Interval (music)
In music theory, an interval is a difference in pitch between two sounds.^{[1]} An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.^{[2]}^{[3]}
In
In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio.
In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the
Size
The size of an interval (also known as its width or height) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.
Frequency ratios
The size of an interval between two notes may be measured by the
Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called
Cents
The standard system for comparing interval sizes is with cents. The cent is a logarithmic unit of measurement. If frequency is expressed in a logarithmic scale, and along that scale the distance between a given frequency and its double (also called octave) is divided into 1200 equal parts, each of these parts is one cent. In twelvetone equal temperament (12TET), a tuning system in which all semitones have the same size, the size of one semitone is exactly 100 cents. Hence, in 12TET the cent can be also defined as one hundredth of a semitone.
Mathematically, the size in cents of the interval from frequency f_{1} to frequency f_{2} is
Main intervals
The table shows the most widely used conventional names for the intervals between the notes of a
Intervals with different names may span the same number of semitones, and may even have the same width. For instance, the interval from D to F♯ is a major third, while that from D to G♭ is a diminished fourth. However, they both span 4 semitones. If the instrument is tuned so that the 12 notes of the chromatic scale are equally spaced (as in equal temperament), these intervals also have the same width. Namely, all semitones have a width of 100 cents, and all intervals spanning 4 semitones are 400 cents wide.
The names listed here cannot be determined by counting semitones alone. The rules to determine them are explained below. Other names, determined with different naming conventions, are listed in a separate section. Intervals smaller than one semitone (commas or microtones) and larger than one octave (compound intervals) are introduced below.
Number of semitones 
Minor, major, or perfect intervals 
Short  Augmented or diminished intervals 
Short  Widely used alternative names 
Short  Audio 

