- Interval vector
In

musical set theory , an**interval vector**(also called an interval-class vector or ic vector) is an array that expresses the intervallic content of apitch-class set.In 12 equal temperament it has six digits, with each digit standing for the number of times an

interval class appears in the set. (Interval classes, not regular intervals, must be used, in order that the interval vector remains the same, regardless of the set's permutation or vertical arrangement.) The interval classes represented by each digit ascend from left to right. That is::1) minor seconds/major sevenths (1/11 semitones):2) major seconds/minor sevenths (2/10 semitones):3) minor thirds/major sixths (3/9 semitones):4) major thirds/minor sixths (4/8 semitones):5) perfect fourths/perfect fifths (5/7 semitones):6) tritones (6 semitones) (The tritone is inversionally related to itself.)Interval class 0 (representing unisons and octaves) is omitted.

The concept was named "intervalic content" by

Howard Hanson in his "The Harmonic Materials of Modern Music", where he introduced themonomial notation d^{a}s^{b}n^{c}m^{d}p^{e}t^{f}for what would now be written . The modern notation, which has considerable advantages and is extendable to anyequal division of the octave was introduced byAllen Forte .While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the sound qualities are created by different collections of pitch classes. That is, sets with high concentrations of conventionally dissonant intervals (i.e. seconds and sevenths) will generally be heard as more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e. thirds and sixths) will be heard as more consonant. (While the actual perception of consonance and dissonance involves many contextual factors, such as register, an interval vector, nevertheless, can be a helpful tool.)

An expanded form of the interval vector is also used in transformation theory, as set out in

David Lewin 's "Generalized Musical Intervals and Transformations".**Z-relation**In

musical set theory , a**Z-relation**, also called**isomeric relation**, is a relation between twopitch-class sets in which the two sets have the same intervallic content (i.e. they have the sameinterval vector ), but they are of different T_{"n"}-type and T_{"n"}/T_{"n"}I-type. That is to say, one set cannot be derived from the other through transposition or inversion.For example, the two sets {0,1,4,6} and {0,1,3,7} have the same interval vector (<1,1,1,1,1,1>) but they are not transpositionally or inversionally related.

The term originated with

Allen Forte , but the notion seems to have first been considered byHoward Hanson . Hanson termed this the isomeric relationship, defining two such sets to be "isomeric".Though it is commonly observed that Z-related sets always occur in pairs,

David Lewin noted that this relation is a result of twelve-toneequal temperament (12-ET). In 16-ET, Z-related sets are found as triplets. Lewin's student Jonathan Wild continued this work for other tuning systems, finding Z-related tuplets with up to 16 members in higher ET systems.Someweal|July 2008 argue that the "relation" is often so remote as to be imperceptible, but certain composersweal|July 2008 have exploited the Z-relation in their work. For instance, the play between {0,1,4,6} and {0,1,3,7} is clear in

Elliot Carter 's second string quartet.**ee also***

Set theory (music) **Further reading***Rahn, John (1980). "Basic Atonal Theory". ISBN 0-02-873160-3.

*Forte, Allen (1973/1977). "Structure of Atonal Music". ISBN 0-300-01610-7/ISBN 0-300-02120-8.

*Hanson, Howard (1960). "The Harmonic Materials of Modern Music". Appleton-Century-Crofts.

*Straus, Joseph N. (1990/2000/2005). "Introduction to Post-Tonal Theory" 3rd Ed. ISBN 0-13-189890-6.**Z-relation references*** Allen Forte, "The Structure of Atonal Music", Yale University Press, 1977. ISBN 0-300-02120-8

* Howard Hanson, "Harmonic Materials of Modern Music", Appleton-Century-Crofts, 1960**External links*** [

*http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/page06/page06.html Set classes and interval-class content*]

* [*http://www.robertkelleyphd.com/atnltrms.htm Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology by Robert T. Kelley*]

* [*http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm Twentieth Century Pitch Theory: Some Useful Terms and Techniques*]

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