In
music theory, the term
interval describes the relationship between the
pitches of two
notes.
Intervals may be described as:
 vertical (or harmonic) if the two notes
sound simultaneously
 linear (or melodic), if the notes sound
successively.
Interval class is a system
of labelling intervals when the order of the notes is left
unspecified, therefore describing an interval in terms of the
shortest distance possible between its two
pitch classes.
Frequency ratios
Intervals may be labelled according to the
ratio of
frequencies of the
two pitches. Important intervals are those using the lowest
integers, such as 1:1 (
unison or prime), 2:1
(
octave), 3:2 (
perfect fifth), 4:3 (
perfect fourth), etc. This system is
frequently used to describe intervals in both Western and
nonWestern music. This method is also often used in
just intonation, and in theoretical
explanations of equaltempered intervals used in European tonal
music, to explain them through their approximation of
just intervals.
Interval number and quality
Number
Interval names
In Western harmonic theory, intervals are labeled according to the
number of
scale steps or
staff positions they encompass, as shown at
right.
Intervals larger than an
octave are called
compound intervals; for example, a tenth is known as a
compound third. The quality of the compound interval is determined
by the quality of the interval on which it is based. For example, a
perfect eleventh is the same as a compound perfect fourth.
Intervals larger than a thirteenth seldom need to be spoken of,
most often being referred to by their compound names, for example
"two octaves plus a fifth" rather than "a 19th".
The name or the label of an interval is determined by counting the
number of degrees between the two
notes
beginning with one for the lower note. The number of degrees
between F and B for example is 4, therefore the interval is a
fourth.
Quality
The name of any interval is further qualified using the terms
perfect,
major,
minor,
augmented, and
diminished. This is called its
interval
quality.
It is possible to have doublydiminished and doublyaugmented
intervals, but these are quite rare.
The name of an interval cannot, in general, be determined by
counting semitones alone. For example, there are four semitones
between
B and
E♭, however this
interval is a diminished fourth—an interval found between the
seventh and third degrees of the
harmonic minor scale—rather than a
major third. In
equaltempered
tuning, as on a piano, these intervals are indistinguishable by
sound, but the
diatonic function
of the notes incorporated might be very different.
Major/minor intervals are so called because rather than containing
one number of semitones they are at a lower, minor, or higher,
major, number. For example, the minor sixth is 8 semitones and the
major sixth is 9 semitones.
Perfect intervals are so called because they are neither minor nor
major (such as thirds, which are either minor or major) but
perfect. For example, the perfect fifth is 7 semitones and may not
be major or minor.
Augmented/diminished intervals are so called because rather than
containing one number of semitones they are at a lower, diminished,
or higher, augmented, number. For example, the diminished second is
0 semitones and the augmented second is 3 semitones, while the
augmented fourth and the diminished fifth are both 6
semitones.
Diatonic and chromatic intervals
A
diatonic interval is an interval formed by two notes of
a
diatonic scale. The table on the
right depicts all diatonic intervals for C major.
Shorthand notation
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 indication M and P are often
omitted. 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. Examples:
 m2: minor second
 M3: major third
 P5: perfect fifth
 m9: minor ninth.
For use in describing
chords, the sign
+ is used for augmented and
° for
diminished. Furthermore the 3 for the third is often omitted, and
for the seventh, the plain form stands for the minor interval,
while the major is indicated by
maj. So for
example:
 m: minor third (with perfect fifth)
 7: minor seventh (with major third and perfect fifth)
 °7: diminished seventh (with minor third and diminished
fifth)
 maj7: major seventh (with major third and perfect fifth)
 +5: augmented fifth (with major third)
 °5: diminished fifth (with minor third).
Enharmonic intervals
Two intervals are considered to be
enharmonic, or
enharmonically
equivalent, if they both contain the same
pitches spelled in different ways; that is, if
the notes in the two intervals are themselves enharmonically
equivalent. Enharmonic intervals span the same number of
semitones. For example, as shown in the matrix
below, F –A (a
major third), G –B (also
a major third), F –B (a
diminished
fourth), and G –A (a double
augmented second) are all enharmonically
equivalent — and they all span four semitones.
Steps and skips
Linear (melodic) intervals may be described as
steps or
skips in a diatonic context. Steps are linear intervals
between consecutive
scale degrees while
skips are not, although if one of the notes is
chromatically altered so that the resulting
interval is three
semitones or more (e.g. C
to D♯), that may also be considered a skip. However, the reverse is
not true: a
diminished third, an
interval comprising two
semitones, is still
considered a skip.
The words
conjunct and
disjunct refer to melodies
composed of steps and skips, respectively.
Pitchclass intervals
Posttonal or
atonal theory, originally
developed for equal tempered European classical music written using
the
twelve tone technique or
serialism,
integer notation is often used, most
prominently in
musical set
theory. In this system intervals are named according to the
number of half steps, from 0 to 11, the largest interval class
being 6.
Ordered and unordered pitch and pitchclass 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. One can also measure the distance between two pitches without
taking into account direction with the unordered pitch interval,
somewhat similar to the interval of tonal theory.
The interval between pitch classes may be measured with ordered and
unordered pitchclass 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. For unordered pitchclass interval see
interval class.
Generic and specific intervals
In
diatonic set theory,
specific and
generic intervals are distinguished.
Specific intervals are the interval class or number of semitones
between scale degrees or collection members, and generic intervals
are the number of scale steps between notes of a collection or
scale.
Cents
The standard system for comparing intervals of different sizes is
with
cents. This is a
logarithmic scale in which the octave is
divided into 1200 equal parts. In
equal temperament, each
semitone is exactly 100 cents.The value in cents
for the interval
f_{1} to
f_{2}
is
1200×log
_{2}(
f_{2}/
f_{1}).
Comparison of different interval naming systems
# semitones

