From a Vibrating String to a Proof
Why can a guitar never be perfectly in tune with itself, and how can we be sure?
You hear whole-number ratios on a monochord, find a pure major third hiding inside a guitar string, and put numbers on why the frets miss it. Twelve pure fifths then overshoot seven octaves, a short odd-versus-even argument proves they always will, and the same move shows that the tempered tritone, the square root of 2, is not a ratio of whole numbers at all. The trail ends with how Euclid's geometry handled such magnitudes and where music sat in the old curriculum.
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Monochord Music & Guitar · interactive
Begin by ear: slide the bridge and hear that where the stopped and open lengths form a simple ratio (half the string for the octave, two-thirds for the fifth) the two notes blend, a fact the page keeps apart from the legend of Pythagoras and the blacksmith's hammers.
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The harmonic series The Science of Sound · interactive
Next, why whole numbers at all: a plucked string vibrates in halves, thirds and quarters at once, and its fifth harmonic, touched just behind the fourth fret of the A string, is a pure major third about 14 cents flat of the fretted C sharp, so the string itself disagrees with the frets.
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Algebra: The Skills That Carry Everything Mathematics, Systems & Languages
Put numbers on that disagreement: an octave multiplies frequency by 2 and each equal semitone by 2^(1/12), so the page asks you to compute seven semitones (about 1.4983) against a pure fifth (1.5), and it labels the link to music a shared structure, not an identity.
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The Pythagorean comma The Science of Sound · interactive
If tempered intervals are slightly off, why not tune in pure fifths? Stack twelve of them on the spiral and they overshoot seven octaves by 23.46 cents, so you see and hear the gap before anyone proves anything.
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Twelve fifths ≠ seven octaves Logic & Proof · interactive
Now prove the gap can never close: if twelve fifths equalled seven octaves, 3^12 would equal 2^19, an odd number equal to an even one, so, as the page concludes, there is no tuning without a loss, only a choice of which loss.
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√2 is irrational Logic & Proof · interactive
The same odd-versus-even move, applied to a fraction in lowest terms, proves the square root of 2 is not a ratio of whole numbers, and the page points out that this is exactly the equal-tempered tritone, an interval the Pythagorean creed that all things are whole-number ratios could not admit.
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Geometry, Euclid and Trigonometry Mathematics, Systems & Languages
How did Greek geometry live with such lengths? This page's Euclid track answers that Book V, traditionally credited to Eudoxus, handles incommensurable lengths without pretending every ratio is a fraction, and it keeps the fact (an ideal string's modes are whole-number multiples) apart from the analogy (that Greek geometry contains the physics).
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The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium Liberal Arts & Reasoning
Close by seeing the whole route as the old quadrivium, where music was number in time beside arithmetic and geometry, and do its music exercise: compute and listen to the just (about 386 cents), Pythagorean (about 408) and tempered (400) major thirds, keeping its warning that such cross-subject parallels are analogies.