Abstract
Several ways to appreciate the diatonicity of a pc-set can be proposed: Anatol Vierù enumerates connected fifths (or semitones, as an indicator of chromaticity), Aline Honing similarly measures ‘interval categories’ against prototype pc-sets [8]; numerous generalizations of the diatonic scales have been advanced, for instance John Clough and Jack Douthett ‘hyperdiatonic’ [5] which supersedes Ethan Agmon’s model [1] and the tetrachordal structure of the usual diatonic, and many others. The present paper purports to show that magnitudes of Fourier coefficients, or ‘saliency’ as introduced by Ian Quinn in [9], provide better measurements of diatonicity, chromaticity, octatonicity...The latter case may help solve the controversies about the octatonic character of slavic music in the beginning of the XX\(^{th}\) century, and generally disambiguate appreciation of hitherto mostly subjective musical characteristics.
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Notes
- 1.
In short, in his theory a poor mode is a subset of several rich modes.
- 2.
Going to extreme cases: is a single note diatonic? What about a minor third?
- 3.
Among other things, it does not integrate the group structure of intervals modulo octave, not to mention subtler features. As G. Mazzolla wryly observes in the preface of [10], it is hopeless to try and apprehend the huge complexity of music with only the simplest mathematical tools – though this complexity can be reconstructed from all its simplifications, if one construes ‘simplification’ as ‘forgetful functor’.
- 4.
The machinery involved, as we will develop below, is actually an algebra structure (with a convolution product) on the vector space of distributions, i.e. vectors describing how much of C, C\(\sharp \), D and so on, are featured in a much generalized pc-set.
- 5.
Up to a constant.
- 6.
For technical reasons that will be made clear below, we do not take into account the symmetries, e.g. \({{\mathrm{{\mathbf {iv}}}}}(n-k) = {{\mathrm{{\mathbf {iv}}}}}(k)\) and consider \({{\mathrm{{\mathbf {iv}}}}}_X\) as a vector in \(\mathbf R^n\).
- 7.
Just check the number of common tones between X and \(X+5\), using the second formula in the definition above.
- 8.
Actually overrated since every tritone is tallied twice.
- 9.
Many other examples can be devised if this one does not sound convincing to you. A more blatant one would be \(\{0,2,7,9\}\) vs. \(\{0,1,7,8\}\), both with \({{\mathrm{{\mathbf {iv}}}}}(5)=2\).
- 10.
- 11.
Vierù had discerned that the two notions are interchanged by multiplication by 5 (or 7) modulo 12, the classical \(M_5\) (or \(M_7\)) operator; and offered thoughtful insights on this dichotomy as expressed by the affine group on \(\mathbf Z_{12}\).
- 12.
In some cases this may not the best for coincidence measurements: the more compact form of a pc-set adresses its chromaticity, not its diatonicity – consider the preceding discussion where the pc-set is first transformed by \(M_5\).
- 13.
One can compute them online at http://canonsrythmiques.free.fr/MaRecherche/styled/.
- 14.
- 15.
The length of a complex number \(x + i y\) is \(\Vert (x, y)\Vert = |x + i y| =\sqrt{x^2+y^2}\).
- 16.
In a convincing study of Ruth Crawford Seeger’s White Moon [17].
- 17.
This is characteristic of DFT up to permutations: see [3], Theorem 1.11.
- 18.
Yust observed that conversely – by inverse DFT – the number of common tones between two pc-sets can be expressed as a sum of products of magnitudes of Fourier coefficients, pondered by cosines of the differences of phases.
- 19.
It has quadratic complexity, while termwise product is linear.
- 20.
It would be even simpler for chromaticity (as suggested by a reviewer) but of less interest for actual analysis.
- 21.
One can use either 5 or 7 as generator of a chain of fifths.
- 22.
But also almost connected chains, like F C G A E B.
- 23.
As a shrewd reviewer noticed, it would also be feasible to correlate interval profiles, but our aim is to find a recipe at once simple, general and efficient.
- 24.
The converse is not true: consider CDE which is undoubtedly diatonic though \({{\mathrm{{\mathbf {iv}}}}}(5)=0\)!
- 25.
It appears that there is little difference when the time-span of the window is expanded from 1 to 2 or even 3 s.
- 26.
Up to the cardinality of pc-sets. On these pictures, the dotted line shows the mean value of a saliency and the solid line a reference value – for \(a_5\), say, it is the mean value found for a Mozart Sonata.
- 27.
Hopefully more exhaustive analyses of saliency of Slavic music of early XX\(^{th}\) century will soon appear, and settle once and for all the question of their octatonicity.
- 28.
His chords systematically include all twelve pcs.
- 29.
Technically this is true since the music can be retrieved from the data of all Fourier coefficients.
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Amiot, E. (2017). Interval Content vs. DFT. In: Agustín-Aquino, O., Lluis-Puebla, E., Montiel, M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science(), vol 10527. Springer, Cham. https://doi.org/10.1007/978-3-319-71827-9_12
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