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# Harmonic Piano Strings

Super flexible strings to beat the inharmonicity
 (+3) [vote for, against]

The ‘problem’ with the piano is with the inharmonicity of stiff strings. That is, the frequencies of the higher overtones of a single vibrating string are not multiples of the fundamental frequency of the string. Therefore, when playing that string at the same time as a much higher-pitch string, the frequencies don’t line up. For example, a low C has an overtone that is close to but slightly higher than the fundamental frequency of a C that is exactly 4 octaves higher. (This has nothing to do with equal temperament, since the octaves are not affected at all by equal temperament tuning - they are perfect octaves.)

The ‘solution’ to this ‘problem’ is the standard Railsback curve, which summarizes that piano tuners typically tune the higher-pitch piano keys slightly sharp (higher frequency), and the lower keys slightly flat (lower frequency). That is great, and gives the piano its beloved ‘crazy’ sound. But it also makes it hard to play along with an orchestra or any other instruments that are not tuned in the Railsback curve. Except in the middle registers, the piano is out of tune with the other instruments, and there is nothing that can be done to avoid this (except to tune the piano with a unstretched curve, instead of Railsback, but this sounds terrible and no one does it.)

My half-baked solution is easy to say but hard to implement: invent a new piano string that does not suffer from inharmonicity. Then the Railsback curve is not needed and all is happy. (Although to be honest, I suspect people so love the ‘crazy’ piano sound that the harmonic piano I am proposing would be rejected. Oh well, there’s no accounting for taste.)

The technical problem then is – how to make a string material that is super flexible (unstiff), which also has the necessary mass density and strength to withstand the high tensions needed (frequency = square root of tension over mass density, right?) Lead-powder-doped polymer? Too weak. Add a thin flexible super-strong ‘wire’ through the center of the polymer? Carbon nanotubes? (I understand these can solve any and all technical problems, at least the half-baked ones). Spider webs? From genetically engineered spiders? I think I need some help here.

 — sqeaketh the wheel, Apr 07 2012

Spider silk spun into violin strings http://www.bbc.co.u...nvironment-17232058
At the BBC [pocmloc, Apr 08 2012]

Wikipedia Piano acoustics http://en.wikipedia...iki/Piano_acoustics
A little background info [scad mientist, Apr 09 2012]

yeah I always wondered how/why that worked (stretch tuning). I originally thought it was just for emphasis.
 — FlyingToaster, Apr 07 2012

 This is, of course, an old problem (with stringed instruments in general), and there have been many attempts to solve it. The main methods these days are over-wound strings, and length.

 This is solvable, but I think you have said it yourself - it's not done because the result wouldn't sound like a piano. Not just because of the stretched octaves, but because the stiffness of the strings gives the characteristic, slightly bell-like in-harmonic plinking sound of individual notes.

The insane approach would be to make the strings so short and thick that the first overtone is tuned to the third harmonic, rather than (slightly sharper than) the second.
 — spidermother, Apr 07 2012

But then the second overtone is WAY off. We get a xylophone then.
 — sqeaketh the wheel, Apr 07 2012

 Well, I did use the word 'insane'. //We get a xylophone then.// sp. glockenwerk.

 The other unwanted side effect of reducing the inharmonicity would be to reduce the loudness that is the instrument's (groan) forte.

Another solution would be to use super-long gut strings. They would have to be taken from one of the larger, non-ruminant ungulates. So, (double groan) horses for courses.
 — spidermother, Apr 07 2012

Gut, yes. I measured the overtones frequency spectrum of a violin with very good gut strings and they were nearly perfectly harmonic! Piano Largo?
 — sqeaketh the wheel, Apr 07 2012

//The insane approach would be to make the strings so short and thick// Well, who says the strings have to be cylindrical? Bell makers have over the last century or so pretty much worked out how to profile a bell to produce pretty much whatever pitch you want from each overtone. I'm sure string makers could do something similar.
 — pocmloc, Apr 07 2012

