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Interval (music)

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 differences 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 or height) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.

Frequency ratios
The size of an interval between two notes may be measured by the 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$7/12$: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.

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

$$n = 1200 \cdot \log_2 \left( \frac{f_2}{f_1} \right)$$

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) 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.

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 C–G is a fifth (denoted P5) because the notes from C to the G above it encompass five letter names (C, D, E, F, G) and 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 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, 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, there are eight notes from C to G: (C–C♯–D–D♯–E–F–F♯–G). 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 C–G♯ (spanning 8 semitones) and C♯–G (spanning 6 semitones) are fifths, like the corresponding natural interval C–G (7 semitones).

Notice that interval numbers represent an inclusive count of encompassed staff positions or note names, not the difference between the endpoints. In other words, one starts counting the lower pitch as one, not zero. For that reason, the interval C–C, a perfect unison, is called a prime (meaning "1"), even though there is no difference between the endpoints. Continuing, the interval C–D is a second, but D is only one staff position, or 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 interval number one less than their sum. For instance, the intervals C–E and E–G are thirds, but joined together they form a fifth (C–G), not a sixth. Similarly, a stack of three thirds, such as C–E, E–G, and G–B, is a seventh (C–B), not a ninth.

This scheme applies to intervals up to an octave (12 semitones). For larger intervals, see 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, 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 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 interval is also perfect. Since the inversion does not change the pitch class of the two notes, it hardly affects their level of consonance (matching of their harmonics). Conversely, other kinds of intervals have the opposite quality with respect to their inversion. The inversion of a major interval is a minor interval, the inversion of an augmented interval is a diminished interval.

Major and minor
As shown in the table, a diatonic scale defines seven intervals for each interval number, each starting from a different note (seven unisons, seven seconds, etc.). The intervals formed by the notes of a diatonic scale are called diatonic. Except for unisons and octaves, the diatonic intervals with a given interval number always occur in two sizes, which differ by one semitone. For example, six of the fifths span seven semitones. The other one spans six semitones. Four of the thirds span three semitones, the others four. If one of the two versions is a perfect interval, the other is called either diminished (i.e. narrowed by one semitone) or augmented (i.e. widened by one semitone). Otherwise, the larger version is called major, the smaller one minor. For instance, since a 7-semitone fifth is a perfect interval (P5), the 6-semitone fifth is called "diminished fifth" (d5). Conversely, since neither kind of third is perfect, the larger one is called "major third" (M3), the smaller one "minor third" (m3).

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

Augmented and diminished
Augmented intervals are wider by one semitone than perfect or major intervals, while having the same interval number (i.e., encompassing the same number of staff positions): they are wider by a chromatic semitone. Diminished intervals, on the other hand, are narrower by one semitone than perfect or minor intervals of the same interval number: they are narrower by a chromatic semitone. 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.

The augmented fourth (A4) and the diminished fifth (d5) are the only augmented and diminished intervals that appear in diatonic scales (see table).

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 B♯, between A and C♯, between A and D♭, and between A♯ and E, but
 * A♭–B♯ is a second, as it encompasses two staff positions (A, B), and it is doubly augmented, as it exceeds a major second (such as A–B) by two semitones.
 * A–C♯ is a third, as it encompasses three staff positions (A, B, C), and it is major, as it spans 4 semitones.
 * A–D♭ is a fourth, as it encompasses four staff positions (A, B, C, D), and it is diminished, as it falls short of a perfect fourth (such as A–D) by one semitone.
 * A♯-E is a fifth, as it encompasses five staff positions (A, B, C, D, E), and it is triply diminished, as it falls short of a perfect fifth (such as A–E) by three semitones.

Shorthand notation
Intervals are often abbreviated with a P for perfect, m for minor, M for major, d for diminished, A for augmented, followed by the interval number. The indications M and P are often omitted. The octave is P8, and a unison is usually referred to simply as "a unison" but can be labeled P1. The tritone, an augmented fourth or diminished fifth is often TT. The interval qualities may be also abbreviated with perf, min, maj, dim, aug. Examples:
 * m2 (or min2): minor second,
 * M3 (or maj3): major third,
 * A4 (or aug4): augmented fourth,
 * d5 (or dim5): diminished fifth,
 * P5 (or perf5): perfect fifth.

Inversion

A simple interval (i.e., an interval smaller than or equal to an octave) may be inverted by raising the lower pitch an octave or lowering the upper pitch an octave. For example, the fourth from a lower C to a higher F may be inverted to make a fifth, from a lower F to a higher C.

There are two rules to determine the number and quality of the inversion of any simple interval: 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.
 * 1) The interval number and the number of its inversion always add up to nine (4 + 5 = 9, in the example just given).
 * 2) 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.

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 the ratio by 2 until it is greater than 1. For example, the inversion of a 5:4 ratio is an 8:5 ratio.

For intervals identified by an integer number of semitones, the inversion is obtained by subtracting that number from 12.

Since an interval class is the lower number selected among the interval integer and its inversion, interval classes cannot be inverted.

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

Melodic and harmonic
An interval can be described as
 * Vertical or harmonic if the two notes sound simultaneously
 * Horizontal, linear, or melodic if they sound successively. Melodic intervals can be ascending (lower pitch precedes higher pitch) or descending.

Diatonic and chromatic
In general,

 * A diatonic interval is an interval formed by two notes of a diatonic scale.
 * A chromatic interval is a non-diatonic interval formed by two notes of a chromatic scale.

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

The distinction between diatonic and chromatic intervals is controversial, as it is based on the definition of diatonic scale, which is variable in the literature. For example, the interval B–E♭ (a diminished fourth, occurring in the harmonic C-minor scale) is considered diatonic if the harmonic minor scales are considered diatonic as well. Otherwise, it is considered chromatic. For further details, see the main article.

