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- Klumpenhouwer_network abstract "A Klumpenhouwer Network, named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is "any network that uses T and/or I operations (transposition or inversion) to interpret interrelations among pcs" (pitch class sets). According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and "this kind of analysis of triadic combinations was implicit in," his "concept of the cyclic set from the beginning", cyclic sets being those "sets whose alternate elements unfold complementary cycles of a single interval.""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 potential of K-network analysis"... "'in great generality: When a system modulates by an operation A, the transformation f ' = A f A -inverse plays the structural role in the modulated system that f played in the original system.'"Given any network of pitch classes, 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 and arrows to interpret pcsets that are of the same set class." "isomorphism of graphs. Two graphs are isomorphic 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 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)(Tn)=Tn; F(1,j)(In) = In+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)(Tn)=T-n; F(11,j)(In)=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)(Tn)=Tn; F(5,j)(In)=In+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)(Tn)=Tn; F(7,j)(In)=In+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.".
- Klumpenhouwer_network thumbnail 7-note_segment_of_C5.svg?width=300.
- Klumpenhouwer_network wikiPageID "25975191".
- Klumpenhouwer_network wikiPageLength "10092".
- Klumpenhouwer_network wikiPageOutDegree "47".
- Klumpenhouwer_network wikiPageRevisionID "610107746".
- Klumpenhouwer_network wikiPageWikiLink Allen_Forte.
- Klumpenhouwer_network wikiPageWikiLink Category:Post-tonal_music_theory.
- Klumpenhouwer_network wikiPageWikiLink Chord_(music).
- Klumpenhouwer_network wikiPageWikiLink Chromatic_scale.
- Klumpenhouwer_network wikiPageWikiLink Complement_(music).
- Klumpenhouwer_network wikiPageWikiLink Cyclic_set.
- Klumpenhouwer_network wikiPageWikiLink David_Lewin.
- Klumpenhouwer_network wikiPageWikiLink Donald_Martino.
- Klumpenhouwer_network wikiPageWikiLink Dyad_(music).
- Klumpenhouwer_network wikiPageWikiLink Flow_network.
- Klumpenhouwer_network wikiPageWikiLink Geometric_transformation.
- Klumpenhouwer_network wikiPageWikiLink George_Perle.
- Klumpenhouwer_network wikiPageWikiLink Harvard.
- Klumpenhouwer_network wikiPageWikiLink Harvard_University.
- Klumpenhouwer_network wikiPageWikiLink Henry_Klumpenhouwer.
- Klumpenhouwer_network wikiPageWikiLink Interval_(music).
- Klumpenhouwer_network wikiPageWikiLink Interval_class.
- Klumpenhouwer_network wikiPageWikiLink Interval_cycle.
- Klumpenhouwer_network wikiPageWikiLink Inversion_(music).
- Klumpenhouwer_network wikiPageWikiLink Isogloss.
- Klumpenhouwer_network wikiPageWikiLink Isograph.
- Klumpenhouwer_network wikiPageWikiLink Isography.
- Klumpenhouwer_network wikiPageWikiLink Isomorphism.
- Klumpenhouwer_network wikiPageWikiLink John_Rahn.
- Klumpenhouwer_network wikiPageWikiLink Music_theorist.
- Klumpenhouwer_network wikiPageWikiLink Music_theory.
- Klumpenhouwer_network wikiPageWikiLink Musical_analysis.
- Klumpenhouwer_network wikiPageWikiLink Network_(mathematics).
- Klumpenhouwer_network wikiPageWikiLink Pitch_class.
- Klumpenhouwer_network wikiPageWikiLink Prolongation.
- Klumpenhouwer_network wikiPageWikiLink Recursion.
- Klumpenhouwer_network wikiPageWikiLink Set_(music).
- Klumpenhouwer_network wikiPageWikiLink Similarity_relation.
- Klumpenhouwer_network wikiPageWikiLink Similarity_relation_(music).
- Klumpenhouwer_network wikiPageWikiLink Tone_row.
- Klumpenhouwer_network wikiPageWikiLink Transformation_(geometry).
- Klumpenhouwer_network wikiPageWikiLink Transformation_(music).
- Klumpenhouwer_network wikiPageWikiLink Transformational_theory.
- Klumpenhouwer_network wikiPageWikiLink Transposition_(music).
- Klumpenhouwer_network wikiPageWikiLink Vertex_(graph_theory).
- Klumpenhouwer_network wikiPageWikiLink File:7-note_segment_of_C5.svg.
- Klumpenhouwer_network wikiPageWikiLink File:Bergs_Lyric_Suite_cyclic_set.png.
- Klumpenhouwer_network wikiPageWikiLink File:K-net_graph.png.
- Klumpenhouwer_network wikiPageWikiLink File:K-net_inversion.png.
- Klumpenhouwer_network wikiPageWikiLink File:K-net_inversion_chord_3.png.
- Klumpenhouwer_network wikiPageWikiLink File:K-net_transposition.png.
- Klumpenhouwer_network wikiPageWikiLinkText "Klumpenhouwer network".
- Klumpenhouwer_network hasPhotoCollection Klumpenhouwer_network.
- Klumpenhouwer_network wikiPageUsesTemplate Template:Cleanup-rewrite.
- Klumpenhouwer_network wikiPageUsesTemplate Template:Reflist.
- Klumpenhouwer_network subject Category:Post-tonal_music_theory.
- Klumpenhouwer_network comment "A Klumpenhouwer Network, named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is "any network that uses T and/or I operations (transposition or inversion) to interpret interrelations among pcs" (pitch class sets).".
- Klumpenhouwer_network label "Klumpenhouwer network".
- Klumpenhouwer_network sameAs Reticulum_Klumpenhouweranum.
- Klumpenhouwer_network sameAs m.0b6dm67.
- Klumpenhouwer_network sameAs Q6421502.
- Klumpenhouwer_network sameAs Q6421502.
- Klumpenhouwer_network wasDerivedFrom Klumpenhouwer_network?oldid=610107746.
- Klumpenhouwer_network depiction 7-note_segment_of_C5.svg.
- Klumpenhouwer_network isPrimaryTopicOf Klumpenhouwer_network.