Dictionary## Interval |

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In Intervals may be described as: - Vertical or
**harmonic**, if the two notes sound simultaneously - Horizontal, linear, or
**melodic**if they sound successively.
Intervals may be roughly classified as: - Diatonic intervals, between the notes of a
**diatonic scale**, - Chromatic intervals, non-diatonic intervals between the notes of a
**chromatic scale**, - Minute intervals (commas, and microtones), sometimes so small that the difference in pitch between the two notes cannot be perceived.
## Interval widthThe size or width of an interval can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents. ## Frequency ratiosMain article: Interval ratio
The size of an interval between two pitches may be measured by the ratio of their frequencies. Important intervals are those measured by fractions of small numbers, such as 1:1 (unison or prime), 2:1 (octave), 3:2 ( ## CentsMain article: Cent (music)
The standard system for comparing interval sizes is with cents. This is a logarithmic scale in which the octave is divided into 1200 equal parts. In The size in cents of the interval from frequency ## Main intervalsThe table shows the most widely used conventional names for the intervals between the notes of a Intervals with different names but spanning the same number of semitones may have the same width, provided that the instrument is tuned so that the 12 notes of the chromatic scale are equally spaced (a commonly used tuning system called Except for the Latin ones, the names listed here cannot be determined by counting semitones alone. The rules to determine them are explained below. Other names, determined with different naming conventions, are listed in a separate section.
## Interval number and qualityIn Western ## NumberAs shown above, intervals are labeled according to the number of staff positions they encompass, counting both the lines and gaps between lines, starting with one at the lower note. The number of staff positions from C to G for example is 5, therefore the interval C-G is a fifth (denoted In a The rule to determine the diatonic number of a compound interval (an interval larger than one octave), based on the diatonic numbers of the simple intervals from which it is built is explained in a separate section. ## QualityThe name of any interval is further qualified using the terms
## ExampleNeither the number, nor the quality of an interval can be determined by counting semitones alone. As explained above, the number of staff positions must be taken into account as well. For example, as shown in the table below, there are four semitones between A and C♯, between A and D♭, between A♯ and D, and between A♭ and B♯, but - the interval A-C♯ is called a major third (as it spans 3 staff positions),
- the intervals A-D♭ and A♯-D are called diminished fourths (as they span 4 staff positions),
- the interval A♭-B♯ is called a doubly augmented second (as it spans 2 staff positions).
The diminished fourth is an interval found between the seventh and third degrees of the harmonic minor scale, while the doubly augmented second only occurs in entirely chromatic contexts. In ## Shorthand notationIntervals are often abbreviated with a - m2 (or min2): minor second,
- M3 (or maj3): major third,
- A4 (or aug4): augmented fourth,
- d5 (or dim5): diminished fifth,
- P5 (or perf5): perfect fifth.
## Intervals in chordsMain articles:
Chord (music) and Chord names and symbols (jazz and pop music)
## Chord qualities and interval qualitiesThe main chord qualities are: dom for dominant (the symbol − alone is not used for diminished).## Deducing component intervals from chord names and symbolsThe main rules to decode chord - For 3-note chords (
**triads**), major or minor always refer to the third interval, while augmented and diminished always refer to the fifth. The same is true for the corresponding symbols (e.g., CM means CM3, and C+ means C+5). Thus, the terms third and fifth and the corresponding symbols 3 and 5 are typically omitted. This rule holds for 4-note chords (tetrads) as well,^{[3]}provided the above mentioned qualities appear immediately after the root note. 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, they should be considered interval qualities, rather than chord qualities. For instance, in Cm/M7 (minor-major seventh chord), m is the chord quality and M refers to the M7 interval. In some cases, the chord quality may refer*not only*to the basic triad (i.e., the third or fifth interval), but*also*to the following interval number. For instance, in CM7 M refers to both M3 and M7 (see specific rules below). - 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 (both the major third and the perfect fifth are implied). In Cm (C minor triad), a minor third is deduced according to rule 1, and a perfect fifth is implied according to this rule. This rule has one exception (see below). - When the fifth interval is diminished, the third must be minor, as a major third would produce a non-tertian chord.
