Anharmonicity in violin strings
This started as a side topic in a different discussion on 'limits of intonation', but I think it deserves a thread on its own.Every now and then, someone states that the partials of a violin string are not at integer multiples of the fundamental frequency. For example David Burgess on April 24, 2009, Adrian Heath on July 9, 2015. (These I could find back with a quick Google search)
This anharmonicity really only applies to plucked strings, not to bowed strings. The most convincing is probably an experiment. I recorded my violin (D'Addario Pro-Arte strings on a student-grade violin), bowed (5 seconds, keeping the sound as constant as I could) and plucked. A Fast Fourier transform resulted in the following frequency peaks (of the first 7 partials), with the offset in cents in parentheses:
Bowed: 440.189 (0.00), 880.436 (0.11), 1320.52 (-0.07), 1760.82 (0.06), 2201.07 (0.10), 2641.09 (-0.03), 3081.52 (0.11).
Plucked: 437.865 (0.00), 882.369 (13.07), 1320.42 (8.97), 1761.32 (9.71), 2215.22 (20.35), 2644.87 (11.61), 3172.93 (59.88)
Clearly, the partials of the bowed string are harmonic, i.e., at integer multiples, whereas the partials of the plucked string are not. The fundamental frequency of the plucked string is 6.3 cents lower than that of the bowed string.
(Technical detail: the sound of the plucked string decayed with about 10 dB per 0.2 seconds; this generates some complications in the interpretation of spectroscopic data that I didn't want to spend too much time on. The main point is that a bowed string generates integer-multiple partials.)
Why is this? It is because the stick-slip process at the point of the string that is in contact with the bow hairs forces all partials to stay synchronized. This is illustrated with this plot (illustration, not measurements) of a wave with overtones with and without anharmonicity. The bow stick phase would be the downward stretches; the quicker upward stretches are the slip phases. You see that after 4 periods, the down phase gets faster than the actual bow motion, which would require the string to slip in the opposite direction. And at periods 5-6, the transition from stick to slip gets chaotic. The anharmonicity is exaggerated for this example; it would take many more than 5 periods before the phase differences between the partials have built up to such an amount. The fact that a proper sustained violin tone does not flutter around indicates that the slip-stick process is stable.
On the plot of the recorded waveforms, you see that the waveform is pretty stable for the bowed string (apart from some 50 Hz background that the microphone seems to have picked up), whereas the waveform changes over time for the plucked string.
(I am an experimental physicist.)
Edit: nomenclature partials/harmonics.
Replies (31)
On the piano it means tuning to different partials. But we don't really tune to this or that partial set. And we probably don't vary our pitch from bowing to plucking. On the upper strings we probably hear few upper partials anyway. For example, on A5, how many partials can an e string generate? Probably not too many. On the lowest strings, we don't perceive different partial pairs to be out of tune, and nor do we even perceive individual partials. On a large grand piano you can hear a prominent minor 7 as part of the sound, an on smaller pianos a strong 5th partial. But we don't hear these on a violin--we hear one coherent sound color instead a multiplicity of partials. Yes, we would expect some kind of tiny imperfection because there is a non-zero amount of string stiffness. But not enough to matter from a practical standpoint, I would think.
Re Scott: it helps me understand what happens when we tune or play chords, especially when it has to match a different instrument. If you play a double-stop fifth on the violin, you are supposed to match the partials. And now I know that it is really just intonation (3:2 ratio), rather than "almost just because that's where the string partials happen to be".
I think a piano is tuned by matching octave-partials and then dividing the "octave" into 12 equal steps. That results in stretched octaves. I think the stretch is about 50 cents over the full range (depending on the type of piano and the intent of the one who tunes it), on average 6 cents per octave, but apparently typically a bit less in the central octaves (example tuning charts for piano).
A violin that is tuned in 3:2 fifths has a stretch of 3.4 cents per octave. I've read on this wonderful forum that when a cello, violin, and piano have to play together, some compromises have to be made in order to match the tuning in just fifths to the equal temper of a piano over the entire range of the instruments. But it seems to be that the problem is different than it would appear because the deviations from equal temper due to octave stretching are actually larger than the difference between equal temper and just fifths.
(Anyway, it's all theory. I can barely get my violin intonation consistently within +/-20 cents, which I unfortunately do hear very well when I hear a recording of myself.)
Way to accurate, alone your bow pressure variation will cause bigger errors. A typical case of pseudo error calculation.
But I agree upon the diffeerences between free and driven oscillator.
