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

we can use the identification via $charChord$ . Likewise we may refine these homogeneous profiles by attributing individual weights to the tone perspectives $f (- Mm,n$ and thus experimentally shaping $locusProfile(Mm,n)$ for each harmonic locus.

Remark 4 This second approach was implemented in the original HarmoRubette on the basis of Noll (1997) and is still avilable as the »Noll-Classic«-method. The Riemannian tonal functions are associated with profiles $locusProfile(Mm,n)$ and can be edited in the »Noll-preferences«-panel. In a future study we intend to embed these methods into harmonic pathway analyses using mathematically »natural« harmonic tensors, like Hausdorff-Metrics on $2ºA$ associated with suitable metrics on $ºA$ .

2.4 Concrete Tone Spaces

We close this section by defining some elementary tone spaces and derived structures in preparation of the subsequent sections. We start with chromatic pitch height $H -~ Z$ encoded according to MIDI, where 60 represents the >middle C<. Further one has octave identification map

oct : H --> Hoct ~= Z12 with oct(H) := H mod 12.

Next, we consider note names of the kind

Na with N (- {F,C,G, D,A, E,B} and a (- {...,bb,b,Ø,#,##,...}.

We indentify note names with integers by arranging them along the line of fifths $~ F= Z$ (i.e. $...,Bb,F, C,G,D, A,E,B, F#,...$ ). Concretely we identify $F$ with $-1$ , $C$ with $0$ , $G$ with $1$ , etc.). Furthermore we consider the enharmonic and the diatonic projections, sending note names to their enharmonic classes and to their diatonic classes:

enh : F --> F ~ Z enh(k) := k mod 12 dia : F --> F enh~=Z 12 enh(k) := k mod 7 dia= 7

The two different music-theoretical interpretations $H$ and $F$ of $Z$ , as well as two interpretations $Hoct$ and $Fenh$ of $Z12$ form the following diagram:

PICT

Remark 5 The simultaneous consideration of all four tone spaces along these maps is a special instance of a well formed tone system (c.f. Carey and Clampitt, 1989). Formally, such a simultaneous view of a note name - pitch height concordance can be described as the limit of the above diagram (consisting of those pairs $(k,H) (- F× H$ of note names $k$ and pitch heights $H$ for which $7 .enh(k) mod 12 = oct(H)$ holds) and fits into the general language of forms and denotators in the sense of Mazzola (2002) (chapter 6).