Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

as the extension $Ext(M )$ of a set $5 5 M = { 8,0}$ of two tone perspectives. The 5-elemented intension $Int(Ext(M )$ (TP-table) is the saturation of $M$ .

5.2 Presence of a Morpheme in a Chord Sequence

Suppose we are given a sequence of chords $S = (X0,X1,...Xn)$ which represents a harmonic extract from a piece of music. One may edit such a sequence by hand or semi-automatically or one may work with an automatically generated sequence of chord slices of simultaneous tones. The latter is what jMorph offers ⁷

⁷: For purposes of semi-automatic pre-editing we recommend the usage of chord-seq-objects in OpenMusic (in connection with the full power of this programming language) and to import a lisp file into $jMorph$

. However, to a given chord-sequence $S$ we may attribute the sequence of their intensions:

( ) ºA(S) = ºA(X0),ºA(X1), ...,ºA(Xn)

The analysis- or slice window is built in full analogy to a pianola window, but the >staff< consists of 144 lines, separated into 12 time 12 horizontal segments, one for each multiplication factor. The slice window thus displays a pianola score of the sequence $ºA(S)$ . The aim of morphological analysis consists in the filtering of this sequence by means of suitable morphemes $(M,U )$ . The presence of $U$ is given by the sequence $xU (S) = (t0,t1,...,tn)$ of boolean truth values $tk (- {T,F}$ , which is the evaluation of the (set theoretic) characteristic function of the chord set $U$ at the individual chords of the sequence $S$ . It simply says where the chords of $U$ are present in $S$ . This works in $jM orph$ simply by choosing a morpheme in the morpheme browser and selecting the command »show in score«. Colored rows and columns then highlight the intension and the extension of the morpheme of interest in the score.