0  Perfect unison 
P1  Diminished second  d2  ^{ⓘ}  
1  Minor second 
m2  Augmented unison  A1  Semitone, half tone, half step  S  ^{ⓘ} 
2  Major second  M2  Diminished third  d3  Tone , whole tone, whole step 
T  ^{ⓘ} 
3  Minor third  m3  Augmented second  A2  ^{ⓘ}  
4  Major third  M3  Diminished fourth  d4  ^{ⓘ}  
5  Perfect fourth  P4  Augmented third  A3  ^{ⓘ}  
6  Diminished fifth  d5  Tritone  TT  ^{ⓘ}  
Augmented fourth  A4  
7  Perfect fifth  P5  Diminished sixth  d6  ^{ⓘ}  
8  Minor sixth  m6  Augmented fifth  A5  ^{ⓘ}  
9  Major sixth  M6  Diminished seventh  d7  ^{ⓘ}  
10  Minor seventh  m7  Augmented sixth  A6  ^{ⓘ}  
11  Major seventh  M7  Diminished octave  d8  ^{ⓘ}  
12  Perfect octave  P8  Augmented seventh  A7  ^{ⓘ} 
Interval number and quality
In Western music theory, an interval is named according to its number (also called diatonic number) and quality. For instance, major third (or M3) is an interval name, in which the term major (M) describes the quality of the interval, and third (3) indicates its number.
Number
The number of an interval is the number of letter names or
There is a
If one adds any accidentals to the notes that form an interval, by definition the notes do not change their staff positions. As a consequence, any interval has the same interval number as the corresponding natural interval, formed by the same notes without accidentals. For instance, the intervals C–G♯ (spanning 8 semitones) and C♯–G (spanning 6 semitones) are fifths, like the corresponding natural interval C–G (7 semitones).
Notice that interval numbers represent an inclusive count of encompassed staff positions or note names, not the difference between the endpoints. In other words, one starts counting the lower pitch as one, not zero. For that reason, the interval C–C, a perfect unison, is called a prime (meaning "1"), even though there is no difference between the endpoints. Continuing, the interval C–D is a second, but D is only one staff position, or diatonicscale degree, above C. Similarly, C–E is a third, but E is only two staff positions above C, and so on. As a consequence, joining two intervals always yields an interval number one less than their sum. For instance, the intervals C–E and E–G are thirds, but joined together they form a fifth (C–G), not a sixth. Similarly, a stack of three thirds, such as C–E, E–G, and G–B, is a seventh (C–B), not a ninth.
This scheme applies to intervals up to an octave (12 semitones). For larger intervals, see § Compound intervals below.
Quality
The name of any interval is further qualified using the terms perfect (P),
Perfect
Perfect intervals are socalled because they were traditionally considered perfectly consonant,^{[6]} although in Western classical music the perfect fourth was sometimes regarded as a less than perfect consonance, when its function was contrapuntal.^{[vague]} Conversely, minor, major, augmented or diminished intervals are typically considered less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or dissonances.^{[6]}
Within a diatonic scale^{[b]} all unisons (P1) and octaves (P8) are perfect. Most fourths and fifths are also perfect (P4 and P5), with five and seven semitones respectively. One occurrence of a fourth is augmented (A4) and one fifth is diminished (d5), both spanning six semitones. For instance, in a Cmajor scale, the A4 is between F and B, and the d5 is between B and F (see table).
By definition, the inversion of a perfect interval is also perfect. Since the inversion does not change the pitch class of the two notes, it hardly affects their level of consonance (matching of their harmonics). Conversely, other kinds of intervals have the opposite quality with respect to their inversion. The inversion of a major interval is a minor interval, the inversion of an augmented interval is a diminished interval.
Major and minor
As shown in the table, a diatonic scale^{[b]} defines seven intervals for each interval number, each starting from a different note (seven unisons, seven seconds, etc.). The intervals formed by the notes of a diatonic scale are called diatonic. Except for unisons and octaves, the diatonic intervals with a given interval number always occur in two sizes, which differ by one semitone. For example, six of the fifths span seven semitones. The other one spans six semitones. Four of the thirds span three semitones, the others four. If one of the two versions is a perfect interval, the other is called either diminished (i.e. narrowed by one semitone) or augmented (i.e. widened by one semitone). Otherwise, the larger version is called major, the smaller one minor. For instance, since a 7semitone fifth is a perfect interval (P5), the 6semitone fifth is called "diminished fifth" (d5). Conversely, since neither kind of third is perfect, the larger one is called "major third" (M3), the smaller one "minor third" (m3).
Within a diatonic scale,^{[b]} unisons and octaves are always qualified as perfect, fourths as either perfect or augmented, fifths as perfect or diminished, and all the other intervals (seconds, thirds, sixths, sevenths) as major or minor.
Augmented and diminished
Augmented intervals are wider by one semitone than perfect or major intervals, while having the same interval number (i.e., encompassing the same number of staff positions): they are wider by a
The augmented fourth (A4) and the diminished fifth (d5) are the only augmented and diminished intervals that appear in diatonic scales^{[b]} (see table).
Example
Neither the number, nor the quality of an interval can be determined by counting
For example, as shown in the table below, there are four semitones between A♭ and B♯, between A and C♯, between A and D♭, and between A♯ and E, but
 A♭–B♯ is a second, as it encompasses two staff positions (A, B), and it is doubly augmented, as it exceeds a major second (such as A–B) by two semitones.
 A–C♯ is a third, as it encompasses three staff positions (A, B, C), and it is major, as it spans 4 semitones.
 A–D♭ is a fourth, as it encompasses four staff positions (A, B, C, D), and it is diminished, as it falls short of a perfect fourth (such as A–D) by one semitone.
 A♯E is a fifth, as it encompasses five staff positions (A, B, C, D, E), and it is triply diminished, as it falls short of a perfect fifth (such as A–E) by three semitones.
Number of semitones 
Interval name  Staff positions
 

