Abstract
It would seem that the notion of musical inversion is one of the simplest and least mysterious: they are just run-of-the-mill symmetries around axes. However, much depends on the context and even more on the model wherein inversions are used. For instance in neo-Riemannian theory, one talks of the local inversion R – turning a triad into its relative –, though its actual effect on pitch-classes depends on which triad R is applied to: the connection with inversions in the circle of pcs is tenuous at best. Other models turn R into a global operation, but at the cost of the essential relation \(R^2=Id\), while still other contexts enable to embed operations on points into the more general operations on (most) pc-sets, in a natural and visual way. This paper purports to synthesize the most important situations and help understand and/or picture what an inversion really is, in its full complexity.
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- 1.
Some authors call it \(\mathcal D_{24}\), pinpointing its cardinality.
- 2.
Already the circle modelization induces this side-effect, that a center of symmetry is turned into an axis: equation \(I_k(x)=x\) has two solutions not one (if k is odd, the fixed points are half-integers).
- 3.
Here R, when applied to \(X_{maj}=k + (0,4,7)\) or \(X_{min} = k+(0,3,7)\), is \(I_{k+4}\) in the major case, and \(I_{10+k}\) in the minor case. Such maps from a local structure – say a manifold – into the linear group of its tangent space – here, its isometries – appear in other domains, theoretical physics of Fields, or pre-sheafs in Category theory. The latter have already been applied to Music Theory in [11], of course.
- 4.
Here we compare PLR with the left-action of T/I on triads, i.e. the image of a triad is the triad of the images of its elements. See [12] for a study of the right-action, when the set of triads is identified with the images of one triad by T/I in the context of G.I.S.
- 5.
Essentially because a group acting simply transitively on major/minor triads while discerning between both kinds – meaning there is a normal subgroup of transpositions – must be \(\mathcal D_{12}\), though there are 48 isomorphic versions. Moreover, there are two ‘good’ ways to define the Tonnetz group, see [7] which pinpoints the extraordinary isomorphism between T/I (acting on pcs) and the PLR group (defined on the Tonnetz) both as subgroups of the 620,448,401,733,239,439,360,000 permutations of all 24 triads!
- 6.
Or, more generally, of the IFunc of two pc-sets. Here we focus on intervals within one pc-set, i.e. \({{\mathrm{\mathbf {IFunc}}}}_A\) is essentially the interval vector \(\mathbf { iv }_A\) up to definition conventions.
- 7.
If necessary, one can easily generalize to any distribution on \(\mathbf Z_{n}\) – for instance multisets, wherein any pc can appear not only 0 or 1 time, but with any real value.
- 8.
Meaning \(\widehat{f*g} = \widehat{f} \times \widehat{g}\), i.e. the Fourier coefficients of convolution product \(f*g\) are obtained by multiplying the corresponding coefficients for f and g. This wonderful feature (noticeably simplifying computation of any convolution-related operation) is essentially a characterization of discrete Fourier transform, cf. [1], Theorem 1.11.
- 9.
Notice that, in general, a chord and its inverse constitute the simplest GIS, with the dihedral group \(\mathcal D_1\) made up of the inversion and identity.
- 10.
The different states are modeled as symmetrical real matrixes with same spectra, hence transitions between them are achieved by way of unitary matrixes, just like the spectral units defined below.
- 11.
Exceptions are pc-sets, or distributions, where one or more Fourier coefficients are nil, i.e. the matrix is singular. These sets are the famous ‘Lewins’s special cases’ whose definition in his seminal paper [10] was so irredeemably obscure. [1], Sect. 2.2.2, shows a way round these singularities when the rank of the matrix is \(n-2\).
- 12.
Though maybe the strategy of exploring the continuous orbit for discrete solutions warrants further exploration.
- 13.
Remember that the whole algebra of circulating matrixes is made of polynomials in \(\mathcal J\). It is deeply satisfying in a sense that in this model every single object or transformation originates in the single transposition by one semitone.
- 14.
Evoking the first bars of R. Strauss’s Also Sprach Zarathoustra.
- 15.
This stands also for compound operations, like the Slide S (exchanging F minor and E major) insofar as they exchange minor and major triads.
- 16.
Actually the values are repeated backwards and conjugated so that only the first 6 are featured.
- 17.
Its topological closure is a subgroup of \({{\mathrm{\mathbf {SU}}}}\) (a finite union of torii with smaller dimension), whose orbit when acting on one triad contains all of them.
- 18.
Unsurprisingly, those pc-sets related to their inverse by an involutive spectral unit are those with a symmetry axis, like major sevenths.
- 19.
Quoting [18]: “...there is a different way of topologically enriching the Tonnetz that preserves the musical insights [...] and leads to a concept of harmonic distance. Such mixing of different-cardinality sets is not possible in voice-leading spaces without forfeiting their basic geometric properties”.
- 20.
J. Yust prefers \(\varphi = 2 \pi \varPhi /12\) where \(\varPhi \), defined modulo 12, is often an integer and is easier to compare with simple values such as those of single pcs.
- 21.
This is because a nil Fourier coefficient does not have a phase. One possibility is to consider that – for instance – an augmented triad has all values of \(\varphi _3\) at the same time and can thus be represented as a vertical line. This enables modulations passing through such a chord, entering any point of the line and getting out at any other point, recalling the flexibility of these chords in Douthett’s chickenwire model. See [2] for an example in Schumann’s Kinderszenen.
- 22.
There is an isomorphism between the induced left-action of T/I on subsets of \(\mathbf Z_{n}\) and (a subgroup of) the dihedral group of translations/central symmetries on the torus.
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Amiot, E. (2017). Strange Symmetries. 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_11
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