Study multi-dimensional musical space through Symmetrical Polytonality, a theoretical framework for polysymmetrical pitch systems and harmonic environments.

This advanced music theory text delves into the following topics:

• Harmonic Symmetry: Details harmonic symmetry and the overtone series, and how they drive tonal harmony.

• Multi-Dimensional Harmonic Vectors: How symmetrical harmonic elements create multi-directional relationships and dimensionality in musical space.

• Harmonic Drive and Resolution: The mechanisms of harmonic tension and resolution, including the role of symmetrical intervals like tritones, augmented triads, and diminished seventh chords.

• Symmetrical Pitch Sets and Chord Pods: The structural characteristics of polysymmetrical scales, including the tetrasymmetrical octotonic, bisymmetrical octotonic, and trisymmetrical nonotonic set, and their unique chord pods.

• Chord Progressions: Analyze progressions within pods and symmsets, guided by Normal Harmonic Progression (NHP) principles.

• Augmented and Diminished Subsets: Subsets like the Augmented Scale and DimHex, and their role in creating rich harmonic textures.

• Overtone Series and Scale Derivatives: The natural harmonic foundation of the overtone series and its connection to major scales and other pitch sets.

• Notation and Analysis: Gain insights into the challenges of chord spelling, roots, and analysis in polysymmetrical environments.

Symmetrical Polytonality is a harmonic environment in which chord structures arising from symmetrically rich pitch sets/scales suggest several simultaneous tonalities. Symmetrical harmonic structures are the drive mechanism for tonal harmony. The symmetrical scales and the chords associated with them are also responsible for common non-diatonicism found in music involving the equally tempered chromatic scale. Harmonic symmetries manifest themselves in all styles throughout musical history, and became a primary force upon adoption of the equal tempered tuning system.

A chord pod is the set of chords built and labeled on the notes of its repeating intervallic scale segments. The chords are derived using all the notes of a particular symmset/scale, and are constructed above each degree of that symmscale. Each pod has an identical set of harmonic intervallic structures, and differs from other pods only by the roots on each tone of the segment.