1  2  3  4  5  
4  doubly augmented second (AA2)  A♭  B♯  
4  major third (M3)  A  C♯  
4  diminished fourth (d4)  A  D♭  
4  triply diminished fifth (ddd5) 
A♯  E 
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 interval number. The indications 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 TT. The interval qualities may be also abbreviated with perf, min, maj, dim, aug. Examples:
 m2 (or min2): minor second,
 M3 (or maj3): major third,
 A4 (or aug4): augmented fourth,
 d5 (or dim5): diminished fifth,
 P5 (or perf5): perfect fifth.
Inversion
A simple interval (i.e., an interval smaller than or equal to an octave) may be inverted by raising the lower pitch an octave or lowering the upper pitch an octave. For example, the fourth from a lower C to a higher F may be inverted to make a fifth, from a lower F to a higher C.
There are two rules to determine the number and quality of the inversion of any simple interval:^{[7]}
 The interval number and the number of its inversion always add up to nine (4 + 5 = 9, 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; the inversion of a doubly augmented interval is a doubly diminished interval, and vice versa.
For example, the interval from C to the E♭ above it is a minor third. By the two rules just given, the interval from E♭ to the C above it must be a major sixth.
Since compound intervals are larger than an octave, "the inversion of any compound interval is always the same as the inversion of the simple interval from which it is compounded".^{[8]}
For intervals identified by their ratio, the inversion is determined by reversing the ratio and multiplying the ratio by 2 until it is greater than 1. For example, the inversion of a 5:4 ratio is an 8:5 ratio.
For intervals identified by an integer number of semitones, the inversion is obtained by subtracting that number from 12.
Since an interval class is the lower number selected among the interval integer and its inversion, interval classes cannot be inverted.
Classification
Intervals can be described, classified, or compared with each other according to various criteria.
Melodic and harmonic
An interval can be described as
 Vertical or harmonic if the two notes sound simultaneously
 Horizontal, linear, or melodic if they sound successively.^{[2]} Melodic intervals can be ascending (lower pitch precedes higher pitch) or descending.
Diatonic and chromatic
In general,
 A diatonic interval is an interval formed by two notes of a diatonic scale.
 A chromatic interval is a nondiatonic interval formed by two notes of a chromatic scale.
The table above depicts the 56 diatonic intervals formed by the notes of the C major scale (a diatonic scale). Notice that these intervals, as well as any other diatonic interval, can be also formed by the notes of a chromatic scale.
The distinction between diatonic and chromatic intervals is controversial, as it is based on the definition of diatonic scale, which is variable in the literature. For example, the interval B–E♭ (a diminished fourth, occurring in the harmonic Cminor scale) is considered diatonic if the harmonic minor scales are considered diatonic as well.^{[9]} Otherwise, it is considered chromatic. For further details, see the main article.
By a commonly used definition of diatonic scale^{}
The distinction between diatonic and chromatic intervals may be also sensitive to context. The abovementioned 56 intervals formed by the Cmajor scale are sometimes called diatonic to C major. All other intervals are called chromatic to C major. For instance, the perfect fifth A♭–E♭ is chromatic to C major, because A♭ and E♭ are not contained in the C major scale. However, it is diatonic to others, such as the A♭ major scale.
Consonant and dissonant
Consonance and dissonance are relative terms that refer to the stability, or state of repose, of particular musical effects. Dissonant intervals are those that cause tension and desire to be resolved to consonant intervals.
These terms are relative to the usage of different compositional styles.
 In 15th and 16thcentury usage, perfect fifths and octaves, and major and minor thirds and sixths were considered harmonically consonant, and all other intervals dissonant, including the perfect fourth, which by 1473 was described (by Johannes Tinctoris) as dissonant, except between the upper parts of a vertical sonority—for example, with a supporting third below ("63 chords").^{[10]} In the common practice period, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously considered dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16thcentury practice was still taught to beginning musicians throughout this period.
 Hermann von Helmholtz (1821–1894) theorised that dissonance was caused by the presence of beats.^{[11]} Helmholtz further believed that the beating produced by the upper partials of harmonic sounds was the cause of dissonance for intervals too far apart to produce beating between the fundamentals.^{[12]} Helmholtz then designated that two harmonic tones that shared common low partials would be more consonant, as they produced less beats.^{[13]}^{[14]} Helmholtz disregarded partials above the seventh, as he believed that they were not audible enough to have significant effect.^{[15]} From this Helmholtz categorises the octave, perfect fifth, perfect fourth, major sixth, major third, and minor third as consonant, in decreasing value, and other intervals as dissonant.
 David Cope (1997) suggests the concept of interval strength,^{[16]} 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: Lipps–Meyer law and #Interval root
All of the above analyses refer to vertical (simultaneous) intervals.
Simple and compound
A simple interval is an interval spanning at most one octave (see Main intervals above). Intervals spanning more than one octave are called compound intervals, as they can be obtained by adding one or more octaves to a simple interval (see below for details).^{[17]}
Steps and skips
Linear (melodic) intervals may be described as steps or skips. A step, or conjunct motion,^{[18]} is a linear interval between two consecutive notes of a scale. Any larger interval is called a skip (also called a leap), or disjunct motion.^{}
For example, C to D (major second) is a step, whereas C to E (major third) is a skip.
More generally, a step is a smaller or narrower interval in a musical line, and a skip is a wider or larger interval, where the categorization of intervals into steps and skips is determined by the tuning system and the pitch space used.
Melodic motion in which the interval between any two consecutive pitches is no more than a step, or, less strictly, where skips are rare, is called stepwise or conjunct melodic motion, as opposed to skipwise or disjunct melodic motions, characterized by frequent skips.
Enharmonic intervals
Two intervals are considered 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, the four intervals listed in the table below are all enharmonically equivalent, because the notes F♯ and G♭ indicate the same pitch, and the same is true for A♯ and B♭. All these intervals span four semitones.
Number of semitones 
Interval name  Staff positions
 

