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Musical Set Theory
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonality, tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone technique, 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 (set (music), sets and permutation (music), permutations) of pitch (music), pitches and pitch classes (pitch-class set theory), which may be Order theory, ordered or unordered, and can be related by musical operations such as Transposition (music), transposition, melodic inversion, and Complement (music), complementation. Some theorists apply the methods of musical set theory to the anal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Rhythm
Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recurrence or pattern in time can apply to a wide variety of cyclical natural phenomena having a periodicity or frequency of anything from microseconds to several seconds (as with the riff in a rock music song); to several minutes or hours, or, at the most extreme, even over many years. The Oxford English Dictionary defines rhythm as ''"The measured flow of words or phrases in verse, forming various patterns of sound as determined by the relation of long and short or stressed and unstressed syllables in a metrical foot or line; an instance of this"''. Rhythm is related to and distinguished from pulse, meter, and beats: In the performance arts, rhythm is the timing of events on a human scale; of musical sounds and silences that occur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trichord
In music theory, a trichord () is a group of three different pitch classes found within a larger group. A trichord is a contiguous three-note set from a musical scale or a twelve-tone row. In musical set theory there are twelve trichords given inversional equivalency, and, without inversional equivalency, nineteen trichords. These are numbered 1–12, with symmetrical trichords being unlettered and with uninverted and inverted nonsymmetrical trichords lettered A or B, respectively. They are often listed in prime form, but may exist in different voicings; different inversions at different transpositions. For example, the major chord, 3-11B (prime form: ,4,7, is an inversion of the minor chord, 3-11A (prime form: ,3,7. 3-5A and B are the Viennese trichord (prime forms: ,1,6and ,5,6. Historical Russian definition In late-19th to early 20th-century Russian musicology, the term trichord (трихорд ()) meant something more specific: a set of three pitches, each at lea ... [...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. The notes of a dyad can be played simultaneously or in succession. Notes played in succession form a melodic interval; notes played simultaneously form a harmonic interval. Dyads can be classified by the interval between the notes. For example, the interval between C and E (four half steps) 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. . See also * Double stop *Interval (music) *Power chord *Harmonic series (music) *Counterpoint In music theory, counterpoint is the relationship of two or more simultaneous musical lines (also called voices) that are harmonically dependent on each other, yet independent in rhythm and melodic contour. The term originates from the Latin ... References Intervals (music) Simultaneities (music) {{Music-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneity (music)
In music, a simultaneity is more than one complete musical texture occurring at the same time, rather than in succession. This first appeared in the music of Charles Ives, and is common in the music of Conlon Nancarrow and others. Types In music theory, a pitch simultaneity is more than one pitch or pitch class all of which occur at the same time, or simultaneously: "A set of notes sounded together." ''Simultaneity'' is a more specific and more general term than chord: many but not all chords or harmonies are simultaneities, though not all but some simultaneities are chords. For example, arpeggios are chords whose tones are not simultaneous. "The practice of harmony typically involves both simultaneity...and linearity."Hijleh, Mark (2012). ''Towards a Global Music Theory: Practical Concepts and Methods for the Analysis of Music Across Human Cultures'', chapter 4, . Ashgate. . A simultaneity succession is a series of different groups of pitches or pitch classes, each of which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naive Set Theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping stone towards more formal treatments. Method A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. Such theory treats sets as platonic absolute o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a ''k''-combination of a set ''S'' is a subset of ''k'' distinct elements of ''S''. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has ''n'' elements, the number of ''k''-combinations, denoted by C(n,k) or C^n_k, is equal to the binomial coefficient \binom nk = \frac, which can be written using factorials as \textstyle\frac whenever k\leq n, and which is zero when k>n. This formula can be derived from the fact that each ''k''-combination of a set ''S'' of ''n'' members has k! permu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Set
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However some a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a ''vertical reflection'') would look like q. Its image by reflection in a horizontal axis (a ''horizontal reflection'') would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term ''reflection'' is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. The set of fixed points (the "mirror") of such an isome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
