Singers retune the third
The third asks for better numbers.
Play the two major thirds below. The first grinds a little on a long hold. 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.
The wide third comes from four pure fifths stacked and two octaves dropped: 81:64. Measured in cents, where a semitone between neighbouring piano keys spans 100, that third is 408 cents. The sweet third is 5:4 at 386 cents.
Their gap is 81:80, the 21.51-cent syntonic comma. Alone it is tiny. In a held triad it is rough.
In 1558 CE Gioseffo Zarlino gave the sweet tuning a full theory and let the number 5 into the sacred ratios.
Play the wide third, the sweet third, then one after the other and inside the pure triad. Listen for the 81:64 third to grind more than 5:4, although notation draws both alike. Their upper tones sit 21.51 cents apart because the ratios tune the same interval differently. The contrast shows how singers can reshape a chord without changing its written notes.