Ideas worth following to their conclusion.

1. 2026-08-30·10 min read

   The Circle of Fifths Does Not Close: A Diophantine Problem in Music

   Part 2 of the series on The Mathematics of Musical Pitch

   Repeated pure fifths generate a remarkably useful map of musical pitch, but the map never closes exactly. This article constructs Pythagorean tuning from the ratio 3/2, compares twelve fifths with seven octaves, and proves that no positive number of pure fifths can equal a stack of octaves. The proof is an elementary application of unique prime factorization. The resulting mismatch, the Pythagorean comma, explains why tuning systems must choose how to distribute small errors rather than preserve every interval perfectly.

2. 2026-08-27·14 min read

   What Is a Musical Interval? Ratios, Tuning, and Consonance

   Part 1 of the series on The Mathematics of Musical Pitch

   This opening article explains what a musical interval is before asking why some intervals have seemed especially important in the history of music. It defines sound, frequency, pitch, and interval in accessible terms; uses an ideal string to derive the octave, fifth, and fourth; introduces ancient Greek harmonic theory and the later history of tuning; and shows why ratios survive transposition while frequency differences do not. It ends by separating exact interval mathematics from the more difficult perceptual question of consonance.

3. 2026-08-22·11 min read

   Till Death Do Us Part: Who Goes First?

   This essay asks why the arithmetic of life expectancy is not enough to predict who outlives whom. It combines survival distributions, spousal age gaps, demographic evidence, and a transparent Monte Carlo experiment to show how the probability and duration of widowhood emerge from overlapping lifetimes rather than fixed dates. The simulation is deliberately stylized: it is designed to clarify the mechanism, not to estimate a particular population's mortality risk. The article also separates population-level patterns from individual predictions and discusses what a serious analysis would need to add, including cause-of-death pathways, dependence between partners, and country-specific life tables.

The Mathematics of Musical Pitch

A sequence on ratios, tuning systems, geometry, and the mathematics behind musical pitch.

3 of 6 notes published · View the sequence →