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 interval is diatonic, except for the augmented fourth and diminished fifth.

The distinction between diatonic and chromatic intervals may be also sensitive to context. The above-mentioned 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.
 * 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 considered dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16th-century practice was still taught to beginning musicians throughout this period.
 * Hermann von Helmholtz (1821–1894) theorised that dissonance was caused by the presence of beats. Helmholtz further believed that the beating produced by the upper partials of harmonic sounds was the cause of dissonance for intervals too far apart to produce beating between the fundamentals. Helmholtz then designated that two harmonic tones that shared common low partials would be more consonant, as they produced less beats. Helmholtz disregarded partials above the seventh, as he believed that they were not audible enough to have significant effect. From this Helmholtz categorises the octave, perfect fifth, perfect fourth, major sixth, major third, and minor third as consonant, in decreasing value, and other intervals as dissonant.
 * David Cope (1997) suggests the concept of interval strength, in which an interval's strength, consonance, or stability is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law and

All of the above analyses refer to vertical (simultaneous) intervals.

Simple and compound

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

Steps and skips
Linear (melodic) intervals may be described as steps or skips. A step, or conjunct motion, is a linear interval between two consecutive notes of a scale. Any larger interval is called a skip (also called a leap), or disjunct motion. In the diatonic scale, a step is either a minor second (sometimes also called half step) or major second (sometimes also called 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 interval in a musical line, and a skip is a wider or larger interval, where the categorization of intervals into steps and skips is determined by the tuning system and the pitch space used.

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

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

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

When played as isolated chords on a piano keyboard, these intervals are indistinguishable to the ear, because they are all played with the same two keys. However, in a musical context, the diatonic function of the notes these intervals incorporate is very different.

The discussion above assumes the use of the prevalent tuning system, 12-tone equal temperament ("12-TET"). But in other historic meantone temperaments, the pitches of pairs of notes such as F♯ and G♭ may not necessarily coincide. These two notes are enharmonic in 12-TET, but may not be so in another tuning system. In such cases, the intervals they form would also not be enharmonic. For example, in quarter-comma meantone, all four intervals shown in the example above would be different.

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

Compound intervals

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

In general, a compound interval may be defined by a sequence or "stack" of two or more simple intervals of any kind. For instance, a major tenth (two staff positions above one octave), also called compound major third, spans one octave plus one major third.

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

The diatonic number DNc of a compound interval formed from n simple intervals with diatonic numbers DN1, DN2, ..., DNn, is determined by:
 * $$DN_c = 1 + (DN_1 - 1) + (DN_2 - 1) + ... + (DN_n - 1), \ $$

which can also be written as:
 * $$DN_c = DN_1 + DN_2 + ... + DN_n - (n - 1), \ $$

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 a 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 that can be considered a multiple of a perfect fifth (7 semitones) as it can be decomposed into four perfect fifths (7 × 4 = 28 semitones), or two octaves plus a major third (12 + 12 + 4 = 28 semitones). Intervals larger than a major seventeenth seldom come up, most often being referred to by their compound names, for example "two octaves plus a fifth" 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 a chord. Chords are classified based on the quality and number of the intervals that define them.

Chord qualities and interval qualities
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,  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 Rules to decode chord names and symbols.
 * 1) 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 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 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 CmM7 (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 = CMM7). However, this is not always true (e.g., Cm6 = CmM6, C+7 = C+m7, CM11 = CMP11). See main article for further details.
 * 2) Without contrary information, a major third interval and a perfect fifth 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 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 that contain only a plain interval number (e.g., "seventh chord") or the chord root and a number (e.g., "C seventh", or C7) are interpreted as follows:
 * 5) *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).
 * 6) *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).
 * 7) *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 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.

Size of intervals used in different tuning systems
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 that starts the interval. This is the art of just intonation. In equal temperament, the intervals are never precisely in tune with each other. This is the price of using equidistant intervals in a 12-tone scale. For 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 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 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. 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.

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 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 intervals).

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. For further details about reference ratios, see.

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

Interval root
Although intervals are usually designated in relation to their lower note, David Cope 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 to the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C–G), is the bottom C, the tonic.

Interval cycles
Interval cycles, "unfold [i.e., repeat] a single recurrent interval in a series that closes with a return to the initial pitch class", and are notated by George Perle using the letter "C", for cycle, with an 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 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 semitonus.

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

Non-diatonic intervals
Intervals in non-diatonic scales can be named using analogs of the diatonic interval names, by using a diatonic interval of similar size and distinguishing it by varying the quality, or by adding other modifiers. For example, the just interval 7/6 may be referred to as a subminor third, since it is ~267 cents wide, which is narrower than a minor third (300 cents in 12-TET, ~316 cents for the just interval 6/5), or as the septimal minor third, since it is a 7-limit interval. These names refer just to the individual interval's size, and the interval number need not correspond to the number of scale degrees of a (heptatonic) scale. This naming is particularly common in just intonation and microtonal scales.

The most common of these extended qualities are a neutral interval, in between a minor and major interval; and subminor and supermajor intervals, respectively narrower than a minor or wider than a major interval. The exact size of such intervals depends on the tuning system, but they often vary from the diatonic interval sizes by about a quarter tone (50 cents, half a chromatic step). For example, the neutral second, the characteristic interval of Arabic music, in 24-TET is 150 cents, exactly halfway between a minor second and major second. Combined, these yield the progression diminished, subminor, minor, neutral, major, supermajor, augmented for seconds, thirds, sixths, and sevenths. This naming convention can be extended to unisons, fourths, fifths, and octaves with sub and super, yielding the progression diminished, sub, perfect, super, augmented. This allows one to name all intervals in 24-TET or 31-TET, the latter of which was used by Adriaan Fokker. Various further extensions are used in Xenharmonic music.