Edit: beeng able to read might help, I thought it was the measurement error in braces... sry
Nevertheless your number of digits suggests a higher precission than optained. Which is a scientific mistake
If bowing-speed fluctuations cause the frequency to fluctuate a bit over the 5s recording, it would affect all partials in the same way and not affect the ratios.
My interest in inharmonicity in in violin strings was stimulated by two observations:
- the four strings have the same vibrating length, but different thicknesses and stiffnesses, and notes which should be exactly opposite are not;
- tuning the second harmonic (first overtone) on one string against the third harmonic (second overtone) on the adjacent lower string does not always give nice smooth fifths on the open strings.
https://www.youtube.com/watch?v=zaNe-9T28vM
In the link, Il Giardino Armonico conducted by Giovanni Antonini were performing in concert three early Haydn symphonies followed by a string symphony by W F Bach. At 1-03-03, after the Haydn symphonies were completed, bows taken and the audience applauding, the brass, woodwind and conductor left the platform.
What happened next over a period of three or four minutes was unexpected - the string players remaining on the platform carried out an intensive re-tune of their instruments, walking round listening to each other carefully. This was not your ordinary quick orchestral re-tune, but something special and instructive, and perhaps a privilege to witness.
I think what was happening was that the string players were getting out of their heads the sound and tuning of the brass and woodwind which they had been hearing for the previous hour and getting re-accustomed to the resonances and harmonics of their period instruments. My opinion in this was supported by the pro concert master of one of my orchestras who remarked it was like a pro string quartet tuning up, but on a larger scale.
After the re-tuning had been completed to everyone's satisfaction the players resumed their places on the platform, Antonini returning to conduct the W F Bach symphony.
Edit added:
I moved this post to this location from the thread "Limit on Intonation" on which it was the final post, because I believe it to be of general interest. This was done shortly before that thread was archived on July 4, 2017.
I don't know what that means in practical term. We tune intervals on the violin as purely as we can in any given instance. Who thinks about partials?
I repeated the analysis on a synthetic data set, consisting of sine waves at different phases and amplitudes at exact harmonic frequency ratios, including a comparable white-noise background.
The resulting frequencies (Hz) and offsets (cents) are: 439.972 (0.00), 879.95 (0.01), 1319.94 (0.04), 1759.98 (0.10), 2200.05 (0.15), 2640.05 (0.15), 3080.04 (0.13)
The error in the frequency analysis itself seems to be about +/-0.03 Hz.
I tried computing the frequencies from weighted averages (rather than interpolation) as well, which brings the accuracy of the FFT frequency estimate to +/-0.001 Hz. The partials of the bowed sample are now at: 440.1886 (0.000), 880.459 (0.161), 1320.356 (0.274), 1760.786 (0.030), 2200.95 (0.005), 2640.486 (0.423), 3081.561 (0.135).
Amazing that they can get all this processing done in a 15 euro clip-on tuner.
Lunch break is over, I'll be back tomorrow.
That beeing said, you estimate the error to 0.03Hz in your generated dataset while I see plenty of data with bigger errors.
Does not take much away from what you conclude, but you may want to CALCULATE the error correctly one time with all error causes included (also think to utilize coupling, which of course makes it much more complex).
From a little short test I can easily change the g string .1Hz by bowing pressure within the range of playing pressure. Easy to force >1Hz when I want to.
I know I might be a bit hard but I have to deal with wrong error estimations on a daily bases made by very important scientists while I think this is the very core of science. If I have a result without exact knowledge how accurate it is, it is not worth more than having the fealing it is like that. That is what shows the difference between research and scientific research and if done correctly there would be no need to discuss anymore if science is right or not! Not in a single case.
The harmonic modes of a bowed string are caused by the sustained effect. We might call these the "elastic" modes.
The frequency of the elastic modes are integer multiples of a base frequency which is a complicated function of the string physical properties and string tension as well as bow speed and pressure.
The enharmonic modes you are observing with a plucked string is due to the transient effect. We might call these the "damping" modes.
Roughly speaking, the frequencies of the damping modes are a function of a difference between a fixed damping frequency and the frequency of the elastic modes.
If the damping of the strings is relatively small, like a piano, the elastic mode frequencies are much higher than the damping frequency, so one would not expect much enharmonic variation among the modes.
For a bowed string, the enharmonic damping modes pale in comparison to the elastic modes.
Is this not a very oversimplified model? The violin consists of more than ten different, complexly damped and coupled oscillators beeing relevant if you try such a model. And even this is a very strong simplification. Thats why those models are usually not choosen to descripe a complex resonator.
