Set theory 3-1 in the chromatic circle fix

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 devel ...

, a Forte number is the pair of

number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

Allen Forte Allen Forte (December 23, 1926 – October 16, 2014) was an American music theorist and musicologist. He was Battell Professor Emeritus of the Theory of Music at Yale University and specialized in 20th-century atonal music and music analysis. ...

assigned to the prime form of each

pitch class In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of three or more members in ''The Structure of

Atonal Atonality in its broadest sense is music that lacks a tonal center, or key. ''Atonality'', in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on ...

Music'' (1973, ). The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches. Parallel tonic chords on C

In the

12-TET 12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 (\sqrt 2/math> ≈ 1.05946). That resul ...

tuning system (or in any other system of tuning that splits the

octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...

into twelve

semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...

s), each pitch class may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers. The prime form of a pitch class set is the most compact (i.e., leftwards packed or smallest in

lexicographic order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...

) of either the normal form of a set or of its inversion. The normal form of a set is that which is transposed so as to be most compact. For example, a

second inversion The second Inverted chord, inversion of a Chord (music), chord is the Voicing (music), voicing of a Triad (music), triad, seventh chord, or ninth chord in which the fifth (chord), fifth of the chord is the bass note. In this inversion, the bass ...

major chord In music theory, a major chord is a chord (music), chord that has a root (chord), root, a major third, and a perfect fifth. When a chord comprises only these three notes, it is called a major Triad (music), triad. For example, the major triad bui ...

contains the pitch classes 7, 0, and 4. The normal form would then be 0, 4, and 7. Its (transposed) inversion, which happens to be the

minor chord In music theory, a minor chord is a chord that has a root, a minor third, and a perfect fifth. When a chord comprises only these three notes, it is called a minor triad. For example, the minor triad built on A, called an A minor triad, has pit ...

, contains the pitch classes 0, 3, and 7; and is the prime form. Diatonic scale on C

The major and minor chords are both given Forte number 3-11, indicating that it is the eleventh in Forte's ordering of pitch class sets with three pitches. In contrast, the Viennese trichord, with pitch classes 0, 1, and 6, is given Forte number 3-5, indicating that it is the fifth in Forte's ordering of pitch class sets with three pitches. The normal form of the

diatonic scale In music theory a diatonic scale is a heptatonic scale, heptatonic (seven-note) scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by eith ...

, such as C major; 0, 2, 4, 5, 7, 9, and 11; is 11, 0, 2, 4, 5, 7, and 9; while its prime form is 0, 1, 3, 5, 6, 8, and 10; and its Forte number is 7-35, indicating that it is the thirty-fifth of the seven-member pitch class sets. Sets of pitches which share the same Forte number have identical

interval vector 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 interv ...

s. Those that have different Forte numbers have different interval vectors with the exception of z-related sets (for example 6-Z44 and 6-Z19).

Calculation

There are two prevailing methods of computing prime form. The first was described by Forte, and the second was introduced in John Rahn's ''Basic Atonal Theory'' and used in Joseph N. Straus's ''Introduction to Post-Tonal Theory'', and is now generally more popular. For example, the Forte prime form for 6-31 is 13589whereas the Rahn algorithm chooses 14579 where adjacency intervals are shown here by subscripts between pitch-class numerals. As seen, both versions of this set class have one of their largest adjacency intervals (3 semitones) at the right—i.e. they both have the smallest possible span—but, within that span, Forte chooses the version that is then most packed towards the left, whereas Rahn chooses the version that is most dispersed away from the right. In the language of

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 ...

, the Forte numbers correspond to the binary

bracelets A bracelet is an article of jewellery that is worn around the wrist. Bracelets may serve different uses, such as being worn as an ornament. When worn as ornaments, bracelets may have a supportive function to hold other items of decoration, ...

of length 12: that is,

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

es of

binary sequence A bitstream (or bit stream), also known as binary sequence, is a sequence of bits. A bytestream is a sequence of bytes. Typically, each byte is an 8-bit quantity, and so the term octet stream is sometimes used interchangeably. An octet may ...

s of length 12 under the operations of

cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k ...

and reversal. In this correspondence, a one in a binary sequence corresponds to a pitch that is present in a pitch class set, and a zero in a binary sequence corresponds to a pitch that is absent. The rotation of binary sequences corresponds to transposition of chords, and the reversal of binary sequences corresponds to inversion of chords. The most compact form of a pitch class set is the lexicographically maximal sequence within the corresponding equivalence class of sequences.

Elliott Carter Elliott Cook Carter Jr. (December 11, 1908 – November 5, 2012) was an American modernist composer who was one of the most respected composers of the second half of the 20th century. He combined elements of European modernism and American " ...

had earlier (1960–1967) produced a numbered listing of pitch class sets, or "chords", as Carter referred to them, for his own use.Carter, Elliott (2002). ''The Harmony Book'', "Appendix 1". .

References

External links

"All About Set Theory: What is a Forte Number?"
''JayTomlin.com''.

, ''ComposerTools.com''.

, ''SolomonsMusic.net''.

, ''MtA.Ca''. {{Set theory (music) Musical set theory

Calculation

See also

References

External links