Abstract
This chapter presents the musical and geometric reasoning behind, and mathematical formulation of, the Spiral Array model, showing how the model successively generates representations for higher level tonal elements as a composite of each entity’s lower level components. The concept of the center of effect is defined, wherein any element in the space can generate a higher level construct, modeled, in the same space, as the centroid, a mathematical sum, of its lower level members. For example, chord representations are generated from their component pitches as the center of effect of their member pitches, and key representations are similarly derived from their defining chords. Intuitive images illustrate the construction of the model, beginning from the “rolling up” of the Harmonic Network, and through the stages of defining higher level entities in the interior of the model. Theorems describe mathematical properties of the different levels of representations. The chapter concludes with a summary of the definitions and a visual depiction of the resulting array of spirals: nested helices comprising the outermost pitch class spiral, the major/minor chord double helix inside the pitch class spiral, and the major/minor key double helix inside the major/minor chord double helix.
This chapter is a revision of The Spiral Array Model (Chapter 3) of “Towards a Mathematical Modeling of Tonality” by Elaine Chew, an MIT PhD dissertation, Cambridge, Massachusetts (2000) https://dspace.mit.edu/handle/1721.1/9139
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Notes
- 1.
Octave equivalence means that pitches of the same letter name are considered to be the same, to belong to the same pitch class.
- 2.
A triad is a chord comprising three pitches: a root, a third, and a fifth. The root and fifth form a perfect fifth. The root and third form a major or minor third for a major or minor triad, respectively. In general, the words chord and triad will be used interchangeably.
- 3.
- 4.
I shall use roman numerals to denote chord function within a key. The number indicates the scale degree of the chord’s root. For example, I represents the tonic chord. I adopt the convention of denoting major chords by upper case roman numerals, and minor chords by lower case ones. For example, a major chord with the tonic as root is I but a minor chord with the same root is i.
References
Bamberger, J.S.: Developing Musical Intuition. Oxford University Press, New York (2000)
Schoenberg, A.: Style and Idea. Philosophical Library, New York (1950)
Schoenberg, A.: Structural Functions of Harmony. Norton & Co., Inc., New York (1954)
Sessions, R.: Harmonic Practice. Harcourt, Brace and Company, New York (1951)
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Chew, E. (2014). The Spiral Array. In: Mathematical and Computational Modeling of Tonality. International Series in Operations Research & Management Science, vol 204. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-9475-1_3
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DOI: https://doi.org/10.1007/978-1-4614-9475-1_3
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