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 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.

Generic and specific intervals
In diatonic set theory, specific and generic intervals are distinguished. Specific intervals are the interval class or number of semitones between scale 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.

Generalizations and non-pitch uses

The term "interval" can also be generalized to other music elements besides pitch. David Lewin's Generalized Musical Intervals and Transformations uses interval as a generic measure of distance between time points, timbres, or more abstract musical phenomena.

For example, an interval between two bell-like sounds, which have no pitch salience, is still perceptible. When two tones have similar acoustic spectra (sets of partials), the interval is just the distance of the shift of a tone spectrum along the frequency axis, so linking to pitches as reference points is not necessary. The same principle naturally applies to pitched tones (with similar harmonic spectra), which means that intervals can be perceived "directly" without pitch recognition. This explains in particular the predominance of interval hearing over absolute pitch hearing.

title: Interval (music)


text:
Interval (mathematics)

In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. An interval can contain neither endpoint, either endpoint, or both endpoints.

For example, the set of real numbers consisting of $0$, $1$, and all numbers in between is an interval, denoted $[0, 1]$ and called the unit interval; the set of all positive real numbers is an interval, denoted $(0, ∞)$; the set of all real numbers is an interval, denoted $(−∞, ∞)$; and any single real number $a$ is an interval, denoted $[a, a]$.

Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc.

Interval arithmetic consists of computing with intervals instead of real numbers for providing a guaranteed enclosure of the result of a numerical computation, even in the presence of uncertainties of input data and rounding errors.

Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.

Definitions and terminology
An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset.

The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers. If the infimum does not exist, one says often that the corresponding endpoint is $$-\infty.$$ Similarly, if the supremum does not exist, one says that the corresponding endpoint is $$+\infty.$$

Intervals are completely determined by their endpoints and whether each endpoint belong to the interval. This is a consequence of the least-upper-bound property of the real numbers. This characterization is used to specify intervals by mean of interval notation, which is described below.

An  does not include any endpoint, and is indicated with parentheses. For example, $$(0, 1) = \{x \mid 0 < x < 1\}$$ is the interval of all real numbers greater than $0$ and less than $1$. (This interval can also be denoted by $]0, 1[$, see below). The open interval $(0, +∞)$ consists of real numbers greater than $0$, i.e., positive real numbers. The open intervals are thus one of the forms
 * $$\begin{align}

(a,b)       &= \{x\in\mathbb R \mid a<x<b\}, \\ (-\infty, b) &= \{x\in\mathbb R \mid x<b\}, \\ (a, +\infty) &= \{x\in\mathbb R \mid a<x\}, \\ (-\infty, +\infty) &= \R, \end{align}$$ where $$a$$ and $$b$$ are real numbers such that $$a\le b.$$ When $$a=b$$ in the first case, the resulting interval is the empty set $$(a,a)=\varnothing,$$ which is a degenerate interval (see below). The open intervals are those intervals that are open sets for the usual topology on the real numbers.

A  is an interval that includes all its endpoints and is denoted with square brackets. For example, $[0, 1]$ means greater than or equal to $0$ and less than or equal to $1$. Closed intervals have one of the following forms in which $a$ and $b$ are real numbers such that $$a\le b\colon$$
 * $$ [a,b] = \{x\in\mathbb R \mid a\le x\le b\} $$

The closed intervals are those intervals that are closed sets for the usual topology on the real numbers. The empty set and $$\R$$ are the only intervals that are both open and closed.

A  has two endpoints and includes only one of them. It is said left-open or right-open depending on whether the excluded endpoint is on the left or on the right. These intervals are denoted by mixing notations for open and closed intervals. For example, $(0, 1]$ means greater than $0$ and less than or equal to $1$, while $[0, 1)$ means greater than or equal to $0$ and less than $1$. The half-open intervals have the form
 * $$\begin{align}

\left(a,b\right]      &= \{x\in\R \mid a<x\le b\}, \\ \left[a,b\right)       &= \{x\in\R \mid a\le x<b\}, \\ \left[a,+\infty\right) &= \{x\in\R \mid a\le x\},  \\ \left(-\infty,b\right] &= \{x\in\R \mid x\le b\}. \end{align}$$ Every closed interval is a closed set of the real line, but an interval that is a closed set need not be a closed interval. For example, intervals $$(-\infty,b]$$ and $$[a,+\infty)$$ are also closed sets in the real line. Intervals $$(a,b]$$ and $$[a,b)$$ are neither an open set nor a closed set. If one allows an endpoint in the closed side to be an infinity (such as $(0,+∞]$, the result will not be an interval, since it is not even a subset of the real numbers. Instead, the result can be seen as an interval in the extended real line, which occurs in measure theory, for example.

In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval.

A  is any set consisting of a single real number (i.e., an interval of the form $[a, a]$). Some authors include the empty set in this definition. A real interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements.

An interval is said to be left-bounded or right-bounded, if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite intervals.

Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as $+∞$, and the size of the empty interval may be defined as $0$ (or left undefined).

The centre (midpoint) of a bounded interval with endpoints $a$ and $b$ is $(a + b)/2$, and its radius is the half-length $|a − b|/2$. These concepts are undefined for empty or unbounded intervals.

