Mapping Tone Helixes to Cylindrical Lattices Using Chiral Angles

The concept of a tone helix has been studied in tone theory and harmonic analysis from a variety of different perspectives. A tone helix represents harmonic relationships between tones in an attempt to model the perception of pitch and harmony in a single form. This paper presents a framework whereby previous helical tone representations can be considered together as one generalization with multiple instantiations. The framework is realized by combining the concept of isomorphic note layouts with cylindrical lattices. The extensively studied geometry of carbon nanotubes is used as a mathematical grounding. Existing tone helix representations are shown to adhere to this new, more general framework, and a process for mapping any flat isomorphism to its corresponding tone helix is presented.