By Jim Moran

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**Example text**

Why Plato chose as the ideal population ﬁgure the product of the ﬁrst seven numbers instead of the ﬁrst six or eight or ten is a moot question, especially because Plato must have been aware that he was courting disaster with this speciﬁcation. If one out of his 5040 citizens—a number with many divisors—died unexpectedly overnight, the philosopher-king would be stranded with 5039 citizens—a catastrophe in numerological terms, because 5039 is a prime number and has only two divisors: one and itself.

It was 1878 before the physical properties of overtones were exhaustively discussed by John Stru , 3rd Baron Rayleigh (1842–1919) in The Theory of Sound, a work that retains its position as a classic of acoustics to this day. The degree to which these overtones enrich their fundamentals is responsible for the speciﬁc timbre of a musical instrument and, indeed, of the human voice, where it is also the decisive factor for keeping the vowel sounds discrete. It was one of the most remarkable achievements of mathematics in the nineteenth century when Joseph Fourier (1768–1830) discovered that 26 Joseph Fourier.

40 The rotoid connects the 12 notes of the chromatic scale along the circle of ﬁ hs A♭–E♭–B♭–F–C–G–D–A– E–B–F♯–C♯ and circles the torus three times in doing so. The 4 circles inscribed on the torus, D–F♯–B♭, A–C♯–F, and B–E♭–G, each connect 3 notes according to the major-third relationship. E B C G B F C♯ A♭ A E F B♭ F♯ A B♭ D E♭ D F♯ C♯ 38 In well-tempered intonation, the inﬁnite number of notes of the Eulerian la ice is reduced to the 12 notes of the chromatic scale. F♯ G E♭ A♭ E A♭ B C C♯ F E A F A C♯ A G F♯ B♭ F♯ D B♭ D B D G G A♭ C C A♭ A B♭ D B♭ B♭ C A♭ G E♭ the shape, broadly speaking, of a doughnut.