Why are pianos traditionally tuned “out of tune” at the extremes?

It wasn't particularly common in Bach's time, though, and the math to do it 'properly' wasn't even known outside of China.

Today, though, it's everywhere. Modern Woodwinds are equal temperament due to the placement of the holes. To match that, the entire orchestra is in equal temperament, regardless of the requirements of the instrument.

Guitars are in equal temperament due to the placement of their frets -- to do otherwise would require frets that bend between strings.

In isolation, people will [attempt to] sing in equal temperament, because that's what 'all' accompaniment is like.

Barbershop quartets, acapella groups, and to some extent string quartets have a tendency to shift harmony notes toward simple fractions of the root note of the chord, on the fly, even as the overall melody runs in equal temperament. This isn't a different 'tuning', per se, but it is a thing.

One of the interesting things about organ tunings is that because you can choose which pipes to perform with, different pipe-sets may be tuned according to slightly different needs on the same instrument. So, you might have one set of pipes that is well suited for equal-tempered, highly chromatic pieces. Another set of pipes may be intended for use with choirs, and another set still tuned to be ideal in an orchestral setting. However, the specific tuning of the instrument is usually maintained over the course of its life, which is part of why organs can be so distinct and characteristically themselves. Most organs are tuned by ear with tooling assists, and generally land in a Wohltemperiert that compromises between equal-temperament and pure 5ths.

Pure 5ths are are especially important on an organ, because many of the pipes produce highly sinusoidal fundamentals, making un-pure 5ths especially obvious and "out of tune" sounding. The complexity and bloom of the harmonics in a piano helps to mask these effects.

But they are still fairly deeply influenced by the fact all our ears are generally tuned to equal temperament, so I think it's still fair to say they use equal temperament + some local deviations, rather than some other tuning. Music that is "truly" not based on equal temperament sounds "wrong" to most people nowadays, or at least requires a significant adjustment period.

I have a lovely blues album by a guy who plays a keyboard in JT. most people who hear it ask me to turn if off "because it sounds wrong" and that includes musicians! it sounds fine to me because I grew up listening to ragas and developed an appreciation for alternate tunings.

Yes and no. It is true that we in orchestras "listen" and intonate in tune (with low major thirds, slightly high fifths and so on), but playing strictly in tune quickly becomes impractical.

If you play many harmonic sequences or modulations "in tune" and then return to you original key through a different path you will often end up nowhere near where you started. We all have the equal temperament as our base and even though we play one or two chords perfectly in tune we adapt tuning not to end up far from the standard equal temperament.

Then we have the problem of melodic intonation. If you play a melody with the intonation of the harmonic series you will sound _very_ low whenever you play a third.

He also makes the claim that equal temperament became wide spread _because_ of the piano specifically - not just to sound the same in every key, but to sound the same on each piano. And so equal temperament would not have saturated European music until the early 19th century when the piano became ubiquitous. The reason being (as mentioned) that many other instruments can be tuned by the musician on the spot, whereas a piano requires some standardization.

It seems like you are talking about what would happen if you tuned a piano using something other than equal temperament. Why is the iterated fifths relevant at all in this case?

However, it doesn't matter. The reason for the stretch-tuning of a piano is due to the physical limitations of a vibrating string being unable to produce the same harmonic intervals at the extremities of the instrument's range, and does not apply to electronic instruments which, particularly in additive synthesis, can generate whatever harmonics are desired, including harmonic configurations that no physical instrument would be able to produce.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.515...

I think you got this backwards. A perfect fifth is precisely 1:1.5 — or, put differently, the third harmonic is 3x times the root/first harmonic's frequency, and, transposed down an octave, that gives us the fifth.

Modern instruments use equal temperament tuning, where the frequency ratio between any two consecutive steps is the same, and you go from any note to a note at double the frequency in 12 equal steps — this means each step has to correspond to multiplying the base frequency by the 12th root of 2. In this tuning, the fifth corresponds to 7 steps, which corresponds to a multiplier of 2^(7/12) = 1.498.

BUT, while all of that is true, it isn't the reason a piano is tuned how it is. It's because the strings don't behave like the mathematical ideal of a string. The harmonics aren't at exact integer multiples of the fundamental frequency.

My source for this is asking a piano technician a bunch of annoying questions while he tuned my piano. He helpfully responded to by explaining the reasons behind stretch tuning, and then showing me the process of using his Sanderson Accu-Tuner to measure the inharmonicity of my piano and compute an individualized stretch tuning for it.

Another source: https://en.wikipedia.org/wiki/Inharmonicity#Inharmonicity_le...

Secondly the explanation is good. A string under tension (such as in a grand piano) has all kinds of weird things going on with the harmonics (mostly, they all go sharp). When tuning pianos this is called stretching.

You can read about it on Wikipedia: https://en.m.wikipedia.org/wiki/Piano_tuning quite far down in that article.

Edit: apparently I got the tension thing wrong. Lower tension and higher stiffness are what makes the lowest octave of the grand piano exhibit inharmonicity. To quote the wikipedia article on inharmonicity:"The less elastic the strings are (that is, the shorter, thicker, smaller tension or stiffer they are), the more inharmonicity they exhibit "

If piano strings were perfect, then we could easily extend the pattern to lower/higher pitches by halving/doubling frequency. The fact that it doesn't work at the extreme is due to the physical reality, as the stackoverflow answer correctly pointed out.

Edit: apologies for adding yet another mention of string inharmonicity before reading the rest of the replies.