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In music theory, the term interval describes the difference in pitch between two notes. Music theory is the field of study that deals with the Mechanics of music and how Music works Pitch represents the perceived Fundamental frequency of a sound In Music, the term note has two primary meanings 1 a sign used in Musical notation to represent the relative duration and pitch of a Sound; Although frequently used in connection with intervals, the term "distance" does not adequately describe the physics and subjective effects of two interacting frequencies.

Intervals may be described as:

Interval class is a system of labelling intervals when the order of the notes is left unspecified, therefore describing an interval in terms of the shortest distance possible between its two pitch classes. In Musical set theory, an interval class (usual abbreviation ic) is the shortest distance in Pitch class space between two unordered Pitch classes In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e [2]

Contents

Frequency ratios

Intervals may be labelled according to the ratio of frequencies of the two pitches. A ratio is an expression which compares quantities relative to each other Frequency is a measure of the number of occurrences of a repeating event per unit Time. Important intervals are those using the lowest integers, such as 1/1, 2/1, 3/2, etc. This system is frequently used to describe intervals in non-Western music. This method is also often used in just intonation, and in theoretical explanations of equal-tempered intervals used in European tonal music which explain their use through their approximation of just intervals. In music just intonation is any Musical tuning in which the frequencies of Notes are related by Ratios of Whole numbers Any interval

Interval number and quality

Interval namesU = unison; 8ve = octave
Interval names
U = unison; 8ve = octave

In Western harmonic theory, intervals are labeled according to the number of scale steps or staff positions they encompass, as shown at right. In Music theory, a scale degree is the name of a particular note of a scale in relation to the tonic (the note of the scale that is considered In standard Western Musical notation, the staff ( AmE) or stave

Intervals larger than an octave are called compound intervals; for example, a tenth is known as a compound third. In Music, an octave ( is the the use of which is "common in most musical systems Intervals larger than a thirteenth are rarely spoken of, since going above this by stacking thirds would result in a double octave (but see 8va for use of 15ma). In Music, an octave ( is the the use of which is "common in most musical systems

The name or the label of an interval is determined by counting the number of degrees between the two notes beginning with one for the lower note. In Music, the term note has two primary meanings 1 a sign used in Musical notation to represent the relative duration and pitch of a Sound; The number of degrees between F and B for example is 4, therefore the interval is a fourth.

The name of any interval is further qualified using the terms perfect, major, minor, augmented, and diminished. In Music and Music theory augmentation is the lengthening or widening of Rhythms melodies, intervals chords The opposite Diminution, from Italian diminuimento, is a musical term used to mean different things in the context of melodies and intervals or chords. This is called its interval quality.

It is possible to have doubly-diminished and doubly-augmented intervals, but these are quite rare.

The name of an interval cannot be determined by counting semitones alone. There are four semitones between B and E♭ but this interval is not a major third, but rather a diminished fourth, a relatively rare interval (but which does appear naturally as part of the harmonic minor scale). Minor Scale was a test conducted by the United States Defense Nuclear Agency (now part of the Defense Threat Reduction Agency) involving the detonation In equal-tempered tuning, as on a piano, these intervals are indistinguishable by sound, but the diatonic function of the notes incorporated might be very different. Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. A diatonic function, in tonal Music theory, is the specific recognized Roles of Notes or chords in relation to the key.

Diatonic and chromatic intervals

The intervals contained in the table are diatonic to C major. All other intervals are chromatic to C major.
The intervals contained in the table are diatonic to C major. All other intervals are chromatic to C major.

A diatonic interval is an interval formed by two notes of a diatonic scale. In Music theory, a diatonic scale (from the Greek διατονικος, meaning " through tones" also known as the heptatonia prima and The table on the right depicts all diatonic intervals for C major. As the diatonic functions are similarly common to each and every major key, the contents of the table can be summarised as follows:

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 diatonic interval number. Major is a Military rank the use of which varies according to country Diminution, from Italian diminuimento, is a musical term used to mean different things in the context of melodies and intervals or chords. The indication M and P are often omitted. The octave is P8, and a unison is usually referred to simply as "a unison" but can be labeled P1. In Music, an octave ( is the the use of which is "common in most musical systems In Music, a unison () is an interval, the ratio of 11 or 0 half steps and zero cents. The tritone, an augmented fourth or diminished fifth is often π or TT. The tritone ( Tri - or three and tone) is a Musical interval that spans three whole tones. Examples:

For use in describing chords, the sign + is used for augmented and ° for diminished. Chord notation refers to the written notation for musical chords. Furthermore the 3 for the third is often omitted, and for the seventh, the plain form stands for the minor interval, while the major is indicated by maj. So for example:

