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Example of Z-relation on two pitch sets analyzable as or

derivable from Z17[1] Play , with intervals between pitch

classes labeled for ease of comparison between the two set

and their common interval vector, 212320.

Interval vector: C major chord

{0,4,7}: 001110.

Interval vectorFrom Wikipedia, the free encyclopedia

n musical set theory, an interval vector(also

alled an interval-class vectoror ic vector) is an

array that expresses the intervallic content of a

pitch-class set. Often referred to as a PIC vector(orpitch-class interval vector), Schuijer suggests that

APIC vector(or absolute pitch-class interval

vector) is more accurate.

One can think of the ICV as the common calculus

derivative of the source material as a discrete

unction; ICV is at root a vector: a non-scalar value

n simple math, and thus subject to the universe of

mathematics. The interval vector is a species of

ntegervector calculus differential, a vectordifferential of the sourcematerial taken as a binary

vector. It can also be calculated via a sort of Discrete Fourier transform

using the Integer function in place of the Exponential function. Exactly

as Fourier transform maps a waveform between time domain and

harmonic-content domain, the ICV maps between an applied musical

domain and an harmonic-reductionist domain.

n 12 equal temperament the ICV has six digits, with each digit standing

orthe number of times an interval class appears in the set. (Interval

lasses, not regular intervals, must be used, in order that the interval vector remains the same, regardless of th

et's permutation or vertical arrangement.) The interval classes represented by each digit ascend from left to

ight. That is:

1) minor seconds/major sevenths (1 or 11 semitones)

2) major seconds/minor sevenths (2 or 10 semitones)

3) minor thirds/major sixths (3 or 9 semitones)

4) major thirds/minor sixths (4 or 8 semitones)

5) perfect fourths/perfect fifths (5 or 7 semitones)

6) tritones (6 semitones) (The tritone is inversionally related to itself.)

nterval class 0 (representing unisons and octaves) is omitted.

The concept was named intervalic contentby Howard Hanson in his The Harmonic Materials of Modern Mu

where he introduced the monomial notation pemdnc.sbdatf[note 1]for what would now be written

The modern notation, which has considerable advantages and is extendable to any equal division of the octav
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was introduced by Allen Forte.

A scale whose interval vector contains six different numbers is said to have the deep scale property. Major,

natural minor and modal scales have this property.

For a practical example, the interval vector for a C major triad in the root position, {C E G} ( Play ), is

. This means that the set has one major third or minor sixth (i.e. from C to E, or E to C), one minor

hird or major sixth (i.e. from E to G, or G to E), and one perfect fifth or perfect fourth (i.e. from C to G, or GC). As the interval vector will not change with transposition or inversion, it belongs to the entire set class, an

is the vector of all major (and minor) triads. It should, however, be noted that some interval vector

orrespond to more than one sets that cannot be transposed or inverted to produce the other. (These are called

elated sets, explained below).

For a set ofxelements, the sum of all the numbers in the set's interval vector equals (x*(x-1))/2.

While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the

ound qualities that are created by different collections of pitch classes. That is, sets with high concentrations

onventionally dissonant intervals (i.e. seconds and sevenths) will generally be heard as more dissonant, whi

ets with higher numbers of conventionally consonant intervals (i.e. thirds and sixths) will be heard as more

onsonant. (While the actual perception of consonance and dissonance involves many contextual factors, suc

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.

Contents

1 Z-relation

1.1 Multiplication

2 See also

3 Further reading

4 Notes

5 Sources

6 External links

Z-relation

n musical set theory, a Z-relation, also called isomeric relation, is a relation between two pitch-class sets i

which the two sets have the same intervallic content (i.e. they have the same interval vector), but they are of

different Tn-type and Tn/TnI-type. That is to say, one set cannot be derived from the other through transposit
http://en.wikipedia.org/wiki/Transposition_(music)http://en.wikipedia.org/wiki/Pitch-classhttp://en.wikipedia.org/wiki/Set_theory_(music)http://en.wikipedia.org/wiki/Interval_vector#External_linkshttp://en.wikipedia.org/wiki/Interval_vector#Sourceshttp://en.wikipedia.org/wiki/Interval_vector#Noteshttp://en.wikipedia.org/wiki/Interval_vector#Further_readinghttp://en.wikipedia.org/wiki/Interval_vector#See_alsohttp://en.wikipedia.org/wiki/Interval_vector#Multiplicationhttp://en.wikipedia.org/wiki/Interval_vector#Z-relationhttp://en.wikipedia.org/wiki/David_Lewinhttp://en.wikipedia.org/wiki/Transformation_theory_(music)http://en.wikipedia.org/wiki/Register_(music)http://en.wikipedia.org/wiki/Pitch_classhttp://en.wikipedia.org/wiki/Interval_vector#Z-relationhttp://en.wikipedia.org/wiki/Set_classhttp://upload.wikimedia.org/wikipedia/commons/5/50/C_major_triad.midhttp://en.wikipedia.org/wiki/File:C_major_triad.midhttp://en.wikipedia.org/wiki/Deep_scale_propertyhttp://en.wikipedia.org/wiki/Allen_Forte


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Successive Z-related hexachords from act 3 of Wozzeck[2]

Play .

or inversion.[1]

For example, the two sets {0,1,4,6} and {0,1,3,7}

have the same interval vector () but

hey are not transpositionally or inversionally

elated.

n the case of hexachords each may be referred to asa Z-hexachord. Any hexachord not of the "Z" type

s its own complement while the complement of a

Z-hexachord is its Z-correspondent, for example 6-

Z3 and 6-Z36.[2]See: 6-Z44, 6-Z17, 6-Z11, and

Forte number.

