Musical set theory
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Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as "sets" and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation.
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Mathematical set theory and musical set theory
Although musical set theory is often assumed to be the application of mathematical set theory to music, there is little coincidence between the terminology and even less between the methods of the two. In fact musical set theory is better characterized as the application of group theory and combinatorics to certain aspects of music theory. In musical set theory what is called a set is often in fact a tuple, an ordered collection of things (such as the term set form for tone row). Musical set theory also uses the terms linear and nonlinear for ordered and unordered sets, which has nothing to do with what these terms mean in mathematics. Allen Forte's book, The Structure of Atonal Music (ISBN 0300021208), one of the primary developments in musical set theory, is sometimes criticised for its supposedly faulty calculations and terminology. Musical set theory is best regarded however as an unrelated field from mathematical set theory, with its own vocabulary, whose only connection to mathematical set theory is in sometimes using the language of naive set theory to talk about finite sets.
Assumptions of atonal theory
In addition to octave and enharmonic equivalency assumed in twelve tone theory and equal tempered tonal theory, set theory also makes use of inversional and transpositional equivalency, though the degree of equivalency varies among theorists. Set theory does not, however, use diatonic functionality that is assumed in tonal theory, and this is the reason for the use of integer notation and modulo 12. Since the structures of tonal theory may then be constructed rather than assumed, tonal theory can be regarded as a specific area of atonal theory.
The set and set types
The fundamental concept of musical set theory is, of course, the set. A set is a collection of any musical materials or qualities, ordered or unordered, although most often sets of pitch classes are considered. Sets may be simultaneities or successions. A set is indicated by being enclosed in brackets: {}, an ordered set is indicated by <>, and an unordered set by (). Thus the set of pitch classes 0, 1, and 2 is {0,1,2}, the ordered set <0,1,2>, and the unordered set (0,1,2).
The domain of all pitch class sets may be partitioned into types or equivalence classes based on cardinality or number of pitch classes, or other criteria. There are thirteen cardinalities from 0-12: the null set, monad, dyad, trichord, tetrachord, pentachord, hexachord, septachord, octachord, nonachord, decachord, undecachord, and aggregate or dodecachord.
Basic operations
The basic operations that may be performed on a set are transposition and inversion and multiplication. Order operations include retrograde and rotation. Compound operations, the result of two basic operations, may be performed and the product of operations X and Y on z is written "Y(X(z))" with X performed on z, and then Y performed on that result. These operations may also be called transformations, mappings, morphisms, or permutations; and in music theory, but not in mathematics, derivations. Taking all combinations of a certain number of basic operations (for example taking all combinations of transposition, inversion, and multiplication by 7) produces permutation groups.
Transposition is moving a set up or down in pitch by a constant interval. If x is the original pitch transposed n semitones, Tn=x+n (mod12). Inversion is turning a set upside-down reversing the order of the intervals between pitch classes. More specifically the compound operation transpositional inversion is TnI(x)=-x+n (mod12). Multiplication is multiplying the pitch class numbers of a set, the most useful multipliers are 1, 5, 7, 11, as multiplication by 1 is the same, multiplication by 11 is inversion, multiplication of the chromatic scale by 5 produces the circle of fourths and multiplication by 7 produces the circle of fifths. Retrograde is reversing the order of the set so the first member is last and the last is first. Rotation is placing the last member of the set first.
Normal form
Another useful concept used in musical set theory is that of normal form. Since sets may be listed in any order without changing their identity, normal form is used as a way to compare sets (sometimes called normal order). Normal order is that which is stacked to the left, rises from left to right, within one octave and fits within the smallest interval. In the event of any ties for what produces the smallest outside interval, one compares the next most outside interval until the tie is broken, or the ordering that starts on the smallest pitch class integer is chosen. Normal order can be used to quickly compare if two sets may be transposed onto each other. For example, it is harder to compare {4,8,1} and {7,0,3} as quickly as {0,3,7} and {1,4,8}.
