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

The intersection of $Ext(MR)$ with the 39 different chords occuring in this passage consists of 9 chords, namely

{{5,9},{0,5,9},{1,5,9},{2,5,9},{0,2,5,9}, {0,3,5,9},{1,4,5,9},{1,5,7,9},{5911}}.

However, there are 19 different chords in the passage, which contain the tones $5$ and $9$ . The intersection $Ext(ML)$ with the 39 chords consists only of three different chords, namely ${{2,5},{2,5,9},{2,5,10}}$ . But there are 7 chords, containing the tones $2$ and $5$ . This shows that the morphemes with intensions $MR$ and $ML$ actually differ from the intensions $5 9 0 { 0,0, 1}$ and $2 5 0 { 0, 0, 1}$ which take only common tones into account.⁸

⁸: A detailed interpretation of this and other presence-analyses is subject of a separate study.

Remark 9 The presence of $M$ can be expressed also in two more refined ways: (both not jet being implemented in $jM orph$ ):

While the visual analysis represents the boolean values $T$ and $F$ (colored column vs. no column) one may also consider a sequence of submonoids of $M$ , namely $º ( º º º ) M /~\ A(S) := M /~\ A(X0),M /~\ A(X1),...,M /~\ A(Xn) .$ The boolean truth value $tk = T$ corresponds to exactly those chords $Xk$ , where $M (_ ºA(Xk)$ , while all proper submonoids $M /~\ ºA(Xk) < M$ correspond to $tk = F$ . One may replace this single boolean value $F$ by quantitatively encoding fuzzy answers as morpheme weights. Instead of attributing a truth value to $Xk$ we attribute to it the ratio $#(M /~\ ºA(X)) kM (Xk) := -----------. #(M )$ A further refinement defines a weight on single tones $x (- X$ within chord slices $X$ by virtue of $--#(M-- /~\ ºA(X,-X))p- wM (x| X) := #(M /~\ ºA(X,X\{x})) .$ The more chord perspectives of $X$ (within $M$ ) fail to map $X$ into $X\{x}$ the more important is the tone $x$ from this morphological point of view. These tone weights can be used for artificial performance purposes or can be compared with other analytical weights (e.g. in the software Rubato - c.f. Anja Volk’s article in this volume).
Alternatively we may use the internal logics of $M$ with the characteristic function $x |M |: T --> _O_$ and may attribute to each tone $t$ in any slice $Xk$ the left ideal $x (t) |M |$ of $M$ . Taking into account the independence of these truth values of the particular chord $Xk$ we mention that this second procedure yields a >refined< analysis only in terms of the specific mixtures of left ideals per chord $Xk$ , but not on the level of tones.