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This post is a follow-up question to this answer:

https://music.stackexchange.com/a/14264/86549

From what I could make of that answer, the need to stretched-tune a piano comes from the strings' less than perfect elasticity, which imperfection is greatest at the two extremities, where the strings are either very thick or thin. Because of that imperfect elasticity, getting the fundamentals exactly right would mean getting the harmonics slightly wrong. By stretching, we get the harmonics that are less wrong but at the expense of fundamentals that have been "stretched." (This may be contrasted with the need, or opportunity, for different temperaments, which to my understanding arises from mathematical reasons.)

Please don't rely on my imperfect summary, but go read the answer itself.

But here comes the question:

If the need for stretched tuning arises from the imperfect nature of the medium the strings are made of, why stretch a (modeled) digital piano? Couldn't the chip generate the frequencies at perfect ratios? By stretching a digital piano, wouldn't you get wrong fundamentals and wrong harmonics?

The only answer I could imagine is that we are so used to stretching in real pianos that we want it replicated in a digital piano. If this were so, could it be that by re-training our ears to octaves without stretching, we could actually begin to hear better music?

By the way, you couldn't replicate a real piano's imperfection in a digital piano simply by multiplying e.g. 0.99 or 1.01 to both the fundamental and all the overtones of a note. You'd have to multiply different factors to the fundamental and each of the overtones, so that, presumably, the fundamental is the most "off" and that the overtones the less off the higher multiple it is of the fundamental (or at least such is the implication of the original answer as I see it).

ADDED LATER

Here I found an explanation for stretching, which if true would apply to a digital piano as much as it does to a real: "I don't see why DPs would benefit less from stretch tuning than acoustics. It's a subjective adjustment meant to compensate for the human ear's relative inefficiency at lower/higher frequencies of the 88-note keyboard."

The same page also includes the following statement, which seems just wrong: "Stretch tuning a piano compensates for two things: 1) the basic fact that our 12 musical intervals are not laid out using perfect fractions (each note is theoretically 1/12 of an octave, but a perfect fifth, which is theoretically at the frequency halfway between the octaves, is actually 6 notes from the lower octave, and only 5 notes from the upper), and . . ." It seems to misunderstand the concept of "half-way." Just because the notes' frequencies are 2:3:4, that doesn't mean the middle note must be "half way" between the other two on a keyboard. And if that were a problem, no "stretching" would fix it.

ADDED STILL LATER

I would also add that the inelastic string and the limited ear ideas are inconsistent and cannot both be right.

The inelastic string idea (by which I mean the answer linked at the top of this post) is committed to saying that stretching only tries to achieve the "right" ratio between the high frequency overtones. For example, if a note and the same note four octaves higher are (meant to be) 1:16, the inelastic string idea says that stretching aims at giving 1:16 to the high overtones rather than to the fundamentals.

In contrast, the limited ear idea is committed to saying that stretching tries to give a ratio other than 1:16, e.g. 1:16.1, to (I suppose) the fundamentals and their respective overtones.

The two ideas are inconsistent, and at least one of them must be wrong. I believe there would be a simple fact of the matter. If you measure the frequencies of well-tuned pianos, fundamentals and overtones, the one or the other idea would prove to be right.

If the inelastic string idea is right, then it would appear that there is in principle no reason for a digital piano to try to emulate a real piano. The piano did not drop down from heaven after all.

If the limited ear idea is right, then even a digital piano must use some sort of stretching.

