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

attributing a tone profile to each harmonic locus. Such a map can practically be edited by hand, especially if one makes further homogeneity assumptions on the structure of tonalities. The HarmoRubette offers a suitable user-interface this (see section 3). Such maps $locusProfile$ can also be drawn from empirical data, where profiles represent statistical results like listeners judgments or relative occurrence rates of tones (c.f. Purwins, 2003, in this volume).

2.3 Perspectival Morpho-Logic

There is a straight forward approach to calculate a $locusProfile$ -map on the basis of the internal logics of harmonic morphemes. We admit, that a proper understanding of this >inserted< subsection requires familiarity with content and notation of Noll and Brand (2004) in this volume (especially Subsections 3.1, 3.2 and 4.2). In this concrete situation we are concerned with the homogeneous 12-tone system $T ON ES = T = {0,1,...,11}$ and we associate harmonic loci with monoids $M (- MON$ of (affine) tone perspectives $f = ba : T --> T$ with $ba(t) = at+ bmod 12$ . More precisely, to each monoid $M$ we have the set $_O_M$ of its left ideals (or cosieves), which represent the truth values for (equivariant) characteristic functions in the internal logics of the $M$ -actions on the chords $X (- Ext(M )$ in the extension $Ext(M )$ of $M$ . In close analogy to profiles $p (- P ro(T)$ we have the characteristic functions $x|M|: T --> _O_M$ . All we have to do, is to >downboil< the left ideals $B (- _O_M$ into numbers (as >fuzzy truth values<). We may choose any map $c : _O_M --> [0, oo )$ preserving the partial ordering within $_O_M$ , i.e. $B1 < B2$ must imply $c(B1) < c(B2)$ , like the cardinality map $c(B) := #(B)$ . Now, suppose we can associate each locus $H (- HARM$ with such a pair $( ) M (H),cH : _O_M(H) --> [0, oo )$ , then we define the map

locusProfile : HARM --> RT with locusProfile(H)[t];= ch(x|H(M) |(t)).

Finally, normalizing the vectors $(locusP rofile(H)[t])t=0,...,11$ leads to proper tone profiles in $P ro(T)$ . As we know from Noll and Brand (2004) (section 4.2) tonal functions in Riemann’s sense are closely related to the bigeneric morphemes

m n m n Mm,n = (Int(Ext( 3, 8),Ext( 3, 8)) with 5m + 2n = ±1.

A purely morphological approach can thus just start with the 24-elemented space $HARM = {(Mm,n, cm,n)|5m + 2n = ± 1}$ , (or the enlarged 48-elemented space including also dissonant morphemes with $5m + 2n = ± 2}$ . The map $cm,n$ can be simply the cardinality map or any user-defined partial-order-preserving map.

We mention also another--closely related--approach, which is based on the setting $T ON ES = ºA = {ba|a,b (- {0,...,11}}$ , i.e. where the 144 tone perspectives replace the role of >ordinary tones< $t (- T$ . Consequently we replace each >ordinary chord< $X (_ T$ by the monoid $ºA(X) = {f (- ºA |f(X) (_ X}$ and use the map $charChord :º A --> Pro(ºA)$ to associate profiles to them. Once we are given a map $locusP rofile : HARM --> P ro(ºA)$ we obtain a Riemann Logic $RL : Pro(ºA)× HARM --> [0, oo )$ as described in the previous subsection. Thus we are done by just identifying the collection $HARM$ of harmonic loci with a suitable subset of $P ro(ºA)$ . For any collection of monoids $M < ºA$ like $HARM = {(Mm,n, cm,n)|5m + 2n = ± 1}$