Pitch class set theory pdf file

Pdf slides of lessons, 2629 october 2009 lemmensinstituut, leuven belgium. Determine the normal order of the following tetrachords using letter names. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. A free visual crossplatform pitch class set theoretical application mika kuuskankare sibelius academy cmt marcus castrdn sibelius academy docmus mikael laurson sibelius academy cmt abstract macset 4 is a computer application intended to assist in learning and teaching basic pitch class set theory and analysis of posttonal music. If you get a number less than 0, add 12to get a valid pitch class number. Important to musical set theory, a pitch class is, all pitches related to each other by octave. The idea of an f is the idea of pitchclass, of notename leaving aside register. An important property of a set is the number of elements it contains. Pitchclass set theory practice quiz paul dell aquila. Chapter 2 simple operations on pitch classes and pitch.

A pitch class pc is the class or set of pitches with the same letter or solfege name that are octave duplications of one another. Schuijer has a keen sense of when the theory seems to take off and live a life of its own. Technically, a pitch class is the group class of all pitches related by octave. The paper suggests a possible intersection between pitchclass set theory theorized by a. Pitch class a pitch without regard to its octave position. The history of pitchclass set pcset theory as a modern and.

Pitchclass set multiplication in theory and practice music. Set theory is a method of musical analysis in which pitch classes are represented by numbers. A string of six numbers that communicates the intervalclass content of a given pitchclass set or set class. As with the latter, it defines a set as a collection of things, and it refers to these things as the elements of the set. View pitch class set theory research papers on academia. See robert morris, composition with pitch classe s. Important to musical set theory, a pitch class is, all pitches related to each other by octave, enharmonic equivalence, or both. You have probably encountered both of these concepts before, even if not by name. Because music employs a set of pitches ranging from low to high, the staff acts like a map for the notesallowing us to hear, read or write them as.

The decimal number is a numeric representation of the set used for identification, much like the forte set class numbers. Greek letters become constant reminders of the works theoretical basis, set theory. Lower lower on the staff higher higher on the staff another way to understand the idea of pitches being lower or higher is to compare it to bears and birds. Set class the abstraction of all particular instances of a class of pcsets that are related to one another either by transposition or inversion. Setclass the abstraction of all particular instances of a class of pcsets that are related to one another either by transposition or inversion. Inversions are marked b sets not marked a or b are symmetrical. This is an incredibly useful concept that forms the basis of much of set theory on the deeper levels of structure. Pitchclass set theory is a musical application of mathematical set theory. Pitch class theory is treated here as part of the musical and cultural landscape of the united states. Set 31 has three possible rotationsinversions, the normal form of which is the smallest pie or most compact form this is a list of pitch class sets by forte number. Pitch class we begin with a distinction introduced in the first term of freshman theory, between pitch and pitch class.

For the purposes of pitchclass set theory, we will not distinguish among. It is the sum of 2 x for all pitch classes x in the set. Set 31 has three possible rotationsinversions, the normal form of which is the smallest pie or most compact form this is a list of pitchclass sets by forte number. If you get a number larger than 11 or smaller than 0, add or subtract 12 to get a valid pitch class number. Notate the following tetrachords using integer notation. Pitch class sets, normal order, and transformations open. Transposition is by ordered pitch class interval, which means that we always count the transposition distance in a positive direction and the distance will be 11 or less. The inversion of a pitch classis the pitch class that is the same numberof half steps away from c, but in the opposite direction. Set theory is a method of musical analysis in which pitch classes are represented by numbers, and any grouping of these pitch classes is called a set. Modern composers will use pitch class sets like chemistry, to mix and create interesting and vibrantly colorful harmonic sounds, which they then use to create works of music. It makes no claims about music itself, but it does make some claims about the basic. Set class 0,1,6 was so popular with schoenberg and his disciples that it has been nicknamed the viennese trichord.

