Interval (music) | Understanding Western Music Theory

For albums named Interval , see Inter-val (disambiguation).

In music theory, an Inter-val is a difference between

Melodic and harmonic intervals. Play

two pitches. An Inter-val 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 differences between notes of a diatonic scale. 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 Inter-val is a 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 Inter-val result in an exponential increase in 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 describe not only the difference in semitones between the upper and lower notes but also how the Inter-val 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

Example: Perfect octave on C in equal temperament and just intonation: 2/1 = 1200 cents. Play

The size of an Inter-val (also known as its width or height) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.

Frequency ratios

The size of an Inter-val 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), 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. To most people, just intervals sound consonant, that is, pleasant and well-tuned.

Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called 12-tone

Inter-val NUMBER AND QUALITY

equal temperament, in which the main intervals are typically perceived as a consonant, but none is justly tuned and as consonant as a just interval, except for the unison (1:1) and octave (2:1). 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 2^7/12:1, 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 section Size in different tuning systems.

Cents

The standard system for comparing Inter-val sizes is with cents. A cent is a logarithmic unit of measurement. If the frequency is expressed in a logarithmic scale, and along with 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 Inter-val 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 a perfect prime) is an Inter-val formed by two identical notes. Its size is zero cents. A semitone is any Inter-val between two adjacent notes in a chromatic scale, a whole tone is an Inter-val spanning two semitones (for example, a major second), and a tritone is an Inter-val spanning three tones, or six semitones (for example, an augmented fourth).

 Rarely, the term ditone is also used to indicate an Inter-val spanning two whole tones (for example, a major third), or more strictly as a synonym of a major third.

Intervals with different names may span the same number of semitones, and may even have the same width. For instance, the Inter-val 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 will also have the same width. Namely, all semitones will have a width of 100 cents, and all intervals spanning 4 semitones will be 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.

Inter-val number and quality

In Western music theory, an Inter-val is named according to its number (also called diatonic number) and quality. For instance, major third (or M3) is an Inter-val name, in which the term major (M) describes the quality of the interval, and third (3) indicates its number.

Number

The number of an Inter-val is the number of staff positions it encompasses. Both lines and spaces (see figure) are counted, including the positions of both notes forming the interval. For instance, the Inter-val C–G is a fifth (denoted P5) because the notes from C to G occupy five consecutive staff positions, including the positions of C and G. The table and the figure above show intervals with numbers ranging from 1 (e.g., P1) to 8 (e.g., P8). 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 a diatonic

scale). This means that Inter-val numbers can be also determined by counting diatonic scale degrees, rather than staff positions, provided that the two notes which form the Inter-val are drawn from a diatonic scale. Namely, C– G is a fifth because in any diatonic scale that contains C and G, the sequence from C to G includes five notes.

For instance, in the A♭-major diatonic scale, the five notes are C–D♭–E♭–F–G (see figure). This is not true for all kinds of scales. For instance, in a chromatic scale, the notes from C to G are eight (C–C♯–D–D♯–E–F–F♯–G). This is the reason Inter-val numbers are also called diatonic numbers, and this convention is called diatonic numbering.

If one takes away any accidentals from the notes which form an interval, by definition the notes do not change their staff positions. As a consequence, any Inter-val has the same Inter-val number as the corresponding natural interval, formed by the same notes without accidentals. For instance, the intervals C–G♯ (spanning 8 semitones) and C♯–G (spanning 6 semitones) are fifths, like the corresponding natural Inter-val C–G (7 semitones).

Inter-val numbers do not represent exactly Inter-val widths. For instance, the Inter-val C–D is a second, but D is only one staff position, or diatonic-scale degree, above C. Similarly, C–E is a third, but E is only two staff positions above C, and so on.

As a consequence, joining two intervals always yields an Inter-val number one less than their sum. For instance, the intervals C–E and E–G are thirds, but joined together they form a fifth (C–G), not a sixth. Similarly, a stack of three thirds, such as C–E, E–G, and G–B, is a seventh (C–B), not a ninth.