^{[4]}This rule overrides rule 2. For instance, in Cdim7 a diminished fifth is deduced according to rule 1, and a minor third is implied according to this rule. - A plain 7 or seventh is equivalent to dom7 or dominant seventh, and stands for an extra minor seventh interval, added to the implied major triad.
The table shows the intervals contained in some of the main chords (
## ClassificationIntervals can be described, classified, or compared with each other according to various criteria. ## Melodic and harmonicAn interval can be described as - Vertical or
**harmonic**if the two notes sound simultaneously - Horizontal, linear, or
**melodic**if they sound successively.^{[5]}
## Diatonic and chromaticMain article: Diatonic and chromatic
A ## Consonant and dissonantMain article:
Consonance and dissonance
These terms are relative to the usage of different compositional styles. - In the Middle Ages, only the unison, octave, perfect fourth, and perfect fifth were considered consonant harmonically.
- 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").^{[6]}In the**common practice period**, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously thought to be dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16th-century practice continued to be taught to beginning musicians throughout this period. - Hermann von Helmholtz (1821–1894) defined a harmonically consonant interval as one in which the two pitches have an overtone in common (specifically
*excluding*the seventh harmonic). This essentially defines all seconds and sevenths as dissonant, while perfect fourths and fifths, and major and minor thirds and sixths, are consonant. - Pythagoras defined a hierarchy of consonance based on how small the numbers are that express the ratio. 20th-century composer and theorist
**Paul Hindemith**'s system has a hierarchy with the same results as Pythagoras's, but defined by fiat rather than by interval ratios, to better accommodate equal temperament, all of whose intervals (except the octave) would be dissonant using acoustical methods. - David Cope (1997) suggests the concept of interval strength
^{[7]}, 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.
All of the above analyses refer to vertical (simultaneous) intervals. ## Simple and compoundA simple interval is an interval spanning at most one octave. Intervals spanning more than one octave are called compound intervals. In general, a compound interval may be defined by a sequence or "stack" of two or more simple intervals of any kind. For instance, a major tenth (two staff positions above one octave), also called Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a seventeenth can be always decomposed into two octaves and one major third, and this is the reason why it is called a compound major third, even when it is built using four fifths. The diatonic number which can also be written as: The quality of a compound interval is determined by the quality of the simple interval on which it is based. For instance, a compound major third is a major tenth (1+(8-1)+(3-1) = 10), or a major seventeenth (1+(8-1)+(8-1)+(3-1) = 17), and a compound perfect fifth is a perfect twelfth (1+(8-1)+(5-1) = 12) or a perfect nineteenth (1+(8-1)+(8-1)+(5-1) = 19). Notice that two octaves are a fifteenth, not a sixteenth (1+(8-1)+(8-1) = 15). Similarly, three octaves are a twentysecond (1+3*(8-1) = 22), and so on. Intervals larger than a seventeenth seldom need to be spoken of, most often being referred to by their compound names, for example "two octaves plus a fifth" ## Steps and skipsLinear (melodic) intervals may be described as The words ## Enharmonic intervalsMain article:
EnharmonicTwo intervals are considered to be 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, as shown in the matrix below, F♯–A♯ (a major third), G♭–B♭ (also a major third), F♯–B♭ (a diminished fourth), and G♭–A♯ (a doubly augmented second) are all enharmonically equivalent. In fact, although they have a different name and staff position, F♯ and G♭ indicate the same pitch, and the same is true for A♯ and B♭. As a consequence, all these intervals span four semitones.