The term, for "zygotic" (yoked or the fusion of two reproductive cells),[3]originated with Allen Forte in 1964

but the notion seems to have first been considered by Howard Hanson. Hanson termed this the isomeric

elationship, defining two such sets to be isomeric.[4]According to Michael Schuijer (2008), "the discovery ohe relation," was, "reported," by David Lewin in 1960.[3][5]

Though it is commonly observed that Z-related sets always occur in pairs, David Lewin noted that this is a

esult of twelve-tone equal temperament (12-ET). In 16-ET, Z-related sets are found as triplets. Lewin's stud

onathan Wild continued this work for other tuning systems, finding Z-related tuplets with up to 16 members

higher ET systems.

Straus argues, "[sets] in the Z-relation will sound similar because they have the same interval content,"[6]wh

has led certain composers to exploit 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.

Multiplication

Some Z-related chords are connected byMorIM(multiplication by 5 or multiplication by 7), due to identica

entries for 1 and 5 on the interval vector.[3]

See also

Interval cycle

Pitch interval

Further reading

Rahn, John (1980).Basic Atonal Theory. ISBN 0-02-873160-3.
http://en.wikipedia.org/wiki/Special:BookSources/0028731603http://en.wikipedia.org/wiki/Pitch_intervalhttp://en.wikipedia.org/wiki/Interval_cyclehttp://en.wikipedia.org/wiki/Interval_vector#cite_note-Schuijer-4http://en.wikipedia.org/wiki/Multiplication_(music)http://en.wikipedia.org/wiki/Interval_vector#Z-relationhttp://en.wikipedia.org/wiki/Elliot_Carterhttp://en.wikipedia.org/wiki/Interval_vector#cite_note-7http://en.wikipedia.org/wiki/Equal_temperamenthttp://en.wikipedia.org/wiki/David_Lewinhttp://en.wikipedia.org/wiki/Interval_vector#cite_note-6http://en.wikipedia.org/wiki/Interval_vector#cite_note-Schuijer-4http://en.wikipedia.org/wiki/David_Lewinhttp://en.wikipedia.org/wiki/Interval_vector#cite_note-5http://en.wikipedia.org/wiki/Howard_Hansonhttp://en.wikipedia.org/wiki/Allen_Fortehttp://en.wikipedia.org/wiki/Interval_vector#cite_note-Schuijer-4http://en.wikipedia.org/wiki/Yokehttp://en.wikipedia.org/wiki/Zygotichttp://en.wikipedia.org/wiki/Forte_numberhttp://en.wikipedia.org/wiki/Sacher_hexachordhttp://en.wikipedia.org/wiki/All-trichord_hexachordhttp://en.wikipedia.org/wiki/Schoenberg_hexachordhttp://en.wikipedia.org/wiki/Interval_vector#cite_note-Forte-3http://en.wikipedia.org/wiki/Complement_(music)http://en.wikipedia.org/wiki/Hexachordhttp://en.wikipedia.org/wiki/Interval_vector#cite_note-Schuijer_99-1http://en.wikipedia.org/wiki/Inversion_(music)http://upload.wikimedia.org/wikipedia/en/c/ce/Z-related_hexachords_from_Wozzeck.midhttp://en.wikipedia.org/wiki/File:Z-related_hexachords_from_Wozzeck.midhttp://en.wikipedia.org/wiki/Interval_vector#cite_note-Forte-3http://en.wikipedia.org/wiki/Wozzeckhttp://en.wikipedia.org/wiki/File:Z-related_hexachords_from_Wozzeck.png


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Notes

1. ^In order to quantify the consonant/dissonant content of a set, Hanson ordered the intervals according to their

dissonance degree, with p=perfect fifth, m=major third, n=minor third, s=major second, d=(more dissonant) minor

second, t=tritone

Sources

1. ^ abSchuijer, Michael (2008).Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.99. ISBN 978-1-

58046-270-9.

2. ^ abForte, Allen (1977). The Structure of Atonal Music, p.79. Yale University Press. ISBN 0-300-02120-8.

3. ^ abcSchuijer (2008), p.98 and 98n18. The meaning of "Z" was finally revealed on Nov. 17, 2004.

4. ^Hanson, Howard (1960).Harmonic Materials of Modern Music,. Appleton-Century-Crofts. ISBN 0-89197-207-2.

5. ^Lewin, David. "The Intervallic Content of a Collection of Notes, Intervallic Relations between a Collection of Note

and its Complement: an Application to Schoenbergs Hexachordal Pieces."Journal of Music Theory4/1 (1960): 98

6. ^Straus (1990).Introduction to Post-Tonal Theory, 67. ISBN 0-13-189890-6. Cited in Schuijer (2008), p.125.

External links

Set classes and interval-class content (http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-

set_new/pages/page06/page06.html)

Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology by Robert T.

Kelley (http://www.robertkelleyphd.com/atnltrms.htm)

Twentieth Century Pitch Theory: Some Useful Terms and Techniques

(http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm)

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Categories: Musical set theory

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