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Sources1 suggest that the frequency resolution of humans—our ability to discriminate differences in pitch—is limited to around five cents.

If this is the case, how can musicians play an excerpt like the following (listen here, and be sure to check out 0:22!), which demands intonation changes of as small as two cents? (The numbers written above/below the pitches are the changes, in cents, for the tuning of each pitch.)

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This is spectralist music, so the intonation changes (at least in the first five measures or so) are based on the harmonic series. As such, the first violin may not be thinking of moving the F down two cents, but rather creating a justly tuned minor third with the cello. In other words, the first violinist isn't just tuning their single pitch down two cents, but they're tuning the interval itself. This all makes sense to me, and explains how a musician could hear this particular difference of under five cents.

But if all of this is true, then it seems to suggest that the first violinist can't actually play their part alone without the aid of a tuner. Am I missing something?

1 Just to limit these sources to SE sites, see here, here, and here.

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    0:28 sounds - weird! When was this written? (And don't say after a few pints!) – Tim Jun 11 at 14:22
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    I got to 30s & just had to switch it off. That hurts. It starts feeling a bit 'just' which is fine by me, but then at 0.22 simply sounds out of tune for no reason. By 0.30 it's no longer bearable. – Tetsujin Jun 11 at 14:28
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    @Tetsujin I'm sorry, but I suspect many people listening to that will share my feeling that although it may sound slightly out of tune from time to time it is still sounds much better than when I practice. – Brian Towers Jun 11 at 14:41
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    It's better than my local karaoke pub, I'll give you that;) I honestly don't understand what it's… for. tbh, I don't know whether I can identify categorically 5 cents - it depends on context as to whether a pitch bothers me - blues, flat 7, fine & dandy. 3rd… gotta be more careful to the overall tonality. I can't abide sharp 3rds, they're just painful & my tolerance for those is pretty small. – Tetsujin Jun 11 at 14:45
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    Wow, I didn't expect such a negative reaction to the piece. I think it's a fantastic work! – Richard Jun 11 at 15:14
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With a bit of training, a good musician can hear differences of 2 cents, and with significant talent and/or a lot of practice, 1 cent.

I base the above statement on my personal experience with developing ear training software for musicians. For example, I have been working on an app for training musicians to tune a guitar or a piano, purely by ear. This software is not published yet, but I can tell you that with some practice both myself and a few other beta testers have been able to routinely reach a precision of 2 cents. On a good day, we can occasionally hit one cent.

So, answering your question: the sources that put the limit at 5 cents are not correct. 5 cents is not a difficult interval to hear (with some training) and I would put the limit at no more than 1 cent for people with a good musical ear.

And by the way, Ben Johnston performers are known to practice for years and years before being able to record a good take of his compositions, so while it's possible, it's not exactly easy... :)

(Should anyone be interested in beta testing the upcoming ear training app, write me privately)

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    To me, it sounds as though they need several more years practice to be able to record a good take... – Tim Jun 11 at 15:54
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I agree with answer given by topo Reinstate Monica. I would also add that in addition to the pitch discrimination threshold being frequency dependent and NOT absolute, this explains a person's ability to judge the difference of two notes played separately. A whole new set of acoustic and mechanical phenomenon occur when harmony is involved. When listening to notes played together one will be better able to judge slight variations in pitch against the harmonic background. The use of contrast here is a very common device in music and art. As an extreme example consider tuning a musical instrument like the guitar. We play the fifth fret harmonic of the low E string and attempt to match it with the seventh fret harmonic of the A string, both being an E. As we tune the A string we will hear beats between the two notes. The two notes are in tune relative to each other when the beat period is infinite (something we never really achieve). The point is, at that point we are not hearing two notes but a fused tone and its envelop. Our ability to judge the envelop as being flat is, as far as I know, only limited by our life span (if you are willing to sit that long, and the acoustic energy does not dissipate). When listening to harmony we will hear similar phenomenon in the interaction of the harmonics of each note. So, hearing A followed by A (+2) in a lab may not be perceived as different. But hearing the interval (A, E) and (A(+2), E) may be noticeable.

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  • That's an interesting way to think of it - but I cannot tune guitars by harmonics, they go out across the fretboard. I find the best, simplest & most reproducible way to tune them is just play 6 open strings one after the other. The wrong ones are obvious to me & I re-tune them as I go; I guess because I "know" those pitches, but I couldn't explain it any better than that. – Tetsujin Jun 11 at 15:50
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    @Tetsujin - I've been tuning guitars and basses using harmonics for decades. It seems to work for me, even though some say it can't be done. What you do is similar to violinists, but using fourths and third in place of their fifths. – Tim Jun 11 at 15:58
  • I get the same if I do the 5th fret/open thing too. I drift out as I cross the fretboard. Just listening to the open strings I can do it in two passes, rough it in, then hone it 2nd time. I've been doing it for so long it's just become 'the way I do it'. – Tetsujin Jun 11 at 16:04
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    The 5th fret will not match the true 5th, but then again the guitar is not a Just instrument. Many can hear the resonance of a 5th or 4th. I was taught to tune my violin by ear, playing two consecutive strings and listening for the resonance of a 5th. At the end of the day we will have to make adjustments for the 12TET nature of fretted instruments. My point about the tuning was to illustrate by example how beats work. – ggcg Jun 11 at 17:06

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