Klumpenhouwer network - AbsoluteAstronomy.com

A "Klumpenhouwer Network", named after its inventor, Canadian music theorist and former doctoral student of 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 at Harvard, Henry Klumpenhouwer

Henry Klumpenhouwer

Henry Klumpenhouwer is a musicologist and professor at the University of Alberta. A former PhD student of David Lewin and the inventor of Klumpenhouwer networks, which are named after him. He is the editor of Music Theory Spectrum.-Bibliography:...

, is, "any network that uses T and/or I operations [ transposition

Transposition (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...

] to interpret interrelations among pcs" [ 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...

sets

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects...

]. According to George Perle

George Perle

George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...

, "a Klumpenhouwer network is a chord

Chord (music)

A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...

analyzed

Musical analysis

Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

in terms of its dyadic

Dyad (music)

In music, a dyad is a set of two notes or pitches. Although most chords have three or more notes, in certain contexts a dyad may be considered to be a chord. The most common two-note chord is made from the interval of a perfect fifth, which may be suggestive of music of the Medieval or Renaissance...

sums and differences

Transposition (music)

," and, "this kind of analysis of triadic combinations was implicit in," his, "concept of the cyclic set

Cyclic set

In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." Those cycles are ascending and descending, being related by inversion since complementary:...

from the beginning", cyclic sets being those, "sets

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects...

whose alternate elements unfold complementary cycles

Interval cycle

In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached In music, an interval cycle is a collection...

of a single interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

."
"Klumpenhouwer's idea, both simple and profound in its implications, is to allow inversional, as well as transpositional, relations into networks like those of Figure 1," showing an arrow down from B to F# labeled T₇, down from F# to A labeled T₃, and back up from A to B, labeled T₁₀ which allows it to be represented by Figure 2a, for example, labeled I₅, I₃, and T₂. In Figure 4 this is (b) I₇, I₅, T₂ and (c) I₅, I₃, T₂.

Lewin asserts the, "recursive

Recursion

Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

potential of K-network analysis"... "'in great generality: When a system modulates by an operation A, the transformation f = A f A -inverse

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...

plays the structural role in the modulated system that f played in the original system.'"

Given any network 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, and given any pc operation A, a second network may be derived from the first, and the relationship thereby derived network isomorphism "arises between networks using analogous configurations of nodes

Vertex (graph theory)

In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

and arrows to interpret pcsets that are of the same set class." "isomorphism of graphs. Two graphs are isomorphic

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...

when they share the same structure of nodes-and-arrows, and when also the operations labeling corresponding arrows correspond under a particular sort of mapping f among T/I."

"To generate isomorphic graphs, the mapping f must be what is called an automorphism of the T/I system. Networks that have isomorphic graphs are called isographic."

"to be isographic, two networks must have these features:

They must have the same configuration of nodes and arrows.
There must be some isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...

F that maps the transformation-system used to label the arrows of one network, into the transformation-system used to label the arrows of the other.
If the transformation X labels an arrow of the one network, then the transformation F(X) labels the corresponding arrow of the other."

"Two networks are positively isographic when they share the same configuration of nodes and arrows, when the T-numbers of corresponding arrows are equal, and when the I-numbers of corresponding arrows differ by some fixed number j mod 12." "We call networks that contain identical graphs 'strongly isographic'". "Let the family of transpositions and inversions on pitch classes be called 'the T/I group.'"

"Any network can be retrograded by reversing all arrows and adjusting the transformations accordingly."

Klumpenhouwer's [true] conjecture: "nodes (a) and (b), sharing the same configuration of arrows, will always be isographic if each T-number of Network (b) is the same as the corresponding T-number of Network (a), while each I-number of Network (b) is exactly j more than the corresponding I, number of Network (a), where j is some constant number modulo 12."

Five Rules for Isography of Klumpenhouwer Networks:

Klumpenhouwer Networks (a) and (b), sharing the same configuration of nodes and arrows, will be isographic under the circumstance that each T-number of Network (b) is the same as the corresponding T-number of Network (a), and each I-number of Network (b) is exactly j more than the corresponding I-number of Network (a). The pertinent automorphism of the T/I group is F(1,j): F(1,j)(T_n)=T_n; F(1,j)(I_n) = I_n+J.
Klumpenhouwer Networks (a) and (b), will be isographic under the circumstance that each T-number of Network (b) is the complement of the corresponding T-number in Network (a), and each I-number of Network (b) is exactly j more than the complement of the corresponding I-number in Network (a)...F(11,j): F(11,j)(T_n)=T_-n; F(11,j)(I_n)=I_-n+j."
Klumpenhouwer Networks (a) and (b), will be isographic under the circumstance each T-number of Network (b) is 5 times the corresponding T-number in Network (a), and each I-number of Network (b) is exactly j more than 5 times the corresponding I-number in Network (a)...F(5,j): F(5,j)(T_n)=T_n; F(5,j)(I_n)=I_n+j.
Klumpenhouwer Networks (a) and (b), will be isographic under the circumstance each T-number of Network (b) is 7 times the corresponding T-number in Network (a), and each I-number of Network (b) is exactly j more than 7 times the corresponding I-number in Network (a)...F(7,j): F(7,j)(T_n)=T_n; F(7,j)(I_n)=I_n+j.
"Klumpenhouwer Networks (a) and (b), even if sharing the same configuration of nodes and arrows, will not be isographic under any other circumstances."

"Any one of Klupmenhouwer's triadic networks may thus be understood as a segment of cyclic set, and the interpretations of these and of the 'networks of networks'...efficiently and economically represented in this way."

If the graphs of chords are isomorphic by way of the appropriate F(u,j) operations, then they may be graphed as their own network.
Other terms include Lewin Transformational Network and strongly isomorphic

Isomorphism

See also

Further reading