[pocmloc] is on to something. Made me think that by varying the mass density along the length of a string, one could tune the higher overtones to drop a little in frequency. That would be done by placing tiny masses on the string at the location of vibration antinodes of the higher overtones. Problem then is - antinodes of many different modes occur at common locations. Solution (?) - vary the mass density linearly along the string to shift the antinodes of different modes away from each other, allowing independent tuning of each overtone by the tiny attached masses.
 — sqeaketh the wheel, Apr 07 2012

I, for one, welcome our Inharmonic Overtones...
 — gnomethang, Apr 07 2012

 // Well, who says the strings have to be cylindrical? //

Not refuting that idea, but keep in mind that structures with irregular profiles have inherent problems under tension. Clever engineering can remedy this to a point.
 — Alterother, Apr 07 2012

I'm dipping my toes into mysterious, dark waters here where I probably have no business swimming, but could the strings be dynamically tuned? In other words, could microprocessors cause the strings be de-tensioned and re- tensioned millisecond by millisecond to achieve the effect you want?
 — AusCan531, Apr 08 2012

You mean like have one end of the music string attached to another string which is attached to the body... the pluck them both ?
 — FlyingToaster, Apr 08 2012

//cause the strings be de-tensioned and re- tensioned millisecond by millisecond// that is a cool idea. totally crazy, but cool.
 — sqeaketh the wheel, Apr 08 2012

 Tensioning actuators controlled by a central processor which is in turn controlled by feedback from the strings themselves, perhaps even by using electric pickups. A digital-analog acoustic piano.

Now all we need is the unobtanium to create variable- profile piano wire. I'd make them thick and flat-topped where the hammers strike for even transfer of energy, then narrow at each end and wide at the waist to carry the most vibrational momentum, if that's an appropriate term. Then again, I'm hardly an acoustic engineer, so maybe a string like that would sound terrible. It's moot, since it would snap on the first note struck if made from conventional materials.
 — Alterother, Apr 08 2012

 I don't see why it needs to be unobtanuim. It is not hard to make a variable-profile thin rod of silver or gold or even brass, all of which work well as music wire and are soft and easy to work. It would however take rather a lot of skilled time. The structural limit is the tensile strength of the smallest cross-section, but the main problem with running strings at less than their maximum working tension is an increase in inharmicity, which we are already dealing with so this is fine. Alternatively, you could work-harden the thinner areas while annealing the thicker areas.

Having the nut end of the string mounted on a vibrating coil like a loudspeaker driver, controlled by the processor, sounds a great idea a s well. Perhaps both systems could be combined.
 — pocmloc, Apr 08 2012

 The thinnest parts of the string aren't just the weakest parts, they're also load-bearing points. They will be flexing back and forth while taking the 'weight' of the thicker center as it vibrates. It's not the best analogy, but anyone who's held the lead rope of a panicky, head-thrashing horse will know what I mean.

Also, I was using 'unobtainium' in the sense of 'specially engineered and bowel-quiveringly expensive alloy', not in our usual sense of 'material that does not exist'. Sorry.
 — Alterother, Apr 08 2012

 Only the very ends of the string touching the bridge and nut flex like that. In fact I think it's the stiffness there that causes most of the inharmicity. Hmmm, how about a tiny roller-bearing universal joint in the string where it touches the bridge and nut? Is a chain less inharmonic than a wire?

//specially engineered and bowel-quiveringly expensive alloy// Oh that, I use it every day on my instruments. One gets a little blasé about it.
 — pocmloc, Apr 08 2012

 // Is a chain less inharmonic than a wire? //

In general, yes, much less, but somebody will probably be along soon to tell me how a chain, or for that matter anything that can be formed into a long strand down to and including goat saliva, can be harmonically tuned.
 — Alterother, Apr 08 2012

//Only the very ends of the string touching the bridge and nut flex like that.// Well, the string flexes along its entire length - you're probably right about the importance of the ends, because the radius of curvature of a theoretical, perfectly harmonic string is zero at the bridge, but that can't be the whole story. In particular, stiffness along the entire length would predict sharpening of the higher partials, whereas the end effect wouldn't, as far as I can tell.
 — spidermother, Apr 08 2012

Adjusting the tuning of the strings super-quickly in real time to correct overtones would simultaneously louse up the prime tonalities, no?
 — RayfordSteele, Apr 09 2012

 I love the stuff I learn on this site! I posted a link to Wikipedia that I ran across that describes the inhamonicity and the Railsback curve. If someone knows of a better description I'd appreciate a link.

