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I was once told that an ensemble that plays in just intonation will always go sharp, and I'm wondering if that's actually true.

The easiest way to disprove this claim is to find an instance where the pitch doesn't go sharp. In this case, let's imagine a descending circle of fifths. If we begin with octave Cs in the ensemble and move down to an F-major triad, it seems we would keep the common-tone C in tune, and the expansion of the perfect fifth required by just intonation would then lower the F by almost 2 cents. If we continue through this descending circle of fifths, the widening of the perfect fifth would continue, and we'd find all the expected commas, resulting in a pitch center moving lower and lower.

With all that said, there are two caveats:

  1. As much as I want to think that my opening premise is false, I really respect the individual that told it to me. He's a college music professor, and one that I would normally expect to always cite his sources (so to speak), so I'd be shocked if he just pulled this out of thin air; I expect he read it somewhere. (Unfortunately, he's not around anymore for me to ask him.) If anyone knows of any source that makes this claim, I'd love to hear it.

  2. Thanks to Pat Muchmore's terrific answer to a recent question, we see that true just intonation is, for all intents and purposes, an impossibility. Nevertheless, there is a "watered-down" version of just intonation that focuses on tuning major and minor triads and connecting common tones, which is what I used in my second paragraph.

So, to repeat: Is it true that a just-intonation ensemble will always go sharp?

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    The prof's assertion seems illogical. Consider performing the same piece in retrograde, starting from the end and ending at the beginning. Would the ensemble still go sharp, or would it go flat? Of course, to maintain the illusion of Just Intonation within a conventional "common-practice diatonic" framework it is necessary to make continual adjustments, otherwise a sequence of notes like C - G - D - A - F does not end on the same pitch as just C - F. This is probably why a capella choirs, etc., tend to drift off pitch when trying to sing "pure intervals". – user19146 Sep 10 '17 at 15:09
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    @alephzero, given the question asker's "descending circle of fifths" example, I suspect that, if we assume that just-intonation ensembles always go sharp, then playing the song in retrograde (and/or inversion) will also make it go sharp. – Dekkadeci Sep 10 '17 at 15:13
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    Well, if you "assume" that JI ensembles always go sharp, then they always go sharp - but that is just a tautology, not a reason! – user19146 Sep 10 '17 at 15:15
  • @alephzero Maybe it's a bit like second law of thermodynamics - things go from a less random to a more random state, regardless of the direction of time, so 'ensembles go from a less sharp to a more sharp state when playing intonation' – marcellothearcane Sep 10 '17 at 15:17
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    @alephzero not tautology because they changed a condition (the 'direction' of play): if we assume that just-intonation ensembles always go sharp, then playing the song in retrograde (and/or inversion) will also make it go sharp. Rather than thinking of them as the same ensemble played backwards, think of it as two different ensembles. – marcellothearcane Sep 10 '17 at 15:20
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The prof's assertion seems illogical. Consider performing the same piece in retrograde, starting from the end and ending at the beginning. Would the ensemble still go sharp, or would it go flat?

Of course, to maintain the illusion of Just Intonation within a conventional "common-practice diatonic" framework it is necessary to make continual adjustments, otherwise a sequence of notes like C - G - D - A - F - C ends on a different pitch from where it began. (C = 1, G = 3/2, D = 9/8, A = 27/16, F = 27/20, C = 81/80 ... oops!)

On the other hand, the retrograde C - F - A - D - G - C gives C = 1, F = 4/3, A = 5/3, D = 10/9, G = 40/27, C = 80/81 ... oops, this is the same sized "comma" as before, but this time we went flat not sharp.

Of course you can "fix" this problem be redefining what you mean by Just Intonation - but I'm assuming that most people would accept that a major third is a pitch ratio of 5:4 (and not some more complicated approximation to that!), a perfect fourth is 4:3, and a perfect fifth is 3:2.

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    This answer is correct. I'd just like to add that in actual practice in choirs, the tendency is rather to go flat than sharp: singers tend to get the just major thirds right, but then make the half step above the major third too small- more like an equal tempered half step than the larger 16/15 just half step. Thus, unless care is taken, the pitch sinks. Of course it can go up as well- it's complicated- but I've experienced this far less frequently in a capella groups. – Scott Wallace Sep 10 '17 at 19:21
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Since the math (as you and @alephzero have pointed out) clearly proves that the statement

an ensemble that plays in just intonation will always go sharp

cannot be supported if taken literally, yet it was made by someone you hold in high esteem, it seems logical not take it literally.

Why not just assume that he meant

ensemble that plays in just intonation will always go out of tune

The professor said "sharp" simply because we generally use the sharp side - moving in the positive direction - to illustrate the point, as we are told Pythagoras did in his experiments regarding the overtone series.

We might add that moving in the positive direction on the number line is more intuitive for most people.

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