1  2  3  4  
4  major third  F♯  A♯  
4  major third  G♭  B♭  
4  diminished fourth  F♯  B♭  
4  doubly augmented second  G♭  A♯ 
When played as isolated chords on a
The discussion above assumes the use of the prevalent tuning system,
Minute intervals
There are also a number of minute intervals not found in the chromatic scale or labeled with a diatonic function, which have names of their own. They may be described as
 A Pythagorean comma is the difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequency ratio 531441:524288 (23.5 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 (21.5 cents).
 A septimal comma is 64:63 (27.3 cents), and is the difference between the Pythagorean or 3limit "7th" and the "harmonic 7th".
 A 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 (41.1 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 (19.6 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 (2.0 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 kleisma is the difference between six minor thirds and one tritave or perfect twelfth (an octave plus a perfect fifth), with a frequency ratio of 15625:15552 (8.1 cents) ( ^{ⓘ}).
 A septimal kleisma is the amount that two major thirds of 5:4 and a septimal major third, or supermajor third, of 9:7 exceeds the octave. Ratio 225:224 (7.7 cents).
 A quarter tone is half the width of a semitone, which is half the width of a whole tone. It is equal to exactly 50 cents.
Compound intervals
A compound interval is an interval spanning more than one octave.^{[17]} Conversely, intervals spanning at most one octave are called simple intervals (see Main intervals below).
In general, a compound interval may be defined by a sequence or "stack" of two or more simple intervals of any kind. For instance, a major tenth (two staff positions above one octave), also called compound major third, spans one octave plus one major third.
Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a major seventeenth can be decomposed into two octaves and one major third, and this is the reason why it is called a compound major third, even when it is built by adding up four fifths.
The diatonic number DN_{c} of a compound interval formed from n simple intervals with diatonic numbers DN_{1}, DN_{2}, ..., DN_{n}, is determined by:
which can also be written as:
The quality of a compound interval is determined by the quality of the simple interval on which it is based. For instance, a compound major third is a major tenth (1+(8−1)+(3−1) = 10), or a major seventeenth (1+(8−1)+(8−1)+(3−1) = 17), and a compound perfect fifth is a perfect twelfth (1+(8−1)+(5−1) = 12) or a perfect nineteenth (1+(8−1)+(8−1)+(5−1) = 19). Notice that two octaves are a fifteenth, not a sixteenth (1+(8−1)+(8−1) = 15). Similarly, three octaves are a twentysecond (1+3×(8−1) = 22), and so on.
Main compound intervals
Number of semitones 
Minor, major, or perfect intervals 
Short  Augmented or diminished intervals 
Short 

