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

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The consonance/dissonance dichotomy⁵

⁵: see Mazzola (1990), Noll (1997)

is then given by the partition of all twelve intervals into two times six:

Cons = {0,1,3,4,8,9} Diss = {2,5,6,7,10,11}.

In order to denote based intervals, i.e. specific intervals $i$ with a concrete base tone $b$ we write them as $ib$ . Based intervals can be mathematically understood as >tangent<- (or >cotangent<)-vectors to $T$ . We write $Int(T) = {ib|i (- Z12,b (- T}$ in order to denote the entire >bundle< of all 144 based intervals. The conceptual >translation< of based intervals into tone perspectives in provided by the map $P$ below. It literally >translates< the based intervals $ib$ to the tone perspectives $b i$ (see figure 1):

b P : Int(T)-- > ºA with P(ib) := i

Figure 1:

Explanation of the map $P : Int(T) --> ºA$ : If we interpret the interval $311$ (major sixth with base tone $F$ ) in relation to the >tonic< fifth $10$ then the tone perspective $P(311) =11 3$ (right graph) is nothing but the unique affine extrapolation of the two attributions $113(0) = 11$ and $113(1) = 2$ (left graph).

There is a remarkable relation between the image $P(K)$ of the 72 consonant intervals $K = {ib| i (- Cons}$ and Riemann’s concept of relative consonance, namely

< > ºA({0,1,4},{1,2,5}) = P(K),

which means that the relative consonances of a tonic triad to its dominant generate the monoid of all 72 consonances. There are four observations to be made:

The 72 consonant tone perspectives form a monoid. This is not the case for the dissonances. The markedness of dissonance is here reflected by the algebraic fact, that some concatenations of dissonances are consonances, but not

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