Interval
class 
Generic
interval 
Common
diatonic name 
Comparable
just interval 
Comparison of interval width in cents 
equal
temperament

just
intonation

quartercomma
meantone

0 
0 
1 
perfect unison 
1:1 
0 
0 
0 
1 
1 
2 
minor second 
16:15 
100 
112 
117 
2 
2 
2 
major second 
9:8 
200 
204 
193 
3 
3 
3 
minor third 
6:5 
300 
316 
310 
4 
4 
3 
major third 
5:4 
400 
386 
386 
5 
5 
4 
perfect fourth 
4:3 
500 
498 
503 
6 
6 
4
5 
augmented fourth
diminished fifth 
45:32
64:45 
600 
590
610 
579
621

7 
5 
5 
perfect fifth 
3:2 
700 
702 
697
wolf fifth 737

8 
4 
6 
minor sixth 
8:5 
800 
814 
814 
9 
3 
6 
major sixth 
5:3 
900 
884 
889 
10 
2 
7 
minor seventh 
16:9 
1000 
996 
1007 
11 
1 
7 
major seventh 
15:8 
1100 
1088 
1083 
12 
0 
1
8 
perfect octave 
2:1 
1200 
1200 
1200 
It is possible to construct just intervals which are closer to the
equaltempered equivalents, but most of the ones listed above have
been used historically in equivalent contexts. In particular the
tritone (augmented fourth or diminished fifth), could have other
ratios; 17:12 (603 cents) is fairly common. The 7:4 interval (the
harmonic seventh) has been a
contentious issue throughout the history of music theory; it is 31
cents flatter than an equaltempered minor seventh. Some assert the
7:4 is one of the
blue notes used in
jazz.
In the diatonic system, every interval has one or more
enharmonic equivalents, such as
augmented second for
minor third.
Consonant and dissonant intervals
Consonance and dissonance
are relative terms referring to the stability, or state of repose,
of particular musical effects. Dissonant intervals would be those
which cause tension and desire to be
resolved to consonant
intervals.
These terms are relative to the usage of different compositional
styles.
 In atonal music all intervals (or
interval classes) are considered
equally consonant melodically and harmonically.
 In the Middle Ages, only the
octave and perfect fifth were considered consonant
harmonically.
 In 16thcentury usage, perfect
fifths and octaves, and major and minor thirds and sixths were
considered harmonically consonant, and all other intervals
dissonant. In the common practice
period, it makes more sense to speak of consonant and dissonant
chords, and certain intervals previously thought to be dissonant
(such as minor sevenths) became acceptable in certain contexts.
However, 16thcentury practice continued to be taught to beginning
musicians throughout this period.
 Hermann von Helmholtz
(1821–1894) defined a harmonically consonant interval as one in
which the two pitches have an overtone in
common (specifically excluding the seventh harmonic). This essentially defines all seconds and
sevenths as dissonant, while perfect fourths and fifths, and major
and minor thirds and sixths, are consonant.
 Pythagoras defined a hierarchy of
consonance based on how small the numbers were which express the
ratio. 20thcentury composer and theorist Paul Hindemith's system has a hierarchy with
the same results as Pythagoras's, but defined by fiat rather than
by interval ratios, to better accommodate equal temperament, all of
whose intervals (except the octave) would be dissonant using
acoustical methods.
 Lucy tuning (1990), uses a system of
ScaleCoding, whereby intervals which are closer on the spiral of
fourths and fifths are considered to be more consonant than those
which are separated by a greater number of steps of fourths and
fifths..
 David Cope (1997) suggests the
concept of interval strength, in
which an interval's strength, consonance, or stability is
determined by its approximation to a lower and stronger, or higher
and weaker, position in the harmonic series. See also: LippsMeyer law.
