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

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of dissonance< with respect to {0, 4, 10} to each tone. Figure 3 shows how to make these assignments in the case $48$ . The five truth values are obtained as follows. A tone $t$ gets the truth value

$T *$: if the identity $01$ maps it into {0, 4, 10}, i.e. if it is an element of {0, 4, 10},
$* P$: if it is mapped into {0, 4, 10} by both tone perspectives $4 8$ and $10 3$ but not by the the identity $0 1$ (the tone $6 = F#/Gb$ is the only one with this truth value),
$R*$: if it is mapped into {0, 4, 10} by $48$ but not by $103$ , (four tones have this truth value, namely $1 = G,3 = A, 7 = D ,9 = E b b$ ),
$L*$: if it is mapped into {0, 4, 10} by $103$ but not by $48$ , which is the case for $2 = D$ and $8 = Ab$ ,
$* C$: if it is mapped into {0, 4, 10} neither by $4 8$ nor by $10 3$ , which is the case for $11 = F$ and $5 = B$ .

Figure 3:

Tone perspectives constitute inner truth values. The left drawing displays the graph of the tone perspective $48$ . It will contribute to the truth value of a tone $t$ if this tone is mapped inside the chord {0 ,4, 10}, i.e. if it >makes the tone $t$ < true. The right figure shows two situations. Obviously, $9 = E b$ is becoming true along $48$ (it is mapped to $4 = E$ ), but $2 = D$ is not becoming true along $48$ (it is mapped to $8 = A b$ which is outside of ${0,4,10}= {C,E,B } b$ )

On the basis of this list we may look at each individual chord of the piece. The segmentation into harmonic units its rather straightforward. It is useful to consider half-bars as the smallest harmonic segments, sometimes full-bar segments are appropriate. Figure 4 gives a representative list of all situations occurring in the piece. The examples are notationally displayed with respect to the concrete transpositions but rotated into the standard picture of ${0,4,10}$ .

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