12  Diminished ninth  d9  
13  Minor ninth 
m9  Augmented octave  A8 
14  Major ninth 
M9  Diminished tenth  d10 
15  Minor tenth 
m10  Augmented ninth 
A9 
16  Major tenth 
M10  Diminished eleventh  d11 
17  Perfect eleventh 
P11  Augmented tenth  A10 
18  Diminished twelfth  d12  
Augmented eleventh 
A11  
19  Tritave 
P12  Diminished thirteenth  d13 
20  Minor thirteenth  m13  Augmented twelfth  A12 
21  Major thirteenth  M13  Diminished fourteenth  d14 
22  Minor fourteenth  m14  Augmented thirteenth  A13 
23  Major fourteenth  M14  Diminished fifteenth  d15 
24  Perfect fifteenth or Double octave  P15  Augmented fourteenth  A14 
25  Augmented fifteenth  A15 
It is also worth mentioning here the major seventeenth (28 semitones)—an interval larger than two octaves that can be considered a multiple of a perfect fifth (7 semitones) as it can be decomposed into four perfect fifths (7 × 4 = 28 semitones), or two octaves plus a major third (12 + 12 + 4 = 28 semitones). Intervals larger than a major seventeenth seldom come up, most often being referred to by their compound names, for example "two octaves plus a fifth"^{[19]} rather than "a 19th".
Intervals in chords
Chords are sets of three or more notes. They are typically defined as the combination of intervals starting from a common note called the
Chord qualities and interval qualities
The main chord qualities are
Deducing component intervals from chord names and symbols
The main rules to decode chord names or symbols are summarized below. Further details are given at
 For 3note chords (main articlefor further details.
 Without contrary information, a major triad) are implied. For instance, a C chord is a C major triad, and the name C minor seventh (Cm^{7}) implies a minor 3rd by rule 1, a perfect 5th by this rule, and a minor 7thby definition (see below). This rule has one exception (see next rule).
 When the fifth interval is diminished, the third must be minor.^{[d]}This rule overrides rule 2. For instance, Cdim^{7} implies a diminished 5th by rule 1, a minor 3rd by this rule, and a diminished 7th by definition (see below).
 Names and symbols that contain only a plain interval number (e.g., "seventh chord") or the chord root and a number (e.g., "C seventh", or C^{7}) are interpreted as follows:
 If the number is 2, 4, 6, etc., the chord is a major names and symbols for added tone chords).
 If the number is 7, 9, 11, 13, etc., the chord is extended chords).
 If the number is 5, the chord (technically not a chord in the traditional sense, but a dyad) is a power chord. Only the root, a perfect fifth and usually an octave are played.
 If the number is 2, 4, 6, etc., the chord is a major
The table shows the intervals contained in some of the main chords (component intervals), and some of the symbols used to denote them. The interval qualities or numbers in
Main chords  Component intervals  

Name  Symbol examples

Third  Fifth  Seventh 
Major triad 
C  M3  P5  
CM, or Cmaj  M3  P5  
Minor triad 
Cm, or Cmin  m3  P5  
Augmented triad  C+, or Caug  M3  A5  
Diminished triad  C°, or Cdim  m3  d5  
Dominant seventh chord  C^{7}, or C^{dom7}  M3  P5  m7 
Minor seventh chord  Cm^{7}, or Cmin^{7}  m3  P5  m7 
Major seventh chord  CM^{7}, or Cmaj^{7}  M3  P5  M7 
Augmented seventh chord  C+^{7}, Caug^{7}, C^{7♯5}, or C^{7aug5} 
M3  A5  m7 
Diminished seventh chord  C°^{7}, or Cdim^{7}  m3  d5  d7 
Halfdiminished seventh chord  C^{ø}^{7}, Cm^{7♭5}, or Cm^{7dim5}  m3  d5  m7 
Size of intervals used in different tuning systems
Number of semitones 
Name  5limit tuning (pitch ratio) 
Comparison of interval width (in cents)  

