We should be able to get astoundingly precise positional and force measurements using the principles by which musical instruments are tuned, especially string instruments.
A friend invited me to a work of musical theater tonight; despite the astoundingly adept dances, I was captivated by the designs of the improvised musical instruments — one being a Blue-Man-Group style PVC flapped pipe organ, where the players activate particular pipes by whacking their upper ends with heavy rubber flaps, and others being a sort of dulcimer or harp with six to ten strings each, in which one end of the string was anchored to the center of the bottom of a topless tin can, which can was screwed down to a wooden table, and the other end of the string was anchored to a hard object fastened to the table some distance away (I’d call it a “barrel”, but in this context that word might be taken literally).
The thing that most captivated me was the extreme pitch bend the player would sometimes extract from the string by squeezing the can a little bit. Perhaps the rim of the can would lift by ten millimeters under this treatment, so the bottom where the string was anchored might be changing its natural position by two millimeters or so, but in all likelihood the can top was substantially more compliant than the string, so perhaps the string end was being displaced by half a millimeter or less out of the 500 millimeters or so of the string’s length: a variation in length on the order of one part in a thousand. Nevertheless, the pitch bend was quite audible and even extreme, maybe more than a semitone.
It occurred to me that this pitch-bending could be the foundation of very precise measurement techniques for measuring distance and thus size.
Electric guitar players commonly do pitch bends by shoving their guitar strings a centimeter or so to the side along the fret, thus lengthening the string. (Or they use the whammy bar, if they have one.) If the string is one meter in length, this would lengthen it to √(1 m² + 1 cm²) = 1.00005 m, about 0.05 millimeters over a meter. Clearly the pitch bend is giving us a measurement of the string length that is sensitive to variations of some 50 parts per million, even better than I estimated in the paragraph above. (But maybe the strings connected to the cans weren’t steel.)
How sensitive should we expect pitch bending to be to the position on the end of the string?
The Wikipedia article on the musical cent says that humans can directly hear pitch differences once they’re larger than 5–25 cents, depending on musical training, on how high the pitch is, and on the harmonic content of the sound. A cent is a variation in frequency of about 578 parts per million, so the just-noticeable difference is on the order of 5000 to 20000 parts per million. The psychoacoustics article claims human frequency resolution is about 3.6 Hz in the 1–2 kHz octave, which is 1800 to 3600 ppm.
The Wikipedia article on classical guitar strings says and the Wikipedia vibrating-string article confirms and explains in more detail that the velocity of waves in strings is √(T/μ), where μ is the linear mass density and T is the tension. We can extrapolate that when the strings approach zero tension, the wave velocity (and thus the frequency) approaches zero, and when the tension varies by 1000 parts per million, the wave velocity varies by 500 parts per million. The frequency should vary by less than 0.5%, since the length of the string and its mass density are also changing, but the difference in those is much smaller, because the tension varies proportional to the difference from the string’s natural length, while the length and mass density vary proportional to its difference from zero length, which is orders of magnitude larger.
So we should expect an audible pitch bend when the string tension changes by something like 5000 to 10000 parts per million, 0.5% to 1%. At most, for a steel string, the tension elongates the string by about 1% — after that, I think music wire will break, though most steels would deform plastically first — so you should always be able to hear an elongation of 2% of that, 0.02% of the total length, 200 parts per million. This is pretty close to the electric guitar number above.
But that’s the worst case! We can do much better by putting the string under less tension. In theory, this should give us arbitrarily precise measurement of the string length, though only over correspondingly arbitrarily short distances. Indeed, I think this is one reason musical instruments are strung tightly, so that they won’t go out of tune easily, and so that the string frequency doesn’t rise for louder notes. In practice, I’m confident you can get one order of magnitude improvement: a length resolution of 20 parts per million.
