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| [[Image:Z-relation Z17 example.png|thumb|right|350px|Example of [[Interval vector#Z-relation|Z-relation]] on two pitch sets analyzable as or derivable from Z17 (Schuijer 2008, p.99), with intervals between pitch classes labeled for ease of comparison between the two sets and their common interval vector, 212320.]]
| | Looking for some simple solutions that will help you have your pole in the water tomorrow? Rest easy, as this activity is fun and simple. Because fishing is a diverse hobby, it accommodates those of varied levels of knowledge and skill. Here are some tips to help make fishing a fun and successful activity, whatever your skill level.<br><br>People that wish to fish in a stream need to cast upstream and allow the current to sweep the lure into the fishing hole. This makes your bait more natural, which increases chances of catching a fish. Try to always keep your line taught, and reduce the slack in your line so you can feel the fish bite.<br><br>Stream fisherman would be wise to start their fishing trips downstream, and make their way upstream as the day progresses. This is simply because fish tend to face against the direction of the current. By moving upstream you are lowering your chances of being seen or heard by your prey.<br><br>Do not ever forget to wear a life jacket when you are fishing. Make sure that it is securely fastened. You also want to make sure that the one you have is the right size. This is very important,because if it is too large it can slip over your head if you fall into the water.<br><br>When fishing alone, be careful not to go into water that is too deep. This is especially true when fishing near large rivers as a sudden undercurrent could pull you under and drown you in even fairly shallow water. It is typically better to fish with at least one other friend.<br><br>If you are going to be using live bait to go fishing, be sure your bait is not [http://www.latestsms.in/indian-political-jokes.htm Friendship SMS] kept in the sun for too long. Most fish prefer bait that is cooler, rather than warmer. Have the bait kept in an insulated container until you are ready to use it.<br><br>A great tip for all fisherman is to stay relatively quite while you are on the water. Everyone has heard that loud voices can scare away fish, and while this is slightly exaggerated, making a lot of noise on the surface, particularly disturbing the surface, can indeed spook fish.<br><br>The introduction stressed the fact that not knowing how to fish isn't a bad thing. Picking up fishing is not difficult at all. Even though there are many levels to it, there are many entry points for the hobby, and if you use the tips from this article, you can start fishing now. |
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| '''Musical set theory''' provides concepts for categorizing [[music]]al objects and describing their relationships. Many of the notions were first elaborated by [[Howard Hanson]] (1960) in connection with [[tonality|tonal]] music, and then mostly developed in connection with [[atonal]] music by theorists such as [[Allen Forte]] (1973), drawing on the work in [[twelve-tone technique|twelve-tone]] theory of [[Milton Babbitt]]. The concepts of set theory are very general and can be applied to tonal and atonal styles in any [[equal temperament|equally-tempered]] 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 class]]es ('''pitch-class set theory'''), which may be [[order (mathematics)|ordered or unordered]], and which can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of [[rhythm]] as well.
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| ==Mathematical set theory versus musical set theory==
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| Although musical set theory is often thought to involve the application of mathematical [[set theory]] to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms [[transposition (music)|transposition]] and [[Inversion (music)|inversion]] where mathematicians would use [[translation (geometry)|translation]] and [[reflection (mathematics)|reflection]].
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| Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of [[ordered set]]s, and these can be seen to include the musical kind in some sense, they are far more involved).
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| Moreover, musical set theory is more closely related to [[group theory]] and [[combinatorics]] than to mathematical set theory, which concerns itself with such matters as, for example, various sizes of infinitely large sets. In combinatorics, an unordered subset of ''n'' objects, such as [[pitch class]]es, is called a [[permutations and combinations|combination]], and an ordered subset a ''permutation''. Musical set theory is best regarded as a field that is not so much related to mathematical set theory, as an application of combinatorics to music theory with its own vocabulary. The main connection to mathematical set theory is the use of [[naive set theory|the vocabulary of set theory]] to talk about finite sets.