The rule to determine the diatonic number of a compound Inter-val (an Inter-val 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 Inter-val is further qualified using the terms perfect (P), major (M), minor (m), augmented (A), and diminished (d). This is called its Inter-val 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 Inter-val is the quality of the simple Inter-val on which it is based.

Perfect

Perfect intervals are so-called because they were traditionally considered perfectly consonant, 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 to be less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or 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. There’s one occurrence of a fourth and a fifth which are not perfect, as they both span six semitones: an augmented fourth (A4), and its inversion, a diminished fifth (d5). For instance, in a C-major scale, the A4 is between F and B, and the d5 is between B and F (see table).

By definition, the inversion of a perfect Inter-val is also perfect. Since the inversion does not change the pitch 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 Inter-val is a minor interval, and the inversion of an augmented Inter-val is a diminished interval.

Major/minor

Major and minor intervals on C. m2 , M2 , m3 , M3 , m6 , M6 , m7 , M7

As shown in the table, a diatonic scale defines seven intervals for each Inter-val number, each starting from a different note (seven unisons, seven seconds, etc.). The intervals formed by the notes of a diatonic scale are called diatonic. Except for unisons and octaves, the diatonic intervals with a given Inter-val number always occur in two

SHORTHAND NOTATION

sizes, which differ by one semitone. For example, six of the fifths span seven semitones. The other one spans six semitones. Four of the thirds span three semitones, the others four. If one of the two versions is a perfect interval, the other is called either diminished (i.e. narrowed by one semitone) or augmented (i.e. widened by one semitone). Otherwise,

the larger version is called major, the smaller one minor. For instance, since a 7-semitone fifth is a perfect Inter-val (P5), the 6-semitone fifth is called a “diminished fifth” (d5). Conversely, since neither kind of third is perfect, the larger one is called “major third” (M3), and the smaller one “minor third” (m3).

Within a diatonic scale, unisons and octaves are always qualified as perfect, fourths as either perfect or augmented, fifths as perfect or diminished, and all the other intervals (seconds, thirds, sixths, sevenths) as major or minor.

Augmented/diminished

Augmented and diminished intervals on C. d2 , A2 , d3 , A3 , d4 , A4 , d5 , A5 , d6 , A6 , d7 , A7 , d8 , A8

Augmented and diminished intervals are so-called because they exceed or fall short of either a perfect Inter-val or a major/minor pair by one semitone while having the same Inter-val number (i.e., encompassing the same number of staff positions). For instance, an augmented third such as C–E♯ spans five semitones, exceeding a major third (C–E) by one semitone, while a diminished third such as C♯–E♭ spans two semitones, falling short of a minor third (C–E♭) by one semitone.

Except for the above-mentioned augmented fourth (A4) and diminished fifth (d5), augmented and diminished intervals do not appear in diatonic scales (see table).

Example

Neither the number nor the quality of an Inter-val 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 B♯, between A and C♯, between A and D♭, and between A♯ and E , but • A♭–B♯ is a second, as it encompasses two staff positions (A, B), and it is doubly augmented, as it exceeds a major second (such as A-B) by two semitones.

1. A–C♯ is a third, as it encompasses three staff positions (A, B, C), and it is major, as it spans 4 semitones.
2. A–D♭ is a fourth, as it encompasses four staff positions (A, B, C, D), and it is diminished, as it falls short of a perfect fourth (such as A-D) by one semitone.
3. A♯-Etions (A, B, C, D, E), and it is triply diminished, asis a fifth, as it encompasses five staff posiit falls short of a perfect fifth (such as A-E) by three semitones.

Shorthand notation

Intervals are often abbreviated with a P for perfect, m for minor, M for major, d for diminished, and A for augmented, followed by the Inter-val number. The indications M and P are often omitted. The octave is P8, and a unison is usually referred to simply as “a unison” but can be labeled P1. The tritone, an augmented fourth or diminished fifth is often TT. The Inter-val qualities may be also abbreviated with perf, min, maj, dim, aug. Examples:

1. m2 (or min2): minor second,
2. M3 (or maj3): major third,
3. A4 (or aug4): augmented fourth, d5 (or dim5): diminished fifth,
4. P5 (or perf5): perfect fifth.