Enharmonic intervals

Two intervals are considered to be 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. In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or Pitch represents the perceived Fundamental frequency of a sound Enharmonic intervals span the same number of semitones. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the For example, as shown in the matrix below, F♯–A♯ (a major third), G♭–B♭ (also a major third), F♯–B♭ (a diminished fourth), and G♭–A♯ (a double augmented second) are all enharmonically equivalent — and they all span four semitones. A major third ( is one of two commonly occurring Musical intervals that span three Diatonic scale degrees the other being the Minor third. In Music, a diminished fourth ( is an interval that spans four Diatonic scale degrees and is narrower than a Perfect fourth by a An augmented second is Enharmonically equivalent to a Minor third ( in Equal temperament, but is not the same interval in other meantone tunings

step 1 2 3 4
major third F♯   A♯  
major third   G♭   B♭
diminished fourth F♯     B♭
double augmented second   G♭ A♯  

Steps and skips

Linear (melodic) intervals may be described as steps or skips in a diatonic context. A major third ( is one of two commonly occurring Musical intervals that span three Diatonic scale degrees the other being the Minor third. In Music, a diminished fourth ( is an interval that spans four Diatonic scale degrees and is narrower than a Perfect fourth by a An augmented second is Enharmonically equivalent to a Minor third ( in Equal temperament, but is not the same interval in other meantone tunings Steps are linear intervals between consecutive scale degrees while skips are not, although if one of the notes is chromatically altered so that the resulting interval is three semitones or more (e. In Music theory, a scale degree is the name of a particular note of a scale in relation to the tonic (the note of the scale that is considered In Music alteration, an example of Chromaticism, is the use of a neighboring pitch in the Chromatic scale in place of its Diatonic neighbor such A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the g. C to D♯), that may also be considered a skip. However, the reverse is not true: a diminished third, an interval comprising two semitones, is still considered a skip. In music a diminished third ( is the interval produced by flattening a Minor third by a Chromatic semitone. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the

The words conjunct and disjunct refer to melodies composed of steps and skips, respectively.

Pitch class intervals

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. Atonality in its broadest sense describes Music that lacks a tonal center, or key. Twelve-tone technique (also dodecaphony, especially in British usage twelve-note composition) is a method of musical composition devised by Arnold In Music, serialism is a technique for composition that uses sets to describe musical elements, and allows the manipulation of those In integer notation, or the Integer model of pitch all Pitch classes and intervals between pitch classes are designated using the numbers 0 through 11 Musical set theory provides concepts for categorizing musical objects and describing their relationships In this system intervals are named according to the number of half steps, from 0 to 11, the largest interval class being 6.

Ordered and unordered pitch and pitch class intervals

In atonal or musical set theory there are numerous types of intervals, the first being ordered pitch interval, the distance between two pitches upward or downward. Musical set theory provides concepts for categorizing musical objects and describing their relationships In Musical set theory, ordered pitch interval is the distance in semitones between two pitches upward or downward For instance, the interval from C to G upward is 7, but the interval from G to C downward is −7. One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, somewhat similar to the interval of tonal theory.

The interval between pitch classes may be measured with ordered and unordered pitch class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. For unordered pitch class interval see interval class. In Musical set theory, an interval class (usual abbreviation ic) is the shortest distance in Pitch class space between two unordered Pitch classes

Generic and specific intervals

In diatonic set theory, specific and generic intervals are distinguished. Diatonic set theory is a subdivision or application of Musical set theory which applies the techniques and insights of Discrete mathematics to properties In Diatonic set theory a specific interval is the shortest possible clockwise distance between Pitch classes on the Chromatic circle ( Interval class In Diatonic set theory a generic interval is the number of scale Steps between notes of a collection or scale. Specific intervals are the interval class or number of semitones between scale degrees or collection members, and generic intervals are the number of scale steps between notes of a collection or scale.

Cents

Main article: Cent (music)

The standard system for comparing intervals of different sizes is with cents. The cent is a logarithmic unit of measure used for musical intervals. The cent is a logarithmic unit of measure used for musical intervals. This is a logarithmic scale in which the octave is divided into 1200 equal parts. Definition and base Logarithmic scales are either defined for ratios of the underlying quantity or one has to agree to measure In equal temperament, each semitone is exactly 100 cents. Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the The value in cents for the interval f1 to f2 is 1200×log2(f2/f1).

