In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
In Western music, intervals are most commonly differencing between notes of a diatonic scale. Intervals between successive notes of a scale are also known as scale steps. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C and D. Intervals can be arbitrarily small, and even imperceptible to the human ear.
In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio.
In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.). Examples include the minor third or perfect fifth. These names identify not only the difference in semitones between the upper and lower notes but also how the interval is spelled. The importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G and G–A.
Size
The size of an interval (also known as its width) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.
Frequency ratios
The size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (unison), 2:1 (octave), 5:3 (major sixth), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Intervals with small-integer ratios are often called just intervals, or pure intervals.
Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called 12-tone equal temperament. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, although it is very close to the size of the corresponding just intervals. For instance, an equal-tempered fifth has a frequency ratio of 2:1, approximately equal to 1.498:1, or 2.997:2 (very close to 3:2). For a comparison between the size of intervals in different tuning systems, see .
Cents
The standard system for comparing interval sizes is with cents. The cent is a logarithmic unit of measurement. If frequency is expressed in a logarithmic scale, and along that scale the distance between a given frequency and its double (also called octave) is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament (12-TET), a tuning system in which all semitones have the same size, the size of one semitone is exactly 100 cents. Hence, in 12-TET the cent can be also defined as one hundredth of a semitone.
Mathematically, 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 scale. A perfect unison (also known as perfect prime)"Prime (ii). See Unison", Grove Music Online. Oxford University Press. Accessed August 2013. ) is an interval formed by two identical notes. Its size is zero cents. A semitone is any interval between two adjacent notes in a chromatic scale, a whole tone is an interval spanning two semitones (for example, a major second), and a tritone is an interval spanning three tones, or six semitones (for example, an augmented fourth). Rarely, the term ditone is also used to indicate an interval spanning two whole tones (for example, a major third), or more strictly as a synonym of major third.
Intervals with different names may span the same number of semitones, and may even have the same width. For instance, the interval from D to F is a major third, while that from D to G is a diminished fourth. However, they both span 4 semitones. If the instrument is tuned so that the 12 notes of the chromatic scale are equally spaced (as in equal temperament), these intervals also have the same width. Namely, all semitones have a width of 100 cents, and all intervals spanning 4 semitones are 400 cents wide.
The names listed here cannot be determined by counting semitones alone. The rules to determine them are explained below. Other names, determined with different naming conventions, are listed in a separate section. Intervals smaller than one semitone (commas or microtones) and larger than one octave (compound intervals) are introduced below.
Number ofsemitones Minor, major,or perfect intervals Short Augmented ordiminished intervals Short Widely usedalternative names Short Audio 0 Perfect unison P1 Diminished second d2 1 Minor second m2 Augmented unison A1 Semitone, half tone, half step S 2 Major second M2 Diminished third d3 Tone, whole tone, whole step T 3 Minor third m3 Augmented second A2 Trisemitone 4 Major third M3 Diminished fourth d4 5 Perfect fourth P4 Augmented third A3 6 Diminished fifth d5 Tritone TT Augmented fourth A4 7 Perfect fifth P5 Diminished sixth d6 8 Minor sixth m6 Augmented fifth A5 9 Major sixth M6 Diminished seventh d7 10 Minor seventh m7 Augmented sixth A6 11 Major seventh M7 Diminished octave d8 12 Perfect octave P8 Augmented seventh A7
Interval number and quality
In Western music theory, an interval is named according to its number (also called diatonic number, interval size or generic interval) and quality. For instance, major third (or M3) is an interval name, in which the term major (M) describes the quality of the interval, and third (3) indicates its number.
Number
The number of an interval is the number of letter names or staff positions (lines and spaces) it encompasses, including the positions of both notes forming the interval. For instance, the interval B–D is a third (denoted m3) because the notes from B to the D above it encompass three letter names (B, C, D) and occupy three consecutive staff positions, including the positions of B and D. The table and the figure above show intervals with numbers ranging from 1 (e.g., P1) to 8 (e.g., d8). Intervals with larger numbers are called compound intervals.
There is a one-to-one correspondence between staff positions and diatonic-scale degrees (the notes of the diatonic scale). This means that interval numbers can also be determined by counting diatonic-scale degrees, rather than staff positions, provided that the two notes that form the interval are drawn from a diatonic scale. Namely, B–D is a third because in any diatonic scale that contains B and D, the sequence from B to D includes three notes. For instance, in the B-natural minor diatonic scale, the three notes are B–C–D. This is not true for all kinds of scales. For instance, in a chromatic scale, there are four notes from B to D: B–C–C–D. This is the reason interval numbers are also called diatonic numbers, and this convention is called diatonic numbering.
If one adds any accidentals to the notes that form an interval, by definition the notes do not change their staff positions. As a consequence, any interval has the same interval number as the corresponding natural interval, formed by the same notes without accidentals. For instance, the intervals B–D (spanning 4 semitones) and B–D (spanning 2 semitones) are thirds, like the corresponding natural interval B–D (3 semitones).
Notice that interval numbers represent an inclusive count of encompassed staff positions or note names, not the difference between the endpoints. In other words, one starts counting the lower pitch as one, not zero. For that reason, the interval E–E, a perfect unison, is also called a prime (meaning "1"), even though there is no difference between the endpoints. Continuing, the interval E–F is a second, but F is only one staff position, or diatonic-scale degree, above E. Similarly, E–G is a third, but G is only two staff positions above E, and so on. As a consequence, joining two intervals always yields an interval number one less than their sum. For instance, the intervals B–D and D–F are thirds, but joined they form a fifth (B–F), not a sixth. Similarly, a stack of three thirds, such as B–D, D–F, and F–A, is a seventh (B–A), not a ninth.
This scheme applies to intervals up to an octave (12 semitones). For larger intervals, see below.
Quality
The name of any interval is further qualified using the terms perfect (P), major (M), minor (m), augmented (A), and diminished (d). This is called its interval quality (or modifier). It is possible to have doubly diminished and doubly augmented intervals, but these are quite rare, as they occur only in chromatic contexts. The combination of number (or generic interval) and quality (or modifier) is called the specific interval, diatonic interval (sometimes used only for intervals appearing in the diatonic scale), or simply interval.
The quality of a compound interval is the quality of the simple interval on which it is based. Some other qualifiers like neutral, subminor, and supermajor are used for non-diatonic intervals.
Perfect
Perfect intervals are so-called because they were traditionally considered perfectly consonant,Definition of perfect consonance from although in Western classical music the perfect fourth was sometimes regarded as a less than perfect consonance, when its function was contrapuntal. Conversely, minor, major, augmented, or diminished intervals are typically considered less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or near-dissonances.
Within a diatonic scale all unisons (P1) and octaves (P8) are perfect. Most fourths and fifths are also perfect (P4 and P5), with five and seven semitones respectively. One occurrence of a fourth is augmented (A4) and one fifth is diminished (d5), both spanning six semitones. For instance, in an E-major scale, the A4 is between A and D, and the d5 is between D and A.
The inversion of a perfect interval is also perfect. Since the inversion does not change the pitch class of the two notes, it hardly affects their level of consonance (matching of their harmonics). Conversely, other kinds of intervals have the opposite quality with respect to their inversion. The inversion of a major interval is a minor interval, and the inversion of an augmented interval is a diminished interval.
Major and minor