An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it contains neither. The interval $0 ≤ x < 1\}$, for example, is left-closed and right-open. The empty set and the set of all reals are both open and closed intervals, while the set of non-negative reals, is a closed interval that is right-open but not left-open. The open intervals are open sets of the real line in its standard topology, and form a base of the open sets.

An interval is said to be left-closed if it has a minimum element or is left-unbounded, right-closed if it has a maximum or is right unbounded; it is simply closed if it is both left-closed and right closed. So, the closed intervals coincide with the closed sets in that topology.

The interior of an interval $I$ is the largest open interval that is contained in $I$; it is also the set of points in $I$ which are not endpoints of $I$. The closure of $I$ is the smallest closed interval that contains $I$; which is also the set $I$ augmented with its finite endpoints.

For any set $X$ of real numbers, the interval enclosure or interval span of $X$ is the unique interval that contains $X$, and does not properly contain any other interval that also contains $X$.

An interval $I$ is a subinterval of interval $J$ if $I$ is a subset of $J$. An interval $I$ is a proper subinterval of $J$ if $I$ is a proper subset of $J$.

However, there is conflicting terminology for the terms segment and interval, which have been employed in the literature in two essentially opposite ways, resulting in ambiguity when these terms are used. The Encyclopedia of Mathematics defines interval (without a qualifier) to exclude both endpoints (i.e., open interval) and segment to include both endpoints (i.e., closed interval), while Rudin's Principles of Mathematical Analysis calls sets of the form [a, b] intervals and sets of the form (a, b) segments throughout. These terms tend to appear in older works; modern texts increasingly favor the term interval (qualified by open, closed, or half-open), regardless of whether endpoints are included.

Notations for intervals
The interval of numbers between $a$ and $b$, including $a$ and $b$, is often denoted $[a, b]$. The two numbers are called the endpoints of the interval. In countries where numbers are written with a decimal comma, a semicolon may be used as a separator to avoid ambiguity.

Including or excluding endpoints
To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Both notations are described in International standard ISO 31-11. Thus, in set builder notation,

 * $$\begin{align}

(a,b) = \mathopen{]}a,b\mathclose{[} &= \{x\in\R \mid a b$, all four notations are usually taken to represent the empty set.

Both notations may overlap with other uses of parentheses and brackets in mathematics. For instance, the notation $(a, b)$ is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra. That is why Bourbaki introduced the notation $]a, b[$ to denote the open interval. The notation $[a, b]$ too is occasionally used for ordered pairs, especially in computer science.

Some authors such as Yves Tillé use $]a, b[$ to denote the complement of the interval $(a, b)$; namely, the set of all real numbers that are either less than or equal to $a$, or greater than or equal to $b$.

Infinite endpoints
In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with $−∞$ and $+∞$.

In this interpretation, the notations $[−∞, b]$, $(−∞, b]$ , $[a, +∞]$ , and $[a, +∞)$ are all meaningful and distinct. In particular, $(−∞, +∞)$ denotes the set of all ordinary real numbers, while $[−∞, +∞]$ denotes the extended reals.

Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction. For example, $(0, +∞)$ is the set of positive real numbers, also written as $$\mathbb{R}_+.$$ The context affects some of the above definitions and terminology. For instance, the interval $(−∞, +∞)$ = $$\R$$ is closed in the realm of ordinary reals, but not in the realm of the extended reals.

Integer intervals
When $a$ and $b$ are integers, the notation ⟦a, b⟧, or $[a .. b]$ or ${a .. b}$ or just $a .. b$, is sometimes used to indicate the interval of all integers between $a$ and $b$ included. The notation $[a .. b]$ is used in some programming languages; in Pascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array.

Another way to interpret integer intervals are as sets defined by enumeration, using ellipsis notation.

An integer interval that has a finite lower or upper endpoint always includes that endpoint. Therefore, the exclusion of endpoints can be explicitly denoted by writing $a .. b − 1$, $a + 1 .. b$ , or  $a + 1 .. b − 1$. Alternate-bracket notations like $[a .. b)$ or $[a .. b[$ are rarely used for integer intervals.

Properties
The intervals are precisely the connected subsets of $$\R.$$ It follows that the image of an interval by any continuous function from $$\mathbb R$$ to $$\mathbb R$$ is also an interval. This is one formulation of the intermediate value theorem.

The intervals are also the convex subsets of $$\R.$$ The interval enclosure of a subset $$X\subseteq \R$$ is also the convex hull of $$X.$$

The closure of an interval is the union of the interval and the set of its finite endpoints, and hence is also an interval. (The latter also follows from the fact that the closure of every connected subset of a topological space is a connected subset.) In other words, we have
 * $$\operatorname{cl}(a,b)=\operatorname{cl}(a,b]=\operatorname{cl}[a,b)=\operatorname{cl}[a,b]=[a,b],$$
 * $$\operatorname{cl}(a,+\infty)=\operatorname{cl}[a,+\infty)=[a,+\infty),$$
 * $$\operatorname{cl}(-\infty,a)=\operatorname{cl}(-\infty,a]=(-\infty,a],$$
 * $$\operatorname{cl}(-\infty,+\infty)=(-\infty,\infty).$$

The intersection of any collection of intervals is always an interval. The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example $$(a,b) \cup [b,c] = (a,c].$$

If $$\R$$ is viewed as a metric space, its open balls are the open bounded intervals $(c + r, c − r)$, and its closed balls are the closed bounded intervals $[c + r, c − r]$. In particular, the metric and order topologies in the real line coincide, which is the standard topology of the real line.