Comparison of different interval naming systems

# semitones
 Interval
class		 Generic
interval		 Common
diatonic name		 Comparable
just interval		 Comparison of interval width in cents
equal
temperament		 just
intonation		 quarter-comma
meantone
0 0 0 perfect unison 1:1 0 0 0
1 1 1 minor second 16:15 100 112 117
2 2 1 major second 9:8 200 204 193
3 3 2 minor third 6:5 300 316 310
4 4 2 major third 5:4 400 386 386
5 5 3 perfect fourth 4:3 500 498 503
6 6 3
4		 augmented fourth
diminished fifth		 45:32
64:45		 600 590
610		 579
621
7 5 4 perfect fifth 3:2 700 702 697
wolf fifth 737
8 4 5 minor sixth 8:5 800 814 814
9 3 5 major sixth 5:3 900 884 889
10 2 6 minor seventh 16:9 1000 996 1007
11 1 6 major seventh 15:8 1100 1088 1083
12 0 0 perfect octave 2:1 1200 1200 1200

It is possible to construct just intervals which are closer to the equal-tempered equivalents, but most of the ones listed above have been used historically in equivalent contexts. In particular the tritone (augmented fourth or diminished fifth), could have other ratios; 17:12 (603 cents) is fairly common. The 7:4 interval (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. The harmonic seventh interval, also known as the septimal minor seventh, is one with an exact 74 ratio (about 969 cents Some assert the 7:4 is one of the blue notes used in jazz. In Jazz and Blues, a blue note (also "worried" note is a Note sung or played at a slightly lower pitch than that of the Jazz is an American Musical art form which originated in the beginning of the 20th century in African American communities in the Southern United States

In the diatonic system, every interval has one or more enharmonic equivalents, such as augmented second for minor third. In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or An augmented second is Enharmonically equivalent to a Minor third ( in Equal temperament, but is not the same interval in other meantone tunings A minor third ( is the smaller of two commonly occurring musical intervals compounded of two steps of the Diatonic scale.

Consonant and dissonant intervals

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

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

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

Inversion

An interval may be inverted, by raising the lower pitch an octave, or lowering the upper pitch an octave (though it is less usual to speak of inverting unisons or octaves). In Music theory, the word inversion has several meanings There are inverted chords, inverted melodies, inverted intervals, and In Music, an octave ( is the the use of which is "common in most musical systems For example, the fourth between a lower C and a higher F may be inverted to make a fifth, with a lower F and a higher C. Here are the ways to identify interval inversions:

Interval inversions
Interval inversions
A full example: E♭ below and C above make a major sixth. By the two rules just given, C natural below and E flat above must make a minor third.

Interval roots

Although intervals are usually designated in relation to their lower note, David Cope and Hindemith both suggest the concept of interval root. David Cope (b San Francisco, California, United States, May 17, 1941) is an American Author, Composer Paul Hindemith (16 November 1895 &ndash 28 December 1963 was a German Composer, Violist, violinist teacher music theorist and conductor. 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). In music a sixth chord is any chord, or meaningful connection of notes that contains the interval of a sixth According the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C–G), is the bottom C, the tonic.

Interval cycles

Interval cycles, "unfold 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. In Music, an interval cycle is the collection of pitches created by starting with a certain note and going up by a certain interval until the original note George Perle (born May 6, 1915 in Bayonne New Jersey) is a Composer and music theorist. 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. (Perle 1990, p. 21)

Other intervals

There are also a number of intervals not found in the chromatic scale or labeled with a diatonic function which have names of their own. Many of these intervals describe small discrepancies between notes tuned according to the tuning systems used. Most of the following intervals may be described as microtones. Microtonal music is Music using microtones — intervals of less than an equally spaced Semitone.

See List of Musical Intervals for more. Bali is an Indonesian Island located at, the westernmost of the Lesser Sunda Islands, lying between Java to the west and Lombok to Background List See also List of meantone intervals

See Musical interval mnemonics at Wikibooks for popular musical fragments that feature common intervals

Generalizations and non-pitch uses

The term "interval" can also be generalized to other elements of music besides pitch. David Lewin's Generalized Musical Intervals and Transformations uses interval as a generic measure of distance in order to show musical transformations which can change, for instance, one rhythm into another, or one formal structure into another. David Lewin ( July 2 1933 - May 5 2003) was an American Music theorist, music Critic and Composer.

See also

Sources

Notes

  1. ^ Lindley, Mark/Campbell, Murray/Greated, Clive. "Interval", Grove Music Online, ed. The Grove Dictionary of Music and Musicians is an Encyclopedic dictionary of music and musicians L. Macy (accessed 27 February 2007), grovemusic.com (subscription access). Events 1560 - The Treaty of Berwick, which would expel the French from Scotland, is signed by England and the Congregation Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century.
  2. ^ Roeder, John. "Interval Class", Grove Music Online, ed. The Grove Dictionary of Music and Musicians is an Encyclopedic dictionary of music and musicians L. Macy (accessed 27 February 2007), grovemusic.com (subscription access). Events 1560 - The Treaty of Berwick, which would expel the French from Scotland, is signed by England and the Congregation Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century.
  3. ^ Kostka, Stephen; Payne, Dorothy. Tonal Harmony. First Edition, 1984.

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