Abstract
Rameau [15] redundantly defines the subdominant (1) as the fifth under the tonic and (2) as the scale degree immediately below the dominant. In the context of scale theory this motivates the interpretation of this definition as an equation. It states that the diazeuxis (the difference between the generator and its octave complement) is a step interval of the scale. The appropriate scale-theoretic concept for the formulation of this equation is that of a Carey–Clampitt Scale (a non-degenerate well-formed scale). The Rameau equation then imposes a constraint on the associated Regener transformation which converts note intervals from generator/co-generator coordinates into step/co-step coordinates. The equation takes two forms depending on the sign of the diazeuxis (positive or negative). The solutions then come either with two (flatward directed) co-steps or two (sharpward directed) steps, accordingly. In addition to this generic characterization the paper closes with a corollary on the specific scale properties of reduced Clough–Myerson scales. These scales solve Rameau’s equation if and only if they are Agmon scales.
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Notes
- 1.
I wish to acknowledge Nicolas Meeùs’ postings to smt-talk from february 24 (23:35:46 CET) and march 11 (14:37:58 CET) 2012 as particularly inspirational for the present paper.
- 2.
Joel Lester [13] cites Dandrieu as the earliest known source, where the term ‘soudominant’ occurs.
- 3.
Proposed term by the author.
- 4.
Proposed term by the author. Clough and Douthett also coined the term hyper-diatonic scale.
- 5.
For the motivation of this term see [17].
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Noll, T. (2017). The Sense of Subdominant: A Fregean Perspective on Music-Theoretical Conceptualization. In: Pareyon, G., Pina-Romero, S., Agustín-Aquino, O., Lluis-Puebla, E. (eds) The Musical-Mathematical Mind. Computational Music Science. Springer, Cham. https://doi.org/10.1007/978-3-319-47337-6_21
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