poppycat
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  • I’ve never heard the "limited ear" theory and I’m not sure it’s valid. The quoted explanation for "limited ear" seems to be describing equal temperament, not stretch tuning. BTW the top strings on a piano are not too thin, they are too thick. The bottom strings are too short. All the strings on a piano are too thick because the idealized resonant string has zero thickness. Also Ableton Live includes a virtual grand piano instrument that is not stretch tuned (along with a stretch tuned one). The one that is not stretch tuned definitely sounds weird to someone used to acoustic pianos. – Todd Wilcox Dec 25 '22 at 10:38
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    "could it be that by re-training our ears to octaves without stretching, we could actually begin to hear better music?" What makes a piano that is not stretch tuned better than one that is? In other words, why *not* stretch tune? The stretch tuned acoustic piano has become what is probably the single most popular musical instrument of all time. People love it. That suggests the sound of stretch tuning is not an acoustic problem to be solved. Rather it might be part of the character that is widely appreciated. – Todd Wilcox Dec 25 '22 at 10:43
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    The stretch tuning *is* a problem if you have pianist playing low notes together with an instrument in standard tuning. Usually the solution is that pianist stays in higher registers and leaves the bass to bassist, but I have the feeling that the non-stretch piano in Ableton is trying to solve the same problem. In that case it wouldn't matter if it doesn't sound good on on its own, as long as it works in the mix. – ojs Dec 25 '22 at 10:48
  • @ojs What would be "an instrument in standard tuning"? If it is a double bass, isn't it also made of a string? Whatever is the reason for stretching a piano (e.g. the string is inelastic), wouldn't it equally apply to the double bass's strings? In other words, wouldn't the bassist see that his notes require stretching and do it (by positioning the left hand fingers at the needed places)? – poppycat Dec 25 '22 at 13:19
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    @poppycat Yes and no. The piano has it worse than other instruments. I’m not certain but i dimly recall hearing that concert harps are also stretch tuned. The thing about the bass is that the tuning for most notes is done by the bassist and so can be adjusted on the fly. I could ask some bassists I know about playing with piano to see if they have to intonate differently. I have noticed that piano stretch tuning creates some challenges for my ear training practice, because I tend to hear the fundamental when I’m singing. But the piano sounds in tune with itself. – Todd Wilcox Dec 25 '22 at 13:49
  • I doubt playing slightly flat with double bass or fretless bass to match stretch tuning would work, because higher harmonics would then be out of tune. With very dark tone, maybe. – ojs Dec 25 '22 at 19:22
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    @ojs bass string instrumentss also have wound wire strings that are stiff and therefore will have stretched harmonics. If you tune a bass's A string so its 8th harmonic matches your 440 Hz tuning fork, it will already be a bit lower than 55 Hz, though perhaps not as low as the corresponding note on the piano (which seems to be around 15 cents flat, but this fan vary widely from one instrument to the next). But interference beating at those frequencies is so slow that you won't notice it anyway. If you play two A's at 55 Hz and 15 cents flat, the beating will have a period of over 2 seconds. – phoog Dec 25 '22 at 23:11
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    @poppycat the strings on a bass guitar or an orchestral/jazz bass are not as stiff as those of a piano so their inharmonicity is far less. But if a piano's low A is 15 cents flat and a bass guitar's low A is say 5 cents flat well then the beating period would be closer to 3 seconds. It's going to be far from obvious. If you have a synthesizer with more harmonic overtones than a real piano it won't sound as much like a normal piano, but it might sound more like [one of David Klavin's pianos](https://youtu.be/X71PY_5XwZ4). – phoog Dec 25 '22 at 23:32
  • @poppycat but even these are going to be somewhat inharmonic. One of Klavin's pianos is a 15 foot upright. A guitar's high E string will be unwound and therefore less stiff. This string might be 60 cm long or longer; to have a piano's low A made of the same string under the same tension, the piano would have to be 12 times longer or over 7.2 meters. That's 24 feet, not 15. – phoog Dec 25 '22 at 23:42
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    @phoog wound strings are used because they are more flexible than unwound. An unwound bass string would be more like solid rod than string. – ojs Dec 26 '22 at 04:07
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    A bowed double bass will have overtones that are extremely close to harmonic. The physics of continually-agitated bowed strings is [very different](https://en.wikipedia.org/wiki/Inharmonicity#Mode-locking) from the physics of strings that are plucked or hammered and then left to resonate on their own. So stretch tuning is not relevant to that case. All the above remarks about basses do of course apply to plucked double basses or to bass guitars. – Micah Dec 27 '22 at 03:33
  • @ojs well, wound strings are used because that's the only practical way to make a string with enough mass. But yes, they are more flexible than an unwound string of the same mass would be. When I said that the unwound string was more flexible I was thinking of a guitar's high E string as compared with a piano's lowest bass strings. – phoog Dec 27 '22 at 09:05

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The reason is that digital piano has the same inharmonicity as real one. If the digital simulation is based on samples from a real piano, it will inherit the inharmonicity from the sampled instrument. If it based on accurate physical modeling, the inharmonicity emerges on its own.

Like it or not, the inharmonicity is part of what a piano sounds like. It is possible to create a digital model with infinitely flexible strings, but it sounds more artificial and less like real instrument than model that includes string stiffness. Generally, instruments marketed as digital pianos attempt to sound and feel like like the acoustic instrument (with varying degrees of success).

ojs
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    @poppycat Even a modeled digital piano must model the inhamonicity or it won’t sound enough like a real piano to compete effectively with other products. Customers want digital pianos to sound as much like real ones as possible, which means they want stretch tuning as well as inharmonicity. – Todd Wilcox Dec 25 '22 at 10:31
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    @poppycat it's actually much easier to model a plucked string without the inharmonicity. The problem is, it sounds thin and weak and digital, and not much like a piano at all. To a large extent, the inharmonicity *is* the sound of a piano. – N. Virgo Dec 26 '22 at 02:06