 Inversion

A simple Inter-val (i.e., an Inter-val smaller than or equal

Inter-val inversions

to an octave) may be inverted by raising the lower pitch an octave, or lowering the upper pitch an octave. For example, the fourth from a lower C to a higher F may be inverted to make a fifth, from a lower F to a higher C.

There are two rules to determine the number and quality of the inversion of any simple interval:

• The Inter-val number and the number of its inversion always add up to nine (4 + 5 = 9, in the example, just given).

Major 13th (compound Major 6th) inverts to a minor 3rd by moving the bottom note up two octaves, the top note down two octaves, or both notes one octave

• The inversion of a major Inter-val is a minor Inter-val and vice versa; the inversion of a perfect Inter-val is also perfect; the inversion of an augmented Inter-val is a diminished interval, and vice versa; the inversion of a doubly augmented Inter-val is a doubly diminished Inter-val and vice versa.

For example, the Inter-val from C to the E♭ above it is a minor third. By the two rules just given, the Inter-val 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 Inter-val is always the same as the inversion of the simple Inter-val 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 Inter-val class is the lower number selected among the Inter-val integer and its inversion, Inter-val classes cannot be inverted.

Classification

Intervals can be described, classified, or compared with each other according to various criteria.

Melodic and harmonic intervals. Play

Melodic and harmonic

An Inter-val can be described as

• Vertical or harmonic if the two notes sound simultaneously
• Horizontal, linear,     or melodic if they sound successive.

Diatonic and chromatic

In general,

• A diatonic Inter-val is an Inter-val formed by two notes of a diatonic scale.
• A chromatic Inter-val is a non-diatonic Inter-val formed by two notes of a chromatic scale.

Ascending and descending chromatic scale on C Play.

The table above depicts the 56 diatonic intervals formed by the notes of the C major scale (a diatonic scale). Notice that these intervals, as well as any other diatonic interval, can be also formed by the notes of a chromatic scale.

The distinction between diatonic and chromatic intervals is controversial, as it is based on the definition of the diatonic scale, which is variable in the literature. For example, the Inter-val B–E♭ (a diminished fourth, occurring in the harmonic C-minor scale) is considered diatonic if the harmonic minor scales are considered diatonic as well. Otherwise, it is considered chromatic. For further details, By a commonly used definition of diatonic scale (which excludes the harmonic minor and melodic minor scales), all perfect, major, and minor intervals are diatonic. Conversely, no augmented or diminished Inter-val is diatonic, except for the augmented fourth and diminished fifth.

The A♭-major scale. Play

The distinction between diatonic and chromatic intervals may be also sensitive to context. The above-mentioned

CLASSIFICATION

56 intervals formed by the C-major scale are sometimes called diatonic to C major. All other intervals are called chromatic to C major. For instance, the perfect fifth A♭– E♭ is chromatic to C major, because A♭ and E♭ are not contained in the C major scale. However, it is diatonic to others, such as the A♭ major scale.

Consonant and dissonant

Consonance and dissonance are relative terms that refer to the stability, or state of repose, of particular musical effects. Dissonant intervals are those that cause tension and desire to be resolved to consonant intervals.

These terms are relative to the usage of different compositional styles.