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| ==Set and set types==
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| {{Main|Set (music)}}
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| The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes (Rahn 1980, 27). More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates) (Forte 1973, 3). The elements of a set may be manifested in music as [[simultaneity (music)|simultaneous]] chords, successive tones (as in a melody), or both.{{Citation needed|date=January 2010}}<!--Must they be "manifested" at all? If so, why are these the only way they may be manifested? And who says so?--> Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {} (Rahn 1980, 28), or square brackets:[] (Forte 1973, 3). Some theorists use angle brackets <math>\langle \rangle</math> to denote ordered sequences (Rahn 1980, 21 & 134), while others distinguish ordered sets by separating the numbers with spaces (Forte 1973, 60–61). Thus one might notate the unordered set of pitch classes 0, 1, and 2 (corresponding in this case to C, C{{Music|#}}, and D) as {0,1,2}. The ordered sequence C-C{{Music|#}}-D would be notated <math>\langle 0,1,2 \rangle</math> or (0,1,2). Although C is considered to be zero in this example, this is not always the case. For example, a piece (whether tonal or atonal) with a clear pitch center of F might be most usefully analyzed with F set to zero (in which case {0,1,2} would represent F, F{{Music|#}} and G. (For the use of numbers to represent notes, see [[pitch class]].)
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| Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes,{{Citation needed|date=September 2010}}<!--Warburton and Cohn discuss beat-classes of rhythmic onsets, but not non-equal-tempered pitch classes.--> rhythmic onsets, or "beat classes" (Warburton 1988, 148; Cohn 1992, 149).
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| Two-element sets are called [[dyad (music)|dyad]]s, three-element sets [[trichord]]s (occasionally "triads", though this is easily confused with the traditional meaning of the word [[Triad (music)|triad]]). Sets of higher cardinalities are called [[tetrachord]]s (or tetrads), [[pentachord]]s (or pentads), [[hexachord]]s (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g., Rahn 1980, 140), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.
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| ==Basic operations==
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| The basic operations that may be performed on a set are [[transposition (music)|transposition]] and [[inversion (music)|inversion]]. Sets related by transposition or inversion are said to be ''transpositionally related'' or ''inversionally related,'' and to belong to the same [[set class]]. Since transposition and inversion are [[isometry|isometries]] of pitch-class space, they preserve the intervallic structure of a set, and hence its musical character. This can be considered the central postulate of musical set theory. In practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece.
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| Some authors consider the operations of [[complement (music)|complementation]] and [[multiplication (music)|multiplication]] as well. The complement of set X is the set consisting of all the pitch classes not contained in X (Forte 1973, 73–74). The product of two pitch classes is the product of their pitch-class numbers modulo 12. Since complementation and multiplication are not [[isometries]] of pitch-class space, they do not necessarily preserve the musical character of the objects they transform. Other writers, such as Allen Forte, have emphasized the [[Z-relation]] which obtains between two sets sharing the same total interval content, or [[interval vector]], but which are not transpositionally or inversionally equivalent (Forte 1973, 21). Another name for this relationship, used by Howard Hanson (1960), is "isomeric" (Cohen 2004, 33).
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| Operations on ordered sequences of pitch classes also include transposition and inversion, as well as [[Permutation (music)|retrograde and rotation]]. Retrograding an ordered sequence reverses the order of its elements. Rotation of an ordered sequence is equivalent to [[cyclic permutation]].
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| Transposition and inversion can be represented as elementary arithmetic operations. If ''x'' is a number representing a pitch class, its transposition by ''n'' semitones is written T''<sub>n</sub>'' = ''x'' + ''n'' (mod12). Inversion corresponds to [[Reflection (mathematics)|reflection]] around some fixed point in [[pitch class space]]. If "x" is a pitch class, the inversion with [[index number]] n is written I''<sub>n</sub>'' = ''n'' - ''x'' (mod12).
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| ==Equivalence relation==
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| "For a relation in set ''S'' to be an [[equivalence relation]] [in [[algebra]]], it has to satisfy three conditions: it has to be [[reflexive relation|reflexive]] [...], [[Symmetry in mathematics|symmetrical]] [...], and [[transitive relation|transitive]] [...]."(Schuijer 2008, p.29-30) "Indeed, an informal notion of equivalence has always been part of music theory and analysis. PC set theory, however, has adhered to formal definitions of equivalence" (Schuijer 2008, 85).