That, though, is assuming we’re trying to detect the length by ear alone. Even a purely acoustic apparatus could improve on that: use two strings — a reference string of fixed length tuned to, say, 3000 Hz, while the measurement string is tuned to 3010 Hz at its default position. These will audibly beat at 10 Hz, more so if the second harmonic is attenuated (for example, by plucking or striking them in the center). It should be easy to hear differences in the beat frequency of 2 Hz or less, allowing an experimenter to hear variations in pitch of some 700 ppm, and thus variations in tension of 1400 ppm and in length of about 1.4 ppm.
But if we’re thinking of an electronic measuring apparatus, rather than a purely acoustic one, we could straightforwardly just use a frequency counter to measure the frequency; these routinely have absolute accuracy of better than 1 ppm, and short-term precision even better than that. I think that would allow you to measure variations in length of about 0.001 ppm, 1 ppb.
At audio frequencies with guitar-sized 1-meter-long strings, 1 ppb is 1 nm, about 10 carbon atoms. If you use the same string under the same tension but only 100 mm of length, you get three octaves higher pitch (on the order of 10 kHz instead of 1 kHz) and resolution of 0.1 nm, about one carbon atom.
You can use an electric guitar pickup to detect very small movements of the string. Linearity isn’t important, since the frequency is what we’re interested in.
The string-stretching mechanism described so far (call it the “whammy bar mechanism”) has one big drawback: the apparatus is very large compared to the displacements being measured. So our hypothetical 100-mm-long, 0.1-nm-resolution sensor described above is only capable of making any measurement at all over the range of about a millimeter before breaking the string, and only about 300 μm with precision in the 0.1-nm range.
As an alternative, instead of altering the pitch of the string by stretching it, we could alter its pitch by sliding a “bottleneck” along it, as in slide-guitar playing. Only the length of the string up to the bottleneck vibrates, so its frequency gives us a proportional measurement of the bottleneck’s position. This way, a meter-long sensor, for example, could read out a location anywhere within a 500-mm length while staying within a single octave.
The precision is correspondingly lower, but if you’re using a 1-ppm-error frequency counter, you still get 1-micron resolution over a meter.
As an alternative to strings, you could use a column of air as your resonant medium, sliding one pipe inside another to continuously vary its length. The only advantage of this that occurs to me is that you can use any material at all. The precision and range should be comparable to the slide-guitar mechanism.
The above resonant mechanisms have certain problems with noise susceptibility and ringdown: it’s quite reasonable to imagine that there might be vibrations in the environment within the range used by the instrument, and it would pick them up and could give erroneous readings as a result. Moreover, once they are resonating at some frequency, that vibration itself could bounce around inside solid bodies and be picked up even after the distance has changed — in effect, the instrument in the past produces its own interference in the future.
Instead of measuring a resonant frequency, though, you could generate random noise, or better still LFSR noise, and feed it into the measurement medium at one point and then read it back out — either at the same point after it’s rebounded from the far end, or electronically at some other point, and measure the time lag with maximum correlation instead of a resonant frequency. In this case, the resonance of the medium is actually undesirable, and you can use the same matched-impedance technique used in electronic signal transmission lines to prevent repeated reflection back and forth and the resulting resonance. That way, the signal you detect is a clean copy of a single lagged version of your input signal, not a sum of many past segments of the signal at different lags, progressively more attenuated.
A particularly interesting approach here is the slide-guitar approach, using non-contact sensing of vibrations in a wire as they travel past, for example using magnetic pickups, so that several sensors can share a single wire, which can end damped by a felt pad or something like that, beyond the sensors; or the trombone approach.
Once resonance is no longer needed, you might be able to dispense with the string or pipe and just transmit the ultrasonic noise signal through free air, permitting distance measurements at many points in space and thus triangulation. Stray reflections may give rise to multipath ghosting, but hopefully they can be kept manageable.
There are a variety of reasons that the temporal measurement from any of the three mechanisms discussed above might vary for reasons other than the displacements that we want to measure. We can try to eliminate these, or we can try to measure them and correct for them. For example, a free-air system should have at least one microphone a known, fixed distance away from the sound source, in order to correct for variations in the speed of sound in air.
