Why no equal division of the octave contains a just perfect fifth
PAPER II — MATHEMATICS · QUESTION 31 OF 35
In plain terms: The free Stanford textbook giving the mathematics the question needs, from the man who wrote the field’s standard texts.
The question as set
Prove that no equal division of the octave contains a just perfect fifth, and say what the continued-fraction expansion of log₂(3/2) recommends in its place.
The question in Paper II · Its section on the reference page
The question in context
This is question 31 of the 35 set in Paper II — Mathematics of The Truly Hardest Exam in the World, the examination of The College of All Minds. In the College’s subject index it is filed under Music theory and tuning. Like every question of the College, it carries three references of record, verified live at publication and described twice — technically, and in plain terms — so the ground can be judged before it is walked. The response required is an argument, not a survey, of at most three thousand words, as technical as the question demands and no more.
The references of record
Mathematics of the Discrete Fourier Transform — with the theory of intervals
The arithmetic of intervals set out: ratios, cents, and the logarithmic map under which equal division becomes a rational-approximation problem.
In plain terms: The free Stanford textbook giving the mathematics the question needs, from the man who wrote the field’s standard texts.
List of intervals and the tuning archive
The tabulated evidence: just ratios against equal-tempered approximations, with the 12-, 19-, 31- and 53-division systems and their errors.
In plain terms: The free reference archive of the world’s tuning systems — where the approximations the theorem forces are set out in numbers.
Continued fractions and equal temperaments
The approximation machinery: convergents of log₂(3/2) generating 12, 41 and 53 as the successively better equal divisions.
In plain terms: A free mathematical treatment showing why twelve notes, and what the next better answers would be.
Background: Equal temperament · Pythagorean comma