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Interval vector

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In <a href="Set_theory_(music)" title="Set theory (music)">musical set theory</a>, an interval vector is an array of <a href="Natural_number" title="Natural number">natural numbers</a> which summarize the <a href="Interval_(music)" title="Interval (music)">intervals</a> present in a <a href="Set_(music)" title="Set (music)">set</a> of <a href="Pitch_class" title="Pitch class">pitch classes</a>. (That is, a set of <a href="Pitch_(music)" title="Pitch (music)">pitches</a> where <a href="Octave" title="Octave">octaves</a> are disregarded.) Other names include: ic vector (or interval-class vector), PIC vector (or pitch-class interval vector) and APIC vector (or absolute pitch-class interval vector, which Michiel Schuijer states is more proper.)<a href="#cite_note-Schuijer-1">[1]</a>: 48

While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the sound qualities that are created by different collections of pitch class. That is, sets with high concentrations of conventionally dissonant intervals (i.e., seconds and sevenths) sound more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e., thirds and sixths) sound more <a href="Consonance_and_dissonance" title="Consonance and dissonance">consonant</a>. While the actual perception of consonance and dissonance involves many contextual factors, such as <a href="Register_(music)" title="Register (music)">register</a>, an interval vector can nevertheless be a helpful tool.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

In <a href="Equal_temperament" title="Equal temperament">twelve-tone equal temperament</a>, an interval vector has six digits, with each digit representing the number of times an <a href="Interval_class" title="Interval class">interval class</a> appears in the set. Because interval classes are used, the interval vector for a given set remains the same, regardless of the set's <a href="Permutation_(music)" title="Permutation (music)">permutation</a> or vertical arrangement. The interval classes designated by each digit ascend from left to right. That is:

<ol><li>minor seconds/major sevenths (1 or 11 semitones)</li>

<li>major seconds/minor sevenths (2 or 10 semitones)</li>

<li>minor thirds/major sixths (3 or 9 semitones)</li>

<li>major thirds/minor sixths (4 or 8 semitones)</li>

<li>perfect fourths/perfect fifths (5 or 7 semitones)</li>

<li>tritones (6 semitones) (The tritone is <a href="Inversional_equivalency" class="mw-redirect" title="Inversional equivalency">inversionally equivalent</a> to itself.)</li></ol>

Interval class 0, representing unisons and octaves, is omitted.

In his 1960 book, The Harmonic Materials of Modern Music, <a href="Howard_Hanson" title="Howard Hanson">Howard Hanson</a> introduced a <a href="Monomial" title="Monomial">monomial</a> method of notation for this concept, which he termed intervallic content: pemdnc.sbdatf for what would now be written ⟨abcdef⟩.<a href="#cite_note-Hanson-1960-2">[2]</a><a href="#cite_note-3">[note 1]</a> The modern notation, introduced by Donald Martino in 1961, has considerable advantages and is extendable to any <a href="Equal_division_of_the_octave" class="mw-redirect" title="Equal division of the octave">equal division of the octave</a>.<a href="#cite_note-Martino-1961-4">[3]</a> Allen Forte in his 1973 work The Structure of Atonal Music notated the interval vector using square brackets, citing Martino;<a href="#cite_note-Forte-5">[4]</a>: 15 subsequent authors, e.g. <a href="John_Rahn" title="John Rahn">John Rahn</a>, use angled brackets.<a href="#cite_note-Rahn-1980-6">[5]</a>: 100

A scale whose interval vector has six unique digits is said to have the <a href="Deep_scale_property" class="mw-redirect" title="Deep scale property">deep scale property</a>. The major scale and its modes have this property.

For a practical example, the interval vector for a C <a href="Major_triad" class="mw-redirect" title="Major triad">major triad</a> (<a href="List_of_set_classes" title="List of set classes">3-11B</a>) in the root position, {C E G} (<span data-nosnippet="" id="ooui-php-2" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/60\/Major_triad_on_C.mid\/Major_triad_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major triad on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/6/60/Major_triad_on_C.mid/Major_triad_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play</a>ⓘ), is ⟨001110⟩. This means that the set has one major third or minor sixth (i.e. from C to E, or E to C), one minor third 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 G to C). As the interval vector does not change with transposition or inversion, it belongs to the entire <a href="Set_class" class="mw-redirect" title="Set class">set class</a>, meaning that ⟨001110⟩ is the vector of all major (and minor) triads. Some interval vectors correspond to more than one sets that cannot be transposed or inverted to produce the other. (These are called <a href="#Z-relation">Z-related sets</a>, explained below).

