Abstract
The description of the Major and Minor modes as fillings of a triadic division of the octave offers the possibility to study them as Pairwise Well-Formed Modes. As a consequence one obtains two projections: the diatonic projection yields the well-known Ionian and Aeolian modes and provides a link between the triadic modes and the pseudo-classical modes. The syntonic projection looks unfamiliar at first sight, but closer inspection shows that these modes provide a common ground for the natural, harmonic, and melodic manifestations of both the Major and the Minor modes.
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Notes
- 1.
For example the generator \(g = (2 - \sqrt{2})/2\) yields a seven-note scale with the step intervals of sizes \(a = 3 - 2 \sqrt{2}\) and \(b = 3/\sqrt{2} - 2\).
- 2.
In consideration of the fact that Dahlhaus ([7], e.g. p. 46), in his critical reflection on Hauptmann’s dialectical concept of the Major and Minor keys, localizes traces of Dualism, it is worthwhile to highlight that these are not implied by the scalar structure. The flattening of scale degrees \(\hat{3}, \hat{6}\) and \(\hat{7}\) (by a lesser augmented prime) in a major scale leads to the corresponding minor scale. No triad needs to be turned upside down. This transformation corresponds to Hindemith’s idea of Trübung (disturbance, turbidation), who emphatically rejects a dualistic concept. ([9], p. 78 and also [10], p. 147).
- 3.
In a separate paper we will have a closer look into this tension, collecting more arguments in favor and against both interpretations of the step pattern of melodic minor.
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Noll, T., Clampitt, D. (2019). Exploring the Syntonic Side of Major-Minor Tonality. 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_10
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