Transpositional and inversional types
Each of the cardinality types listed above may be further partitioned into transpositional type (Tn type) and/or inversional type (Tn/TnI type). A list of all sets which are in the same transpositional type as a given set may be found by transposing the original set by all intervals. Thus the trichord {8,4,1}, {1,4,8} in normal form, is in the same transpositional type as {1,4,8}+1={2,5,9}, {1,4,8}+2={3,6,10}, {4,7,11}, {5,8,0}, {6,9,1}, {7,10,2}, {8,11,3}, {9,0,4}, {10,1,5}, {11,2,6}, and {0,3,7}. All of the above are in the transpositional type {0,3,7}Tn, as the representative set is that which is in the most normal form. {0,3,7} is equivalent under transposition and/or inversion with twenty four rather than twelve sets, the twelve above and their inversions. It happens to be the representative set for its class: {0,3,7}Tn/TnI, as it is the most normal ordered form between the most normal ordered form uninverted, {0,3,7}, and the most normal ordered transposition of its inversion, {0,4,7} (T7I{0,3,7}={0,4,7}).
Thus, to find the type of a set:
- List the set in normal form.
- Transpose the set so that the first pitch class is zero.
This is the representative form of the Tn type.
- Perform TnI and repeat the steps above.
This is the representative form of the inversions Tn type.
- Compare the Tn type representative forms.
The most normal form of the two representative types above is the representative form of the set's TnI type.
Given any set of numbers from zero to eleven, there is a corresponding indexing integer ranging from 0 to 4095, defined as the sum of the numbers 2i for each number i in the set. Transpositions, or transpositions with inversion, are examples of permutation groups. Given any such group on the numbers from 0 to 11, we can find a corresponding representative form by finding the smallest index in the orbit of the set under the transformations of the group.
Symmetry
The number of times which a set may be mapped onto itself through different operations is its degree of symmetry. Every set has at least one degree of symmetry, as it maps onto itself under the identity operation T0. Transpositional symmetry is the property of set which maps onto itself for Tn where n does not equal 0. Inversional symmetry is the property of a set which maps into itself under TnI. For any given Tn/TnI 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 Tn/TnI type.
Inversionally symmetrical sets have a canonical ordering. The canonical ordering is the ordering such that the interval series of the set is its own retrograde or rather is retrograde symmetrical. For each of these canonical orderings a set will map onto itself under TxI, x being the inversional index which is the sum of the first and last members of each canonical ordering. The first and last members, and each pair of members farther in, of sets of even cardinality will all equal the inverional index. For odd cardinality sets the middle number is a 1/2 index and the center of inversional symmetry.
Transpositionally symmetrical sets in normal form may be partitioned into segments which under transposition map onto each other cyclically, so that the last segment maps onto the first.
Theorists and books
- John Rahn: Basic Atonal Theory (ISBN 0028731603)
- Allen Forte: Structure of Atonal Music (ISBN 0300021208)
- David Lewin: Musical Form and Transformation: 4 Analytic Essays (ASIN 0300056869), Generalized Musical Intervals and Transformations (ASIN 0300034938)
- Joseph N. Straus: Introduction to Post-Tonal Theory (ISBN 0130143316)
- Important terms: pitch (music), Z-relation, interval vector, permutation, identity (music), identity function
External links
- A Brief Introduction to Pitch-Class Set Analysis (http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/introduction/toc.html)
- Nick Collins : Uniqueness of pitch class spaces, minimal bases and Z partners (http://www.sonic.mdx.ac.uk/research/nickpitch.html)
- Twentieth Century Pitch Theory: Some Useful Terms and Techniques (http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm)
- Introduction to Set Theory by Larry Solomon (http://music.theory.home.att.net/setheory.htm)
- Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology by Robert T. Kelley (http://www-student.furman.edu/users/r/rkelley/atnltrms.htm)