Interval (music)
Encyclopedia
In music theory
, an interval is a combination of two note
s, or the ratio between their frequencies. Two-note combinations are also called dyads
. Although chords
are often defined as sets of three or more notes, intervals are sometimes considered to be the simplest kind of chord.
Intervals may be described as:
In Western culture
, the most common method to classify and name intervals is based on their quality (perfect, major, minor, etc.) and number (unison, second, third, etc.). For instance, two frequently used types of interval are called minor third
and major third
(see below for further details). Intervals may be also classified as:
Minute intervals (commas
, and microtones) can be formed by the notes of musical scales containing more than 12 pitches (e.g., by the notes A and G found in some extended scales), or by two notes having the same name, but tuned a different way (e.g., the syntonic comma
is sometimes defined as the difference between an F tuned using the D-based Pythagorean system
, and another F tuned using the D-based quarter-comma meantone
system). They are sometimes so small that the difference in pitch between the two notes cannot be perceived.
of their frequencies
. Important intervals are those measured by small-number ratios, such as 1:1 (unison
or prime), 2:1 (octave
), 3:2 (perfect fifth
), 4:3 (perfect fourth
), 5:4 (major third
) etc. (see: Limit in music
). This system is frequently used to describe intervals in both Western and non-Western music. This method is also often used in just intonation
, and in theoretical explanations of equal-tempered intervals used in European tonal music, to explain them through their approximation of just intervals.
. 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 size in cents of the interval from frequency f1 to frequency f2 is
. A semitone
is any interval between two adjacent notes in a chromatic scale, a whole tone is any interval spanning two semitones, and a tritone
is any interval spanning three tones, or six semitones. In Latin
nomenclature, the term ditonus was originally used to indicate any interval composed of two tones, but in current English the corresponding term ditone
is used with a much more specific meaning, to refer only to a major third with frequency ratio 81:64.
Intervals with different names but spanning the same number of semitones may have the same width, provided that the instrument
is tuned so that the 12 notes of the chromatic scale are equally spaced (a commonly used tuning system called equal temperament
). However, they are defined by different notes. For instance, the interval from D to F is a major third
, while that from D to G is a diminished fourth
, although they have identical width in equal temperament (4 semitones, or 400 cents
).
Except for the Latin ones, 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.
, to name an interval we typically need to determine 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.
In a diatonic scale
, the number of staff positions coincides with the number of scale steps. This is not true for other kinds of scales, such as the chromatic scale
. However, if you take away any accidental
s from the notes which form an interval, by definition the notes do not change their staff position. As a consequence, the interval number for any chromatic interval is the same as the interval number of the diatonic interval formed by the two notes without accidentals. This is the reason why interval numbers are also called diatonic interval numbers, and this convention is called diatonic interval numbering.
The rule to determine the diatonic number of a compound interval (an interval larger than one octave), based on the diatonic numbers of the simple intervals from which it is built is explained in a separate section.
(M), minor
(m), augmented
(A), and diminished
(d). This is called its interval quality. It is possible to have doubly diminished and doubly augmented intervals, but these are quite rare, as they occur only in chromatic contexts. The quality of a compound interval is the quality of the simple interval on which it is based (see Interval (music)#Simple and compound for details).
Major/minor
Major and minor intervals are so-called because certain diatonic intervals (seconds, thirds, sixths, sevenths, and their compounds) may occur in two sizes in the diatonic scale
. The larger of the two versions is called major, the smaller one minor. For example, the third occurs both as three semitones away from Re, Mi, La, and Si in the major scale (or in the C major diatonic scale, three semitones above D, E, A, and B), and four semitones away from Do, Fa, and Sol, (or C, F, and G). The smaller, three-semitone version is called the "minor third" and the larger, four-semitone one is called the "major third". Major intervals invert to minor ones, and vice-versa. For example, a major second inverts to a minor seventh, and the reverse.
Perfect
Perfect intervals are so-called because of their high levels of consonance
, and because the inversion
of a perfect interval is also perfect. Other kinds of intervals have an 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. Within the diatonic scale all fourths and fifths are perfect, with five and seven semitones respectively, except for one occurrence each of six semitones: the fourth between Fa and Si (an augmented fourth), and its inversion, i.e., the fifth between Si and Fa (a diminished fifth).
Augmented/diminished
Augmented and diminished intervals are so called because they exceed or fall short of either a perfect interval, or a major/minor pair by one semitone, while having the same interval number (i.e., encompassing the same number of staff positions). Except for the augmented fourth (Fa-Ti) and diminished fifth (Ti-Fa), they do not appear in the diatonic scale. For instance, there is no three-semitone interval in the diatonic scale that functions as a second, and the augmented second (e.g., E–F) is three semitones wide.
For example, as shown in the table below, there are four semitones between A and C, between A and D, between A and D, and between A and B, but
The diminished fourth is an interval found between the seventh and third degrees of the harmonic minor scale, while the doubly augmented second only occurs in entirely chromatic contexts. In equal-tempered tuning
, as on a piano, these intervals are indistinguishable by sound when played in isolation, but in musical context the diatonic function
of the notes incorporated is very different.
, M for major
, d for diminished
, A for augmented
, followed by the 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 TT. The interval qualities may be also abbreviated with perf, min, maj, dim, aug. Examples:
are sets of three or more notes. They are typically defined as the combination of intervals starting from a common note called the root of the chord. For instance a major triad is a chord containing three notes defined by the root and two intervals (major third and perfect fifth). Sometimes even a single interval (dyad
) is considered to be a chord. Chords are classified based on the quality and number of the intervals which define them.
, minor
, augmented, diminished
, half-diminished
, and dominant.
The symbols used for chord quality are similar to those used for interval quality (see above). In addition, + or aug is used for augmented, ° or dim for diminished, for half diminished, and dom for dominant (the symbol − alone is not used for diminished).
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 boldface font can be deduced from chord name or symbol by applying rule 1. In symbol examples, C is used as chord root.
. The table on the right depicts all diatonic intervals for C major. A chromatic interval is a non-diatonic interval formed by two notes of a chromatic scale
.