For a set of n pitch classes, the sum of all the numbers in the set's interval vector equals the <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{2}}={\tfrac {n(n-1)}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{2}}={\tfrac {n(n-1)}{2}}}</annotation>

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</math><img src="./bd968aa9578b40963d342ae78ceebc08659ab9ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.403ex; height:4.176ex;" alt="{\displaystyle {\tbinom {n}{2}}={\tfrac {n(n-1)}{2}}}" loading="lazy">, since the interval vector elements are computed comparing each pair of pitch classes from the set consisting of n elements. This corresponds also to the <a href="Triangular_number" title="Triangular number">triangular number</a>

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<annotation encoding="application/x-tex">{\displaystyle T_{n-1}}</annotation>

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</math><img src="./1d4bb2662bd4951087d40e217559205971339e54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.677ex; height:2.509ex;" alt="{\displaystyle T_{n-1}}" loading="lazy">.

An expanded form of the interval vector is also used in <a href="Transformation_theory_(music)" class="mw-redirect" title="Transformation theory (music)">transformation theory</a>, as set out in <a href="David_Lewin" title="David Lewin">David Lewin</a>'s Generalized Musical Intervals and Transformations.

<div class="mw-heading mw-heading2"><h2 id="Z-relation">Z-relation</h2></div>

In musical set theory, a Z-relation, also called isomeric relation, is a relation between two pitch class sets in which the two sets have the same intervallic content (and thus the same interval vector) but they are not <a href="Transposition_(music)" title="Transposition (music)">transpositionally</a> related (are of different Tn-type ) or <a href="Inversion_(music)" title="Inversion (music)">inversionally</a> related (are of different Tn/TnI-type).<a href="#cite_note-Schuijer-1">[1]</a>: 99 For example, the two sets 4-z15A {0,1,4,6} and 4-z29A {0,1,3,7} have the same interval vector ⟨111111⟩ but one can not transpose and/or invert the one set onto the other.

In the case of <a href="Hexachord" title="Hexachord">hexachords</a> each may be referred to as a Z-hexachord. Any hexachord not of the "Z" type is its own <a href="Complement_(music)" title="Complement (music)">complement</a> while the complement of a Z-hexachord is its Z-correspondent, for example 6-Z3 and 6-Z36.<a href="#cite_note-Forte-5">[4]</a>: 79 See: <a href="Schoenberg_hexachord" title="Schoenberg hexachord">6-Z44</a>, <a href="All-trichord_hexachord" title="All-trichord hexachord">6-Z17</a>, <a href="Sacher_hexachord" title="Sacher hexachord">6-Z11</a>, and <a href="Forte_number" title="Forte number">Forte number</a>.

The symbol "Z", standing for "<a href="Zygotic" class="mw-redirect" title="Zygotic">zygotic</a>" (from the Greek, meaning paired or <a href="Yoke" title="Yoke">yoked</a>, such as the fusion of two reproductive cells),<a href="#cite_note-Schuijer-1">[1]</a>: 98 originated with Allen Forte in 1964, but the notion seems to have first been considered by Howard Hanson. Hanson called this the isomeric relationship, and defined two such sets as isomeric.<a href="#cite_note-Hanson-1960-2">[2]</a>: 22 See: <a href="Isomer" title="Isomer">isomer</a>.

According to Michiel Schuijer (2008), the hexachord theorem, that any two pitch-class complementary hexachords have the same interval vector, even if they are not equivalent under transposition and inversion, was first proposed by <a href="Milton_Babbitt" title="Milton Babbitt">Milton Babbitt</a>, and, "the discovery of the relation," was, "reported," by <a href="David_Lewin" title="David Lewin">David Lewin</a> in 1960 as an example of the complement theorem: that the difference between pitch-class intervals in two complementary pitch-class sets is equal to the difference between the cardinal number of the sets (given two hexachords, this difference is 0).<a href="#cite_note-Schuijer-1">[1]</a>: 96–7 <a href="#cite_note-Lewin-7">[6]</a> Mathematical proofs of the hexachord theorem were published by Kassler (1961), Regener (1974), and Wilcox (1983).<a href="#cite_note-Schuijer-1">[1]</a>: 96–7

Though it is commonly observed that Z-related sets always occur in pairs, <a href="David_Lewin" title="David Lewin">David Lewin</a> noted that this is a result of twelve-tone <a href="Equal_temperament" title="Equal temperament">equal temperament</a> (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.

