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

$sum x2 = 1 t (- TONES t$ .

The set of all tone profiles (with respect to $TON ES$ ) is denoted by $Pro(TON ES)$ . Geometrically speaking, tone profiles are points on the positive quadrant of the unit sphere in $TONES R$ with finitely many non-vanishing coordinates. Now we introduce a pair of maps

PICT

such that tone sets can be considered as special tone profiles, or--more precisely--such that the composition $toneSet o charChord$ yields the identity on $F in(T ON ES)$ (in other words: $F in(T ON ES)$ becomes a retract in $Pro(TON ES)$ ). These maps are defined as follows: The $toneSet$ map sends each tone profile $X = (Xt)t (- TONES$ to its carrier set

toneSet(X) = |X |:= {t (- T ON ES |xi /= 0},

i.e. to the finite set of those tones $t$ for which the coordinates $x t$ do not vanish. The $charChord$ map sends each finite tone set $T < T ON ES$ to its normalized characteristic function $x : T ON ES --> [0,1] T$ with

{ card(T)-2 for t (- T xT(t) = 0 otherwise.

which can be interpreted as a tone profile.

In this setup it is useful to simply set $CHORDS = P ro(Tones)$ and to define a $comparison$ -map for tone profiles. A natural solution is the canonical scalar product

<.,.> : RTONES × RT ONES --> R

according to which the vector basis $T ON ES$ becomes an orthonomal base. If one restricts this scalar product to tone profiles $X,Y (- P ro(T ON ES)$ , $<X,Y >$ yields always values between $0$ and $1$ namely the cosinus of the angle between $X$ and $Y$ . In addition one may concatenate the scalar product of profiles with a suitably chosen monotone function $f : [0,1]-- > [0, oo )$ . Hence, we define the comparison map as follows:

prof = f o <.,.> : P ro(T ON ES) × Pro(TON ES) --> [0, oo ).

A monotone function $f$ would not change the essential quality of the resulting Riemann Logic, but in connection with the best path calculation it is nevertheless a sensitive ingredience. In particular, the deformation $f$ may be non-linear. Recall from our considerations in subsection 2.1, that--in order to complete the definition of the Riemann Logic $RL$ --we need a map

locusP rof ile : HARM --> Pro(Tones)