Aside generic interpretations, set theory has a standard kind of interpretations into itself, where each set is interpreted by the class meta set. The abstract, universal, or essential class of a pitch. The modulo operator takes the remainder of an integer divided by some other integer. A set is any unordered group of unique norepeats pitch classes. A concept like pitch, for example, is very concrete, while pitch class is somewhat more abstract. To transpose a set down by n half steps, subtract n from each pitch class in the set. This article expands on a paper presented at the annual meeting of the society for music theory, kansas city, 1992, and is primarily drawn from stephen heinemann, pitchclass set multiplication in boulezs le marteau sans maitre d.

Welcome to ams open math notes, a repository of freely downloadable mathematical works in progress hosted by the american mathematical society as a service to researchers, teachers and students. Basic set theory a set is a many that allows itself to be thought of as a one. Pitch class set theory and its contexts combines thorough discussions of musical concepts with an engaging historical narrative. To transpose a set by t n, add n to each integer of the set. This class of sets is represented by the prime form.

It is not a theory about music in any common sense that is, it is not some set of ideas about music that may or may not be true. Cardinal number refers to the number of elements in a set. Pitch class sets, normal order, and transformations. Interpreting inserting set theory into itself, involves articulating two interpretations models of set theory, which will be distinguished by giving the meta prefix to the one used as framework. Every pitch that can be called an f, say, is collected together into the pitch class that we just call f.

A pitch class set is a list of pitch class numbers. In set theory, inversionmeans the same number of half stepsin the opposite direction. Basic concepts of set theory, functions and relations. Set theory concepts the abstract and the particular. When manipulating pitch classes, you will use a special operator, called the modulo operator. When two setsaandbconsist of the same elements, they are.

Introduction to pitchclass set theory flashcards quizlet. This is the first in a series of posts introducing pitch class set theory at a very basic level. Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. Should examples from manuals and textbooks of music analysis or the.

The pitch class c stands for all possible cs, in whatever octave position. Determine the normal order of the following trichords using letter names. These draft works include course notes, textbooks, and research expositions in progress. In this post ill say a few things about what the theory s for and why its useful, and give some of the most basic definitions. Given a pitch class set in normal form, the transposition number is the pitch class interval that must be added to all the pitch classes to obtain the second set. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. Dec 31, 2008 pc pitch class set theory was part of a programme to lift the importance of music theory in american universities. This intersection turns out to be an expansion of the concept of pitchclass set, so that it can express additional information concerning the frequencies under consideration. Pitch class sets are the chemistry of harmonic color.

The group is a pitch and all other pitches related by. Sets are usually denoted by capital letters, likea,b,c, etc. Pitch class sets 1 1202007 pitch class sets paul nelson revised. These notations and methods can describe and manipulate any type of chord that can be created within a 12tone equally tempered scale. In set theory, inversion means the same number of half steps in the opposite direction.

For example, the pitch class set consisting of the notes c, e, and g would be written as 0,4,7. Common tones are examples of invariance, but the concept is larger. Xenakis uses in eonta three basic sets, which he names 5, 6, and i. But avoid asking for help, clarification, or responding to other answers. Set theory music theory for the 21stcentury classroom. For those of you new to abstract mathematics elementary does not mean simple though much of the material. As you can imagine from the name of the theory, pitch class set theory is a theory about pitch classes. If not specified, the elements are denoted by lowercase letters. We can perform a pitch, but we cant really perform a pitch class.

Pitch class set theory is a musical application of mathematical set theory. A pitch class set pc set is a group of pitch classes. Normal order is the smallest possible arrangement of pitch classes, in ascending order. For example, the set 0, 1, 4, 6 is represented by the number. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Composers treat sets with varying amounts of freedom when applying the set class method to their atonal music. Set class and prime form 1 lots of concepts in pitchclass set theory are best viewed along a sliding scale of concreteness or abstractness. Set classes john paul ito general overview pitchclass set theory is not well named. Set theory \a set is a many that allows itself to be thought of as a one. Usually it isa motive that occurs melodically, harmonically, or both. Pdf from pitchclass set theory to musical composition.