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| ==Transpositional and inversional set classes==
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| Two transpositionally related sets are said to belong to the same transpositional set class (T<sub>''n''</sub>). Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written T<sub>''n''</sub>I or I<sub>''n''</sub>). Sets belonging to the same transpositional set class are very similar-sounding; while sets belonging to the same transpositional/inversional set class are fairly similar sounding. Because of this, music theorists often consider set classes to be basic objects of musical interest.
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| There are two main conventions for naming equal-tempered set classes. One, known as the [[Forte number]], derives from Allen Forte, whose ''The Structure of Atonal Music'' (1973), is one of the first works in musical set theory. Forte provided each set class with a number of the form ''c''-''d'', where ''c'' indicates the cardinality of the set and ''d'' is the ordinal number (Forte 1973, 12). Thus the chromatic trichord {0, 1, 2} belongs to set-class 3-1, indicating that it is the first three-note set class in Forte's list (Forte 1973, 179–81). The augmented trichord {0, 4, 8}, receives the label 3-12, which happens to be the last trichord in Forte's list.
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| The primary criticisms of Forte's nomenclature are: (1) Forte's labels are arbitrary and difficult to memorize, and it is in practice often easier simply to list an element of the set class; (2) Forte's system assumes equal temperament and cannot easily be extended to include diatonic sets, pitch sets (as opposed to pitch-class sets), [[multisets]] or sets in other tuning systems; (3) Forte's original system considers inversionally related sets to belong to the same set-class.This means that, for example a major triad and a minor triad are considered the same set. Western tonal music for centuries has regarded major and minor as significantly different. Therefore there is a limitation in Forte's theory.{{Citation needed|date=September 2010}}<!--Who has made these criticisms? Who draws this conclusion?--> However, the theory was not created to fill a vacuum in which existing theories inadequately explained tonal music. Rather, Forte's theory is used to explain atonal music, where the composer has invented a system where the distinction between {0, 4, 7} (called 'major' in tonal theory) and its inversion {0, 8, 5} (called 'minor' in tonal theory) may not be relevant.
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| The second notational system labels sets in terms of their [[Set (music)#Non-serial|normal form]], which depends on the concept of ''normal order''. To put a set in ''normal order,'' order it as an ascending scale in pitch-class space that spans less than an octave. Then permute it cyclically until its first and last notes are as close together as possible. In the case of ties, minimize the distance between the first and next-to-last note. (In case of ties here, minimize the distance between the first and next-to-next-to-last note, and so on.) Thus {0, 7, 4} in normal order is {0, 4, 7}, while {0, 2, 10} in normal order is {10, 0, 2}. To put a set in normal form, begin by putting it in normal order, and then transpose it so that its first pitch class is 0 (Rahn 1980, 33–38). Mathematicians and computer scientists most often order combinations using either alphabetical ordering, binary (base two) ordering, or [[Gray coding]], each of which lead to differing but logical normal forms.{{Citation needed|date=September 2010}}
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| Since transpositionally related sets share the same normal form, normal forms can be used to label the T<sub>''n''</sub> set classes.
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| To identify a set's T<sub>''n''</sub>/I<sub>''n''</sub> set class:
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| * Identify the set's T<sub>''n''</sub> set class.
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| * Invert the set and find the inversion's T<sub>''n''</sub> set class.
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| * Compare these two normal forms to see which is most "left packed."
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| The resulting set labels the initial set's T<sub>''n''</sub>/I<sub>''n''</sub> set class. | |
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| ==Symmetry==
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| The number of distinct operations in a system that map a set into itself is the set's [[degree of symmetry]] (Rahn 1980, 90). Every set has at least one symmetry, as it maps onto itself under the identity operation T<sub>''0''</sub> (Rahn 1980, 91). Transpositionally symmetric sets map onto themselves for T<sub>''n''</sub> where ''n'' does not equal 0. Inversionally symmetric sets map onto themselves under T<sub>''n''</sub>I. For any given T<sub>''n''</sub>/T<sub>''n''</sub>I type all sets will have the same degree of symmetry. The number of distinct sets in a type is 24 (the total number of operations, transposition and inversion, for n = 0 through 11) divided by the degree of symmetry of T<sub>''n''</sub>/T<sub>''n''</sub>I type.