The equivalence relationship of `having the same interval content', allowing the trivial isometric case, was initially studied in crystallography and is known as <a href="Homometric_structures" title="Homometric structures">Homometry</a>. For instance the complement theorem is known to physicists as <a href="Babinet's_principle" title="Babinet's principle">Babinet's principle</a>. For a recent survey see.<a href="#cite_note-Mandereau_et_alii-8">[7]</a>

Straus argues, "[sets] in the Z-relation will sound similar because they have the same interval content,"<a href="#cite_note-9">[8]</a><a href="#cite_note-Schuijer-1">[1]</a>: 125 which 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 <a href="Elliott_Carter" title="Elliott Carter">Elliott Carter</a>'s <a href="String_Quartet_No._2_(Carter)" title="String Quartet No. 2 (Carter)">Second String Quartet</a>.

<div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3></div>

Some <a href="#Z-relation">Z-related</a> chords are connected by M or IM (<a href="Multiplication_(music)" title="Multiplication (music)">multiplication</a> by 5 or multiplication by 7), due to identical entries for 1 and 5 on the interval vector.<a href="#cite_note-Schuijer-1">[1]</a>: 83, 110

<ul><li><a href="Interval_cycle" title="Interval cycle">Interval cycle</a></li>

<li><a href="Pitch_interval" title="Pitch interval">Pitch interval</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> 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.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Schuijer-1">^ <a href="#cite_ref-Schuijer_1-0">a</a> <a href="#cite_ref-Schuijer_1-1">b</a> <a href="#cite_ref-Schuijer_1-2">c</a> <a href="#cite_ref-Schuijer_1-3">d</a> <a href="#cite_ref-Schuijer_1-4">e</a> <a href="#cite_ref-Schuijer_1-5">f</a> <a href="#cite_ref-Schuijer_1-6">g</a> <a href="#cite_ref-Schuijer_1-7">h</a> Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. University of Rochester. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-58046-270-9</bdi>.

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<li id="cite_note-Hanson-1960-2">^ <a href="#cite_ref-Hanson-1960_2-0">a</a> <a href="#cite_ref-Hanson-1960_2-1">b</a> Hanson, Howard (1960). Harmonic Materials of Modern Music New York: Appleton-Century-Crofts. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89197-207-2</bdi>.

</li>

<li id="cite_note-Martino-1961-4"><a href="#cite_ref-Martino-1961_4-0">^</a> <cite id="CITEREFMartino1961" class="citation journal cs1">Martino, Donald (1961). "The Source Set and Its Aggregate Formations". Journal of Music Theory. 5 (2). New Haven: Yale University Press: 224-273. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F843226">10.2307/843226</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/843226">843226</a>.</cite>

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<li id="cite_note-Forte-5">^ <a href="#cite_ref-Forte_5-0">a</a> <a href="#cite_ref-Forte_5-1">b</a> <a href="#cite_ref-Forte_5-2">c</a> <cite id="CITEREFForte1973" class="citation book cs1"><a href="Allen_Forte" title="Allen Forte">Forte, Allen</a> (1973). The Structure of Atonal Music. New Haven: <a href="Yale_University_Press" title="Yale University Press">Yale University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-300-01610-7</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/72091295">72091295</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/861792420">861792420</a>. <a href="OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL5307893M">5307893M</a>. <a href="WDQ_(identifier)" class="mw-redirect" title="WDQ (identifier)">Wikidata</a> <a href="https://www.wikidata.org/wiki/Q130092153" class="extiw external" title="d:Q130092153">Q130092153</a>.</cite>

</li>

<li id="cite_note-Rahn-1980-6"><a href="#cite_ref-Rahn-1980_6-0">^</a> <a href="John_Rahn" title="John Rahn">Rahn, John</a> (1980). Basic Atonal Theory. New York: Longman. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780582281172</bdi>. Reprinted 1987, New York: Schirmer Books; London: Collier Macmillan. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-02-873160-3</bdi>.

