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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.

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  • 2
    "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. Dec 25, 2022 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, 2022 at 10:48
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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. Dec 25, 2022 at 13:49
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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, 2022 at 23:11
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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, 2022 at 4:07

1 Answer 1

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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).

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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. Dec 25, 2022 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, 2022 at 2:06

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