Factoids about tuning

December 15, 2021, 9:17 PM · Not many people think about what they're listening to when they tune their instruments. Violinists tend to be pretty good (eventually) at tuning fourths, fifths, and octaves.

But what are we listening to when tuning a violin? Maybe not what you think.

If we tune, for example, our open D and A strings for a perfect fifth, are we tuning the D4 and the A4? Well, not exactly. When we listen to an interval, we're not really listening to the D and the A, but rather partials on the two strings that happen to produce a higher but exact same pitch. These are called "coincident partials." The D string produces a set of partials--the harmonic series--and so does the A. This is because strings naturally subdivide themselves into smaller and smaller vibrating units. Some on each string should match.

In the case of D4 and A4, both strings produce a higher A5 which is an octave and a fifth up from D and an octave above A. These are the 3rd partial of D and the 2nd partial of A (we say we are tuning a "3:2 fifth").

These partials should match, and if it doesn't, we hear a wavering beat. It's not the only coinciding partial pair--the D and A strings also produce a another A an octave above that, which is 2 octaves and a fifth above the D and 2 octaves above the A. In that case, the 6th and 4th partials (a 6:4 partial pair).

Octave, likewise, aren't tuned by listening and comparing the lowest tones. On the violin, at least in the lower registers, not much of the fundamental--the note for which the string is named-- is audible. For example, the G string produces very little actual G3. Our brain just thinks it does. It's the same on many other instruments. Our brains are fooled into perceiving fundamental tone that is very weak, or practically non-existent.

We tune octaves to higher coinciding partials, such as the 6th partial of the low note and the 3rd partial of the higher note. Again, there are other matching higher partial pairs, but which are used for tuning may vary depending on the particular notes in question. On the piano, tuners must chose different partial pairs depending on the range, whether bass, middle, or high treble. And length of piano.

You ask, "why can't we just match the D and the A? Or a low G3 and a high G4?"
The reason is that the beating sound we use to match would be too fast to perceive. We need to match two exact pitches that are very close together. We can perceive and judge a max of about 9 or 10 beats per second. Any faster and it's just a buzz.

Since you asked, the fourth is also tuned by matching higher partials: the 4th partial of lower note must match the 3rd partial of the higher note.

A note about terminology: partials and overtones (and harmonics) are the same phenomena.
The difference is just the numbering system. When we number overtones, the lowest tone is the fundamental, and the next highest tone is 1, then 2, etc.

However, when we call the tones "partials," we call the lowest tone the 1st partial, then the 2nd, etc. So, the two systems just differ in numbering by one number.

On the violin, many levels of coincident partials should, theoretically, line up. If you get one pair, then other pairs above it should also be in tune, as long as the strings aren't false.

On the piano, however, we have a different situation: the higher the partials, the sharper they are relative to the fundamental. We can only pick one pair of partials to be in tune. The others must be out of tune. How does the tuner decide? We generally use whatever is the most prominent pair.

Replies (14)

December 15, 2021, 9:33 PM · Thank you for explaining so clearly how the sweet ring of 5ths in violin tuning is really about the unison of some shared overtones!
December 15, 2021, 9:59 PM · Interesting. Do these partials have an effect on timbre, or is that another phenomenon? I guess I'm curious if when I'm playing, I've learned to associate this with another stimulus, which is what I then perceive when I'm playing to keep me on-track? It seems like I can listen for the beating when I really focus and have some time to perceive, but when I'm playing, I just kind of "know" that a doublestop sounds in-tune, and I don't seem to have the time and mental bandwidth to listen for beating or not, so my perception seems to hook onto a resonance (which perhaps I'm incorrectly using interchangeably with timbre), so in learning to play in tune, am I coupling the phenomenon of hearing the beating with it's particular resonance, so that I can use the resonance as the shorthand when I'm playing?

Maybe I'm overcomplicating the whole thing....

Edited: December 16, 2021, 3:10 AM · An exercise my teacher showed me was to put my first finger on the A string where I guessed the D was, put my 2nd finger on the D string where I guessed the A was and then play them double-stopped to see how close the fourth was. Then play both strings open to see how close my guesses were. Ditto for other pairs of strings.
December 16, 2021, 4:16 AM · Thank you Scott for writing that down for us, very clear and interesting. Can you (or someone else) recommend a book that gives a clear physical explanation *why* these partials are produced in addition to the fundamental? I know that is nontrivial physics, and I vaguely recall from physics classes (some time ago :-) that it has to do with eigenvalues.
Edited: December 16, 2021, 5:46 AM · Unless you can bow a string to produce a sine wave, you will produce harmonics. Fourier analysis is the starting point if you want to get technical. Eigenvalues are something very specific, and I can't remember exactly what, or how relevant, they are. They are not really necessary for a lay understanding.
Edited: December 16, 2021, 9:28 AM · Fourier's work, "The Analytical Theory of Heat," was published in 1822. Obviously, his idea that mathematical functions can be expressed as sums of sine functions of several frequencies (sometimes an infinite sum) was an earth-shaking contribution to mathematics with immediate and striking implications in physics. It seems just as obvious to me, however, that violinists likely were tuning their instruments just fine before 1822. The physics of vibrations is still fun to think about. I do think a little of this kind of knowledge can be helpful when thinking about why some notes "ring" and others don't, and for rationalizing Pythagorean (horizontal) vs. just (vertical) intonation. But in the end, we learn to shrink our major intervals a little and expand our minor ones to stay in tune with ourselves and with others.

