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

and hence obtain the following simplified >classic< Riemann Logic:

RLdirect(X |\ H) := prof(X,locusProfile(H))

with the monotonous function

{ exp(t) t /= 0, f(t) := 0 otherwise.

The user has now access to the monotonous function $f$ and may also experiment with the identity, the normal exponential function or a custom function.

In Mazzola’s original implementation the formula for $RLdirect(X |\H)$ was not directly applied to each given chord $X$ but only to (realisations of) third chains. A third-chain is a sequence $K = (x1,x2,...,xk)$ with $xi (- {3,4} < Z12$ such that the corresponding sequence of partial sums $S(K) = (x1,x1 + x2,...,x1 + x2 + ...+ xk)$ contains no dublicates. The realisation of a third chain $K$ at pitch class $x0 (- Hoct$ is the tone set $Rx0(K) := |x0 + s(K)|$ . For any tone set $X$ we consider the collection $T C(X)$ of all minimal third chain realisations containing X, namely

T C(X) := {Rx0(K) |X < Rx0(K), length(K) minimal}.

The present version of the HarmoRubette offers two ways of taking the (n-elemented) set $T C(X)$ into account (the second of which is the original proposal), namely

to accumulate an average profile $p = c. sum Y (- T C(X) charChord(Y)$ and to calculate $RLThirdChain(X |\H) := RLdirect(p |\H)$ (c = normalizing factor),
to average the results of separately applying $RLdirect$ to all characteristic profiles for minimal third chains $1- sum RLMazzola- Classic(X |\H) := n RLdirect(charChord(Y ),H). Y (- TC(X)$