Aside from tritones, all intervals that are either augmented or diminished are chromatic, and the rest are diatonic.
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.
All of the above analyses refer to vertical (simultaneous) intervals.
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. A major seventeenth (two staff positions above two octaves) is another example of compound major third, and can be built either by adding up two octaves and one major third, or four perfect fifths.
Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a seventeenth can be always 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 using four fifths.
The diatonic number DNc of a compound interval formed from n simple intervals with diatonic numbers DN1, DN2, ..., DNn, 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 twenty-second (1+3*(8-1) = 22), and so on.
Intervals larger than a seventeenth 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".
s 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 semitone
s, is still considered a skip.
The words conjunct and disjunct refer to melodies composed of steps and skips, respectively.
, 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 semitone
s. 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 doubly augmented second
) are all enharmonically equivalent. In fact, although they have a different name and staff position, F and G indicate the same pitch, and the same is true for A and B. As a consequence, all these intervals span four semitones.
In this table, the interval widths used in four different tuning systems are compared. To facilitate comparison, just intervals as provided by 5-limit tuning (see symmetric scale n.1) are shown in bold font, and the values in cents are rounded
to integers. Notice that in each of the non-equal
tuning systems, by definition the width of each type of interval (including the semitone) changes depending on the note from which the interval starts. This is the price paid for seeking just intonation
. However, for the sake of simplicity, for some types of interval the table shows only one value (the most often observed
one).
In 1/4-comma meantone, by definition 11 perfect fifths have a size of approximately 697 cents (700−ε cents, where ε ≈ 3.42 cents); since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of about 700+11ε cents, which is about 738 cents (the wolf fifth); 8 major thirds have size about 386 cents (400−4ε), 4 have size about 427 cents (400+8ε), and their average size is 400 cents. In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε (the difference between the 1/4-comma meantone fifth and the average fifth). A more detailed analysis is provided at 1/4-comma meantone#Size of intervals. Note that 1/4-comma meantone was designed to produce just major thirds, but only 8 of them are just (5:4, about 386 cents).
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 temperament
s, including 1/4-comma meantone, it is tempered to a size smaller than 700. A more detailed analysis is provided at Pythagorean tuning#Size of intervals.
The 5-limit tuning system uses just tones and semitones as building blocks, rather than a stack of perfect fifths, and this leads to even more varied intervals throughout the scale (each kind of interval has three or four different sizes). A more detailed analysis is provided at 5-limit tuning#Size of intervals. Note that 5-limit tuning was designed to maximize the number of just intervals, but even in this system some intervals are not just (e.g., 3 fifths, 5 major thirds and 6 minor thirds are not just; also, 3 major and 3 minor thirds are wolf interval
s).
The above mentioned symmetric scale 1, defined in the 5-limit tuning system, is not the only method to obtain just intonation
. It is possible to construct juster intervals or just intervals closer to the equal-tempered equivalents, but most of the ones listed above have been used historically in equivalent contexts. In particular, the asymmetric version of the 5-limit tuning scale provides a juster value for the minor seventh (9:5, rather than 16:9). Moreover, the tritone
(augmented fourth or diminished fifth), could have other just ratios; for instance, 7:5 (about 583 cents) or 17:12 (about 603 cents) are possible alternatives for the augmented fourth (the latter is fairly common, as it is closer to the equal-tempered value of 600 cents). The 7:4 interval (about 969 cents), also known as the harmonic seventh
, has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh. Some assert the 7:4 is one of the blue note
s used in jazz
. For further details about reference ratios, see 5-limit tuning#The justest ratios.
In the diatonic system, every interval has one or more enharmonic
equivalents, such as augmented second
for minor third
.
are approximate, and they can be also classified as comma
s, as they describe small discrepancies, observed in some tuning systems, between enharmonically equivalent
notes.
See Musical interval mnemonics at Wikibooks for popular musical fragments that feature common intervals
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:
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."
For intervals identified by their 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.
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.
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 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.
s, "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 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.
, five-limit tuning
, or meantone temperament tuning systems such as quarter-comma meantone
. Notice that ditone and semiditone are specific for Pythagorean tuning, while tone and tritone are used generically for all tuning systems. Interestingly, despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents). All the intervals with prefix sesqui- are justly
tuned, and their frequency ratio
, shown in the table, is a superparticular number (or epimoric ratio). The same is true for the octave.
The diminished second
is a comma
, but some commas are not diminished seconds. For instance, the Pythagorean comma
(531441:524288) is the opposite of a diminished second. 5-limit 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 Five-limit tuning#Diminished seconds for further details.
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
.
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.
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 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. For unordered pitch-class intervals, see interval class
.
, specific
and generic interval
s are distinguished. Specific intervals are the interval class or number of semitones between scale steps or collection members, and generic intervals are the number of diatonic scale steps (or staff positions) between notes of a collection or scale.
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.
's Generalized Musical Intervals and Transformations uses interval as a generic measure of distance to show musical transformations that can change—for instance—one rhythm into another, or one formal structure into another.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, an interval is a combination of two note
Note
In music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....
s, or the ratio between their frequencies. Two-note combinations are also called dyads
Dyad (music)
In music, a dyad is a set of two notes or pitches. Although most chords have three or more notes, in certain contexts a dyad may be considered to be a chord. The most common two-note chord is made from the interval of a perfect fifth, which may be suggestive of music of the Medieval or Renaissance...
. Although chords
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...
are often defined as sets of three or more notes, intervals are sometimes considered to be the simplest kind of chord.
Intervals may be described as:
- Vertical or harmonicHarmonyIn music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
, if the two notes sound simultaneously - Horizontal, linear, or melodicMelodyA melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...
if they sound successively.
In Western culture
Western culture
Western culture, sometimes equated with Western civilization or European civilization, refers to cultures of European origin and is used very broadly to refer to a heritage of social norms, ethical values, traditional customs, religious beliefs, political systems, and specific artifacts and...
