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For example, I try to press the G1 key down without it producing a sound, and when I abruptly play the C1 beneath it, the G, an octave above, is clearly audible, although the string isn't free to vibrate. Why does that happen? Do strings an octave apart vibrate sympathetically?

Andy Bonner
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Luke
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    On a digital piano?? – Tetsujin Dec 06 '21 at 09:19
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    yes, Yamaha YDP-164 – Luke Dec 06 '21 at 10:11
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    Note that, if the G1 key is pressed, technically the strings are free to vibrate (the damper is not in contact) so they could vibrate sympathetically (at least on a real piano). Maybe Yamaha pushed the modelisation that far… – Tom Dec 06 '21 at 10:17
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    They were working on modelled sympathetic resonances when I was there in the 90s - I can only assume it's in the products these days. – Tetsujin Dec 06 '21 at 10:42
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    It's not clear to me whether your question is just "why do I get sympathetic vibrations on a digital piano" (answered well) or: "But I held down *G1*, and heard *G2*." Since it's a digital piano, there's always a chance it's a quirk of the programming, but if you heard the same phenomenon from an acoustic piano, it would be because [G1 "contains" G2](https://en.wikipedia.org/wiki/Harmonic_series_(music)). – Andy Bonner Dec 06 '21 at 14:45
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    So just to make it clear, there are of course no strings in a *digital* piano. Whatever you hear is synthetized by software. – JohannesD Dec 06 '21 at 22:00
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    Funnily enough, the Technology Connections channel on YouTube had a video about Yamaha digital pianos a couple of weeks ago where exactly this feature was mentioned. ([Link to timestamp](https://youtu.be/JXYMdxwTf8s?t=885)) – TylerW Dec 06 '21 at 22:49
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    FWIW, Roland digital pianos have had also sympathetic resonance for years (since the V-Piano in 2009, if not before). – gidds Dec 07 '21 at 12:10
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    I think people are confused by the phrasing of your question, because it sounds like you're saying that you have a digital piano, and the digital piano has strings, and those strings are restricted from vibrating. I think you meant to say "when it doesn't have any strings that are free to vibrate." – Tanner Swett Dec 07 '21 at 15:53
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    @TannerSwett Actually, the OP's original title and question never mentioned digital pianos. I edited it into the title early on, in the same misunderstanding that many of us have rushed to, that the question was about why there are sympathetics at all. See phoog's answer: The core of the OP's question is "Why do I hear G2, *which isn't being held down*, when I hold down G1 and hit C1," which would be equally true on an acoustic piano. – Andy Bonner Dec 07 '21 at 16:58

2 Answers2

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Tim's answer is of course correct, but I'd like to offer a more quantitative way of saying the same thing. As others have noted, digital pianos emulate the sympathetic vibrations of acoustic pianos (well, some do, at least), so this answer describes how this works in an acoustic piano.

To restate the question, you press the G1 key slowly enough that the hammer does not strike the string, raising the damper from the G1 strings. Then you strike the C1 key sharply and release it. You hear G2 until you release the G1 key, meaning that the G1 strings for some reason were vibrating at the G2 frequency. Why do they vibrate at that frequency instead of at G1?

This has to do with overtones. A string's fundamental frequency is produced by a standing wave along the length of the string, but the string has other modes of vibration, at integer multiples of the fundamental frequency. These overtones correspond more or less to other notes in the scale. When you play C1, the first few overtones of the string are C2, G2, C3, E3, G3, B♭3, C4, D4, E4, F♯4, and G4. The seventh and 11th harmonics, B♭3 and F♯4, are particularly far from their equal-tempered counterparts, and the fifth and tenth harmonics, E3 and E4, are also not particularly close.

You don't necessarily hear these harmonic modes as individual tones; normally they blend to form a single complex tone.

Now the first few overtones of G1 are G2, D3, G3, B3, D4, F4, and G4. When you play the C1, the G1 strings are set in sympathetic vibration by the overtones of the C1 string. The third harmonic of C1 stimulates the second harmonic of G1. The sixth harmonic of C1 stimulates the fourth harmonic of G1. The ninth harmonic of C1 stimulates the sixth harmonic of G1, and so on. You're left with a harmonic series comprising G2, G3, D4, G4, B5, and so on. This is only a subset of the harmonics of the G1 string, but, notably, it is the harmonic series of G2, and that is precisely how the ear hears it.

Elements in Space
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phoog
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Any piano string (or other, for that matter) will produce harmonics from that root note. The first few harmonics produced from a C string are the octave (C), the 5th above that (G) and the next octave (C).

So when the damper on the higher G string is opened, that G string will 'hear' the harmonic from the lower C and begin to vibrate in sympathy.

There are no strings attached, but possibly some clever software in their place, to get as close to what a piano does sonically.

PiedPiper
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Tim
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    @PiedPiper - I wondered about that - and conclude, therefore, that, as you suppose, there is indeed very clever softwre. Afte all, it's trying hard to emulate a real piano! – Tim Dec 06 '21 at 10:25
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    Yes, modern digital pianos emulate string sympathetic vibrations. – user1079505 Dec 06 '21 at 15:58
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    Some digital pianos do this by sampling the actual sound from a real piano, so the reason you're hearing a sympathetic vibration is that you're listening to a playback of the sound recorded from an actual piano. – Darrel Hoffman Dec 06 '21 at 17:51
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    This level of sophistication has been around a long time. I've had my FP7 for 10+ years and it does all those tricks. Even stuff like just pressing and releasing the damper pedal without pressing any keys makes an audible kind of "sigh". Plus the highest strings not having dampers. It's all simulated. – Steve Bennett Dec 07 '21 at 03:31
  • This site: https://usa.yamaha.com/products/musical_instruments/pianos/p_series/index.html mentions "Virtual Resonance Modeling (VRM)" for the P-515 while this feature is missing in the description of the P-125. So: yes, it certainly is an advertised feature and no, it is not in every model. – piet.t Dec 08 '21 at 09:21
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    @DarrelHoffman no, sampling from a real piano is not a feasible way to obtain sympathetic resonances. You'd need to have a sample for every key with _every possible combination_ of other keys depressed, as well as at every dynamic level. That's a combinatorial explosion that would necessitate at least a whole datacenter for storing all the audio. Sympathetic resonances are generally implemented by physical modelling, even if the individual notes are sample-based. – leftaroundabout Dec 08 '21 at 09:30
  • Technically this is only true to the extent that the string diameter and density are consistent. From a physics point of view, a wonky string will have a non-linear set of resonances (like a bell). This is, by the way, why old strings need to be replaced. They develop inconsistencies over time from being under tension and getting whacked. – dmedine Dec 09 '21 at 05:52
  • @dmedine - a piano would have to be many decades old, and played hard every day for that to be the reasonable case. – Tim Dec 10 '21 at 10:58
  • @Tim that is not exactly true. In fact the upper partials of any non-theoretical string are always going to be 'out of tune' due to the fact that there are inevitably some inconsistencies in the string. These are definitely audible and indeed advanced physical modeling algorithms for piano synthesis take this into account. It lends a characteristic timbre to the sound. Yes, piano wire needn't be replaced as frequently as, say, violin strings, but the bending of upper partials is a feature of real vibrating strings. – dmedine Dec 11 '21 at 22:57