes, one of which is double or half the pitch of the other. 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 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 is is 7 and is 700 cents in equal temperament, which is a ratio 2 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, 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, 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.
- 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 minor seventh are the most dissonant intervals with the possible exception of the tritone, below. The minor second is also know 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 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 devils interval." It plays an important role in the dominant seventh chord.
Non-equal temperament intervals
There are also a number of intervals not found in the chromatic scale which have names of their own. 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 centss
- 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
- 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 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.
- Additionally, some cultures around the world have their own names for intervals found in their music. See: sargam, Bali
For the mathematical use of the word "interval", see interval (mathematics)