Morphisms of generalized interval systems andPR-groups

Abstract

We begin the development of a categorical perspective on the theory of generalized interval systems (GIS's). Morphisms of GIS's allow the analyst to move between multiple interval systems and connect transformational networks. We expand the analytical reach of the Sub Dual Group Theorem of Fiore-Noll [8] and the generalized contextual group of Fiore-Satyendra [9] by combining them with a theory of GIS morphisms. Concrete examples include an analysis of Schoenberg, String Quartet in D minor, op. 7, and simply transitive covers of the octatonic set. This work also lays the foundation for a transformational study of Lawvere-Tierney upgrades in the topos of triads of Noll [21].

Hexatonic systems and dual groups in mathematical music theory

“…For perspective, we mention that simply transitive group actions correspond to the generalized interval systems of Lewin, see the very influential original source [16], or see [10,Section 2] for an explanation of some aspects. Dual groups correspond to dual generalized interval systems: the transpositions of one system are the interval preserving bijections of the other.…”

Shall We (Math and) Dance?

“…Musical gestures allow one to shape physical sound parameters for expressive reasons: a purely 'technical' gesture such as pressing a piano key can be shaped into, let us say, a delicate, caressing gesture. 7 The most basic gesture is probably breathing, that can be seen in general as a couple of (inverted) curves between two points -a 'tense' state and a 'relaxed' state. In piano playing, we have the general idea of preparation and key-pressing.…”

Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality

“…If Y is a pitch-class segment, then RICH(Y ) is that retrograde inversion of Y which has the first two notes y 2 and y 3 , in that order. This transformation was used in our analysis of Schoenberg in [7].…”

“…More specifically, in Theorem 3.2, we take X to be (0, 4, 7) and G to be the T /I-group, so that Σ 3 (T /I)(0, 4, 7) is the 144 = 6 × 24 possible orderings of major and minor triads, and ρ(Σ 3 (T /I)) is the internal direct product of ρ(Σ 3 ) and the P LR-group. The group ρ(Σ 3 (T /I)) is also the subgroup of 7 As we remarked earlier, the formulas in equation (1) for P , L, and R are only valid for major triads in root position, or minor triads in the ordering I n (0, 4, 7). For other orderings of consonant triads, conjugation must be used, as in Example 3.3.…”

“…This octatonically restricted RICH-transformation involves two (and only two) Flip-Flop Cycles of length 8 in the sense of John Clough [3]. Analogous orbits can be obtained for pitch-class segments of jet and shark triads in [7]. The last P R-cycle in Figure 2 contains the cello motive in measures 8-10, which is pictured in [7,Figures 12 and 13], and located in the octatonic scale {2, 3, 5, 6, 8, 9, 11, 0}.…”