5limit tuning

Pythagorean tuning 
1⁄4comma meantone 
Equal temperament  
0  Perfect unison  1:1  0  0  0  0 
1  Minor second  16:15 27:25 
112 133 
90  117  100 
2  Major second  9:8 10:9 
204 182 
204  193  200 
3  Minor third  6:5 32:27 
316 294 
294 318 
310 (wolf) 269 
300 
4  Major third  5:4  386  408 384 
386 (wolf) 427 
400 
5  Perfect fourth  4:3 27:20 
498 520 
498 (wolf) 522 
503 (wolf) 462 
500 
6  Augmented fourth Diminished fifth 
45:32 25:18 
590 569 
612 588 
579 621 
600 
7  Perfect fifth  3:2 40:27 
702 680 
702 (wolf) 678 
697 (wolf) 738 
700 
8  Minor sixth  8:5  814  792  814  800 
9  Major sixth  5:3 27:16 
884 906 
906  890  900 
10  Minor seventh  16:9 9:5 
996 1018 
996  1007  1000 
11  Major seventh  15:8 50:27 
1088 1067 
1110  1083  1100 
12  Perfect octave  2:1  1200  1200  1200  1200 
In this table, the interval widths used in four different tuning systems are compared. To facilitate comparison,
In
The Pythagorean tuning is characterized by smaller differences because they are multiples of a smaller ε (ε ≈ 1.96 cents, the difference between the Pythagorean fifth and the average fifth). Notice that here the fifth is wider than 700 cents, while in most meantone temperaments, including 1⁄4comma meantone, it is tempered to a size smaller than 700. A more detailed analysis is provided at Pythagorean tuning § Size of intervals.
The
The abovementioned symmetric scale 1, defined in the 5limit tuning system, is not the only method to obtain
In the diatonic system, every interval has one or more enharmonic equivalents, such as augmented second for minor third.
Interval root
Although intervals are usually designated in relation to their lower note, David Cope^{[16]} and Hindemith^{[21]} 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^{}
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.^{[22]}
Alternative interval naming conventions
As shown below, some of the abovementioned intervals have alternative names, and some of them take a specific alternative name in
Typically, a comma is a diminished second, but this is not always true (for more details, see Alternative definitions of comma). For instance, in Pythagorean tuning the diminished second is a descending interval (524288:531441, or about −23.5 cents), and the Pythagorean comma is its opposite (531441:524288, or about 23.5 cents). 5limit tuning defines four kinds of comma, three of which meet the definition of diminished second, and hence are listed in the table below. The fourth one, called syntonic comma (81:80) can neither be regarded as a diminished second, nor as its opposite. See Diminished seconds in 5limit tuning for further details.
Number of semitones 
Generic names  Specific names  

Quality and number  Other naming convention  Pythagorean tuning  5limit tuning 
1⁄4comma meantone  
Full  Short  
0  perfect unison or perfect prime 
P1  
diminished second  d2  descending Pythagorean comma (524288:531441) 
lesser diesis (128:125)  
diaschisma (2048:2025) greater diesis (648:625) 

1  minor second 
m2  semitone, half tone, half step 
diatonic semitone, major semitone 
limma (256:243) 

augmented unison or augmented prime 
A1  chromatic semitone, minor semitone 
apotome (2187:2048) 

2  major second  M2  tone, whole tone , whole step 
sesquioctavum (9:8) 

3  minor third  m3  sesquiquintum (6:5) 

4  major third  M3  sesquiquartum (5:4)
 
5  perfect fourth  P4  sesquitertium (4:3) 

6  diminished fifth 
d5  tritone^{[a]}  
augmented fourth 
A4  
7  perfect fifth  P5  sesquialterum (3:2)  
12  perfect octave  P8  duplex (2:1) 
Additionally, some cultures around the world have their own names for intervals found in their music. For instance, 22 kinds of intervals, called shrutis, are canonically defined in Indian classical music.
Latin nomenclature
Up to the end of the 18th century, Latin was used as an official language throughout Europe for scientific and music textbooks. In music, many English terms are derived from Latin. For instance, semitone is from Latin semitonus.
The prefix semi is typically used herein to mean "shorter", rather than "half".^{[23]}^{[24]}^{[25]} Namely, a semitonus, semiditonus, semidiatessaron, semidiapente, semihexachordum, semiheptachordum, or semidiapason, is shorter by one semitone than the corresponding whole interval. For instance, a semiditonus (3 semitones, or about 300 cents) is not half of a ditonus (4 semitones, or about 400 cents), but a ditonus shortened by one semitone. Moreover, in Pythagorean tuning (the most commonly used tuning system up to the 16th century), a semitritonus (d5) is smaller than a tritonus (A4) by one Pythagorean comma (about a quarter of a semitone).
Number of semitones 
Quality and number  Short  Latin nomenclature 

