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Interval Class
In musical set theory, an interval class (often abbreviated: ic), 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''. Use of interval classes The concept of interval class accounts for octave, enharmonic, and inversional equivalency. Consider, for instance, the following passage: (To hear a MIDI realization, click the following: In the example above, all four labeled pitch-pairs, or dyads, share a common "intervallic color." In atonal theory, this simil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Class
In musical set theory, an interval class (often abbreviated: ic), 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''. Use of interval classes The concept of interval class accounts for octave, enharmonic, and inversional equivalency. Consider, for instance, the following passage: (To hear a MIDI realization, click the following: In the example above, all four labeled pitch-pairs, or dyads, share a common "intervallic color." In atonal theory, this simil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory (music)
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. One branch of musical set theory deals with collections ( sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, melodic inversion, and complementation. Some theorists apply the methods of musical set theory to the analysis of rhythm as well. Mathematical set theory versus musical set theory Although musical set theory is often thought to involv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pitch Class Space
In music theory, pitch-class space is the circular space representing all the notes (pitch classes) in a musical octave. In this space, there is no distinction between tones that are separated by an integral number of octaves. For example, C4, C5, and C6, though different pitches, are represented by the same point in pitch class space. Since pitch-class space is a circle, we return to our starting point by taking a series of steps in the same direction: beginning with C, we can move "upward" in pitch-class space, through the pitch classes C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, and B, returning finally to C. By contrast, pitch space is a linear space: the more steps we take in a single direction, the further we get from our starting point. Tonal pitch-class space , and Lerdahl and Jackendoff (1983) use a "reductional format" to represent the perception of pitch-class relations in tonal contexts. These two-dimensional models resemble bar graphs, using height to represent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pitch Class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, 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 : = . Although there is no formal upper or lower limit to this sequence, only a few of these pitches are audible to humans. Pitch class is important because human pitch-perception is periodic: pitches belonging to the same pitch class are perceived as having a similar quality or color, a property called " octave equivalence". Psychologists refer to the quality of a pitch as its "chroma". A ''chroma'' is an attribute of pitches (as opposed to ''tone height''), just like hue is an attribute of color. A ''pitch class'' is a set of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulo Operation
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is the remainder of the Euclidean division of by , where is the dividend and is the divisor. For example, the expression "5 mod 2" would evaluate to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to inclusive ( mod 1 is always 0; is undefined, possibly resulting in a division by zero error in some programming languages). See Modular arithmetic for an older and related c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) 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 Western music notation, notes separated by an octave (or multiple octaves) have the same name and are of the same pitch class. To emphasize that it is one of the perfect intervals (including unison, perfect fourth, and perfect fifth), the octave is designated P8. Other interval qualities are also possible, though rare. The octave above or below an indicated note is sometimes abbreviated ''8a'' or ''8va'' ( it, all'ottava), ''8va bassa'' ( it, all'ottava bassa, sometimes also ''8vb''), or simply ''8'' for the octave in the direction indicated by pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin ''enharmonicus'', from Late Latin ''enarmonius'', from Ancient Greek ἐναρμόνιος (''enarmónios''), from ἐν (''en'') and ἁρμονία (''harmonía''). Definition For example, in any twelve-tone equal temperament (the predominant system of musical tuning in Western music), the notes C and D are ''enharmonic'' (or ''enharmonically equivalent'') notes. Namely, they are the same key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony and chord progressions. Arbitrary amounts of accidentals can produce further enharmonic equivalents, such as B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversional Equivalency
In music theory, an inversion is a type of change to intervals, chords, voices (in counterpoint), and melodies. In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in musical set theory. Intervals An interval is inverted by raising or lowering either of the notes by one or more octaves so that the positions of the notes reverse (i.e. the higher note becomes the lower note and vice versa). For example, the inversion of an interval consisting of a C with an E above it (the third measure below) is an E with a C above it – to work this out, the C may be moved up, the E may be lowered, or both may be moved. : The tables to the right show the changes in interval quality and interval number under inversion. Thus, perfect intervals remain perfect, major intervals become minor and vice versa, and augmented intervals become diminished and vice versa. (Doubly diminished intervals become doubly augmen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octatonic Ic7
An octatonic scale is any eight-note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetrical scale is commonly called the ''octatonic scale'' (or the ''octatonic collection''), although there are a total of 42 enharmonically non-equivalent, transpositionally non-equivalent eight-note sets. The earliest systematic treatment of the octatonic scale was in Edmond de Polignac's unpublished treatise "Étude sur les successions alternantes de tons et demi-tons (Et sur la gamme dite majeure-mineure)" (''Study of the Succession of Alternating Whole Tones and Semitones (and of the so-called Major-Minor Scale)'') from c. 1879, which preceded Vito Frazzi's ''Scale alternate per pianoforte'' of 1930 by a full half-century. Nomenclature In Saint Petersburg at the turn of the 20th century, this scale had become so familiar in the circle of comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyad (music)
In music, a dyad (less commonly, diad) is a set of two notes or pitches that, in particular contexts, may imply a chord. Dyads can be classified by the interval between the notes. For example, the interval between C and E is a major third, which can imply a C major chord, made up of the notes C, E and G.Young, Doug (2008). ''Mel Bay Presents Understanding DADGAD'', p.53. . When the pitches of a dyad occur in succession, they form a melodic interval. When they occur simultaneously, they form a harmonic interval. The harmonic series is built over a fundamental pitch, and the rest of the partials in the series are called overtones. The second partial is an octave above the fundamental and the third pitch is a fifth, so if C is the fundamental pitch the second note is C an octave higher and then the next pitch would be G. The harmonic series has more fifths than just this one, for example the fourth to the sixth, the sixth to the ninth and the seventh to the eleventh partial are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atonal
Atonality in its broadest sense is music that lacks a tonal center, or key. ''Atonality'', in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on a single, central triad is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term ''atonality'' describes music that does not conform to the system of tonal hierarchies that characterized European classical music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments". The term is also occasionally used to describe music that is neither tonal nor serial, especially the pre-twelve-tone music of the Second Viennese School, principally Alban Berg, Arnold Schoenberg, and Anton Webern. However, "as a categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
