Last updated Traditional interval complementation: P4 + P5 = P8
In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.
Music theory is the study of the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory":
The first is what is otherwise called "rudiments", currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, and so on. [...] The second is the study of writings about music from ancient times onwards. [...] The third is an area of current musicological study that seeks to define processes and general principles in music—a sphere of research that can be distinguished from analysis in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built.
In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were also working to establish serialism as a form of post-tonal thinking. Twelve-tone technique orders the twelve notes of the chromatic scale, forming a row or series and providing a unifying basis for a composition's melody, harmony, structural progressions, and variations. Other types of serialism also work with sets, collections of objects, but not necessarily with fixed-order series, and extend the technique to other musical dimensions, such as duration, dynamics, and timbre.
In interval complementation a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is also known as its inverse or inversion. Note that the octave and the unison are each other's complements and that the tritone is its own complement (though the latter is "re-spelt" as either an augmented fourth or a diminished fifth, depending on the context).
In music theory, an interval is the 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 music, an octave or perfect octave is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". The interval between the first and second harmonics of the harmonic series is an octave.
In music, unison is two or more musical parts sounding the same pitch or at an octave interval, usually at the same time.
In the aggregate complementation of twelve-tone music and serialism the complement of one set of notes from the chromatic scale contains all the other notes of the scale. For example, A-B-C-D-E-F-G is complemented by B♭-C♯-E♭-F♯-A♭.
The chromatic scale or twelve-tone scale is a musical scale with twelve pitches, each a semitone above or below its adjacent pitches. As a result, in 12-tone equal temperament, the chromatic scale covers all 12 of the available pitches. Thus, there is only one chromatic scale.
Note that musical set theory broadens the definition of both senses somewhat.
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.
Interval complementation
Rule of nine
The rule of nine is a simple way to work out which intervals complement each other.[1] Taking the names of the intervals as cardinal numbers (fourth etc. becomes four), we have for example 4 + 5 = 9. Hence the fourth and the fifth complement each other. Where we are using more generic names (such as semitone and tritone) this rule cannot be applied. However, octave and unison are not generic but specifically refer to notes with the same name, hence 8 + 1 = 9.
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 most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half.
Perfect intervals complement (different) perfect intervals, major intervals complement minor intervals, augmented intervals complement diminished intervals, and double diminished intervals complement double augmented intervals.
Rule of twelve
Integer interval complementation: 5 + 7 = 0 mod 12
Using integer notation and modulo 12 (in which the numbers "wrap around" at 12, 12 and its multiples therefore being defined as 0), any two intervals which add up to 0 (mod 12) are complements (mod 12). In this case the unison, 0, is its own complement, while for other intervals the complements are the same as above (for instance a perfect fifth, or 7, is the complement of the perfect fourth, or 5, 7 + 5 = 12 = 0 mod 12).
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth is the fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones.
In musical set theory or atonal theory, complement is used in both the sense above (in which the perfect fourth is the complement of the perfect fifth, 5+7=12), and in the additive inverse sense of the same melodic interval in the opposite direction – e.g. a falling 5th is the complement of a rising 5th. [citation needed]
In twelve-tone music and serialism complementation (in full, literal pitch class complementation) is the separation of pitch-class collections into complementary sets, each containing pitch classes absent from the other[2] or rather, "the relation by which the union of one set with another exhausts the aggregate".[3] To provide, "a simple explanation...: the complement of a pitch-class set consists, in the literal sense, of all the notes remaining in the twelve-note chromatic that are not in that set."[4]
In the twelve-tone technique this is often the separation of the total chromatic of twelve pitch classes into two hexachords of six pitch classes each. In rows with the property of combinatoriality, two twelve-note tone rows (or two permutations of one tone row) are used simultaneously, thereby creating, "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."[2] In other words, the first and second hexachord of each series will always combine to include all twelve notes of the chromatic scale, known as an aggregate, as will the first two hexachords of the appropriately selected permutations and the second two hexachords.
Hexachordal complementation is the use of the potential for pairs of hexachords to each contain six different pitch classes and thereby complete an aggregate.[5]
The difference is always 11. The first set may be called P0 (see tone row), in which case the second set would be P1.
In contrast, "where transpositionally related sets show the same difference for every pair of corresponding pitch classes, inversionally related sets show the same sum."[7] For example, given the inversionally related sets (P0 and I11):
The sum is always 11. Thus for P0 and I11 the sum of complementation is 11.
Abstract complement
[clarification needed]In set theory the traditional concept of complementation may be distinguished as literal pitch class complement, "where the relation obtains between specific pitch-class sets",[3] while, due to the definition of equivalent sets, the concept may be broadened to include "not only the literal pc complement of that set but also any transposed or inverted-and-transposed form of the literal complement,"[8] which may be described as abstract complement,[9] "where the relation obtains between set classes".[3] This is because since P is equivalent to M, and M is the complement of M, P is also the complement of M, "from a logical and musical point of view,"[10] even though not its literal pc complement. Originator Allen Forte[11] describes this as, "significant extension of the complement relation," though George Perle describes this as, "an egregious understatement".[12]
As a further example take the chromatic sets 7-1 and 5-1. If the pitch-classes of 7-1 span C–F♯ and those of 5-1 span G–B then they are literal complements. However, if 5-1 spans C–E, C♯–F, or D–F♯, then it is an abstract complement of 7-1.[9] As these examples make clear, once sets or pitch-class sets are labeled, "the complement relation is easily recognized by the identical ordinal number in pairs of sets of complementary cardinalities".[3]
In music, a tone row or note row, also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or even actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is, "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set
In musical set theory, an interval class, also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'", is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on modulo 12. The largest interval class is 6 since any greater interval n may be reduced to 12 − n.
The term "partition" is also French for the sheet music of a transcription.
In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."
In music, transposition refers to the process or operation of moving a collection of notes up or down in pitch by a constant interval.
The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.
In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words, a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached. In other words, interval cycles "unfold a single recurrent interval in a series that closes with a return to the initial pitch class". See: wikt:cycle.
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals, it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.
A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.
In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. Other names include: ic vector, PIC vector and APIC vector
An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords, when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].
A trope or tropus may refer to a variety of different concepts in medieval, 20th-, and 21st-century music.
In music theory, the word inversion has distinct, but related, meanings when applied to intervals, chords, voices, and melodies. The concept of inversion also plays an important role in musical set theory.
In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." Those cycles are ascending and descending, being related by inversion since complementary:
In music, the all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is {012478} and its Forte number is 6-Z17. Its complement is 6-Z43 and they share the interval vector of <3,2,2,3,3,2>.
The Tone Clock, and its related compositional theory Tone-Clock Theory, is a post-tonal music composition technique, developed by composers Peter Schat and Jenny McLeod. Music written using tone-clock theory features a high economy of musical intervals within a generally chromatic musical language. This is because tone-clock theory encourages the composer to generate all their harmonic and melodic material from a limited number of intervallic configurations. Tone-clock theory is also concerned with the way that the three-note pitch-class sets can be shown to underlie larger sets, and considers these triads as a fundamental unit in the harmonic world of any piece. Because there are twelve possible triadic prime forms, Schat called them the 'hours', and imagined them arrayed in a clock face, with the smallest hour in the 1 o'clock position, and the largest hour in the 12 o'clock position. A notable feature of Tone-Clock Theory is 'tone-clock steering': transposing and/or inverting hours so that each note of the chromatic aggregate is generated once and once only.
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