, the most common method to classify and name intervals is based on their quality (perfect, major, minor, etc.) and number (unison, second, third, etc.). For instance, two frequently used types of interval are called minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
and major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
(see below for further details). Intervals may be also classified as:
- Diatonic intervals, between the notes of a diatonic scaleDiatonic scaleIn music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
, or - Chromatic intervals, non-diatonic intervals between the notes of a chromatic scaleChromatic scaleThe chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
.
Minute intervals (commas
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
, and microtones) can be formed by the notes of musical scales containing more than 12 pitches (e.g., by the notes A and G found in some extended scales), or by two notes having the same name, but tuned a different way (e.g., the syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
is sometimes defined as the difference between an F tuned using the D-based Pythagorean system
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, and another F tuned using the D-based quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
system). They are sometimes so small that the difference in pitch between the two notes cannot be perceived.
Interval width
The size or width of an interval 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 pitches may be measured by the ratioRatio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
of their frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
. Important intervals are those measured by small-number ratios, such as 1:1 (unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
or prime), 2:1 (octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
), 3:2 (perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
), 4:3 (perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
), 5:4 (major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
) etc. (see: Limit in music
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
). This system is frequently used to describe intervals in both Western and non-Western music. This method is also often used in just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
, and in theoretical explanations of equal-tempered intervals used in European tonal music, to explain them through their approximation of just intervals.
Cents
The standard system for comparing interval sizes is with centsCent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
. This is a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...
in which the octave is divided into 1200 equal parts. In equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, each semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
is exactly 100 cents.
The size in cents of the interval from frequency f1 to frequency f2 is
Main intervals
The table shows the most widely used conventional names for the intervals between the notes of a chromatic scaleChromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
. A semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
is any interval between two adjacent notes in a chromatic scale, a whole tone is any interval spanning two semitones, and a tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
is any interval spanning three tones, or six semitones. In Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
nomenclature, the term ditonus was originally used to indicate any interval composed of two tones, but in current English the corresponding term ditone
Ditone
In music, a ditone is the interval 81:64, which is 407.82 cents. It is the major third in Pythagorean tuning. The ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma...
is used with a much more specific meaning, to refer only to a major third with frequency ratio 81:64.
Intervals with different names but spanning the same number of semitones may have the same width, provided that the instrument
Musical instrument
A musical instrument is a device created or adapted for the purpose of making musical sounds. In principle, any object that produces sound can serve as a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates back to the...
is tuned so that the 12 notes of the chromatic scale are equally spaced (a commonly used tuning system called equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
). However, they are defined by different notes. For instance, the interval from D to F is a major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
, while that from D to G is a diminished fourth
Diminished fourth
In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...
, although they have identical width in equal temperament (4 semitones, or 400 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
).
Except for the Latin ones, 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.
| Number of semitone Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... s |
Diatonic interval | Short | Chromatic interval | Short | Latin Latin Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and... nomenclature |
Short | Audio |
|---|---|---|---|---|---|---|---|
| 0 | Perfect Unison Unison In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance... |
P1 | Diminished second Diminished second In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C... |
d2 | |||
| 1 | Minor second Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
m2 | Augmented unison Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
A1 | Semitone Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
S | |
| 2 | Major second Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions... |
M2 | Diminished third Diminished third In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,... |
d3 | Whole tone | T | |
| 3 | Minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... |
m3 | Augmented second Augmented second In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning... |
A2 | |||
| 4 | Major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
M3 | Diminished fourth Diminished fourth In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,... |
d4 | |||
| 5 | Perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there... |
P4 | Augmented third Augmented third In classical music from Western culture, an augmented third is an interval produced by widening a major third by a chromatic semitone. For instance, the interval from C to E is a major third, four semitones wide, and both the intervals from C to E, and from C to E are augmented thirds, spanning... |
A3 | |||
| 6 | Diminished fifth Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
d5 | Tritone Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
TT | |||
| Augmented fourth Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
A4 | ||||||
| 7 | Perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... |
P5 | Diminished sixth Diminished sixth In classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning... |
d6 | |||
| 8 | Minor sixth Minor sixth -Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7.... |
m6 | Augmented fifth Augmented fifth In classical music from Western culture, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C to G, and from C to G are augmented fifths,... |
A5 | |||
| 9 | Major sixth Major sixth In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two... |
M6 | Diminished seventh Diminished seventh In classical music from Western culture, a diminished seventh is an interval produced by narrowing a minor seventh by a chromatic semitone. For instance, the interval from A to G is a minor seventh, ten semitones wide, and both the intervals from A to G, and from A to G are diminished sevenths,... |
d7 | |||
| 10 | Minor seventh Minor seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the... |
m7 | Augmented sixth Augmented sixth In classical music from Western culture, an augmented sixth is an interval produced by widening a major sixth by a chromatic semitone. For instance, the interval from C to A is a major sixth, nine semitones wide, and both the intervals from C to A, and from C to A are augmented sixths, spanning... |
A6 | |||
| 11 | Major seventh Major seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two... |
M7 | Diminished octave Diminished octave In classical music from Western culture, a diminished octave is an interval produced by narrowing a perfect octave by a chromatic semitone. As such, the two notes are denoted by the same letter but have different accidentals... |
d8 | |||
| 12 | Perfect octave Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"... |
P8 | Augmented seventh Augmented seventh In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths,... |
A7 | |||
Interval number and quality
In Western music theoryMusic theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, to name an interval we typically need to determine 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
As shown above, intervals are labeled according to the number of staff positions they encompass. Both the lines and the gaps between lines are counted, including the position of the lower note. For instance, the number of staff positions from C to G is 5, therefore the interval C-G is a fifth (denoted P5 in the figure above).In a diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
, the number of staff positions coincides with the number of scale steps. This is not true for other kinds of scales, such as the chromatic scale
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
. However, if you take away any accidental
Accidental (music)
In music, an accidental is a note whose pitch is not a member of a scale or mode indicated by the most recently applied key signature. In musical notation, the symbols used to mark such notes, sharps , flats , and naturals , may also be called accidentals...
s from the notes which form an interval, by definition the notes do not change their staff position. As a consequence, the interval number for any chromatic interval is the same as the interval number of the diatonic interval formed by the two notes without accidentals. This is the reason why interval numbers are also called diatonic interval numbers, and this convention is called diatonic interval numbering.
