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Is the "comma pump" an empirical problem, or merely theoretical?

It's a standard music theory exercise to show that certain chord progressions don't actually quite work. For example, in the I-IV-ii-V-I progression, following the "naive" method of finding each pitch (tuning each major chord as a pure 4:5:6 ratio and the minor chord as a pure 1/6:1/5:1/4 ratio, and keeping common notes between adjacent chords constant) will lead to the final I chord being flatter than the initial one by a syntonic comma.

Is this actually part of the reason why choirs or singers singing a capella (or anyone else making music without a pervasive absolute pitch reference) tend to drift in pitch unless they're careful and/or skilled? Or is anyone whose pitch sense is good enough for this not to be dominated by other errors also capable of instinctively compensating for it?

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I would say yes, and yes. You've explained the problem pretty clearly, and explained its consequence. Choirs frequently find that they sing everything internally, consistently in-tune throughout a piece, but then at the end of the piece, they discover that they are no longer in tune with the reference pitches upon which they started the piece. The frame of reference has drifted. Inexperienced singers don't understand that this is in fact a natural occurance. In fact many choir directors don't understand it either, and expect their choirs to sing in 12-tone equal temperament as if there was a piano playing along all the time.

However, singers and string players can compensate for the drift in pitch and try to keep pitches in line with the starting pitches, and end the piece in the same pitch framework where they started. It is a struggle, but it is often possible.

Update

Reference this previously asked question, about Arnold Schoenberg's observation about why choirs get "off pitch": Natural vs. Tempered Semi-Tones

As I said previously in my answer to that question:

As a singer with perfect pitch, when I am in this situation, particularly in performance, it presents a dilemma: if I perceive the other choral singers around me drifting away from the starting pitch, tonic and key (taking into account the modulations that may be written into the piece), do I fight against the other singers by keeping my pitches up, relative to the starting pitch and tonic and key, or do I "go with it" and follow the gradual shift in tonic that everyone else is making? In most cases, it's better to "go with it" rather than "fight it", because it produces a more internally consistent sound. While I have perfect pitch, I must concede that if a choir were only to sing the 12 pitches on the piano, that would not be a good choir, because that is not really what choirs are supposed to do.

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