0  Perfect unison  P1  unisonus 
1  Minor second  m2  semitonus 
Augmented unison  A1  unisonus superflua  
2  Major second  M2  tonus 
Diminished third  d3  
3  Minor third  m3  semiditonus 
Augmented second  A2  tonus superflua  
4  Major third  M3  ditonus 
Diminished fourth  d4  semidiatessaron  
5  Perfect fourth  P4  diatessaron 
Augmented third  A3  ditonus superflua  
6  Diminished fifth  d5  semidiapente, semitritonus 
Augmented fourth  A4  tritonus  
7  Perfect fifth  P5  diapente 
Diminished sixth  d6  semihexachordum  
8  Minor sixth  m6  hexachordum minus, semitonus maius cum diapente, tetratonus 
Augmented fifth  A5  diapente superflua  
9  Major sixth  M6  hexachordum maius, tonus cum diapente 
Diminished seventh  d7  semiheptachordum  
10  Minor seventh  m7  heptachordum minus, semiditonus cum diapente, pentatonus 
Augmented sixth  A6  hexachordum superflua  
11  Major seventh  M7  heptachordum maius, ditonus cum diapente 
Diminished octave  d8  semidiapason  
12  Perfect octave  P8  diapason 
Augmented seventh  A7  heptachordum superflua 
Nondiatonic intervals
Intervals in nondiatonic scales can be named using analogs of the diatonic interval names, by using a diatonic interval of similar size and distinguishing it by varying the quality, or by adding other modifiers. For example, the just interval 7/6 may be referred to as a subminor third, since it is ~267 cents wide, which is narrower than a minor third (300 cents in 12TET, ~316 cents for the just interval 6/5), or as the
The most common of these extended qualities are a
Pitchclass intervals
In posttonal or
In atonal or musical set theory, there are numerous types of intervals, the first being the
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 intervals, see interval class.^{[27]}
Generic and specific intervals
In
Notice that staff positions, when used to determine the conventional interval number (second, third, fourth, etc.), are counted including the position of the lower note of the interval, while generic interval numbers are counted excluding that position. Thus, generic interval numbers are smaller by 1, with respect to the conventional interval numbers.
Comparison
Specific interval

