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

We consider the natural--translation invariant--distance between integers on the line of fifths $d : F× F --> N U {0}$ with $d(x,y) = |x - y|$ . Further we consider the induced enharmonic and the diatonic distance between note names, by virtue of:

denh : F× F --> {0,1,...,6} denh(x,y) := min{d(x,z)|enh(y) = enh(z)} ddia : F× F --> {0,1,2,3} ddia(x,y) := min{d(x,z)|dia(y) = dia(z)}

The following definitions introduce elementary music-theoretical objects (c.f. sections 4 and 5)

The sequence $Dia(k) := k+ (-1,0,1,2,3,4,5) < F$ of seven consecutive elements of the fifth line beginning from the note $k- 1$ is called diatonic collection with reference $k$ .
An alteration of $Dia(k)$ is any sequence $D = (n- 1,...,n5)$ of note names such that $dia(ni) = dia(k+ i)$ for $i = -1,...,5$ . The signature of this alteration is the sequence $1 Sig(D) = 7(D - Dia(k))$ . We write $D = Dia(k)Sig(D)$ .
The tripel $tMaj(k) = (k,k+ 1,k + 4)$ is called major triad with base note $k$ .
The tripel $t (k) = (k,k+ 1,k- 3) min$ is called minor triad with base note $k$ .
The tripel $tdim(k) = (k,k- 6,k- 3)$ is called diminished triad with base note $k$ .

3 HarmoRubette and Re-Design

The HarmoRubette was implemented as a plug-In of the RUBATO-software. The original software was created by Guerino Mazzola and Oliver Zahorka for the operating system NEXTSTEP and has been ported to Mac OSX and further extended by the second author.⁴

⁴: Further information on this OpenSource-Project see http://www. rubato.org.

The main concern of this paper is the harmonic path analysis for chord-sequences. We skip several practical aspects such as the translation of a score or its parts into a chord-sequence, the interpretation and usage of harmonic weights in performance experiments and refer to Fleischer (2003) and the RUBATO-Documentation. In this section we recapitulate the original approach and motivate the current extensions in the re-design of this tool.

3.1 Harmonic Configuration Space in the >Classic< HarmoRubette

The original version of the HarmoRubette implements a single 72-elemented space $HARM = R$ of tonal functions with respect to the 12 enharmonic classes as keys. It is a cartesian product

R = Fenh× {T,D, S}× {M aj,min}