The rule to determine the diatonic number of a compound interval (an interval larger than one octave), based on the diatonic numbers of the simple intervals from which it is built is explained in a separate section.
Quality
The name of any interval is further qualified using the terms perfect (P), majorMajor and minor
In Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval.Major and minor are frequently referred to in the titles of classical compositions, especially in reference to the key of a piece.-Intervals and chords:With regard...
(M), minor
Major and minor
In Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval.Major and minor are frequently referred to in the titles of classical compositions, especially in reference to the key of a piece.-Intervals and chords:With regard...
(m), augmented
Augmentation (music)
In Western music and music theory, the word augmentation has three distinct meanings. Augmentation is a compositional device where a melody, theme or motif is presented in longer note-values than were previously used...
(A), and diminished
Diminution
In Western music and music theory, diminution has four distinct meanings. Diminution may be a form of embellishment in which a long note is divided into a series of shorter, usually melodic, values...
(d). This is called its interval quality. It is possible to have doubly diminished and doubly augmented intervals, but these are quite rare, as they occur only in chromatic contexts. The quality of a compound interval is the quality of the simple interval on which it is based (see Interval (music)#Simple and compound for details).
Major/minor
Major and minor intervals are so-called because certain diatonic intervals (seconds, thirds, sixths, sevenths, and their compounds) may occur in two sizes in the diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
. The larger of the two versions is called major, the smaller one minor. For example, the third occurs both as three semitones away from Re, Mi, La, and Si in the major scale (or in the C major diatonic scale, three semitones above D, E, A, and B), and four semitones away from Do, Fa, and Sol, (or C, F, and G). The smaller, three-semitone version is called the "minor third" and the larger, four-semitone one is called the "major third". Major intervals invert to minor ones, and vice-versa. For example, a major second inverts to a minor seventh, and the reverse.
Perfect
Perfect intervals are so-called because of their high levels of consonance
Consonance
Consonance is a stylistic device, most commonly used in poetry and songs, characterized by the repetition of the same consonant two or more times in short succession, as in "pitter patter" or in "all mammals named Sam are clammy".Consonance should not be confused with assonance, which is the...
, and because the inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
of a perfect interval is also perfect. Other kinds of intervals have an 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. Within the diatonic scale all fourths and fifths are perfect, with five and seven semitones respectively, except for one occurrence each of six semitones: the fourth between Fa and Si (an augmented fourth), and its inversion, i.e., the fifth between Si and Fa (a diminished fifth).
Augmented/diminished
Augmented and diminished intervals are so called because they exceed or fall short of either a perfect interval, or a major/minor pair by one semitone, while having the same interval number (i.e., encompassing the same number of staff positions). Except for the augmented fourth (Fa-Ti) and diminished fifth (Ti-Fa), they do not appear in the diatonic scale. For instance, there is no three-semitone interval in the diatonic scale that functions as a second, and the augmented second (e.g., E–F) is three semitones wide.
Example
Neither the number, nor the quality of an interval can be determined by counting semitones alone. As explained above, the number of staff positions must be taken into account as well.For example, as shown in the table below, there are four semitones between A and C, between A and D, between A and D, and between A and B, but
- the interval A-C is called a major third (as it spans 3 staff positions),
- the intervals A-D and A-D are called diminished fourths (as they span 4 staff positions),
- the interval A-B is called a doubly augmented second (as it spans 2 staff positions).
| Number of semitones |
Interval name | Staff positions | |||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | ||
| 4 | major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
A | C | ||
| 4 | diminished fourth Diminished fourth In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,... |
A | D | ||
| 4 | diminished fourth Diminished fourth In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,... |
A | D | ||
| 4 | doubly augmented second Augmented second In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning... |
A | B |
The diminished fourth is an interval found between the seventh and third degrees of the harmonic minor scale, while the doubly augmented second only occurs in entirely chromatic contexts. In equal-tempered tuning
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, as on a piano, these intervals are indistinguishable by sound when played in isolation, but in musical context the diatonic function
Diatonic function
In tonal music theory, a diatonic function is the specific, recognized role of each of the 7 notes and their chords in relation to the diatonic key...
of the notes incorporated is very different.
Shorthand notation
Intervals are often abbreviated with a P for perfect, m for minorMajor and minor
In Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval.Major and minor are frequently referred to in the titles of classical compositions, especially in reference to the key of a piece.-Intervals and chords:With regard...
, M for major
Major and minor
In Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval.Major and minor are frequently referred to in the titles of classical compositions, especially in reference to the key of a piece.-Intervals and chords:With regard...
, d for diminished
Diminution
In Western music and music theory, diminution has four distinct meanings. Diminution may be a form of embellishment in which a long note is divided into a series of shorter, usually melodic, values...
, A for augmented
Augmentation (music)
In Western music and music theory, the word augmentation has three distinct meanings. Augmentation is a compositional device where a melody, theme or motif is presented in longer note-values than were previously used...
, followed by the interval number. The indication M and P are often omitted. The octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
is P8, and a unison
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
is usually referred to simply as "a unison" but can be labeled P1. The tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
, 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.
Intervals in chords
ChordsChord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...
are sets of three or more notes. They are typically defined as the combination of intervals starting from a common note called the root of the chord. For instance a major triad is a chord containing three notes defined by the root and two intervals (major third and perfect fifth). Sometimes even a single interval (dyad
Dyad (music)
In music, a dyad is a set of two notes or pitches. Although most chords have three or more notes, in certain contexts a dyad may be considered to be a chord. The most common two-note chord is made from the interval of a perfect fifth, which may be suggestive of music of the Medieval or Renaissance...
) is considered to be a chord. Chords are classified based on the quality and number of the intervals which define them.
Chord qualities and interval qualities
The main chord qualities are: majorMajor chord
In music theory, a major chord is a chord having a root, a major third, and a perfect fifth. When a chord has these three notes alone, it is called a major triad...
, minor
Minor chord
In music theory, a minor chord is a chord having a root, a minor third, and a perfect fifth.When a chord has these three notes alone, it is called a minor triad....
