The Tessellation Interference Principle: Consonance as Aligned Grids, Dissonance as Churn, Music Rendered by Overlapping Triangles
Abstract
Two overlapping rings on the Fibonacci wheel each carry an inscribed triangle, and each triangle can be cut into finer and finer rows, the tessellation levels. This paper reads the pair of grids as a pair of frequencies. The center distance sets a ratio between the two grids, 2(4 − d/R)/3, and at the wheel's named settings that ratio lands on musical ground. The Vesica distance d = R gives exactly 2, the octave and the only exact alignment, the Wide setting gives 1.512, the geometric shadow of the perfect fifth 3:2, and the k = 3 setting gives 1.399, the septimal tritone 7:5, at the same setting whose crossing notes are also a tritone, the same interval arriving through two unrelated systems at once. Where the grids align, the figure is still and consonant. Where they cannot align, and for the deep settings k ≥ 9 they never align at any level, so that region is tessellation-silent, the crossing pattern churns without repeating, the geometric picture of dissonance. Following Tetryonics (Abraham 2021), where the triangular diamond is the shape of a quantized frequency, the overlapping grids are read as interference, an interpretation labeled as such, while the geometry beneath it is verified by direct computation throughout.
Paper 8 of 10 in the music series of The Strassner Series: Fibonacci Geometry and Harmony. The program that checks every number in the paper is printed in its appendix.
Related
- Too Useful to Avoid, Too Uneasy to Accept: Cognitive Dissonance Among Language Majors in the GenAI Era
- Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems
- Measuring destination image dissonance: A compositional data analysis of cross-media representations in Sanya’s Tourism
- Unifying Stern-Brocot, Farey Sequences, and Tessellation via PSL(2,Z) — E8 Intelligence Research
- Niven's Theorem on the Cosine Ruler: Why Its Dial and the Tessellation Grids Share Only the Vesica Piscis
BibTeX
@misc{strassner2026tessellation,
title = {{The Tessellation Interference Principle: Consonance as Aligned Grids, Dissonance as Churn, Music Rendered by Overlapping Triangles}},
author = {Neal Strassner},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23605191},
url = {https://zenodo.org/records/23605191},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Tessellation Interference Principle: Consonance as Aligned Grids, Dissonance as Churn, Music Rendered by Overlapping Triangles
AU - Neal Strassner
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23605191
UR - https://zenodo.org/records/23605191
AB - Two overlapping rings on the Fibonacci wheel each carry an inscribed triangle, and each triangle can be cut into finer and finer rows, the tessellation levels. This paper reads the pair of grids as a pair of frequencies. The center distance sets a ratio between the two grids, 2(4 − d/R)/3, and at the wheel's named settings that ratio lands on musical ground. The Vesica distance d = R gives exactly 2, the octave and the only exact alignment, the Wide setting gives 1.512, the geometric shadow of the perfect fifth 3:2, and the k = 3 setting gives 1.399, the septimal tritone 7:5, at the same setting whose crossing notes are also a tritone, the same interval arriving through two unrelated systems at once. Where the grids align, the figure is still and consonant. Where they cannot align, and for the deep settings k ≥ 9 they never align at any level, so that region is tessellation-silent, the crossing pattern churns without repeating, the geometric picture of dissonance. Following Tetryonics (Abraham 2021), where the triangular diamond is the shape of a quantized frequency, the overlapping grids are read as interference, an interpretation labeled as such, while the geometry beneath it is verified by direct computation throughout. Paper 8 of 10 in the music series of The Strassner Series: Fibonacci Geometry and Harmony. The program that checks every number in the paper is printed in its appendix.
ER -