Any element $x$ of an interval $I$ defines a partition of $I$ into three disjoint intervals $I$1, $I$2, $I$3: respectively, the elements of $I$ that are less than $x$, the singleton $$[x,x] = \{x\},$$ and the elements that are greater than $x$. The parts $I$1 and $I$3 are both non-empty (and have non-empty interiors), if and only if $x$ is in the interior of $I$. This is an interval version of the trichotomy principle.

Dyadic intervals
A dyadic interval is a bounded real interval whose endpoints are $$\tfrac{j}{2^n}$$ and $$\tfrac{j+1}{2^n},$$ where $$j$$ and $$n$$ are integers. Depending on the context, either endpoint may or may not be included in the interval.

Dyadic intervals have the following properties:

 * The length of a dyadic interval is always an integer power of two.
 * Each dyadic interval is contained in exactly one dyadic interval of twice the length.
 * Each dyadic interval is spanned by two dyadic intervals of half the length.
 * If two open dyadic intervals overlap, then one of them is a subset of the other.

The dyadic intervals consequently have a structure that reflects that of an infinite binary tree.

Dyadic intervals are relevant to several areas of numerical analysis, including adaptive mesh refinement, multigrid methods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for $p = 2$).

Balls
An open finite interval $$(a, b)$$ is a 1-dimensional open ball with a center at $$\tfrac12(a + b)$$ and a radius of $$\tfrac12(b - a).$$ The closed finite interval $$[a, b]$$ is the corresponding closed ball, and the interval's two endpoints $$\{a, b\}$$ form a 0-dimensional sphere. Generalized to $$n$$-dimensional Euclidean space, a ball is the set of points whose distance from the center is less than the radius. In the 2-dimensional case, a ball is called a disk.

If a half-space is taken as a kind of degenerate ball (without a well-defined center or radius), a half-space can be taken as analogous to a half-bounded interval, with its boundary plane as the (degenerate) sphere corresponding to the finite endpoint.

Multi-dimensional intervals
A finite interval is (the interior of) a 1-dimensional hyperrectangle. Generalized to real coordinate space $$\R^n,$$ an axis-aligned hyperrectangle (or box) is the Cartesian product of $$n$$ finite intervals. For $$n=2$$ this is a rectangle; for $$n=3$$ this is a rectangular cuboid (also called a "box").

Allowing for a mix of open, closed, and infinite endpoints, the Cartesian product of any $$n$$ intervals, $$I = I_1\times I_2 \times \cdots \times I_n$$ is sometimes called an $$n$$-dimensional interval.

A facet of such an interval $$I$$ is the result of replacing any non-degenerate interval factor $$I_k$$ by a degenerate interval consisting of a finite endpoint of $$I_k.$$ The faces of $$I$$ comprise $$I$$ itself and all faces of its facets. The corners of $$I$$ are the faces that consist of a single point of $$\R^n.$$

Convex polytopes
Any finite interval can be constructed as the intersection of half-bounded intervals (with an empty intersection taken to mean the whole real line), and the intersection of any number of half-bounded intervals is a (possibly empty) interval. Generalized to $$n$$-dimensional affine space, an intersection of half-spaces (of arbitrary orientation) is (the interior of) a convex polytope, or in the 2-dimensional case a convex polygon.

Domains
An open interval is a connected open set of real numbers. Generalized to topological spaces in general, a non-empty connected open set is called a domain.

Complex intervals
Intervals of complex numbers can be defined as regions of the complex plane, either rectangular or circular.

Definitions
The concept of intervals can be defined in arbitrary partially ordered sets or more generally, in arbitrary preordered sets. For a preordered set $$(X,\lesssim)$$ and two elements $$a,b\in X,$$ one similarly defines the intervals
 * $$(a,b)           =\{x\in X \mid a<x<b\},$$
 * $$[a,b]           =\{x\in X \mid a\lesssim x\lesssim b\},$$
 * $$(a,b]           =\{x\in X \mid a<x\lesssim b\},$$
 * $$[a,b)           =\{x\in X \mid a\lesssim x<b\},$$
 * $$(a,\infty)      =\{x\in X \mid a<x\},$$
 * $$[a,\infty)      =\{x\in X \mid a\lesssim x\},$$
 * $$(-\infty,b)     =\{x\in X \mid x<b\},$$
 * $$(-\infty,b]     =\{x\in X \mid x\lesssim b\},$$
 * $$(-\infty,\infty) =X,$$

where $$x<y$$ means $$x\lesssim y\not\lesssim x.$$ Actually, the intervals with single or no endpoints are the same as the intervals with two endpoints in the larger preordered set
 * $$\bar X=X\sqcup\{-\infty,\infty\}$$
 * $$-\infty<x<\infty\qquad(\forall x\in X)$$

defined by adding new smallest and greatest elements (even if there were ones), which are subsets of $$X.$$ In the case of $$X=\mathbb R$$ one may take $$\bar\mathbb R$$ to be the extended real line.

Convex sets and convex components in order theory
A subset $$A\subseteq X$$ of the preordered set $$(X,\lesssim)$$ is (order-)convex if for every $$x,y\in A$$ and every $$x\lesssim z\lesssim y$$ we have $$z\in A.$$ Unlike in the case of the real line, a convex set of a preordered set need not be an interval. For example, in the totally ordered set $$(\mathbb Q,\le)$$ of rational numbers, the set
 * $$\mathbb Q=\{x\in\mathbb Q \mid x^2<2\}$$

is convex, but not an interval of $$\mathbb Q,$$ since there is no square root of two in $$\mathbb Q.$$

Let $$(X,\lesssim)$$ be a preordered set and let $$Y\subseteq X.$$ The convex sets of $$X$$ contained in $$Y$$ form a poset under inclusion. A maximal element of this poset is called an convex component of $$Y.$$  By the Zorn lemma, any convex set of $$X$$ contained in $$Y$$ is contained in some convex component of $$Y,$$ but such components need not be unique. In a totally ordered set, such a component is always unique. That is, the convex components of a subset of a totally ordered set form a partition.