I'm curious to learn how much pros "tighten" their viola or cello fifths to play with quartets or orchestras. If I tune my best perfect fifths on my viola, my open G and C always sound annoyingly flat to the point where I need to bring up the pitch by putting my finger on the nut.

Edited: December 16, 2021, 1:05 PM · Jean, why overtones are produced is actually not very hard to understand (hint: Think about why we can play harmonics on the violin). Every intro book on physics should give you the basics if it is any good.
Edited: December 16, 2021, 1:51 PM · My "go to" reference book on this sort of topic is Helmholtz's "On the Sensation of Tone", a title that aptly says what this 580 page encyclopedic work is all about. Part I is "On the composition of vibrations", Part II "On the interruptions of harmony" about combinational tones and beats, Part III is about scales (and a lot more), and the whole work is topped by 200+ pages of appendices that go into the underlying theory in greater detail, often mathematical.

Helmholtz's book was originally in German, but I use the English translation published by Dover.

December 16, 2021, 9:43 PM · Christian,
The relative strengths of various partials is what makes one tone (or even person's voice) differ from another. It's the defining difference in timbres.

Strings have a curious tendency to subdivide themselves, but I'm not sure why. That's way above my pay grade...

December 16, 2021, 11:32 PM · In fact I was confusing eigenvalues with eigenvectors. You don't need to worry about either. "Fourier series" might interest you, but this page doesn't mention music: -
Edited: December 17, 2021, 10:28 AM · For my money, Fournier's series of cello suites is the best.

The subdivision of the strings comes out of the lowest energy states for strings to resonate at when acted upon by a constant external force - This is the part that I can't fully close the loop on conceptually. The constraint of the string being fixed at either end drives the basic math, and the fundamental depends on the length between these fixed ends. Since all the overtones are simple divisions of the string by integers (it needs to be a division by integers in order to give standing waves and place the nodes at fixed distances along the string (I'm not trying to imply a chain of causation)), that puts a node in the middle of the string for all overtones that split the string into an even number of parts, so all those overtones reinforce each other, while divisions of prime numbers above 2 don't have any friends to play with. I think on an ideal string, you could say there is an infinite amount of overtones, all comprised of various divisions of the string length by increasing integers, but I'm not sure what the constraints of the materials and various energy losses mean for the system in practice.

The string (if we want to approximate it as an ideal (without stiffness) string) is going to resonate just according to the simple math on all its overtones, with the strongest being the fundamental. I believe it's the violin then acting as a resonating cavity that boosts various overtones, the construction of which can, I suppose, explain why the fundamental on the G string is lost - It's not because of the property of the string, but rather the construction of the violin.

Anyway, I like to talk out of turn, so someone give me a smack if what I just wrote is full of errors.


Here's kind of a nice rundown (I believe the condition described for getting a single vibrational mode going by placing a finger down at a node is actually describing a harmonic, so it wouldn't be the placing of the finger all the way down, but just lightly enough to drive that particular overtone without driving them all (putting the finger down fully is akin to changing the free string length, as if you moved the nut))

Bah, I just realized there is a big gap in my explanation. The real-world version of determining the fundamental of the string depends on the string thickness and the tension that string is under - the thickness is a manufacturer spec, and the tension is the amount of force we subject the string to when we turn our peg. The manufacturer thusly chooses string materials and string thickness so that under a given tension, the string fundamental lines up with the expected frequency. This could mean that the missing G fundamental is a function of the manufacturer not wanting to keep increasing string thickness to match that strongly, and instead matching for the first overtone (but that's a big conjecture on my part), or the string itself could match the fundamental, but again, the violin construction doesn't boost that signal. It seems unlikely that a lot of this math was done formally by string manufacturers back in the day, so they probably tried different thicknesses of cat gut, strung their violins up, decided it sounded good, and called it a day. So there may be some effect coming from the string construction as well.

December 17, 2021, 10:07 AM · The fundamental is not missing from the open G string on the violin, nor the open C string on a viola. The instruments simply do not do a very good job of amplifying these low frequencies.

Some sites display responses using a linear scale for the amplitude of the sound. This can give the impression of zero response for the lowest open strings.

But the ear does not operate linearly. It is closer to logarithmic, that is, the ear can hear sounds that are reduced by powers of 10 as opposed to 1/2, 1/3, etc.

When viewing test results using a logarithmic scale, the fundamental of the lowest open string is easily seen. But it is much smaller than the first overtone (octave) of the note.

Violin amplification really starts to kick in at around 250hz, or around the fingered B on the G string. Luckily, the first octave (first overtone) of the G and the A notes on the G string have frequencies in the powerhouse of the violin's amplification range.

A similar thing exists for a viola. Larger violas actually have a modest, but obvious, amplification mode near the D on the C string, but it does not seem to add much to the amplification.

December 17, 2021, 12:19 PM · Great info, thanks all. The Helmholz book must be what I was asking for specifically, thanks Trevor!
December 17, 2021, 6:51 PM · Carmen,
Thanks for the clarification. Same is true on most pianos--my software will usually recognize every pitch, but in the 2nd and 3rd octaves it can easily be fooled into thinking it's hearing the 3rd partial. The fundamental is really missing, but just relatively weak.

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