, augmented, diminished
Diminished chord
A diminished triad chord or diminished chord is a triad consisting of two minor thirds above the root — if built on C, a diminished chord would have a C, an E and a G. It resembles a minor triad with a lowered fifth....
, half-diminished
Half-diminished seventh chord
In music theory, the half-diminished seventh chord is created by taking the root, minor third, diminished fifth and minor seventh of any major scale; for example, C half-diminished is . Its consecutive intervals are minor 3rd, minor 3rd, major 3rd...
, and dominant.
The symbols used for chord quality are similar to those used for interval quality (see above). In addition, + or aug is used for augmented, ° or dim for diminished, for half diminished, and dom for dominant (the symbol − alone is not used for diminished).
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 Chord names and symbols (jazz and pop music)#Rules to decode chord names and symbols.- For 3-note chords (triadsTriad (music)In music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:* the Root...
), majorMajor thirdIn classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
or minorMinor thirdIn classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
always refer to the third interval, while augmentedAugmented fifthIn classical music from Western culture, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C to G, and from C to G are augmented fifths,...
and diminished always refer to the fifth. The same is true for the corresponding symbols (e.g., Cm means Cm3, and C+ means C+5). Thus, the terms thirdThird (music)In music and music theory third may refer to:*major third*minor third*augmented third/perfect fourth*diminished third/major second*Third , chord member a third above the root*Mediant, third degree of the diatonic scale...
and fifth and the corresponding symbols 3 and 5 are typically omitted. This rule can be generalized to all kinds of chords, provided the above mentioned qualities appear immediately after the root note, or at the beginning of the chord name or symbol. For instance, in the chord symbols Cm and Cm7, m refers to the interval m3, and 3 is omitted. When these qualities do not appear immediately after the root note, or at the beginning of the name or symbol, they should be considered interval qualities, rather than chord qualities. For instance, in Cm/M7 (minor-major seventh chord), m is the chord quality and refers to the m3 interval, while M refers to the M7 interval. When the number of an extra interval is specified immediately after chord quality, the quality of that interval may coincide with chord quality (e.g. CM7 = CM/M7). However, this is not always true (e.g. Cm6 = Cm/M6, C+7 = C+/m7, CM11 = CM/P11). See main article for further details. - Without contrary information, a major thirdMajor thirdIn classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
interval and a perfect fifthPerfect fifthIn classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
interval (major triad) are implied. For instance, a C chord is a C major triad, and the name C minor seventh (Cm7) implies a minor 3rd by rule 1, a perfect 5th by this rule, and a minor 7thMinor seventhIn classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the...
by definition (see below). This rule has one exception (see next rule). - When the fifth interval is diminished, the third must be minor. This rule overrides rule 2. For instance, Cdim7 implies a diminished 5th by rule 1, a minor 3rd by this rule, and a diminished 7th by definition (see below).
- Names and symbols which contain only a plain interval number (e.g. “Seventh chord”) or the chord rootRoot (chord)In music theory, the root of a chord is the note or pitch upon which a triadic chord is built. For example, the root of the major triad C-E-G is C....
and a number (e.g. “C seventh”, or C7) are interpreted as follows:-
- If the number is 2, 4, 6, etc., the chord is a major added tone chordAdded tone chordAn added tone chord is a non-tertian chord composed of a tertian triad and an extra "added" note. The added note is not a seventh , but typically a non-tertian note, which cannot be defined by a sequence of thirds from the root, such as the added sixth or fourth...
(e.g. C6 = CM6 = Cadd6) and contains, together with the implied major triad, an extra major 2ndMajor secondIn Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...
, perfect 4thPerfect fourthIn classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
, or major 6thMajor sixthIn classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two...
(see names and symbols for added tone chords). - If the number is 7, 9, 11, 13, etc., the chord is dominant (e.g. C7 = Cdom7) and contains, together with the implied major triad, one or more of the following extra intervals: minor 7th, major 9th, perfect 11th, and major 13th (see names and symbols for seventh and extended chords).
- If the number is 2, 4, 6, etc., the chord is a major added tone chord
-
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 boldface font can be deduced from chord name or symbol by applying rule 1. In symbol examples, C is used as chord root.
| Main chords Chord (music) A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords... |
Component intervals | |||
|---|---|---|---|---|
| Name | Symbol examples | Third | Fifth | Seventh |
| Major triad | C | maj3 | perf5 | |
| CM, or Cmaj | maj3 | perf5 | ||
| Minor triad | Cm, or Cmin | min3 | perf5 | |
| Augmented triad Augmented triad In music, an augmented triad is a triad, or chord, consisting of two major thirds . The term augmented triad arises from an augmented triad being a three note chord, or triad, whose top note is raised, or augmented... |
C+, or Caug | maj3 | aug5 | |
| Diminished triad | C°, or Cdim | min3 | dim5 | |
| Dominant seventh chord Dominant seventh chord In music theory, a dominant seventh chord, or major minor seventh chord,is a chord composed of a root, major third, perfect fifth, and minor seventh. It can be also viewed as a major triad with an additional minor seventh... |
C7, or Cdom7 | maj3 | perf5 | min7 |
| Minor seventh chord Minor seventh chord In music, a minor seventh chord is any nondominant seventh chord where the "third" note is a minor third above the root.Most typically, minor seventh chord refers to where the "seventh" note is a minor seventh above the root... |
Cm7, or Cmin7 | min3 | perf5 | min7 |
| Major seventh chord Major seventh chord In music, a major seventh chord is any nondominant seventh chord where the "third" note is a major third above the root.Most typically, major seventh chord refers to where the "seventh" note is a major seventh above the root . This is more precisely known as the major/major seventh chord, and it... |
CM7, or Cmaj7 | maj3 | perf5 | maj7 |
| Augmented seventh chord Augmented seventh chord The augmented seventh chord , or seventh augmented fifth chord,or seventh sharp five chord is a dominant seventh chord consisting of an augmented triad with a minor seventh. Thus, it consists of a root, major third, augmented fifth, and the minor seventh.Thus in the key of C major it would be C,... |
C+7, or Caug7 | maj3 | aug5 | min7 |
| Diminished seventh chord Diminished seventh chord A diminished seventh chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh above the root. Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically... |
C°7, or Cdim7 | min3 | dim5 | dim7 |
| Half-diminished seventh chord Half-diminished seventh chord In music theory, the half-diminished seventh chord is created by taking the root, minor third, diminished fifth and minor seventh of any major scale; for example, C half-diminished is . Its consecutive intervals are minor 3rd, minor 3rd, major 3rd... |
C7 | min3 | dim5 | min7 |
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 harmonicHarmonyIn music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
if the two notes sound simultaneously - Horizontal, linear, or melodicMelodyA melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...
if they sound successively.
