Fourier Oracles for Computer-Aided Improvisation
Emmanuel Amiot, Thomas Noll, Moreno Andreatta, and Carlos Agon
CPGE Perpignan, Esmuc Barcelona, Ircam/CNRS
manu.amiot@free.fr, noll@cs.tu-berlin.de, {andreatta, agonc}@ircam.fr
Abstract
A as a FT of its characteristic function:
The mathematical study of the diatonic and chromatic universes in the tradition of David Lewin [9] and John Clough
[6] is a point of departure for several recent investigations.
Surprisingly, Lewin 's original idea to applyfinite Fourier transform to musical structures has not been further investigated
for four decades (until [12]). It turns out that several musictheoretically interesting properties ofcertain types ofmusical
structures, as the partial symmetry ofFourier balances, Maximal Evenness [5] and Well-Formedness [3], allow alternative characterisations in terms of their Fourier transforms.
The paper explores two particularly interesting cases: vanishing Fourier coefficients as an expression of chord symmetry and maximal Fourier coefficients as a reinterpretation of
maximally even scales. In order to experimentally explore
the Fourier approach we design an interactive playground
for rhythmic loops. We propose a Fourier-based approach
to be integrated as an "Scratching"-interface in the OMAX
environment (built on OpenMusic and MaxMSP) which allows to interactively change a rhythm through a gestural control of its Fourier image. A collection of theoretical tools in
OpenMusic visual programming language helps the improviser to explore some new musical situations by inspecting
mathematical and visual characteristics ofthe Fourier image.
TA tF Y -2i~rkt/c
keA
For example, the FT of a diminished seventh D7 =(O 3 6 9)
is
3
TD7 t Y, 6- (3k)2ikt/12
k=0
- 1- 2i3t
1 -eixt/2
k=O
Notice TfD7(4) = 4 and.FD7 (1, 2,3) = 0. This prominence
of this 4th Fourier coefficient means precisely that set A is
12/4 = 3-periodic.
The usual properties of FT apply and are indeed elementary in this simple discrete case. It is worthwhile to note
that the usual musical operations (transposition, retrogradation, even complementation) do not change the modulus (i.e.
length) of the FT: this means already that |FA is a good musical invariant.
The most interesting property is that the FT of the relative intervallic content of two chords (a function stating the
number of occurences of any interval between A, B) is simply the product of their respective Fourier transforms, as FT
turns convolution product into ordinary product:
F(IC(A, B)) = FA X F-B
David Lewin's call for Fourier
In a few lines at the end of his very first paper [9], David
Lewin cryptically alludes to Fourier transforms and convolution products to explain how he was led to the special symmetries he considers in his paper about the relative intervallic
content of two chords.
In the present paper we elaborate some of Lewin's ideas
by applying Fourier Transform (FT) to several discrete musical structures, which may be (periodic) rhythms or scales. We
begin in this section with the set of pitch classes, modeled by
Z, (c = 12 in the equal division of the octave in twelve parts)
and define, following Lewin, the Fourier transform of a subset
Interestingly, this line of thought leads to a "one line" proof of
Babbitt's hexacord theorem on which Babbitt and Lewin were
both working hard at the same time. It was also the basis of
Dan Tudor Vuza's original work [13] on 'Vuza canons' which
has been also implemented in OpenMusic.
1.1 Vanishing Fourier Transform of Characteristic Functions
Lewin's interest though was in a kind of inverse problem reconstructing (say) A from B and the intervallic content. This is possible when TF-B is non vanishing, and this
excludes precisely the five cases that Lewin put forward as
'augmented-triad property' and the like in [9], and simply
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