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<li id="cite_note-Lewin-7"><a href="#cite_ref-Lewin_7-0">^</a> Lewin, David. "The Intervallic Content of a Collection of Notes, Intervallic Relations between a Collection of Notes and its Complement: an Application to Schoenberg’s Hexachordal Pieces", Journal of Music Theory 4/1 (1960): 98–101.

</li>

<li id="cite_note-Mandereau_et_alii-8"><a href="#cite_ref-Mandereau_et_alii_8-0">^</a> John Mandereau, Daniele Ghisi, Emmanuel Amiot, Moreno Andreatta, Carlos Agon. Z-relation and homometry in musical distributions. Journal of Mathematics and Music, Taylor & Francis (2011), 5 (2), 83-98.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Straus, Joseph Nathan (1990). Introduction to Post-Tonal Theory, p.67. 1st ed. Prentice Hall: Englewood Cliffs, New Jersey. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-189890-6</bdi>. Cited in Schuijer (2008), p.125.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/page06/page06.html">Set classes and interval-class content</a></li>

<li><a rel="nofollow" class="external text" href="https://www.robertkelleyphd.com/home/atnltrms.htm">Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology, by Robert T. Kelley</a></li>

<li><a rel="nofollow" class="external text" href="http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm">Twentieth Century Pitch Theory: Some Useful Terms and Techniques</a></li></ul>

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</style><div id="Musical_set_theory89" style="font-size:114%;margin:0 4em"><a href="Set_theory_(music)" title="Set theory (music)">Musical set theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="All-interval_tetrachord" title="All-interval tetrachord">All-interval tetrachord</a></li>

<li><a href="All-trichord_hexachord" title="All-trichord hexachord">All-trichord hexachord</a></li>

<li><a href="Complement_(music)" title="Complement (music)">Complement</a></li>

<li><a href="Forte_number" title="Forte number">Forte number</a></li>

<li><a href="Identity_(music)" title="Identity (music)">Identity</a></li>

<li><a href="Interval_class" title="Interval class">Interval class</a></li>

<li><a href="Multiplication_(music)" title="Multiplication (music)">Multiplication</a></li>

<li><a href="Permutation_(music)" title="Permutation (music)">Permutation</a></li>

<li><a href="Pitch_class" title="Pitch class">Pitch class</a></li>

<li><a href="Pitch_interval" title="Pitch interval">Pitch interval</a></li>

<li><a href="Pitch_interval#Pitch-interval_class" title="Pitch interval">Pitch-interval class</a></li>

<li><a href="Similarity_relation_(music)" title="Similarity relation (music)">Similarity relation</a></li>

<li><a href="Transformation_(music)" title="Transformation (music)">Transformation</a></li>

<li><a class="mw-selflink-fragment" href="#Z-relation">Z-relation</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:tan;"><a href="Diatonic_set_theory" title="Diatonic set theory">Diatonic set theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bisector_(music)" title="Bisector (music)">Bisector</a></li>

<li><a href="Cardinality_equals_variety" title="Cardinality equals variety">Cardinality equals variety</a></li>

<li><a href="Common_tone_(scale)" title="Common tone (scale)">Common tone</a> (Deep scale property)</li>

<li><a href="Diatonic_scale" title="Diatonic scale">Diatonic scale</a></li>

<li><a href="Generated_collection" title="Generated collection">Generated collection</a></li>

<li><a href="Generic_and_specific_intervals" title="Generic and specific intervals">Generic and specific intervals</a> (Myhill's property)</li>

<li><a href="Maximal_evenness" title="Maximal evenness">Maximal evenness</a></li>

<li><a href="Rothenberg_propriety" title="Rothenberg propriety">Rothenberg propriety</a></li>

<li><a href="Structure_implies_multiplicity" title="Structure implies multiplicity">Structure implies multiplicity</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-11-19" href="https://en.wikipedia.org/wiki/?title=Interval_vector&oldid=1258398599">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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