 I enjoy playing the piano, but I'm not too great. It's wierd that I've always been annoyed by the fact that the low and high notes sounded out of tune, but for some reason I always had the impression that it was because my hearing wasn't quite right at the extremes or the piano tuner didn't do his job well enough. It never occured to me that it was a "flaw" in the way pianos are designed and constructed.

 I am going to question [spidermother]'s statement // it's not done because the result wouldn't sound like a piano. // mainly because she give good answers when questioned. The Wikipedia link says that a larger piano will generally have less inharmonicity, so it's best to get the largest one that budget/space allows. If that's true, it seems like if it was possible to reduce inhomonicity further, it would be done. Or maybe it is good to reduce inharmonicity up to the point where the inharmonicity of the upper and lower ranges are the same as the middle ranges.

I wonder if it would be possible to synthesize a inharmonicity-free piano by taking a normal piano sound, converting the signal into the frequency domain and adjusting the location of the peaks before converting back to time domain? I'm guessing there would be a lot of risk of adding strange artifacts to the sound. Maybe create a simulation of a piano. Tweak the simulation until you can reproduce a sound that very closely matches the sound from a real piano. With that simulation you can then experiment with parameters for the string that may be impossible with available materials.
 — scad mientist, Apr 09 2012

 Those are very good questions. It's certainly true that bigger pianos have less inharmonicity, and also more harmonics. But my impression is that the best grand pianos are widely regarded as 'just right', and that smaller pianos are worse, but larger ones still wouldn't be better. I know that's a weasel argument, so I'd better explain in more detail.

 First, I fairly recently put a set of the lightest (i.e. thinnest) strings available on my steel-string acoustic guitar. There were two reasons; firstly, I was given the instrument with a damaged neck, so I wanted to keep the string tension low, and secondly, I play with a slide, and wanted to keep the intonation linear and the partials harmonic, because (natch) I mostly try to play in just intonation. It worked, and I quite like the effect, but it's lost some of the character of an acoustic guitar. The sound is purer, but quieter and with less subtle jangliness.

 I'm also somewhat familiar with some relatives of the modern piano, including earlier pianos, harpsichords, and clavichords. These are also more harmonic, but again, their sound is pure, but kind of thin and quiet.

 Inharmonicity, and other non-linear effects, create interest and movement in notes. If you play a single note on a good, well-tuned piano, you can hear it change over time - it's 'alive', so to speak. I'm sure the nature of the strings both creates inharmonicity and contributes to this aliveness; they are two sides of the one coin. In fact, good piano tuners deliberately make the individual wires making up each of the lower courses very slightly out of tune with each other. This, like the wire's stiffness, also creates a subtle out-of-tuneness within each note; when it's done well, the strings phase-lock as the note progresses, so the beginning of a note is more jangly than the end. All of which suggests that you can make relatively pure, harmonic pianos, but you lose some piano character if you do.

 I suspect that another reason it all works is that when absolutely everything is out of tune the tempered intervals don't stick out so much.

This is also why I don't like pianos anywhere near choirs. Not only is every single note on a piano out of tune with every other note, but the individual notes are OUT OF TUNE WITH THEMSELVES! Pianos are beautiful things, but they're more percussive than harmonic instruments.
 — spidermother, Apr 09 2012

Perhaps the solution to the original “problem” is to dump the piano and get a harpsichord?
 — pocmloc, Apr 13 2012

Which is a bit like Hoffstadter's tongue-in-cheek suggestion that the best English 'translation' of a Dostoyevsky novel is a Dickens novel.
 — spidermother, Apr 14 2012

 Couldn't you increase string length and/or decrease tension until inharmonicity effectvely goes to zero? Perhaps it's not so much a matter of thin strings, but rather the ratio between string diameter and the distance the string can oscillate away from its center while resonating.

 It occurs to me that the tension of the piano strings might in part arise out of a trade off between harmonic integrity and loudness.

In other words: Loudness, dynamics and harmonic integrity. Pick two.
 — fishboner, Dec 24 2013

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