All of the above analyses refer to vertical (simultaneous)
intervals.
Inversion
An interval may be
inverted, by raising the lower pitch
an
octave, or lowering the upper pitch an
octave (though it 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.
Here are the ways to identify interval inversions:
 For diatonicallynamed intervals there are two
rules which apply to all simple (i.e., noncompound) intervals:
 The number of any interval and the number of its
inversion always add up to nine (four + five = nine, in the example
just given).
 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♭ below and C above make a major
sixth. By the two rules just given, C natural below and E flat
above must make a minor third.
 For intervals identified by ratio, the
inversion is determined by reversing the ratio and multiplying by
2. For example, the inversion of a 5:4 ratio is an 8:5 ratio.
 Intervals identified by integer can be simply
subtracted from 12. However, since an interval
class is the lower of the interval integer or its
inversion, interval classes cannot be inverted.
Interval roots
Although intervals are usually designated in relation to their
lower note,
David Cope and
Hindemith both suggest the concept of
interval root. 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 sixfive 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.
Interval cycles
Interval cycles, "unfold [i.e.
repeat] 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 intervalclass integer to distinguish the interval. Thus
the diminishedseventh 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.
Other intervals
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 3limit
"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 diaschisma is the
difference between three octaves and four justly tuned perfect
fifths plus two justly tuned major thirds. It is expressed by the
ratio 2048:2025, and is equal to about 19.5 cents.
 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 from a just major
third, eight fifths down and five octaves up, F♭ in C.
 A quarter tone is half the
width of a semitone, which is half the width of a whole tone. It is
equal to 50 cents.
 A kleisma is six major thirds
up, five fifths down and one octave up, or, sometimes, the septimal kleisma 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
List of Musical
Intervals for more.
See
Musical
interval mnemonics at Wikibooks for popular musical fragments
that feature common intervals
Generalizations and nonpitch uses
The term "interval" can also be generalized to other elements of
music besides pitch.
David Lewin's
Generalized Musical Intervals and Transformations uses
interval as a generic measure of distance in order to show musical
transformations which can change, for instance, one rhythm into
another, or one formal structure into another.
See also
Notes
 Aikin, Jim (2004). A Player's Guide to Chords and Harmony:
Music Theory for RealWorld Musicians, p.24. ISBN
0879307986.
 Cope, David
(1997). Techniques of the Contemporary Composer, p.40–41.
New York, New York: Schirmer Books. ISBN 0028647378.
 Kostka, Stephen; Payne, Dorothy (2008). Tonal Harmony,
p.21. First Edition, 1984.
 Hindemith, Paul (1934). The Craft of Musical
Composition. New York: Associated Music Publishers. Cited in
Cope (1997), p.4041.
 Perle,
George (1990). The Listening Composer, p.21.
California: University of California Press. ISBN
0520069919.
 Lewin, David
(1987). Generalized Musical Intervals and Transformations,
for example sections 3.3.1 and 5.4.2. New Haven: Yale University
Press. Reprinted Oxford University Press, 2007. ISBN
9780195317138
 Ockelford, Adam (2005). Repetition in Music: Theoretical
and Metatheoretical Perspectives, p.7. ISBN 0754635732.
External links