The natural length of the string will vary as it expands and contracts under the influence of temperature, potentially altering its tension, but in itself this need not introduce a large error — if the frame it’s stretched on expands and contracts by the same proportion, its tension should remain constant. This may be difficult, since making the frame from music wire is probably not practical, and even if it is, the properties of music wire should vary by diameter.
The slide-guitar mechanism will, however, have a temperature-proportional error in position: if a movement causes the frequency to change by 0.1%, that represents a movement of 0.1% of its total length. But if it has expanded from 1000 mm to 1000.5 mm due to temperature, that 0.1% is now 0.10005% of its original length.
I think the whammy-bar mechanism will also have an analogous error: the Young’s modulus of the string material will not remain constant with temperature, and indeed I think should have roughly the same thermal coefficient as the material’s natural length.
Steel’s coefficient of thermal expansion is about 10.8 ppm/° around room temperature, so we’re easily looking at a 100 ppm error here if temperature is not controlled.
The trombone mechanism, however, suffers greatly from temperature drift, since the speed of sound in gases varies as the square root of the temperature. A variation of 2% in the temperature (5.5°) thus changes the tuning or time lag of the tube by 1%.
A potentially much bigger problem for the whammy-bar mechanism is that, when the surrounding temperature changes, the string will reach the new temperature much sooner than the frame will; and if a human handles the device, they will warm it up where they touch it. If the string is steel 10° warmer than the steel frame, it will be ≈108 ppm longer, but if its strain was only 1000 ppm, that’s an error of 11%, 110’000 ppm, in the strain.
The use of Invar or some similar material might be worthwhile if the apparatus cannot be protected from such variations in temperature.
Solid materials should be pretty immune to pressure (at least until the mass of the air around the string becomes significant compared to that of the string — steel weighs about 8000 times as much as the same volume of air, and air’s density varies only proportional to pressure, so this should be a source of uncontrolled variation of frequency in the 1 ppm range). Steel is pretty immune to humidity, too, but other possible string materials might be hygroscopic.
The trombone mechanism, including the free-air version, suffers the most here: although to first order the speed of sound in air doesn’t vary with pressure, it does vary with humidity; since water replaces some air molecule with water molecules of roughly half the weight, it speeds up sound transmission by up to about half a percent, introducing a half-percent error (5000 ppm) in the distance measurement.
Creep can reduce tension over time if a string is under constant tension. Steel doesn’t creep much at room temperature, which is how pianos can stay in tune for months or years at a time, but other possible string materials (and, even more so, frame materials) might creep rather badly. And it might be that these mechanisms are sensitive enough to detect creep phenomena in steel that usually go undetected.
It’s desirable for position transducers to respond as rapidly as possible to help keep control loops stable.
The resonant approaches suffer from the need for vibrations to build up over potentially several round trips; the noise-correlation approach avoids this, but still requires at least one round trip if the microphone is colocated with the speaker. Sound in air travels at only 343 m/s, so measuring a distance of a meter is going to take about 3 ms. Music wire can transmit vibrations faster but you’re still potentially looking at milliseconds of latency if you wait for the reflection.
However, with the noise-correlation approach, if you locate the sensor at the place whose position you’re measuring, you don’t suffer this latency (except in the whammy-bar mechanism, which inherently averages the propagation time over the time the signal is traveling through the string). You just need enough signal to correlate so that you don't get misled by noise. So you could have a feedback latency measured in microseconds (assuming the vibration frequencies are sub-MHz) instead of milliseconds.
The wire we’re talking about here is elastic, which is how sound waves can travel over it in the first place. Up to now, the force needed to stretch it has been nothing more than a nuisance — hopefully we can use a thin enough wire that it doesn’t exert too much force and disturb the thing we’re trying to measure. But what if we use that very elasticity? Instead of stretching the string on a frame, hang an unknown mass from it and weigh it by way of measuring the sound propagation speed on the string. It’s like the whammy-bar mechanism, but now our guitar has no neck.
Again, we should expect to be able to measure the lag with an error of about 1 ppm, translating to 2 ppm error in the tension, with (I'm guessing) 11 ppm/° of thermal error.