1. In the Middle Ages, only the unison, octave, perfect fourth, and perfect fifth were considered consonant harmonically.
2. In 15th- and 16th-century usage, perfect fifths and octaves, and major and minor thirds and sixths were considered harmonically consonant, and all other intervals dissonant, including the perfect fourth, which by 1473 was described (by Johannes Tinctoris) as dissonant, except between the upper parts of a vertical sonority—for example, with a supporting third below (“6-3 chords”). In the common practice period, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously thought to be dissonant (such as minor sevenths) became acceptable in certain contexts. However, the 16th-century practice continued to be taught to beginning musicians throughout this period.
3. Hermann von Helmholtz (1821–1894) defined a harmonically consonant Inter-val as one in which the two pitches have an upper part (an overtone) in common (specifically excluding the seventh harmonic). This essentially defines all seconds and sevenths as dissonant, and the above thirds, fourths, fifths, and sixths as a consonant.
4. Pythagoras defined a hierarchy of consonance based on how small the numbers are that express the ratio. 20th-century composer and theorist Paul Hindemith’s system has a hierarchy with the same results as Pythagoras’s but defined by fiat rather than by Inter-val ratios, to better accommodate equal temperament, all of whose intervals (except the octave) would be dissonant using acoustical methods.
5. David Cope (1997) suggests the concept of Inter-val strength, in which an interval’s strength, consonance, or stability is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps– Meyer law.

Simple and compound

Simple and compound major third. Play

A simple Inter-val is an Inter-val spanning at most one octave (see Main intervals above). Intervals spanning more than one octave are called compound intervals, as they can be obtained by adding one or more octaves to a simple Inter-val (see below for details).

Steps and skips

Linear (melodic) intervals may be described as steps or skips. A step, or conjunct motion, is a linear Inter-val between two consecutive notes of a scale. Any larger Inter-val is called a skip (also called a leap), or disjunct motion. In the diatonic scale, a step is either a minor second (sometimes also called half step) or major second (sometimes also called the whole step), with all intervals of a minor third or larger being skips.

For example, C to D (major second) is a step, whereas C to E (major third) is a skip.

More generally, a step is a smaller or narrower Inter-val in a musical line, and a skip is a wider or larger interval, with the categorization of intervals into steps and skips determined by the tuning system and the pitch space used.

Melodic motion in which the Inter-val between any two consecutive pitches is no more than a step, or, less strictly, where skips are rare, is called stepwise or conjunct melodic motion, as opposed to skipwise or disjunct melodic motions, characterized by frequent skips.

Enharmonic intervals

Two intervals are considered to be enharmonic, or en-

Enharmonic tritones: A4 = d5 on C Play .

harmonically equivalent, if they both contain the same pitches spelled in different ways; that is if the notes in the two intervals are themselves enharmonically equivalent. Enharmonic intervals span the same number of semitones.

For example, the four intervals listed in the table below are all enharmonically equivalent, because the notes F♯ and G♭ indicate the same pitch, and the same is true for A♯ and B♭. All these intervals span four semitones.

When played on a piano keyboard, these intervals are indistinguishable as they are all played with the same two keys, but in a musical context, the diatonic function of the notes incorporated is very different.

Minute intervals

Pythagorean comma on C. Play . The note depicted as lower on the staff (B♯+++) is slightly higher in pitch (than C♮).

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, and some of them can be also classified as commas, as they describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes. In the following list, the Inter-val sizes in cents are approximate.

1. A Pythagorean comma is a difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequency ratio 531441:524288 (23.5 cents).
2. A syntonic comma is a difference between four justly tuned perfect fifths and two octaves plus a major third. It is expressed by the ratio of 81:80 (21.5 cents).
3. A septimal comma is 64:63 (27.3 cents), and is the difference between the Pythagorean or 3-limit “7th” and the “harmonic 7th”.
4. A diesis is generally used to mean the difference between three justly tuned major thirds and one octave. It is expressed by the ratio of 128:125 (41.1 cents). However, it has been used to mean other small intervals: see diesis for details.
5. A diaschisma is the difference between three octaves and four justly tuned perfect fifths plus two justly tuned major thirds. It is expressed by the ratio 2048:2025 (19.6 cents).
6. A schisma (also skhisma) is the difference between five octaves and eight justly tuned fifths plus one justly tuned major third. It is expressed by the ratio 32805:32768 (2.0 cents). It is also the difference between the Pythagorean and syntonic commas. (A schismatic major third is a schisma different from a just major third, eight fifths down and five octaves up, F♭ in C.)
7. A kleisma is the difference between six minor thirds and one tritave or perfect twelfth (an octave plus a perfect fifth), with a frequency ratio of 15625:15552 (8.1 cents) ( Play ).
8. A septimal kleisma is six major thirds up, five fifths down, and one octave up, with a ratio of 225:224 (7.7 cents).
9. A quarter tone is half the width of a semitone, which is half the width of a whole tone. It is equal to exactly 50 cents.

