Well Temperaments (Part 3)

Last time we saw the importance of some tiny musical intervals: irritating but inevitable glitches in our search for perfectly beautiful harmonies. Today I want to talk about a truly microscopic interval called the ‘atom of Kirnberger’.

It was discovered by Bach’s student Johann Kirnberger, and it has a frequency ratio absurdly close to 1:

2161 · 3-84 · 5-12 ≈ 1.0000088728601397

It arose naturally in Kirnberger’s attempt to find a tuning system close to equal temperament with only rational fre...