Abstract
The paper offers an integration of the theory of structural modes, functional theory and diatonic scale degrees. In analogy to the parsimonious voice leading between generic diatonic triads we study parsimonious function leading between embedded structural modes. A combinatorics of diatonic embeddings of structural modes is given. In four analytical examples we study the interaction of relative minor and major modes within an encompassing diatonic collection. Finally we discuss alternative possibilities for the interpretation of the diminished fifth as a fundament progression.
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Notes
- 1.
The first and second modes are refinements of the authentic division of the octave into fifth and fourth, while the third mode is not. Algebraically, this corresponds to the fact that \(a \mapsto ba, b \mapsto b\) and \(a \mapsto ab, b \mapsto b\) are both automorphisms of the free group \(F_2\), while \(a \mapsto bb, b \mapsto a\) is not. This is in strict analogy to the fact, that the sixth-fourth chord is the bad conjugate among the three triadic modes. The root position triad and the sixth chord are refinements of the division of the octave in third and sixth, while the sixth-fourth chord is not. Algebraically, \(a \mapsto a, b \mapsto ab\) and \(a \mapsto a, ba \mapsto b\) are both automorphisms, while \(a \mapsto b, b \mapsto aa\) is not.
- 2.
Of course we also adapt the chord tones in the other voices. But our concept of embedding applies to the horizontal dimension of fundament progressions, which may or may not coincide with the vertical dimension of the chords.
- 3.
Explicitly he writes: “Eine Theorie aber, die gerade dort versagt, wo auch das Phänomen, das sie erklären soll, ins Vage und Unbestimmte gerät, darf als adäquat gelten” ([3], p. 50). With this epistemological faux pas Dahlhaus threw the ball into Mazzola’s court, who illustrates the collapse of functional parallelism through the combinatorial description of the seven diatonic triads in the form of a Moebius strip, which is known to be non-orientable [10].
- 4.
for reference and further discussion see Sect. 5.
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We wish to thank Jason Yust, David Clampitt and the anonymous reviewers for valuable feedback.
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Noll, T., De Jong, K. (2019). Embedded Structural Modes: Unifying Scale Degrees and Harmonic Functions. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_11
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