Compound intervals

A compound Inter-val is an interval spanning more than one octave. Conversely, intervals spanning at most one octave are called simple intervals (see Main intervals above).

In general, a compound interval may be defined by a sequence or “stack” of two or more simple intervals of any

5.9. INTERVALS IN CHORDS

kind. For instance, a major tenth (two staff positions above one octave), also called a compound major third, spans one octave plus one major third.

Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a major seventeenth can be decomposed into two octaves and one major third, and this is the reason why it is called a compound major third, even when it is built by adding up four-fifths.

The diatonic number DN of a compound interval formed from n simple intervals with diatonic numbers DN₁, DN₂, …, DN, is determined by:

DN_c = 1+(DN₁−1)+(DN₂−1)+…+(DN_n−1),

which can also be written as:

DN_c = DN₁ + DN₂ + … + DN_n − (n − 1),

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 twenty-second (1+3*(8–1) = 22), and so on.

Main compound intervals

It is also worth mentioning here the major seventeenth (28 semitones), an interval larger than two octaves which can be considered a multiple of a perfect fifth (7 semitones) as it can be decomposed into four perfect fifths (7 * 4 = 28 semitones), or two octaves plus a major third (12 + 12 + 4 = 28 semitones). Intervals larger than a major seventeenth seldom 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”.

Intervals in chords

Chords are sets of three or more notes. They are typically defined as the combination of intervals starting from a common note called the 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.

Chordqualitiesandintervalqualities

The main chord qualities are major, 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, ^ø 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 in Rules to decode chord names and symbols.

For 3-note chords (triads), major or minor always refer to the interval of the third above the root note, while augmented and diminished always refer to the interval of the fifth above the root. The same is true for the corresponding symbols (e.g., Cm means Cm3, and C+ means C+5). Thus, the terms third 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 inter-val 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 inter-val qualities, rather than chord qualities.

For instance, in Cm/M7 (minor major seventh chord), m is the chord quality and refers to the m3 inter-val, while M refers to the M7 inter-val. When the number of an extra interval is specified immediately after chord quality, the quality of that inter-val 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). S

2. Without contrary information, a major third inter-val and a perfect fifth inter-val (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 7th by definition (see below). This rule has one exception (see next rule).
3. 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).
4. Names and symbols which contain only a plain inter-val number (e.g., “Seventh chord”) or the chord root 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 chord (e.g., C6 = CM6 = Cadd6) and contains, together with the implied major triad, an extra major 2nd, perfect 4th, or major 6th (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 5, the chord (technically not a chord in the traditional sense, but a dyad) is a power chord. Only the root, a perfect fifth, and usually an octave are played.

The table shows the inte-rvals contained in some of the main chords (component inte-rvals), and some of the symbols used to denote them. The inte-rval 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 a chord root.

Size of inter-vals used in different tuning systems

In this table, the inte-rval widths used in four different tuning systems are compared. To facilitate comparison, just inter-vals 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 nonequal tuning systems,

by definition the width of each type of inter-val (including the semitone) changes depending on the note from which the interv-al starts. This is the price paid for seeking just intonation. However, for the sake of simplicity, for some types of inter-val, 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 738 cents (700+11ε, the wolf fifth or diminished sixth); 8 major thirds have size about 386 cents (400−4ε), 4 have size about 427 cents (400+8ε, actually diminished fourths), and their average size is 400 cents.

In short, similar differences in width are observed for all inter-val 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 inter-vals. 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 temperaments, 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 inter-vals.

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 inte-rvals throughout the scale (each kind of interv-al has three or four different sizes). A more detailed analysis is provided at 5-limit tuning#Size of inter-vals. Note that 5-limit tuning was designed to maximize the number of just intervals, but even in this system, some inte-rvals 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 inter-vals).

