Interval vector
Encyclopedia
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 a pitch-class set
. Often referred to as a PIC vector (or pitch-class interval vector), Schuijer suggests that APIC vector (or absolute pitch-class interval vector) is more accurate.
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:
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 the monomial
notation pemdnc.sbdatf 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 for what would now be written . The modern notation, which has considerable advantages and is extendable to any equal division of the octave 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} , 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 will not change with transposition or inversion, it belongs to the entire set class, and <001110> is the vector of all major (and minor) triads. It should, however, be noted that some interval vectors correspond to more than one sets that cannot be transposed or inverted to produce the other. (These are called Z-related sets, explained below).
For a set of x elements, 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 sound qualities that are created by different collections of pitch class
es. 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.
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.
In the case of hexachord
s each may be referred to as a Z-hexachord. Any hexachord not of the "Z" type is its own complement
while the complement of a Z-hexachord is its Z-correspondent, for example 6-Z3 and 6-Z36.
The term, for "zygotic" (yoke
d or the fusion of two reproductive cells), originated with Allen Forte
in 1964, but the notion seems to have first been considered by Howard Hanson
. Hanson termed this the isomeric relationship, defining two such sets to be isomeric. According to Michael Schuijer (2008), "the discovery of the relation," was, "reported," by David Lewin
in 1960.
Though it is commonly observed that Z-related sets always occur in pairs, David Lewin
noted that this is a result of twelve-tone equal 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.
Straus argues, "[sets] in the Z-relation will sound similar because they have the same interval content." Some argue that the "relation" is often so remote as to be imperceptible, but certain composers 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.
by 5 or multiplication by 7), due to identical entries for 1 and 5 on the interval vector.
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
. Often referred to as a PIC vector (or pitch-class interval vector), Schuijer suggests that APIC vector (or absolute pitch-class interval vector) is more accurate.
In 12 equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
it has six digits, with each digit standing for the number of times an interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
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
Permutation (music)
In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...
or vertical arrangement.) The interval classes represented by each digit ascend from left to right. 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.)
Interval class 0 (representing unisons and octaves) is omitted.
The concept was named intervalic content by Howard Hanson
Howard Hanson
Howard Harold Hanson was an American composer, conductor, educator, music theorist, and champion of American classical music. As director for 40 years of the Eastman School of Music, he built a high-quality school and provided opportunities for commissioning and performing American music...
in his The Harmonic Materials of Modern Music, where he introduced the monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
notation pemdnc.sbdatf 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 for what would now be written . The modern notation, which has considerable advantages and is extendable to any equal division of the octave was introduced by Allen Forte
Allen Forte
Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...
.
A scale whose interval vector contains six different numbers is said to have the deep scale property
Deep scale property
In diatonic set theory, the deep scale property is the quality of pitch class collections or scales containing each interval class a unique number of times. Examples include the diatonic scale...
. 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} , 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 will not change with transposition or inversion, it belongs to the entire set class, and <001110> is the vector of all major (and minor) triads. It should, however, be noted that some interval vectors correspond to more than one sets that cannot be transposed or inverted to produce the other. (These are called Z-related sets, explained below).
For a set of x elements, 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 sound qualities that are created by different collections of pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
es. 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
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
's Generalized Musical Intervals and Transformations.
Z-relation
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 (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 transpositionTransposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
or inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
.
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.
In the case of hexachord
Hexachord
In music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...
s each may be referred to as a Z-hexachord. Any hexachord not of the "Z" type is its own complement
Complement (music)
In music the term complement refers to two distinct concepts.In traditional music theory a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is also known as its...
while the complement of a Z-hexachord is its Z-correspondent, for example 6-Z3 and 6-Z36.
The term, for "zygotic" (yoke
Yoke
A yoke is a wooden beam, normally used between a pair of oxen or other animals to enable them to pull together on a load when working in pairs, as oxen usually do; some yokes are fitted to individual animals. There are several types of yoke, used in different cultures, and for different types of oxen...
d or the fusion of two reproductive cells), originated with Allen Forte
Allen Forte
Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...
in 1964, but the notion seems to have first been considered by Howard Hanson
Howard Hanson
Howard Harold Hanson was an American composer, conductor, educator, music theorist, and champion of American classical music. As director for 40 years of the Eastman School of Music, he built a high-quality school and provided opportunities for commissioning and performing American music...
. Hanson termed this the isomeric relationship, defining two such sets to be isomeric. According to Michael Schuijer (2008), "the discovery of the relation," was, "reported," by David Lewin
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
in 1960.
Though it is commonly observed that Z-related sets always occur in pairs, David Lewin
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
noted that this is a result of twelve-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
(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.
Straus argues, "[sets] in the Z-relation will sound similar because they have the same interval content." Some argue that the "relation" is often so remote as to be imperceptible, but certain composers 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.
Multiplication
Some Z-related chords are connected by M or IM (multiplicationMultiplication (music)
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals , it has been used in other ways for twelve-tone technique, and musical set theory...
by 5 or multiplication by 7), due to identical entries for 1 and 5 on the interval vector.