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| Transpositionally symmetrical sets either divide the octave evenly, or can be written as the union of equally-sized sets that themselves divide the octave evenly. Inversionally-symmetrical chords are invariant under reflections in pitch class space. This means that the chords can be ordered cyclically so that the series of intervals between successive notes is the same read forward or backward. For instance, in the cyclical ordering (0, 1, 2, 7), the interval between the first and second note is 1, the interval between the second and third note is 1, the interval between the third and fourth note is 5, and the interval between the fourth note and the first note is 5. One obtains the same sequence if one starts with the third element of the series and moves backward: the interval between the third element of the series and the second is 1; the interval between the second element of the series and the first is 1; the interval between the first element of the series and the fourth is 5; and the interval between the last element of the series and the third element is 5. Symmetry is therefore found between T<sub>0</sub> and T<sub>2</sub>I, and there are 12 sets in the T<sub>''n''</sub>/T<sub>''n''</sub>I equivalence class (Rahn 1980, 148).
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| ==See also==
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| * [[Identity (music)]]
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| * [[Pitch interval]]
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| * [[Tonnetz]]
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| * [[Transformational music theory]]
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| ==References==
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| * Cohen, Allen Laurence. 2004. ''Howard Hanson in Theory and Practice''. Contributions to the Study of Music and Dance 66. Westport, Conn. and London: Praeger. ISBN 0-313-32135-3.
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| * Cohn, Richard. 1992. "Transpositional Combination of Beat-Class Sets in Steve Reich's Phase-Shifting Music". ''Perspectives of New Music'' 30, no. 2 (Summer): 146–77.
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| * Forte, Allen (1973). ''The Structure of Atonal Music''. New Haven and London: Yale University Press. ISBN 0-300-01610-7 (cloth) ISBN 0-300-02120-8 (pbk).
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| * [[Howard Hanson|Hanson, Howard]] (1960). ''Harmonic Materials of Modern Music: Resources of the Tempered Scale''. New York: Appleton-Century-Crofts.
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| * Rahn, John (1980). ''Basic Atonal Theory''. New York: Schirmer Books; London and Toronto: Prentice Hall International. ISBN 0-02-873160-3.
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| *Schuijer, Michael (2008). ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts''. ISBN 978-1-58046-270-9.
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| * Warburton, Dan. 1988. "A Working Terminology for Minimal Music". ''Intégral'' 2:135–59.
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| ==Further reading==
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| *[[Elliott Carter|Carter, Elliott]] (2002). ''Harmony Book'', edited by Nicholas Hopkins and John F. Link. New York: Carl Fischer. ISBN 0-8258-4594-7.
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| * [[David Lewin|Lewin, David]] (1993). ''Musical Form and Transformation: Four Analytic Essays''. New Haven: Yale University Press. ISBN 0-300-05686-9. Reprinted, with a foreword by Edward Gollin, New York: Oxford University Press, 2007. ISBN 978-0-19-531712-1
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| * Lewin, David (1987). ''Generalized Musical Intervals and Transformations''. New Haven: Yale University Press. ISBN 0-300-03493-8. Reprinted, New York: Oxford University Press, 2007. ISBN 978-0-19-531713-8
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| * [[Robert Morris (composer)|Morris, Robert]] (1987). ''Composition With Pitch-Classes: A Theory of Compositional Design''. New Haven: Yale University Press. ISBN 0-300-03684-1.
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| * [[George Perle|Perle, George]] (1996). ''Twelve-Tone Tonality'', second edition, revised and expanded. Berkeley: University of California Press. ISBN 0-520-20142-6. (First edition 1977, ISBN 0-520-03387-6)
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| * Starr, Daniel (1978). "Sets, Invariance and Partitions". ''Journal of Music Theory'' 22, no. 1 (Spring): 1–42.