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 inter-vals or just inter-vals 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 inte-rval (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 that 7:4 is one of the blue notes used in jazz. For further details about reference ratios, see 5-limit tuning#The justest ratios.

In the diatonic system, every inter-val has one or more enharmonic equivalents, such as augmented second for minor third.

PITCH-CLASS INTER-VALS

Interval root

Although inter-vals are usually designated in relation to their lower note, David Cope and Hindemith both suggest the concept of interval root. To determine an interval’s root, one locates its nearest approximation in the harmonic series.

The root of a perfect fourth, then, is its top note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered inter-val are the roots, as are the tops of all even-numbered inter-vals. The root of a collection of intervals or a chord is thus determined by the interv-al root of its strongest interv-al.

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 to the inter-val root of the strongest inter-val of the chord (in first inversion, CEGA), the perfect fifth (C–G), is the bottom C, the tonic.

Interval cycles

Interval cycles, “unfold [i.e., repeat] a single recurrent interval in a series that closes with a return to the initial pitch class”, and are notated by George Perle using the letter “C”, for cycle, with an 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 tuning, five-limit tuning, or meantone temperament tuning systems such as quarter-comma meantone. All the intervals with the 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.

Typically, a comma is a diminished second, but this is not always true (for more details, see Alternative definitions of comma). For instance, in Pythagorean tuning the diminished second is a descending interval (524288:531441, or about −23.5 cents), and the Pythagorean comma is its opposite (531441:524288, or about 23.5 cents).

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 Diminished seconds in 5-limit tuning 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.

Latin nomenclature

Up to the end of the 18th century, Latin was used as an official language throughout Europe for scientific and music textbooks. In music, many English terms are derived from Latin. For instance, semitone is from Latin semitones.

The  prefix   semi-   is   typically  used   herein  to  mean

“shorter”, rather than “half”.Namely, a semitones, semitones, semidiatessaron, semidiapente, semihexachordum, semiheptachordum, or semidiapason, is shorter by one semitone than the corresponding whole interval. For instance, a semiditonus (3 semitones, or about 300 cents) is not half of a ditonus (4 semitones, or about 400 cents), but a ditonus shortened by one semitone. Moreover, in Pythagorean tuning (the most commonly used tuning system up to the 16th century), a semitritonus (d5) is smaller than a tritonus (A4) by one Pythagorean comma (about a quarter of a semitone).

pitch-class intervals

In post-tonal or atonal theory, originally developed for equal-tempered European classical music written using the twelve-tone technique or serialism, integer notation is often used, most prominently in musical set theory. In this system, intervals are named according to the number of half steps, from 0 to 11, the largest interval class being 6.

In atonal or musical set theory, there are numerous types of intervals, the first being the ordered pitch interval, the distance between two pitches upward or downward. For instance, the interval from C upward to G is 7, and the interval from G downward to C 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 inter-val 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.

Generic and specific intervals

In diatonic set theory, specific and generic inter-vals are distinguished. Specific inte-rvals are the inter-val class or a number of semitones between scale steps or collection members, and generic inte-rvals 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 inte-rval number (second, third, fourth, etc.), are counted including the position of the lower note of the inte-rval, while generic inte-rval numbers are counted excluding that position. Thus, generic inter-val numbers are smaller by 1, with respect to the conventional inte-rval numbers.

Comparison

Generalizations   and   non pitch uses

The term “inter-val” can also be generalized to other music elements besides pitch. David Lewin’s Generalized Musical Inte-rvals and Transformations uses inte-rval as a generic measure of distance between time points, timbres, or more abstract musical phenomena.

Know more:

Pitch | Understanding Western Music Theory

Musical notation | Understanding Western Music Theory | part-1

Articulation | Form or structure | Performance and style | Fundamentals of music | Music theory

Music subjects | Understanding Western Music Theory

Texture | Timbre | Expression | Dynamics | Fundamentals of music | Music theory

–

Harmony and Form: Chapter 5 – Cadence (Understanding Basic Music Theory)