Generic interval

Diatonic name  

Number of semitones  Interval class  
0  0  0  Perfect unison 
1  1  1  Minor second 
2  2  1  Major second 
3  3  2  Minor third 
4  4  2  Major third 
5  5  3  Perfect fourth 
6  6  3 4 
Augmented fourth Diminished fifth 
7  5  4  Perfect fifth 
8  4  5  Minor sixth 
9  3  5  Major sixth 
10  2  6  Minor seventh 
11  1  6  Major seventh 
12  0  7  Perfect octave 
Generalizations and nonpitch uses
The term "interval" can also be generalized to other music elements besides pitch.
For example, an interval between two belllike sounds, which have no pitch salience, is still perceptible. When two tones have similar acoustic spectra (sets of partials), the interval is just the distance of the shift of a tone spectrum along the frequency axis, so linking to pitches as reference points is not necessary. The same principle naturally applies to pitched tones (with similar harmonic spectra), which means that intervals can be perceived "directly" without pitch recognition. This explains in particular the predominance of interval hearing over absolute pitch hearing.^{[30]}^{[31]}
See also
 Circle of fifths
 Ear training
 List of meantone intervals
 List of pitch intervals
 Music and mathematics
 Pseudooctave
 Regular temperament
Notes
 ^ ^{a} ^{b} The term tritone is sometimes used more strictly as a synonym of augmented fourth (A4).
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g}
The expression melodic minor and the harmonic minor scales (see also Diatonic and chromatic).
 ^ ^{a} ^{b} General rule 1 achieves consistency in the interpretation of symbols such as CM^{7}, Cm^{6}, and C+^{7}. Some musicians legitimately prefer to think that, in CM^{7}, M refers to the seventh, rather than to the third. This alternative approach is legitimate, as both the third and seventh are major, yet it is inconsistent, as a similar interpretation is impossible for Cm^{6} and C+^{7} (in Cm^{6}, m cannot possibly refer to the sixth, which is major by definition, and in C+^{7}, + cannot refer to the seventh, which is minor). Both approaches reveal only one of the intervals (M3 or M7), and require other rules to complete the task. Whatever is the decoding method, the result is the same (e.g., CM^{7} is always conventionally decoded as C–E–G–B, implying M3, P5, M7). The advantage of rule 1 is that it has no exceptions, which makes it the simplest possible approach to decode chord quality.
According to the two approaches, some may format the major seventh chord as CM^{7} (general rule 1: M refers to M3), and others as C^{M7} (alternative approach: M refers to M7). Fortunately, even C^{M7} becomes compatible with rule 1 if it is considered an abbreviation of CM^{M7}, in which the first M is omitted. The omitted M is the quality of the third, and is deduced according to rule 2 (see above), consistently with the interpretation of the plain symbol C, which by the same rule stands for CM.
 ^ All triads are tertian chords (chords defined by sequences of thirds), and a major third would produce in this case a nontertian chord. Namely, the diminished fifth spans 6 semitones from root, thus it may be decomposed into a sequence of two minor thirds, each spanning 3 semitones (m3 + m3), compatible with the definition of tertian chord. If a major third were used (4 semitones), this would entail a sequence containing a major second (M3 + M2 = 4 + 2 semitones = 6 semitones), which would not meet the definition of tertian chord.
References
 ISBN 9780781207836
 ^ .
 Grove Music Online. Oxford University Press. Accessed August 2013. (subscription required))
 ^ ^{a} ^{b} Definition of Perfect consonance in Godfrey Weber's General music teacher, by Godfrey Weber, 1841.
 ^ Kostka, Stefan; Payne, Dorothy (2008). Tonal Harmony, p. 21. First edition, 1984.
 ISBN 0403003261.
 ^ Drabkin, William (2001). "Fourth". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan.
 ^ Helmholtz 1895, p. 172: "The roughness from sounding two tones together depends... the number of beats produced in a second.".
 ^ Helmholtz 1895, p. 178: "The cause of this phenomenon must be looked for in the beats produced by the high upper partials of such compound tones".
 ^ Helmholtz 1895, p. 182.
 ISBN 0486607534, p. 182d: "Just as the coincidences of the two first upper partial tones led us to the natural consonances of the Octave and Fifth, the coincidences of higher upper partials would lead us to a further series of natural consonances."
 ^ Helmholtz 1895, p. 183: "Here I have stopped, because the 7th partial tone is entirely eliminated, or at least much weakened.".
 ^ ISBN 0028647378.
 ^ .
 ^ ^{a} ^{b}
Bonds, Mark Evan (2006).
A History of Music in Western Culture, p.123. 2nd ed. ISBN 0131931040.
 .
 .
 ^ Hindemith, Paul (1934). The Craft of Musical Composition. New York: Associated Music Publishers. Cited in Cope (1997), p. 40–41.
 ISBN 0520069919.
 ^ Gioseffo Zarlino, Le Istitutione harmoniche ... nelle quali, oltre le materie appartenenti alla musica, si trovano dichiarati molti luoghi di Poeti, d'Historici e di Filosofi, si come nel leggerle si potrà chiaramente vedere (Venice, 1558): 162.
 ISBN 9004047948.
 .
 ^ ^{a} ^{b} "Extendeddiatonic interval names". Xenharmonic wiki.
 ISBN 9781561592395.
 . "Lewin posits the notion of musical 'spaces' made up of elements between which we can intuit 'intervals'....Lewin gives a number of examples of musical spaces, including the diatonic gamut of pitches arranged in scalar order; the 12 pitch classes under equal temperament; a succession of timepoints pulsing at regular temporal distances one time unit apart; and a family of durations, each measuring a temporal span in time units....transformations of timbre are proposed that derive from changes in the spectrum of partials..."
 .
 .
Sources
 Longmans, Green, and Co.
External links
 Gardner, Carl E. (1912): Essentials of Music Theory, p. 38
 "Interval", Encyclopædia Britannica
 Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings
 Elements of Harmony: Vertical Intervals
 Just intervals, from the unison to the octave, played on a drone note on YouTube