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One difficulty when tuning a piano is to sense the difference between equal temperament and just intonation. Let's consider A=440Hz and E=(660Hz in just temperament), a perfect 5th above A. After calculations, the third harmonic of A and the second harmonic of E has frequency difference 1.5Hz, which means I can hear beating of that frequency.

However, this is really difficult to judge, and it seems complicated, because I need to calculate this beating frequency for every fifth I tune.

Does anyone know how people judge this two cent difference in practice? I am learning how to tune and need a tip on this.

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    "because I need to calculate this beating frequency for every fifth I tune": with the traditional approach of setting the temperament in the middle of the keyboard and then tuning the rest in octaves, you only have to do this calculation 12 times. Also note that the actual beating frequency will be slightly different because the overtones are slightly inharmonic, which leads to stretched tuning, but I don't know whether that leads to a perceptible difference in the middle octaves. Piano tuning books surely talk about this (mine isn't handy or I'd check); you probably ought to get some.
    – phoog
    Commented Jul 8 at 10:30
  • See Inharmonicity between upper partials, which has a graph showing that (on one piano) the third partial of A3 was 4 cents sharp while the second of E4 was one cent sharp. This will get in the way of trying to tune the fundamentals 700 cents apart through the acoustical interference of the overtones. It also shows that inharmonicity does not vary smoothly, which is no doubt a large part of why piano tuning remains an art: it takes a lot of experience to be able to adapt the temperament to the varying characteristics of individual pianos.
    – phoog
    Commented Jul 8 at 11:31
  • If I remember correctly, my piano tuning book is a modern reprint of J. Cree Fischer's Piano Tuning, first published in 1907.
    – phoog
    Commented Jul 8 at 11:33
  • I'd normally say "Get a tuner" since a 2-cent difference is really hard to distinguish by ear if there's no beating, but I actually don't quite think the last piano tuner we brought in used a tuner extensively.
    – Dekkadeci
    Commented Jul 8 at 14:51
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    @Tim pianos are also polyphonic! When you're setting the temperament, the reference pitch is another string a fifth below or above the string you're adjusting. Typically you would tune only one of the hundreds of strings to a reference pitch and then tune the circle of fifths relative to that string, tempering the fifths by ear relative to one another. You use tuning mutes -- wedges between two strings -- to ensure that only one of the strings is vibrating for each key. Once the temperament is set, you tune the rest of the strings as unisons or octaves with the strings you've already tuned.
    – phoog
    Commented Jul 9 at 12:50

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I suggest that you not try to measure what you asked about.

Because calculating theoretical frequencies and attempting to discern beat rate values will not put you on a good path to learning how to tune pianos. It may seem correct to a rational mind with an awareness of string physics, but is not practical.

There are many ways to learn piano tuning, but I would suggest using the "ladder of thirds" technique of setting temperament. It uses comparisons of coincident partials of different musical intervals in order to accurately account for both the requirements of equal temperament and piano string inharmonicity. Once you are able to set temperament, you expand it to the rest of the piano. That's the approach, in a nutshell.

There are many resources for learning this. The Piano Technicians Guild is one. There are many schools that will sell courses, and many are great, but you seem curious and resourceful, and they may not be necessary.

Good luck!

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