By Michael H. Brennan
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Mathematik ist eine vielseitige und lebendige Wissenschaft. Von den großen Themen wie Zahlen, Unendlichkeiten, Dimensionen und Wahrscheinlichkeiten spannen die Autoren einen Bogen zu den aktuellen mathematischen Anwendungen in der Logistik, der Finanzwelt, der Kryptographie, der Medizin und anderen Gebieten.
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However, that relation can be inferred by the temporal relations that the SPPseq pattern do enforce. That is, referring back to Figure 2ii, whenever the relations m(a, b), m(d, e), sw(b, e) and ov(a, d) are present, then sw(e, a) can be inferred. This inference is also indicated in Figure 6iii by a dashed edge. Temporal Patterns in Polyphony 7 41 Discussion This paper has presented three approaches that can accurately represent networks of temporal relations. Alternative approaches to polyphonic patterns often lack that accuracy.
A SPP pattern is interpreted as follows with respect to the temporal relations it enforces: Layers of sequences Layers SPP SPP R Sequences of layers R SPP R a c st(a, c) a ; ... c ; ... st(a, c) a b ; c d m(a, b), m(c, d) −a c sw(c, a) −a ; . . c ; ... sw(c, a) −a b ; c d m(a, b), m(c, d) a −c sw(a, c) a ; ... −c ; . . sw(a, c) a b ; −c d m(a, b), m(c, d) −a −c ov(a, c) −a ; . . −c ; . . ov(a, c) Sequences SPP R a; b m(a, b) −a ; b m(a, b) a ; −b m(a, b) −a ; −b m(a, b) a −b ; c d m(a, b), m(c, d) ..
I) A 4-3 suspension between bass and alto voices in bars 16-17 of Bach’s chorale BWV 283 and (ii) A piano-roll representation of the alto and bass voices The Humdrum toolkit is widely-used for pattern matching in symbolic music data. Although Humdrum supports polyphony, it can be diﬃcult to use for even simple patterns [5,2]. For example, Figure 3ii shows a Humdrum suspension pattern expressed with regular expressions. These typically do two things : i) match the beginning of events, the continuation of events or possibly no event at all (a “don’t care” option) and ii) match features of those events by matching corresponding values in additional columns (this is the purpose of the “t” tokens at the end of the each lines, the ﬁrst one matching a consonance feature and the second a dissonance feature).