Properties
A generalization of the characterizations of the real intervals follows. For a non-empty subset $$I$$ of a linear continuum $$(L,\le),$$ the following conditions are equivalent.
 * The set $$I$$ is an interval.
 * The set $$I$$ is order-convex.
 * The set $$I$$ is a connected subset when $$L$$ is endowed with the order topology.

For a subset $$S$$ of a lattice $$L,$$ the following conditions are equivalent.
 * The set $$S$$ is a sublattice and an (order-)convex set.
 * There is an ideal $$I\subseteq L$$ and a filter $$F\subseteq L$$ such that $$S=I\cap F.$$

In general topology
Every Tychonoff space is embeddable into a product space of the closed unit intervals $$[0,1].$$ Actually, every Tychonoff space that has a base of cardinality $$\kappa$$ is embeddable into the product $$[0,1]^\kappa$$ of $$\kappa$$ copies of the intervals.

The concepts of convex sets and convex components are used in a proof that every totally ordered set endowed with the order topology is completely normal or moreover, monotonically normal.

Topological algebra
Intervals can be associated with points of the plane, and hence regions of intervals can be associated with regions of the plane. Generally, an interval in mathematics corresponds to an ordered pair $(x, y)$ taken from the direct product $$\R \times \R$$ of real numbers with itself, where it is often assumed that $y > x$. For purposes of mathematical structure, this restriction is discarded, and "reversed intervals" where $y − x < 0$ are allowed. Then, the collection of all intervals $[x, y]$ can be identified with the topological ring formed by the direct sum of $$\R$$ with itself, where addition and multiplication are defined component-wise.

The direct sum algebra $$( \R \oplus \R, +, \times)$$ has two ideals, { [x,0] : x ∈ R } and { [0,y] : y ∈ R }. The identity element of this algebra is the condensed interval $[1, 1]$. If interval $[x, y]$ is not in one of the ideals, then it has multiplicative inverse $[1/x, 1/y]$. Endowed with the usual topology, the algebra of intervals forms a topological ring. The group of units of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component of this group is quadrant I.

Every interval can be considered a symmetric interval around its midpoint. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" $[x, −x]$ is used along with the axis of intervals $[x, x]$ that reduce to a point. Instead of the direct sum $$R \oplus R,$$ the ring of intervals has been identified with the hyperbolic numbers by M. Warmus and D. H. Lehmer through the identification
 * $$z = \tfrac12(x + y) + \tfrac12(x - y)j,$$

where $$j^2 = 1.$$

This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Interval (mathematics)


text:
Talk:Transport in Egypt

When did Egypt Railways start?
An article in "The Middle East" magazine of April 2005 credits Egyptian Railways as having started in 1825.

This beats even the ground breaking Liverpool and Manchester Railway opened in 1830. The date surely cannot be right.

When did Egyptian Railways start. 1835 or 1845 would sound more reasonable.

 * It started in 1852. --Meno25 12:57, 6 November 2006 (UTC)

"Egypt has one of the highest incidence of road fatalities per miles driven in the world"
This statement of fact cites the US Department of State website on international travel. That website does not put forward any source for the claim. After a quick search, I was unable to find any data or estimates on the number of miles driven in Egypt. Without that data it would be impossible to measure the number of road fatalities per mile driven. On a per capita basis however, Egypt has a much lower road fatality rate than the US according to this paper: http://www.transport-links.org/transport_links/filearea/publications/1_771_Pa3568.pdf — Preceding unsigned comment added by 137.94.118.15 (talk) 21:17, 13 September 2011 (UTC)

'Egypt has one of the highest incidence of road fatalities per miles driven in the world.' ...this sentence is grammatically wrong in so many ways that I can't be bothered to list them. If people writing these articles paid more attention to correct communication, their articles would be easier to read and they wouldn't look illiterate. — Preceding unsigned comment added by 87.194.238.31 (talk) 11:29, 17 November 2012 (UTC)

Information on roadways and the Nile
I am surprised that there is not much information on the usage of roads or the Nile as transport. I am currently researching these topics, so will add anything I find. Please contribute if you can. --62.49.39.202 (talk) 14:52, 14 November 2011 (UTC)

title: Talk:Transport in Egypt


text:
Conjugacy class

In mathematics, especially group theory, two elements $$a$$ and $$b$$ of a group are conjugate if there is an element $$g$$ in the group such that $$b = gag^{-1}.$$ This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under $$b = gag^{-1}$$ for all elements $$g$$ in the group.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

Functions that are constant for members of the same conjugacy class are called class functions.

Definition
Let $$G$$ be a group. Two elements $$a, b \in G$$ are conjugate if there exists an element $$g \in G$$ such that $$gag^{-1} = b,$$ in which case $$b$$ is called of $$a$$ and $$a$$ is called a conjugate of $$b.$$

In the case of the general linear group $$\operatorname{GL}(n)$$ of invertible matrices, the conjugacy relation is called matrix similarity.

It can be easily shown that conjugacy is an equivalence relation and therefore partitions $$G$$ into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes $$\operatorname{Cl}(a)$$ and $$\operatorname{Cl}(b)$$ are equal if and only if $$a$$ and $$b$$ are conjugate, and disjoint otherwise.) The equivalence class that contains the element $$a \in G$$ is $$\operatorname{Cl}(a) = \left\{ gag^{-1} : g \in G \right\}$$ and is called the conjugacy class of $$a.$$ The  of $$G$$ is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same order.

Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1. In some cases, conjugacy classes can be described in a uniform way; for example, in the symmetric group they can be described by cycle type.

Examples
The symmetric group $S_3,$ consisting of the 6 permutations of three elements, has three conjugacy classes:

 * 1) No change $$(abc \to abc)$$.  The single member has order 1.
 * 2) Transposing two $$(abc \to acb, abc \to bac, abc \to cba)$$.  The 3 members all have order 2.
 * 3) A cyclic permutation of all three $$(abc \to bca, abc \to cab)$$.  The 2 members both have order 3.

These three classes also correspond to the classification of the isometries of an equilateral triangle.

The symmetric group $S_4,$ consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description, cycle type, member order, and members:

 * 1) No change.  Cycle type = [14].  Order = 1.  Members = { (1, 2, 3, 4) }.  The single row containing this conjugacy class is shown as a row of black circles in the adjacent table.
 * 2) Interchanging two (other two remain unchanged).  Cycle type = [1221].  Order = 2.  Members = { (1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (4, 2, 3, 1), (3, 2, 1, 4), (2, 1, 3, 4) }).  The 6 rows containing this conjugacy class are highlighted in green in the adjacent table.
 * 3) A cyclic permutation of three (other one remains unchanged).  Cycle type = [1131].  Order = 3.  Members = { (1, 3, 4, 2), (1, 4, 2, 3), (3, 2, 4, 1), (4, 2, 1, 3), (4, 1, 3, 2), (2, 4, 3, 1), (3, 1, 2, 4), (2, 3, 1, 4) }).  The 8 rows containing this conjugacy class are shown with normal print (no boldface or color highlighting) in the adjacent table.
 * 4) A cyclic permutation of all four.  Cycle type = [41].  Order = 4.  Members = { (2, 3, 4, 1), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 3, 1, 2) }).  The 6 rows containing this conjugacy class are highlighted in orange in the adjacent table.
 * 5) Interchanging two, and also the other two.  Cycle type = [22].  Order = 2.  Members = { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) }).  The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent table.

The proper rotations of the cube, which can be characterized by permutations of the body diagonals, are also described by conjugation in $$S_4.$$

In general, the number of conjugacy classes in the symmetric group $$S_n$$ is equal to the number of integer partitions of $$n.$$ This is because each conjugacy class corresponds to exactly one partition of $$\{ 1, 2, \ldots, n \}$$ into cycles, up to permutation of the elements of $$\{ 1, 2, \ldots, n \}.$$

In general, the Euclidean group can be studied by conjugation of isometries in Euclidean space.

Example

Let G = $$S_3$$

a = ( 2 3 )

x = ( 1 2 3 )

x-1 = ( 3 2 1 )

Then xax-1

= ( 1 2 3 ) ( 2 3 ) ( 3 2 1 ) = ( 3 1 )

= ( 3 1 ) is Conjugate of ( 2 3 )

Properties

 * The identity element is always the only element in its class, that is $$\operatorname{Cl}(e) = \{ e \}.$$
 * If $$G$$ is abelian then $$gag^{-1} = a$$ for all $$a, g \in G$$, i.e. $$\operatorname{Cl}(a) = \{ a \}$$ for all $$a \in G$$ (and the converse is also true: if all conjugacy classes are singletons then $$G$$ is abelian).
 * If two elements $$a, b \in G$$ belong to the same conjugacy class (that is, if they are conjugate), then they have the same order. More generally, every statement about $$a$$ can be translated into a statement about $$b = gag^{-1},$$ because the map $$\varphi(x) = gxg^{-1}$$ is an automorphism of $$G$$ called an inner automorphism. See the next property for an example.
 * If $$a$$ and $$b$$ are conjugate, then so are their powers $$a^k$$ and $$b^k.$$ (Proof: if $$a = gbg^{-1}$$ then $$a^k = \left(gbg^{-1}\right)\left(gbg^{-1}\right) \cdots \left(gbg^{-1}\right) = gb^kg^{-1}.$$) Thus taking $k$th powers gives a map on conjugacy classes, and one may consider which conjugacy classes are in its preimage. For example, in the symmetric group, the square of an element of type (3)(2) (a 3-cycle and a 2-cycle) is an element of type (3), therefore one of the power-up classes of (3) is the class (3)(2) (where $$a$$ is a power-up class of $$a^k$$).
 * An element $$a \in G$$ lies in the center $$\operatorname{Z}(G)$$ of $$G$$ if and only if its conjugacy class has only one element, $$a$$ itself. More generally, if $$\operatorname{C}_G(a)$$ denotes the of $$a \in G,$$ i.e., the subgroup consisting of all elements $$g$$ such that $$ga = ag,$$ then the index $$\left[G : \operatorname{C}_G(a)\right]$$ is equal to the number of elements in the conjugacy class of $$a$$ (by the orbit-stabilizer theorem).
 * Take $$\sigma \in S_n$$ and let $$m_1, m_2, \ldots, m_s$$ be the distinct integers which appear as lengths of cycles in the cycle type of $$\sigma$$ (including 1-cycles). Let $$k_i$$ be the number of cycles of length $$m_i$$ in $$\sigma$$ for each $$i = 1, 2, \ldots, s$$ (so that $$\sum\limits_{i=1}^s k_i m_i = n$$). Then the number of conjugates of $$\sigma$$ is: $$\frac{n!}{\left(k_1!m_1^{k_1}\right) \left(k_2!m_2^{k_2}\right) \cdots \left(k_s!m_s^{k_s}\right)}.$$

Conjugacy as group action
For any two elements $$g, x \in G,$$ let $$g \cdot x := gxg^{-1}.$$ This defines a group action of $$G$$ on $$G.$$ The orbits of this action are the conjugacy classes, and the stabilizer of a given element is the element's centralizer.

