Music is based off of ratios in between each other. An Octave is 1:2 (For example, A1 = 440 Hz, A2 - 880Hz, etc.). But the fundamental ratio for modern music is the Fifth. Thr first eight fifths of F is C, G, D, A, E, B. Re arranged, that is also C, D, E, F, G, A, B (a Major Scale). If you go to 12 Fifths from F, you get the chromatic scale. This will work for any starting note, and is why we have 12 Major Scales.
However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take your instrument), it vibrates at a fundamental frequency and several harmonics above it as well. I cannot find it exactly, but I believe it is closer to 1:1.48 to make overtones work. This means the math does not work out between a Fifth and an octave. To solve this issue, we use the approximation on the piano so that the fifth and Octave line up exactly for the ratios (we tune fifths slightly sharp and octaves slightly flat).
Now remember how I said music is all based off of ratios? Due lot this, we need a common fundamental frequency that everybody can agree to tune to. The musical world as decided on the A=440 Hz (which is close to the middle of the piano). Due to the fact that the reference frequency is in the middle of the piano, as you get out further, the approximation rears its ugly head.
This is why pianos are tuned "out of tune" at extremes.
> However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take your instrument), it vibrates at a fundamental frequency and several harmonics above it as well. I cannot find it exactly, but I believe it is closer to 1:1.48 to make overtones work.
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.
I very well could have, I just got annoyed at Edomund's explanation. It isn't a limitation of the piano, it's the physics of how music is (Cunningham's Law at play).
I can look it up when I get home and dig up my music theory books.
You are right that there is an unavoidable mathematical compromise. A pure 5th would be a ratio of 3:2 or 1.5:1. But in equal temperament, semitones are 2^(1/12), so a 5th is 2^(7/12), which is about 1.4983:1, so just slightly off from 1.5:1. This issue applies to all instruments.
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.
Firstly, you got the ratio thing backwards. A perfect fifth is 1:1.5. an equally tempered fifth is 1:1498...
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.
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 "
Hmm, not really. If we look at the "Railsback curve" someone linked below ( https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_... ), it looks clear that around 440 Hz an octave is realized by a nearly perfect 1:2 ratio. Of course, it means the fifth (C-G, etc.) is never perfect 2:3, but that's an orthogonal issue.
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.
Both explanations are correct factors in the overall messy compromise that is tuning. The explanation they gave was attempting to describe string inharmonicity, which is worse for metal strings than more flexible ones and only applies when they are plucked or struck. So violins do not have this problem, because the bowing constantly corrects the frequencies as it slips and grabs. I love your description of the Pythagorean comma, though. The difficult compromise of mathematical purity is what led me to an obsession with pedal steel guitar, since it's one of the only things that can play full chords in all keys with perfect intonation.
Edit: apologies for adding yet another mention of string inharmonicity before reading the rest of the replies.
As a fun fact too, the overtone series is also why you have different size pianos. The longer string gives you a different overtone, and if i recall right, the overtone is "more in tune" overtone series (if there are any piano folks in here and I got that wrong, please forgive me, I am a mere trombone player).
Inharmonicity is a trade off between the ratio of string length to fundamental and stiffness (which is related to thickness.)
You could make a piano with really, really long bass strings, and the overtones would be more linear. There would also be a lot more fundamental, which is almost non-existent on the lowest octave or so of a grand piano.
Native Instruments actually sells a sample pack called The Giant which is taken from an experimental long-string piano.
I don't think it sounds all that good, because there's something just right about the colour of the "imperfect" bass strings on a fine grand. (That could just be acculturation, but I'm not completely convinced that's the case.)
Contrariwise, uprights tend to sound boxy and constrained in the bass because the strings are shorter than on a full-sized grand, and the overtones are even louder and even less linear.
You can't do much at the top end, because nicely linear strings - like the ones on a steel guitar - would have to be very thin and they'd be too fragile to survive piano hammers.
The closest approximation would be a hammer dulcimer, which has a much sweeter and more open top end than the percussive plink of a piano, but doesn't go quite as high.
Deeper tones have a longer wavelength. If the resonator is too small compared to the wavelength, it can't transmit the sound efficiently. I guess it has something to do with the sound wave wrapping into itself and causing distortions or something like that.
Deep tones sound better on a concert grand piano than on a small upright where they tend to be somewhat blurry.
Compare with big subwoofers for low-frequency sound reproduction.
Caveat emptor: I'm not really in the business, this is just off the top of my head. :)
Music is based off of ratios in between each other. An Octave is 1:2 (For example, A1 = 440 Hz, A2 - 880Hz, etc.). But the fundamental ratio for modern music is the Fifth. Thr first eight fifths of F is C, G, D, A, E, B. Re arranged, that is also C, D, E, F, G, A, B (a Major Scale). If you go to 12 Fifths from F, you get the chromatic scale. This will work for any starting note, and is why we have 12 Major Scales.
However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take your instrument), it vibrates at a fundamental frequency and several harmonics above it as well. I cannot find it exactly, but I believe it is closer to 1:1.48 to make overtones work. This means the math does not work out between a Fifth and an octave. To solve this issue, we use the approximation on the piano so that the fifth and Octave line up exactly for the ratios (we tune fifths slightly sharp and octaves slightly flat).
Now remember how I said music is all based off of ratios? Due lot this, we need a common fundamental frequency that everybody can agree to tune to. The musical world as decided on the A=440 Hz (which is close to the middle of the piano). Due to the fact that the reference frequency is in the middle of the piano, as you get out further, the approximation rears its ugly head.
This is why pianos are tuned "out of tune" at extremes.