Diatonic and chromatic
A diatonic interval is an interval formed by two notes of a diatonic scaleDiatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
. The table on the right depicts all diatonic intervals for C major. A chromatic interval is a non-diatonic interval formed by two notes of a chromatic scale
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
.
Aside from tritones, all intervals that are either augmented or diminished are chromatic, and the rest are diatonic.
Consonant and dissonant
Consonance and dissonanceConsonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
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 the Middle AgesMedieval musicMedieval music is Western music written during the Middle Ages. This era begins with the fall of the Roman Empire and ends sometime in the early fifteenth century...
, only the unison, octave, perfect fourth, and perfect fifth were considered consonant harmonically. - In 15th- and 16th-centuryRenaissance musicRenaissance music is European music written during the Renaissance. Defining the beginning of the musical era is difficult, given that its defining characteristics were adopted only gradually; musicologists have placed its beginnings from as early as 1300 to as late as the 1470s.Literally meaning...
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 TinctorisJohannes TinctorisJohannes Tinctoris was a Flemish composer and music theorist of the Renaissance. He is known to have studied in Orléans, and to have been master of the choir there; he also may have been director of choirboys at Chartres...
) as dissonant, except between the upper parts of a vertical sonority—for example, with a supporting third below ("6-3 chords"). In the common practice periodCommon practice periodThe common practice period, in the history of Western art music , spanning the Baroque, Classical, and Romantic periods, lasted from c. 1600 to c. 1900.-General characteristics:...
, 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, 16th-century practice continued to be taught to beginning musicians throughout this period. - Hermann von HelmholtzHermann von HelmholtzHermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
(1821–1894) defined a harmonically consonant interval as one in which the two pitches have an overtoneOvertoneAn overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
in common (specifically excluding the seventh harmonicHarmonicA harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
). This essentially defines all seconds and sevenths as dissonant, while perfect fourths and fifths, and major and minor thirds and sixths, are consonant. - PythagorasPythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
defined a hierarchy of consonance based on how small the numbers are that express the ratio. 20th-century composer and theorist Paul HindemithPaul HindemithPaul Hindemith was a German composer, violist, violinist, teacher, music theorist and conductor.- Biography :Born in Hanau, near Frankfurt, Hindemith was taught the violin as a child...
'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. - David CopeDavid CopeDavid Cope is an American author, composer, scientist, and professor emeritus of music at the University of California, Santa Cruz...
(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 seriesHarmonic series (music)Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
. See also: Lipps-Meyer lawLipps-Meyer lawThe Lipps–Meyer law, named for Theodor Lipps and Max F. Meyer , hypothesizes that the closure of melodic intervals is determined by "whether or not the end tone of the interval can be represented by the number two or a power of two", in the frequency ratio between notes."The 'Lipps-Meyer' Law...
.
All of the above analyses refer to vertical (simultaneous) intervals.
Simple and compound
A simple interval is an interval spanning at most one octave. Intervals spanning more than one octave are called compound intervals.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. A major seventeenth (two staff positions above two octaves) is another example of compound major third, and can be built either by adding up two octaves and one major third, or four perfect fifths.
Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a seventeenth can be always 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 using four fifths.
The diatonic number DNc of a compound interval formed from n simple intervals with diatonic numbers DN1, DN2, ..., DNn, 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 twenty-second (1+3*(8-1) = 22), and so on.
Intervals larger than a seventeenth 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".
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 semitoneSemitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s or more (e.g., C to D♯), that may also be considered a skip. However, the reverse is not true: a diminished third
Diminished third
In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,...
, an interval comprising two semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s, is still considered a skip.
The words conjunct and disjunct refer to melodies composed of steps and skips, respectively.
Enharmonic intervals
Two intervals are considered to be enharmonicEnharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
, or enharmonically equivalent, if they both contain the same pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
spelled in different ways; that is, if the notes in the two intervals are themselves enharmonically equivalent. Enharmonic intervals span the same number of semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s. For example, as shown in the matrix below, F–A (a major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
), G–B (also a major third), F–B (a diminished fourth
Diminished fourth
In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...
), and G–A (a doubly augmented second
Augmented second
In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...
) are all enharmonically equivalent. In fact, although they have a different name and staff position, F and G indicate the same pitch, and the same is true for A and B. As a consequence, all these intervals span four semitones.
| Number of semitones |
Interval name | Staff positions | |||
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | ||
| 4 | major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
F | A | ||
| 4 | major third | G | B | ||
| 4 | diminished fourth Diminished fourth In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,... |
F | B | ||
| 4 | doubly augmented second Augmented second In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning... |
G | A |
Size of intervals used in different tuning systems
| Number of semitones |
Name | 5-limit tuning (pitch ratio) |
Comparison of interval width (in cents) | |||
|---|---|---|---|---|---|---|
| 5-limit tuning | Pythagorean tuning Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant... |
1/4-comma meantone |
Equal temperament Equal temperament An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for... |
|||
| 0 | Perfect unison | 1:1 | 0 | 0 | 0 | 0 |
| 1 | Minor second | 16:15 | 112 | 90 | 117 | 100 |
| 2 | Major second | 9:8 | 204 | 204 | 193 | 200 |
| 3 | Minor third | 6:5 75:64 32:27 |
316 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 275 294 |
294 318 |
310 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 269 |
300 |
| 4 | Major third | 5:4 512:405 32:25 81:64 |
386 406 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 427 408 |
408 384 |
386 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 427 |
400 |
| 5 | Perfect fourth | 4:3 675:512 27:20 |
498 478 520 |
498 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 522 |
503 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 462 |
500 |
| 6 | Augmented fourth Diminished fifth |
45:32 64:45 |
590 610 |
612 588 |
579 621 |
600 |
| 7 | Perfect fifth | 3:2 40:27 1024:675 |
702 680 722 |
702 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 678 |
697 (wolf) Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... 738 |
700 |
| 8 | Minor sixth | 8:5 | 814 | 792 | 814 | 800 |
| 9 | Major sixth | 5:3 | 884 | 906 | 890 | 900 |
| 10 | Minor seventh | 9:5 16:9 |
1018 996 |
996 | 1007 | 1000 |
| 11 | Major seventh | 15:8 | 1088 | 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, just intervals as provided by 5-limit tuning (see symmetric scale n.1) are shown in bold font, and the values in cents are rounded
Rounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...
