Singers retune the third
The third asks for better numbers.
The Greeks built every note from the numbers 2 and 3 alone. Stack a fifth, stack another, and the chain reaches all the notes of a scale with every fifth pure. Count up that scale to its third note and you have a third. Four stacked fifths make that one 81:64. The chain climbs to 1113.75 Hz, and folding that tone down two octaves puts it at 278.44 Hz against the open string. For centuries nobody minded, because nobody held a third for long.
Then polyphony gave the third long holds. Play Wide third, then The sweet third. The first grinds a little. The second is the sweet one. English choirs around 1300 CE sang thirds and let their voices drift toward that sweet place. Walter Odington described where they settled: close to 5:4.
A cent is a twelve-hundredth of an octave, and neighbouring piano keys sit 100 cents apart. The wide third measures 408 cents, the sweet third 386. Their gap is 81:80, the 21.51-cent syntonic comma. Alone it is tiny. Three notes held together make a triad, and inside one the gap turns rough. The grinding lives in the upper partials, so every control here uses a reedy tone.
In 1558 CE Gioseffo Zarlino gave the sweet tuning a full theory and let the number 5 into the sacred ratios.
Stack the four fifths, drop the top tone two octaves, then hear the wide third it lands on against the sweet third. Listen for 81:64 to throb about fourteen times a second while 5:4 holds still.