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| * Straus, Joseph N. (2005). ''Introduction to Post-Tonal Theory'', 3rd edition. Upper Saddle River, NJ: Prentice-Hall. ISBN 0-13-189890-6.
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| ==External links==
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| * Tucker, Gary (2001) [http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/ "A Brief Introduction to Pitch-Class Set Analysis"], ''Mount Allison University Department of Music''.
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| * Nick Collins [http://www.sonic.mdx.ac.uk/research/nickpitch.html "Uniqueness of pitch class spaces, minimal bases and Z partners"], ''Sonic Arts''.
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| * [http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm "Twentieth Century Pitch Theory: Some Useful Terms and Techniques"], ''Form and Analysis: A Virtual Textbook''.
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| * Solomon, Larry (2005). [http://solomonsmusic.net/setheory.htm "Set Theory Primer for Music"], ''SolomonMusic.net''.
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| * Kelley, Robert T (2001). [http://www.robertkelleyphd.com/atnltrms.htm "Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology"], ''RobertKelleyPhd.com''.
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| * Kelley (2002). [http://www.robertkelleyphd.com/12-tone.htm "Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method"].
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| * [http://www.flexatone.net/athenaSCv.html "SetClass View (SCv)"], ''Flexatone.net''. An athenaCL netTool for on-line, web-based pitch class analysis and reference.
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| * Tomlin, Jay. [http://www.jaytomlin.com/music/settheory/help.html "All About {Musical} Set Theory"], ''JayTomlin.com''.
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| * [http://www.jaytomlin.com/music/settheory/ "Java Set Theory Machine"] or Calculator
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| * Helmberger, Andreas (2006). [http://www.andreas-helmberger.de/projekte/pcsetcalculator.aspx "Projekte: Pitch Class Set Calculator"], ''www.Andreas-Helmberger.de''. {{de icon}}
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| * [http://dactyl.som.ohio-state.edu/Gibson/research.summary.html "Pitch-Class Set Theory and Perception"], ''Ohio-State.edu''. {{Dead link|date=September 2010}}
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| * [http://www.composertools.com/Tools/ "Software Tools for Composers"], ''ComposerTools.com''. Javascript PC Set calculator, two-set relationship calculators, and theory tutorial.
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| {{Music theory}}
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| {{Set theory (music)}}
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| {{DEFAULTSORT:Set Theory (Music)}}
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| [[Category:Musical set theory|*]]
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Looking for some simple solutions that will help you have your pole in the water tomorrow? Rest easy, as this activity is fun and simple. Because fishing is a diverse hobby, it accommodates those of varied levels of knowledge and skill. Here are some tips to help make fishing a fun and successful activity, whatever your skill level.
People that wish to fish in a stream need to cast upstream and allow the current to sweep the lure into the fishing hole. This makes your bait more natural, which increases chances of catching a fish. Try to always keep your line taught, and reduce the slack in your line so you can feel the fish bite.
Stream fisherman would be wise to start their fishing trips downstream, and make their way upstream as the day progresses. This is simply because fish tend to face against the direction of the current. By moving upstream you are lowering your chances of being seen or heard by your prey.
Do not ever forget to wear a life jacket when you are fishing. Make sure that it is securely fastened. You also want to make sure that the one you have is the right size. This is very important,because if it is too large it can slip over your head if you fall into the water.
When fishing alone, be careful not to go into water that is too deep. This is especially true when fishing near large rivers as a sudden undercurrent could pull you under and drown you in even fairly shallow water. It is typically better to fish with at least one other friend.
If you are going to be using live bait to go fishing, be sure your bait is not Friendship SMS kept in the sun for too long. Most fish prefer bait that is cooler, rather than warmer. Have the bait kept in an insulated container until you are ready to use it.
A great tip for all fisherman is to stay relatively quite while you are on the water. Everyone has heard that loud voices can scare away fish, and while this is slightly exaggerated, making a lot of noise on the surface, particularly disturbing the surface, can indeed spook fish.
The introduction stressed the fact that not knowing how to fish isn't a bad thing. Picking up fishing is not difficult at all. Even though there are many levels to it, there are many entry points for the hobby, and if you use the tips from this article, you can start fishing now.