Music theory and mathematics

Beat-Class Set Classes and the Power Group Enumeration Theorem

“…Considering that pitch-class set theory as a discipline did not emerge until the 1960s and early 1970s, the enumeration of what are essentially pitch-class set classes has a surprisingly long tradition, beginning in the late 19th and early 20th centuries. Jonathan Bernard [8] and Catherine Nolan [9,10] discuss the history of early efforts in this endeavor, citing the work of Heinrich Vincent, Anatole Loquin, and Ernst Bacon. Julian Hook [11] presents a detailed tutorial of classical combinatorial enumeration techniques, including Burnside's Lemma and Pólya's Enumeration Theorem, as applied to a host of music-theoretical topics, including the counting of T n , T n /T n I, and T n /T n I/T n M set classes in modular pitch-class spaces of various sizes, row classes of twelve-tone series, equivalence classes of beat-class sets, and the like.…”

Communication And Time In The Philosophy Of Music A. Losev

“…This time of the present moment is both illogical and eidetic. It is associated with a human being who implements a symbol, which stays not only beyond the structure of time he has created but also beyond the meaning, which is the essence of musical utterance (Nolan, 2008). Therefore, the number is the very structure of music, "the number that sounds," rather than a category of quantity.…”

ON HORACE'S PYRAMIDS (C. 3.30.1–2)

“…p. 62 Ast (p. 84 de Falco) ἔν τε ἐπιπέδοις καὶ στερεοῖς πρῶτά ἐστι ταῦτα· στιγμὴ γραμμὴ τρίγωνον πυραμίς· ἔχει δὲ ταῦτα τὸν τῶν δέκα ἀριθμὸν καὶ τέλος ἴσχει, by which he means that the limit of a point is 1, the limit of a line is 2, the limit of a triangle is 3 and the limit of a pyramid is 4, all of which add up to 10, thereby allowing the τετρακτύς to accommodate all these forms; see Tarán (1981) 281–4; Waterfield (1988) 113 n. 20, 114 n. 23. On the τετρακτύς in general, see Burkert (1972) 72–3, 186–8, 427; Nolan (2002) 272–4. That Speusippus' book (or at least the edition to which pseudo-Iamblichus had access) was entitled Περὶ Πυθαγορικῶν ἀριθμῶν demonstrates that the τετρακτύς of the decad was under discussion, even though Speusippus working in the Platonic tradition seems to prefer the less mystical term ὁ τῶν δέκα ἀριθμός; see Tarán (1981) 262–3.…”