You are confusing the physics of a damped, vibrating string with the physics of the entire violin. We are only talking about the string here.
The end conditions of the string at the nut and the bridge cause all the string modes to be phase locked with one-another and, in a well made and setup violin, mostly independent of what the rest of the violin is doing.
The rest of the violin will be driven to vibrate at the frequencies of the string modes.
The amplitude of those vibrations at each of the string modes will vary dramatically from violin to violin depending, as you suggest, on a very complicated set of interactions.
The phases of those vibrational modes will also vary, unlike the string itself.
Thus, the make of the violin will color the sound of the pure string, but will basically reflect the frequencies of the string itself: enharmonic if plucked or harmonic if bowed.
Nothing to do with inharmonic strings, though...
There is way more coupling going on in the violin, between the strings, even the bridge has to be looked at it as an own oscillator and the bow is very important for the string oscilattion too (soundpost propably not, it is pretty much just coupling top and back and not oscillating on its own). Taking anything out of the system and look at it alone will not descripe its behaviour in the system. Its always a two way track.
anharmonic?
enharmonic?
inharmonic"
unharmonic?
????
E.g. is a point a line of zero length, a circle of zero diameter etc etc.
"Let the body resonances be very small compared to the string vibrations" - dsicuss....
Sure it is important to simplify systems, but the violin is not to be descriped in this second semester mechanical approach.
Before neglecting parts you have to look at the influence and in this case its a big one (also for well setup violins, look at vibrating the open g string on d for example).
If you want to simplify it this way however you get comperable simple DE that shows predictable non chaotic behaviour. That is just a very academic discussion.
Just because two things are coupled in some way does not mean that coupling is significant to the operation of both things.
We probably have exceeded the level of technical detail the members are willing to consume. If you are truly interested in a scientific discussion of the physics of energy transfer between the strings and the violin body, feel free to drop me a personal note and we can take that discussion offline.
If somehow your stopped notes are based on open string pitches instead of the sound you want to make, vibrato further obscures them. Also, this is assuming you have truly inhuman intonation.
(I tune pianos.)
The hypothesis was: all partials are phase-locked on a bowed string. If the fundamental frequency wanders a bit, the partials will follow. The frequencies that I get from the FFT are the average frequencies of each partial over the time interval that I sampled. The sources of measurement errors are: (1) background noise and (2) the limited time window of the sample. It could very well be that the fundamental frequency wandered between 439.7 and 440.7 Hz (and the second partial between 879.4 and 881.4 Hz, and so on). The averages would still be 440.2 and 880.4 Hz, with an near-exact 1:2 ratio.
(I do have a Ph.D. degree in molecular spectroscopy and have had jobs in physics research and engineering for 15 years after, so I think I do know how to analyze my data)
Scott,
Re "I don't know what that means in practical term. We tune intervals on the violin as purely as we can in any given instance. Who thinks about partials?" -- I mean: if a violin string did have anharmonic partials and you tried to tune a violin in fifths, the lower string could have partials at 100, 201, 304 Hz and the higher one at 150, 301.5, 456 Hz. If you tune the fundamental at an exact 2:3 ratio, there will be an ugly beating between the 304 and 301.5 Hz partials. Alternatively, you could tune the second string a bit higher to partials 151.2, 304, 459.6 Hz, which will probably sound more pleasing.
Andrew, re anharmonic/inharmonic:
So, the anharmonicity of a string causes inharmonicity in the partials. I wasn't aware of this subtle distinction, but the latter Wikipedia article actually reads: "However, this inharmonicity disappears when the strings are bowed, because the bow's stick-slip action is periodic, so it drives all of the resonances of the string at exactly harmonic ratios, even if it has to drive them slightly off their natural frequency."
Han, while you know how to analyze data, you don't seem to understand how to explain to readers on this forum what you're talking about or why it should matter.
We simply tune our violins so they sound acceptable. We don't use numbers. We only use our ears.
I now understand better why some things don't work as planned.
This discussion has been archived and is no longer accepting responses.
Violinist.com is made possible by...
Dimitri Musafia, Master Maker of Violin and Viola Cases
Johnson String Instrument/Carriage House Violins
Subscribe
Laurie's Books
Discover the best of Violinist.com in these collections of editor Laurie Niles' exclusive interviews.
Violinist.com Interviews Volume 1, with introduction by Hilary Hahn
Violinist.com Interviews Volume 2, with introduction by Rachel Barton Pine