{"content":[{"text":"2","bbox":{"left":0.166667,"top":0.061144,"width":0.011531000000000013,"height":0.015374000000000006}},{"text":"Rossiter and Heather","bbox":{"left":0.619308,"top":0.061144,"width":0.21401099999999995,"height":0.015374000000000006}},{"text":"to music theory into a transformational one. Earlier the emphasis \n had been on the pitch classes being treated as sets of elements, \n each element being a note within the item. This enabled chords \n and other combinations of notes to be modelled. The transfor- \n mational approach extended this technique by adding a transition \n from one pitch class to another to capture the dynamic possibilities \n within a musical piece. Such a transition from, say, K1 to K2 was \n tightly structured, with the target K2 being isographic to a source \n K1. Isographism is similar to isomorphism but within a graphical \n context. The cardinalities of the source and target nodes must be \n the same. K-nets are therefore a very disciplined but restricted \n way of moving from one pitch class to another: they can handle \n the situation where the labelling of nodes is consistent from one \n system to another and where the transformations are classical as, \n for example, within the 48 preludes and fugues (Well-Tempered \n Clavier) of JS Bach, handling all 24 keys. \n A more general form of K-nets was deﬁned by Lewin [19] in \n an extension of Klumpenhouwer’s work, attempting to make the \n graphs and their transformations more general. These have been \n termed L-nets by O’Donnell [32]. L-nets extend a node from being \n a static collection of pitch classes to a network of transpositions, \n giving a transformational model, allowing numerous graphic possi- \n bilities for representing a single pitch-class set.","bbox":{"left":0.166667,"top":0.114955,"width":0.666778,"height":0.42180799999999996}},{"text":"2","bbox":{"left":0.166667,"top":0.570073,"width":0.018662999999999985,"height":0.022160999999999986}},{"text":"Previous Attempts with Category \n Theory","bbox":{"left":0.222706,"top":0.570073,"width":0.555865,"height":0.04983499999999996}},{"text":"L-nets still have their origin in set-based graph theory. It was \n not long before the potential was seen for a move to category the- \n ory with the nodes in the graph becoming categories and the edges \n becoming functors. Category theory should facilitate the develop- \n ment of a logical approach to music, which can be mapped into one \n of the physical approaches for implementation. Towards this aim \n a massive tome The Topos of Music, 1335pp long, was produced \n by Mazzola [23], bringing together many of the recent advances in \n the theory of music. The title is, however, misleading with a for- \n mal topos approach, based on the Cartesian closed category, not \n attempted. In the preface (p.v) it is stated that the word topos is \n used in the style of Aristotle’s or Kant’s topic. Chapter 19 Topoi \n of Music gives an overview of the Grothendieck Topology but does \n not relate the topology to music. In later sections the word topology","bbox":{"left":0.166667,"top":0.637517,"width":0.666733,"height":0.25521499999999997}}],"images_bbox":[],"images_bbox_not_text_overlap":[]}