## 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 In 1/4-comma meantone, by definition 11 perfect fifths have a size of approximately 697 cents (700−ε cents, where ε ≈ 3.42 cents); since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of about 700+11ε cents, which is about 738 cents (the wolf fifth); 8 major thirds have size about 386 cents (400−4ε), 4 have size about 427 cents (400+8ε), and their average size is 400 cents. In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε (the difference between the 1/4-comma meantone fifth and the average fifth). A more detailed analysis is provided here. Note that 1/4-comma meantone was designed to produce just major thirds, but only 8 of them are just (5:4, about 386 cents). The Pythagorean tuning is characterized by smaller differences because they are multiples of a smaller ε (ε ≈ 1.96 cents, the difference between the Pythagorean fifth and the average fifth). Notice that here the fifth is wider than 700 cents, while in most meantone temperaments, including 1/4-comma meantone, it is tempered to a size smaller than 700. A more detailed analysis is provided here. 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 here. Note that 5-limit tuning was designed to maximize the number of just intervals, but even in this system some intervals are not just (e.g., 3 fifths, 5 major thirds and 6 minor thirds are not just; also, 3 major and 3 minor thirds are wolf 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 In the diatonic system, every interval has one or more ## Minute intervalsThere are also a number of minute intervals not found in the chromatic scale or labeled with a diatonic function, which have names of their own. They may be described as microtones. Except for the quarter tone, the equivalents in cents are approximate, and they can be also classified as commas, as they describe small discrepancies, observed in some tuning systems, between - 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) ( Play (help·info)). - A
*septimal kleisma*is six major thirds up, five fifths down and one octave up, with 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.
See Musical interval mnemonics at Wikibooks for popular musical fragments that feature common intervals ## InversionMain article:
Inversion (music)A simple interval (i.e., an interval shorter than or equal to an octave) may be There are two rules to determine the number and quality of the inversion of any simple interval: - The interval number and the number of its inversion always add up to nine (4 + 5 = 9, in the example just given).
- The inversion of a major interval is a minor interval, and vice versa; the inversion of a perfect interval is also perfect; the inversion of an augmented interval is a diminished interval, and vice versa; the inversion of a doubly augmented interval is a doubly diminished interval, and vice versa.
For example, the interval from C to the E♭ above it is a minor third. By the two rules just given, the interval from E♭ to the C above it must be a major sixth. Since compound intervals are larger than an octave, "the inversion of any compound interval is always the same as the inversion of the simple interval from which it is compounded." For intervals identified by their ratio, the inversion is determined by reversing the ratio and multiplying by 2. For example, the inversion of a 5:4 ratio is an 8:5 ratio. For intervals identified by an integer number of semitones, the inversion is obtained by subtracting that number from 12. Since an ## Interval rootAlthough intervals are usually designated in relation to their lower note, David Cope As to its usefulness, Cope ## Interval cyclesMain articles: Interval cycle and Identity (music)
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 conventionsAs 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. Notice that The diminished second is a comma, but some commas are not diminished seconds. For intance, the Pythagorean comma (531441:524288) is the opposite of a diminished second. 5-limit tuning defines four kinds of comma, three of which meet the definition of diminished second, and hence are listed in the table below. The fourth one, called syntonic comma (81:80) can neither be regarded as a diminished second, nor as its opposite. See here 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. ## Pitch-class intervalsMain articles:
Interval class and Ordered pitch intervalPost-tonal or In atonal or musical set theory there are numerous types of intervals, the first being ordered pitch interval, the distance between two pitches upward or downward. For instance, the interval from C to G upward is 7, but the interval from G to C downward is −7. One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, somewhat similar to the interval of tonal theory. The interval between pitch classes may be measured with ordered and unordered pitch-class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. For unordered pitch-class intervals, see ## Generic and specific intervalsMain articles: Specific interval and Generic interval
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. ## Comparison
## Generalizations and non-pitch usesThe term "interval" can also be generalized to other music elements besides pitch. David Lewin's ## See also## Notes- ^
^{a}^{b}Károlyi, Otto (1965),*Introducing Music*, p. 63. Hammondsworth (England), and New York: Penguin Books. ISBN 0140206590. **^**Lindley, Mark/Campbell, Murray/Greated, Clive. "Interval",*Grove Music Online*, ed. L. Macy (accessed 27 February 2007), grovemusic.com (subscription access).**^**General rule 1 achieves consistency in the interpretation of symbols such as CM7, Cm6, and Caug7. Some musicians legitimately prefer to think that, in CM7, the interval quality M refers to the seventh, rather than to both the third and seventh. However, this approach is inconsistent, as a similar interpretation is impossible for Cm6 and Caug7, where m cannot possibly refer to the sixth, which is major by definition, and aug cannot refer to the seventh, which is minor. Both approaches reveal only one of the intervals (M3 or M7), and require other rules to complete the task. Whatever is the decoding method, the result is the same (e.g., CM7 is always conventionally decoded as C-E-G-B, implying M3, P5, M7). The advantage of rule 1 is that it has no exceptions, which makes it the simplest possible approach to decode chord quality. According to the two approaches, some may format CM7 as CM^{7}(general rule 1), and others as C^{M7}(alternative approach). Fortunately, even C^{M7}becomes compatible with rule 1 if it is considered an abbreviation of CM^{M7}, in which the first M is omitted. The omitted M is the chord quality and is deduced according to rule 2 (see above), consistently with the interpretation of the plain symbol C, which by the same rule stands for CM.**^**The diminished fifth spans 6 semitones, thus it may be decomposed into a sequence of two minor thirds each spanning 3 semitones (m3 + m3), compatible with the definition of tertian chord. If a major third were used (4 semitones), a major second (2 semitones) would be necessary to reach the diminished fifth (4 + 2 = 6 semitones), but this sequence (M3 + M2) would not meet the definition of tertian chord.**^**Lindley, Mark/Campbell, Murray/Greated, Clive . " Interval ",*Grove Music Online*, ed. L. Macy (accessed 27 February 2007), grovemusic.com (subscription access).**^**Drabkin, William (2001). "Fourth".*The New Grove Dictionary of Music and Musicians*, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.- ^
^{a}^{b}^{c}Cope, David (1997).*Techniques of the Contemporary Composer*, pp. 40–41. New York, New York: Schirmer Books. ISBN 0-02-864737-8. **^**Wyatt, Keith (1998).*Harmony & Theory…*. Hal Leonard Corporation. pp. 77. ISBN 0793579910.**^**Aikin, Jim (2004).*A Player's Guide to Chords and Harmony: Music Theory for Real-World Musicians*, p. 24. ISBN 0-87930-798-6.**^**Kostka, Stephen; Payne, Dorothy (2008).*Tonal Harmony*, p. 21. First Edition, 1984.**^**Prout, Ebenezer (1903).*Harmony: Its Theory and Practice*, 16th edition. London: Augener & Co. (facsimile reprint, St. Clair Shores, Mich.: Scholarly Press, 1970), p. 10. ISBN 0-403-00326-1.**^**Hindemith, Paul (1934).*The Craft of Musical Composition*. New York: Associated Music Publishers. Cited in Cope (1997), p. 40-41.**^**Perle, George (1990).*The Listening Composer*, p. 21. California: University of California Press. ISBN 0-520-06991-9.**^**Roeder, John. "Interval Class",*Grove Music Online*, ed. L. Macy (accessed 27 February 2007), grovemusic.com (subscription access).**^**Lewin, David (1987).*Generalized Musical Intervals and Transformations*, for example sections 3.3.1 and 5.4.2. New Haven: Yale University Press. Reprinted Oxford University Press, 2007. ISBN 978-0-19-531713-8**^**Ockelford, Adam (2005).*Repetition in Music: Theoretical and Metatheoretical Perspectives*, p. 7. ISBN 0-7546-3573-2.
## External links- Interval conversion: Frequency ratio to cents and cents to frequency ratio
- Encyclopaedia Britannica, Interval
- Morphogenesis of chords and scales Chords and scales classification
- Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings
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