Does anyone know whether the inharmonicity in an interval between upper partials is expected to be greater or lesser than the inharmonicity in same interval between a fundamental and a partial?

Here's an example. Suppose I tune A4 on a piano to 440. I then tune the A3 to the A4, which requires tuning A3 a bit flat of 220, so that its second harmonic rings at 440 Hz, matching A4. Now, both A3 and A4 will have A5 as a harmonic (fourth harmonic and second harmonic, respectively). Which of those two A5's would I expect to be sharper?

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    Are you asking about piano specifically? I’m not sure but I suspect that the answer varies from instrument to instrument. Sep 27 '21 at 16:33
  • @ToddWilcox: I mainly meant piano but left it a bit vague since I'm interested in the topic generally.
    – cruthers
    Sep 27 '21 at 21:09

In general, the higher partials on a piano tend to be more inharmonic than the lower partials, so I would expect that in the circumstance you indicated, the 4th partial of A3 would be sharper than the 2nd partial of A4.

But, see Figure 5, page 9 of this paper: Inharmonicity of Piano Strings, Simon Hendry, October 2008 : Figure 5, Inharmonicity of Piano Strings, Simon Hendry, October 2008

Although the graph only spans one octave, we see that it is not as regular and smooth as the theoretical prediction, so presumably there could be a circumstance where the 4th partial of one note is flat compared to the 2nd partial of the note an octave above.

It's also worth noting that there is a big difference in inharmonicity between the wound strings of the lowest register and the plain strings of the middle and upper registers of a piano (see Figure 12, p. 20 of that paper). A comparison of one note in the "wound" register to a note an octave above in a "plain" register will highlight this difference:

Figure 12, Inharmonicity of Piano Strings, Simon Hendry, October 2008 This is why octave stretching has to be a compromise, and is unique to each instrument.

  • Thanks Theodore--I'll need to study this more. What motivates this all is a note that I've seen in a few different places stating that, in tuning an equal-temperament, you'd expect the F3-A3 major 3rd to beat slower than the F3-A4 major 10th. In both cases, the beating comes from A5, as the F3 is expected to produce an A5 partial somewhere around 873Hz but in any case flat of the A5 partial produced by A3 or A4. So if I'm tuning A3 to match A4 (at 440, its 2nd harmonic), I'd expect the A5 generated by A3 to overshoot the one produced by A4, producing a faster (not slower) beat. Any thoughts?
    – cruthers
    Sep 27 '21 at 21:08
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    @cruthers for major thirds, the error due to 12-edo temperament is always going to be much more than the one due to inharmonicity, so I don't think there's much point paying too much consideration to that. If you want nice thirds, tune the piano to a well temperament like Werckmeister Ⅱ, instead of 12-edo. Sep 27 '21 at 23:30
  • Thanks @leftaroundabout but I don't understand. That is - I get that ET won't give you nice, pure thirds, but that's not what I want. I'm trying to tune to ET and would like to understand the reason why these sources tell me that I should typically expect F3-A3 to beat slower than F3-A4, when the study provided by Theodore would suggest the opposite trend.
    – cruthers
    Sep 28 '21 at 2:15
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    @Theodore, thanks, I think I've hit upon the answer. See youtube.com/watch?v=P5wG6Pnfryc at around 14 min. This tuner explains that (unlike what I thought earlier), A3 is typically tuned not to create a 2:1 octave with A4 (i.e. tuning A3 so the second partial hits 440 on the nose), but rather between 4:2 and 6:3, (i.e. closer to having a 6th partial that matches the 3rd partial of A4, which is E5). The 2:1 octave, if the math I've worked out is correct (using the theoretical inharmonicity), would indeed produce a faster F3-A3 than F3-A4... (continued)
    – cruthers
    Sep 28 '21 at 18:09
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    ... and flattening the A3 to get a 4:2 octave would produce equal beats, since both A3 and A4 would be tuned to produce the same A5 overtone (the coincidental partial that beats against the F3). However, if you flatten the A3 any further, to approach 6:3, the A3 produces a flatter A5 than does the A4, which would be closer to the one produced by F3 - resulting in a relatively slower beat. So, thank you!
    – cruthers
    Sep 28 '21 at 18:10

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