to integers. Notice that in each of the non-equal
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
tuning systems, by definition the width of each type of interval (including the semitone) changes depending on the note from which the interval starts. This is the price paid for seeking just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
. However, for the sake of simplicity, for some types of interval the table shows only one value (the most often observed
Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
one).
In 1/4-comma meantone, by definition 11 perfect fifths have a size of approximately 697 cents (700−ε cents, where ε ≈ 3.42 cents); since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of about 700+11ε cents, which is about 738 cents (the wolf fifth); 8 major thirds have size about 386 cents (400−4ε), 4 have size about 427 cents (400+8ε), and their average size is 400 cents. In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε (the difference between the 1/4-comma meantone fifth and the average fifth). A more detailed analysis is provided at 1/4-comma meantone#Size of intervals. Note that 1/4-comma meantone was designed to produce just major thirds, but only 8 of them are just (5:4, about 386 cents).
The Pythagorean tuning
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
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 temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
s, including 1/4-comma meantone, it is tempered to a size smaller than 700. A more detailed analysis is provided at Pythagorean tuning#Size of intervals.
The 5-limit tuning system uses just tones and semitones as building blocks, rather than a stack of perfect fifths, and this leads to even more varied intervals throughout the scale (each kind of interval has three or four different sizes). A more detailed analysis is provided at 5-limit tuning#Size of intervals. Note that 5-limit tuning was designed to maximize the number of just intervals, but even in this system some intervals are not just (e.g., 3 fifths, 5 major thirds and 6 minor thirds are not just; also, 3 major and 3 minor thirds are wolf interval
Wolf interval
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament...
s).
The above mentioned symmetric scale 1, defined in the 5-limit tuning system, is not the only method to obtain just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
. It is possible to construct juster intervals or just intervals closer to the equal-tempered equivalents, but most of the ones listed above have been used historically in equivalent contexts. In particular, the asymmetric version of the 5-limit tuning scale provides a juster value for the minor seventh (9:5, rather than 16:9). Moreover, the tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
(augmented fourth or diminished fifth), could have other just ratios; for instance, 7:5 (about 583 cents) or 17:12 (about 603 cents) are possible alternatives for the augmented fourth (the latter is fairly common, as it is closer to the equal-tempered value of 600 cents). The 7:4 interval (about 969 cents), also known as the harmonic seventh
Harmonic seventh
The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of...
, has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh. Some assert the 7:4 is one of the blue note
Blue note
In jazz and blues, a blue note is a note sung or played at a slightly lower pitch than that of the major scale for expressive purposes. Typically the alteration is a semitone or less, but this varies among performers and genres. Country blues, in particular, features wide variations from the...
s used in jazz
Jazz
Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...
. For further details about reference ratios, see 5-limit tuning#The justest ratios.
In the diatonic system, every interval has one or more enharmonic
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
equivalents, such as augmented second
Augmented second
In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...
for minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
.
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 microtones. Except for the quarter tone, the equivalents in centsCent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
are approximate, and they can be also classified as comma
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
s, as they describe small discrepancies, observed in some tuning systems, between enharmonically equivalent
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
notes.
- A Pythagorean commaPythagorean commaIn musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
is the difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequencyFrequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
ratio 531441:524288 (23.5 cents). - A syntonic commaSyntonic commaIn music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
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 commaSeptimal commaA septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63....
is 64:63 (27.3 cents), and is the difference between the Pythagorean or 3-limit "7th" and the "harmonic 7th". - A diesisDiesisIn classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
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 diesisDiesisIn classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
for details. - A diaschismaDiaschismaThe diaschisma is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds . It can be represented by the ratio 2048:2025 and is about 19.5 cents...
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 schismaSchismaIn music, the schisma is the ratio between a Pythagorean comma and a syntonic comma and equals 32805:32768, which is 1.9537 cents...
(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 kleismaKleismaIn music theory and tuning, the kleisma, or semicomma majeur, is a minute and barely perceptible comma type interval important to musical temperaments. It is the difference between six justly tuned minor thirds and one justly tuned tritave or perfect twelfth...
is the difference between six minor thirdMinor thirdIn classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
s and one tritave or perfect twelfth (an octaveOctaveIn music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
plus a perfect fifthPerfect fifthIn classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
), with a frequency ratio of 15625:15552 (8.1 cents) . - A septimal kleismaSeptimal kleismaIn music, the ratio 225/224 is called the septimal kleisma .It is a minute comma type interval of approximately 7.7 cents. Factoring it into primes gives 2−5 32 52 7−1, which can be rewritten 2−1 2 . That says that it is the amount that two major thirds of 5/4 and a septimal...
is six major thirds up, five fifths down and one octave up, with ratio 225:224 (7.7 cents). - A quarter toneQuarter toneA quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole tone....
is half the width of a semitoneSemitoneA semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
, which is half the width of a whole toneMajor secondIn Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...
. It is equal to exactly 50 cents.
See Musical interval mnemonics at Wikibooks for popular musical fragments that feature common intervals
Inversion
A simple interval (i.e., an interval shorter than or equal to an octave) may be invertedInversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
by raising the lower pitch an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, 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:
- 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."
