|
Z-relation Z17 Example
In Set theory (music), musical set theory, an interval vector is an array of natural numbers which summarize the Interval (music), intervals present in a Set (music), set of pitch classes. (That is, a set of Pitch (music), pitches where octaves are disregarded.) Other names include: ic vector (or interval-class vector), PIC vector (or pitch-class interval vector) and APIC vector (or absolute pitch-class interval vector, which Michiel Schuijer states is more proper.) While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the sound qualities that are created by different collections of pitch class. That is, sets with high concentrations of conventionally dissonant intervals (i.e., seconds and sevenths) sound more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e., thirds and sixths) sound more consonance and dissonance, consonant. While the actual perception of consonance and dissonance involv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversional Equivalency
In music theory, an inversion is a rearrangement of the top-to-bottom elements in an interval, a chord, a melody, or a group of contrapuntal lines of music. 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 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 augme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Lewin
David Benjamin Lewin (July 2, 1933 – May 5, 2003) was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation", he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to music. Biography Lewin was born in New York City and studied piano from a young age and was for a time a pupil of Eduard Steuermann. He graduated from Harvard in 1954 with a degree in mathematics. Lewin then studied theory and composition with Roger Sessions, Milton Babbitt, Edward T. Cone, and Earl Kim at Princeton University, earning an M.F.A. in 1958. He returned to Harvard as a Junior Fellow in the Harvard Society of Fellows from 1958 to 1961. After holding teaching positions at the University of California, Berkeley (1961–67), the State University of New York at Stony Brook (1967–79), and Yale University (1979–85), he returned to Harv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation Theory (music)
Transformational theory is a branch of music theory developed by David Lewin in the 1980s, and formally introduced in his 1987 work ''Generalized Musical Intervals and Transformations''. The theory—which models musical transformations as elements of a mathematical group—can be used to analyze both tonal and atonal music. The goal of transformational theory is to change the focus from musical objects—such as the "C major chord" or "G major chord"—to relations between musical objects (related by transformation). Thus, instead of saying that a C major chord is followed by G major, a transformational theorist might say that the first chord has been "transformed" into the second by the " Dominant operation." (Symbolically, one might write "Dominant(C major) = G major.") While traditional musical set theory focuses on the makeup of musical objects, transformational theory focuses on the intervals or types of musical motion that can occur. According to Lewin's description of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-relation
In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interval-class vector), PIC vector (or pitch-class interval vector) and APIC vector (or absolute pitch-class interval vector, which Michiel Schuijer states is more proper.) While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the sound qualities that are created by different collections of pitch class. That is, sets with high concentrations of conventionally dissonant intervals (i.e., seconds and sevenths) sound more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e., thirds and sixths) sound more consonant. While the actual perception of consonance and dissonance involves many contextual factors, such as register, an interval vector can nevertheless be a he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Class
A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) 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.Wittlich, Gary (1975). "Sets and Ordering Procedures in Twentieth-Century Music", ''Aspects of Twentieth-Century Music'', p.475. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall. . A set by itself does not necessarily possess any additional structure, such as an ordering or permutation. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called ''segments''); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis. Two-element sets are called dyads, three-e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Set Classes
This is a list of set classes, by Forte number. In music theory, a ''set class'' (an abbreviation of ''pitch-class-set class'') is an ascending collection of pitch classes, transposed to begin at zero. For a list of ordered collections, see this list of tone rows and series. Sets are listed with links to their complements. For unsymmetrical sets, the prime form is marked with "A" and the inversion with "B"; sets without either are symmetrical. Sets marked with a "Z" refer to a pair of different set classes with identical interval class content unrelated by inversion, with one of each pair listed at the end of the respective list when they occur. ("Z" is derived from the prefix ''zygo-'', from Ancient Greek ''ζυγόν'', "yoke". Hence: zygosets.) "T" and "E" are conventionally used in sets to notate ten and eleven, respectively, as single characters. The ordering of sets in the lists is based on the string of numerals in the interval vector treated as an integer, decre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