Similarly, we can define a group action of $$G$$ on the set of all subsets of $$G,$$ by writing $$g \cdot S := gSg^{-1},$$ or on the set of the subgroups of $$G.$$

Conjugacy class equation
If $$G$$ is a finite group, then for any group element $$a,$$ the elements in the conjugacy class of $$a$$ are in one-to-one correspondence with cosets of the centralizer $$\operatorname{C}_G(a).$$ This can be seen by observing that any two elements $$b$$ and $$c$$ belonging to the same coset (and hence, $$b = cz$$ for some $$z$$ in the centralizer $$\operatorname{C}_G(a)$$) give rise to the same element when conjugating $$a$$: $$bab^{-1} = cza(cz)^{-1} = czaz^{-1}c^{-1} = cazz^{-1}c^{-1} = cac^{-1}.$$ That can also be seen from the orbit-stabilizer theorem, when considering the group as acting on itself through conjugation, so that orbits are conjugacy classes and stabilizer subgroups are centralizers. The converse holds as well.

Thus the number of elements in the conjugacy class of $$a$$ is the index $$\left[ G : \operatorname{C}_G(a)\right]$$ of the centralizer $$\operatorname{C}_G(a)$$ in $$G$$; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element $$x_i$$ from every conjugacy class, we infer from the disjointness of the conjugacy classes that $$|G| = \sum_i \left[ G : \operatorname{C}_G(x_i)\right],$$ where $$\operatorname{C}_G(x_i)$$ is the centralizer of the element $$x_i.$$ Observing that each element of the center $$\operatorname{Z}(G)$$ forms a conjugacy class containing just itself gives rise to the class equation: $$|G| = |{\operatorname{Z}(G)}| + \sum_i \left[G : \operatorname{C}_G(x_i)\right],$$ where the sum is over a representative element from each conjugacy class that is not in the center.

Knowledge of the divisors of the group order $$|G|$$ can often be used to gain information about the order of the center or of the conjugacy classes.

Example
Consider a finite $p$-group $$G$$ (that is, a group with order $$p^n,$$ where $$p$$ is a prime number and $$n > 0$$). We are going to prove that.

Since the order of any conjugacy class of $$G$$ must divide the order of $$G,$$ it follows that each conjugacy class $$H_i$$ that is not in the center also has order some power of $$p^{k_i},$$ where $$0 < k_i < n.$$ But then the class equation requires that $|G| = p^n = |{\operatorname{Z}(G)}| + \sum_i p^{k_i}.$ From this we see that $$p$$ must divide $$|{\operatorname{Z}(G)}|,$$ so $$|\operatorname{Z}(G)| > 1.$$

In particular, when $$n = 2,$$ then $$G$$ is an abelian group since any non-trivial group element is of order $$p$$ or $$p^2.$$ If some element $$a$$ of $$G$$ is of order $$p^2,$$ then $$G$$ is isomorphic to the cyclic group of order $$p^2,$$ hence abelian. On the other hand, if every non-trivial element in $$G$$ is of order $$p,$$ hence by the conclusion above $$|\operatorname{Z}(G)| > 1,$$ then $$|\operatorname{Z}(G)| = p > 1$$ or $$p^2.$$ We only need to consider the case when $$|\operatorname{Z}(G)| = p > 1,$$ then there is an element $$b$$ of $$G$$ which is not in the center of $$G.$$ Note that $$\operatorname{C}_G(b)$$ includes $$b$$ and the center which does not contain $$b$$ but at least $$p$$ elements. Hence the order of $$\operatorname{C}_G(b)$$ is strictly larger than $$p,$$ therefore $$\left|\operatorname{C}_G(b)\right| = p^2,$$ therefore $$b$$ is an element of the center of $$G,$$ a contradiction. Hence $$G$$ is abelian and in fact isomorphic to the direct product of two cyclic groups each of order $$p.$$

Conjugacy of subgroups and general subsets
More generally, given any subset $$S \subseteq G$$ ($$S$$ not necessarily a subgroup), define a subset $$T \subseteq G$$ to be conjugate to $$S$$ if there exists some $$g \in G$$ such that $$T = gSg^{-1}.$$ Let $$\operatorname{Cl}(S)$$ be the set of all subsets $$T \subseteq G$$ such that $$T$$ is conjugate to $$S.$$

A frequently used theorem is that, given any subset $$S \subseteq G,$$ the index of $$\operatorname{N}(S)$$ (the normalizer of $$S$$) in $$G$$ equals the cardinality of $$\operatorname{Cl}(S)$$:
 * $$|\operatorname{Cl}(S)| = [G : N(S)].$$

This follows since, if $$g, h \in G,$$ then $$gSg^{-1} = hSh^{-1}$$ if and only if $$g^{-1}h \in \operatorname{N}(S),$$ in other words, if and only if $$g \text{ and } h$$ are in the same coset of $$\operatorname{N}(S).$$

By using $$S = \{ a \},$$ this formula generalizes the one given earlier for the number of elements in a conjugacy class.

The above is particularly useful when talking about subgroups of $$G.$$ The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are isomorphic, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.

Geometric interpretation
Conjugacy classes in the fundamental group of a path-connected topological space can be thought of as equivalence classes of free loops under free homotopy.

Conjugacy class and irreducible representations in finite group
In any finite group, the number of nonisomorphic irreducible representations over the complex numbers is precisely the number of conjugacy classes.

title: Conjugacy class