For intervals identified by their 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.
For intervals identified by an integer number of semitones, the inversion is obtained by subtracting that number from 12.
Since an interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
is the lower number selected among the interval integer and its inversion, interval classes cannot be inverted.
Interval root
Although intervals are usually designated in relation to their lower note, David CopeDavid Cope
David Cope is an American author, composer, scientist, and professor emeritus of music at the University of California, Santa Cruz...
and Hindemith
Paul Hindemith
Paul Hindemith was a German composer, violist, violinist, teacher, music theorist and conductor.- Biography :Born in Hanau, near Frankfurt, Hindemith was taught the violin as a child...
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 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.
Interval cycles
Interval cycleInterval cycle
In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached In music, an interval cycle is a collection...
s, "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
George Perle
George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...
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.
Alternative interval naming conventions
As shown below, some of the above mentioned intervals have alternative names, and some of them take a specific alternative name in Pythagorean tuningPythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, five-limit tuning
Five-limit tuning
Five-limit tuning, or 5-limit tuning is a method to obtain a justly tuned musical scale. The frequencies of the notes of such a scale are obtained by multiplying the frequency of a given reference note by powers of 2, 3, or 5, or a combination of them...
, or meantone temperament tuning systems such as quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
. Notice that ditone and semiditone are specific for Pythagorean tuning, while tone and tritone are used generically for all tuning systems. Interestingly, despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents). All the intervals with prefix sesqui- are justly
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
tuned, and their frequency ratio
Interval ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2 , 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12, 1.498...
, shown in the table, is a superparticular number (or epimoric ratio). The same is true for the octave.
The diminished second
Diminished second
In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C...
is a comma
Comma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
, but some commas are not diminished seconds. For instance, the Pythagorean comma
Pythagorean comma
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
(531441:524288) is the opposite of a diminished second. 5-limit 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
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
(81:80) can neither be regarded as a diminished second, nor as its opposite. See Five-limit tuning#Diminished seconds for further details.
| Number of semitones | |Specific names | ||||||
|---|---|---|---|---|---|---|---|
| Quality and number | Other naming conventions | Pythagorean tuning Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant... | |1/4-comma meantone |
||||
| Full | Short | ||||||
| 0 | diminished second Diminished second In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C or B and C... |
d2 | descending Pythagorean comma Pythagorean comma In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C... (524288:531441) |
lesser diesis Diesis In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents... (128:125) |
|||
| diaschisma Diaschisma The diaschisma is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds . It can be represented by the ratio 2048:2025 and is about 19.5 cents... (2048:2025) greater diesis (648:625) |
|||||||
| 1 | minor second Minor second In modern Western tonal music theory a minor second is the interval between two notes on adjacent staff positions, or having adjacent note letters, whose alterations cause them to be one semitone or half-step apart, such as B and C or C and D.... |
m2 | semitone, half tone, half step |
diatonic semitone, minor semitone |
limma (256:243) | ||
| augmented unison Augmented unison In modern Western tonal music theory an augmented unison is the interval between two notes on the same staff position, or having the same note letter, whose alterations cause them, in ordinary equal temperament, to be one semitone apart. In other words, it is a unison where one note has been raised... |
A1 | chromatic semitone, major semitone |
apotome (2187:2048) | ||||
| 2 | diminished third Diminished third In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,... |
d3 | tone, whole tone, whole step | ||||
| major second Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions... |
M2 | sesquioctavum (9:8) | |||||
| 3 | minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... |
m3 | semiditone (32:27) | sesquiquintum (6:5) | |||
| 4 | major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
M3 | ditone Ditone In music, a ditone is the interval 81:64, which is 407.82 cents. It is the major third in Pythagorean tuning. The ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma... (81:64) |
sesquiquartum (5:4) | |||
| 5 | perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there... |
P4 | diatessaron | sesquitertium (4:3) | |||
| 6 | diminished fifth | d5 | tritone (TT) | ||||
| 6 | augmented fourth | A4 | |||||
| 7 | perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... |
P5 | diapente | sesquialterum (3:2) | |||
| 12 | (perfect) octave Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"... |
P8 | diapason | 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
Indian classical music
The origins of Indian classical music can be found in the Vedas, which are the oldest scriptures in the Hindu tradition. Indian classical music has also been significantly influenced by, or syncretised with, Indian folk music and Persian music. The Samaveda, one of the four Vedas, describes music...
.
Pitch-class intervals
Post-tonal or atonalAtonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...
theory, originally developed for equal tempered European classical music written using the twelve tone technique or serialism
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...
, 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.
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 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. For unordered pitch-class intervals, see interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
.
Generic and specific intervals
In diatonic set theoryDiatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and...
, specific
Specific interval
In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal...
and generic interval
Generic interval
In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members...
s are distinguished. Specific intervals are the interval class or number of semitones between scale steps or collection members, and generic intervals are the number of diatonic scale steps (or staff positions) between notes of a collection or scale.
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 Specific interval In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal... |
Generic interval Generic interval In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members... |
Diatonic name | |
|---|---|---|---|
| Number of semitones | Interval class Interval class In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod... |
||
| 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 non-pitch uses
The term "interval" can also be generalized to other music elements besides pitch. David LewinDavid Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
's Generalized Musical Intervals and Transformations uses interval as a generic measure of distance to show musical transformations that can change—for instance—one rhythm into another, or one formal structure into another.
See also
- Circle of fifthsCircle of fifthsIn music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
- List of musical intervals
- List of meantone intervals
- Ear trainingEar trainingEar training or aural skills is a skill by which musicians learn to identify, solely by hearing, pitches, intervals, melody, chords, rhythms, and other basic elements of music. The application of this skill is analogous to taking dictation in written/spoken language. Ear training may be...
- Pseudo-octavePseudo-octaveA pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...
- Regular temperamentRegular temperamentRegular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios...
External links
- Interval conversion: Frequency ratio to cents and cents to frequency ratio
- Encyclopaedia Britannica, Interval
- Morphogenesis of chords and scales Chords and scales classification
- Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings