# Why You Can Never Tune a Piano

Grab a cup of coffee, put on your thinking cap, and start working through this newly-released video from Minute Physics, which explains why guitars, violins and other instruments can be tuned to a tee. But when it comes to pianos, it’s an entirely differently story, a mathematical impossibility. Pianos are slightly but necessarily out of tune.

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• Bob Dvorak says:

In addition to all of the above, the human ear is imperfect.

In theory, one should be able to start with, say, middle C (called MC from here on in) you should be able to achieve at least a start by tuning all the “C” notes first. (Most old-fashioned tuning forks used by piano technicians were either A or C — mine happens to be an A.)

As pointed out in the presentation, you hear “beat” frequencies between different pairs of notes. Since the C above MC is (in theory) exactly twice the frequency of MC, there should be no beat frequency. The octave should produce a pure, crisp, pair of notes. One can continue going up the piano by doubling the frequency each time and comparing the note pairs in each octave for purity of sound. And the same from MC down, again comparing the purity in halving the frequency each time.

And then you check your work. And you compare MC to, say, three octaves up. And though there is no beat frequency (exactly eight times the frequency) it sounds flat. I don’t understand the mechanism in the ear or the brain. But a properly tuned/tempered piano will actually introduce a slight error in each octave so that it is not exactly correct, but ever-so-slightly sharp going up each octave from bottom to top. Unfortunately I don’t remember the exact number of very-high-frequency beats from MC to that three-octaves-up C. It’s been 20+ years since I tuned any piano, and 30+ since I tuned my own. Maybe someone else on this site can help. As well as explaining why the ear/brain does this.

Another note. The 12th root of 2 (which is what physics expects) is a nice number to reproduce in an electronic tuner. It’s nigh impossible for a human to do that. One tunes by looking for those extra beats in the 4th (C to F) and 5th (C to G), which can actually be heard, and counted. As noted, you can’t do this twelve times (C-G, G-D, D-A, A-E, etc., staying withing the octave) and arrive back at C. So the piano is tempered. That pure fifth in the violin is not quite a pure fifth in the piano. How does one do it? Piano technicians have the experience to “know” exactly how a given pair or triad of notes should sound (assuming that the pair is a reasonable chord) and adjusts accordingly — without changing the octave pair, if the octave lies in the chord.

Hint: For anyone who goes to try tuning a piano — you’ll have better success in checking the tune of a given pair of notes by checking the 10th rather than the 3rd. I.e., instead of MC to ME, use MC to the E ten white notes up. Check an 11th instead of a 4th. Why? Again, I don’t know.

Acoustic (i.e., non-electronic) tuning of a piano leans toward the C scale. C, F, G, D, and Bb will sound really good. As one moves toward the keys with more sharps and/or flats, the “beats” in a chord will sound progressively more insistent. But this is what gives the “character” to a work written in a given key — a work written in F# and transposed to G for beginning players will NOT sound the same. F# will have somewhat more irregularities in its scale and chords…

• Arthur says:

Very interesting observations. I think the correct explanation for octaves sounding flat if tuned at precisely 2x the frequency of the lower note is not to do with the human ear and the brain, but with a physical phenomenon called inharmonicity. It basically means that because the higher piano strings are too thick for their length, they behave more like metal objects, rather than strings – as such their overtones are lower than ideal, giving the impression of a lower pitch. This calls for what is called stretched tuning for pianos and the ideal “stretch” varies with every different octave, note and instrument, depending on many factors. There are some very good wikipedia articles: Inharmonicty and Piano Acoustics which explain this in detail.

I can also try to suggest an explanation as to why tenths and larger intervals are easier to tune than 3rds and 4ths. If you consider the series of natural overtones you will notice that the third for example appears for the first time at two octaves and a third above the fundamental note. i.e. for C1, E3 is the pitch of the 4th overtone. As such, playing the actual key E3 against C1 will result in the note E3 beating against the 4th overtone of C1, which would be roughly at the same pitch. In contrast, playing the third C1-E1 will give beatings between E1s 3rd overtone (E3) and C1s 4th overtone (also E3) which is much weaker – overtones get weaker as rule, the further away they get from the fundamental.

• Edgar Saramago says:

I happen to be a choir conductor who started playing violin and piano before the specific study of choir conducting.
A choir sings not tempered because the human hear does not seem to know much about mathematics and does not seem to care much about it either.
As you can see in this piece, if you tune only by the mathematic rules you never come back to the beginning note, simply because nature is endless.
But we need a system that goes round, or if you prefer, a system that closes a circle. To get there, you need to consider certain notes equal to some others, what the musicians call “enharmonics”. Mathemacally they are not equal, but the sistem needs a b sharp to be equal to a c, or a f sharp to be equal to a g flat so that the circle closes.
Put it this way: the circle of fifths goes, for instance from c to g to d to a to e to b to f sharp to c sharp to g sharp to d sharp to a sharp to e sharp to b sharp. Well the circle is closed if you consider b sharp equal to c. Which means that e sharp is equal to f, a sharp equal to b flat and so on.
Why is that so?
In the seventeenth century the music theory only had seven notes, but the singers would sometimes sing diferently some notes which didn’t have a name, they were sharp or flat in relationship to what the “normal note” should be. So they called it b flat or c sharp… And the discussion flourished because another system was needed.
There have been many propositions, among which the division of the major second or a whole tone in five equal parts, which led to the thirty one tone system in which there are thirty one diferent notes within the ocatve. (I may develop this item a little further because there is a thirty one tone organ that I happen to know, on some other ocasion).
But this and other propositions were simply not practical.
Until a harpsychord builder, in the eighteenth century proposed to tune the harpsychord slightly out of tune, which would solve all the theory problems. The first reaction of the musicians was of course NO! We are trying to get a better tuning, not a worse one! But he insisted: He built a harpsychord, tuned it slightly out of tune, which he called tempered, and asked Johan Sebastian Bach to try it. And so he did.
And Bach’s reaction was: Well it is really a little out of tune but this is what I need to compose. And he wrote the very famous twenty four “Prelude and fugue for well tempered harpsychord”
Why twenty four? Well he now had twelve major and twelve minor independent tonalities to work with. Actually he could now modulate from a tonality to another, simply by considering a certain note equal to another one, or giving a certain note a diferent function, by considering it another note.
And so was the twelve note system accepted that we still use. And if you dislike the fact that it is slightly out of tune, don’t forget that it gave us the modern harmony through great classic composers like Haydnn, Mozart and Beethoven, all the romantics and so on. This system has actually made the simphony orchestra a possibility.
Shal we go back to the mathematics?
The sequence of harmonics is: n=root, 2n=octave, 3n=fifth, … 7n=minor seventh…9n=major ninth.
If the root is for instance 100Hz then the ninth wiil be 900Hz.
Take now the number 500n, which corresponds to the major third and divide by 2 and again by 2, and you have the major third two octaves lower, in the case the number 125n. From this number, the seventh harmonic will be 875n. Let’s now think in terms of notes. Imagine that 100n would be for instance the note c. Then the major third would be 500, or 250 (an octave lower, or 125 (another octave lower). So the note e, a major third above c would be 125,
Starting from 125, the note e, as the root, the sequence of harmonics will give us a seventh harmonic of 875, the note d, a minor seventh above the root. But the note d is also the ninth harmonic of c and this equals 900! So we have two different notes that we call d, one of 900 Hz and another of 875 Hz. On the piano the key for the note d is somewhere bertween 875 and 900, being slightly out of tune but able to perform the function of a ninth upon c as wel as a seventh upon e. And every musician knows that every note must be given a different intonation for a different context, except the pianist or other instrumentist of a tempered instrument that has only one possibility for each note.

So does every singer know that each note must be sung in its harmonic context and properly intonated to perform its harmonic function. This is the reason why a good choir has a specific pitch when singing “a capella” and can adapt it when singing with instruments or orchestra.

And this is also the reason why we musicians don’t care too much about mathematics unless it helps us understand the necessity of working with our sense of beauty.

• Litter Picker says:

As for the mechanism, I suppose that our hearing is biased in favour of those frequencies which have the greatest survival value. The sound of your child screaming in your cave would require that you drop your spears along with the day’s kill and run home immediately, whereas, barring natural disasters, there’s not really much of value happening in the bass prior to Paul McCartney’s picking up a Rickenbacker.

• Matt says:

Too much smashed into too small a time frame for the average Joe to comprehend. I think this screed was more about the author’s knowledge rather than the technique. Look at me type stuff. 1/10.

• Henry Basnacite says:

Though completely neglected by high school and college teachers music theory is the point of departure for an understanding of Quantum Mechanics (‘Forbidden States’). The best explanation of this phenomena rests in the monograph written by Sir James Jeans (1937) SCIENCE AND MUSIC.

https://archive.org/details/sciencemusic00jean

Especially pages: 66 – 68

The essays in this marvelous work should be a part of the math and physics curriculum at every college and university.

• Maria Swingle says:

Being a pianist I agree with our views. The natural tone and harmony of a piano can’t be changed or tuned. You simple need the perfect product to get what you want

• Dave Irving says:

Interesting, if a bit glib and slightly inaccurate. No-one who knows what they’re doing tunes a guitar using harmonics either. (The pythagorean comma will always get you.) These days I use an electronic tuner, but I can, with slightly more effort, get as good a result tuning by fretted fifths and counting the beats.

Additionally, a real vibrating string (as opposed to the ideal string implied by the video) doesn’t vibrate exactly from the bridge / nut at either end. Because it’s slightly rigid, the actual vibrating length is slightly shorter than the distance between the bridge and the nut. This vibrating length is also slightly different for each of the harmonics.

• As a piano tuner for 41 years, I find this quite amusing as the subject is too dense for mere paragraphs; suffice it to say the Chinese figured out equal tempering 5,000 years ago, the church refused to go along with defiling pure fifths, and the modern piano and its high tension causes stretching of octaves in varying degrees due to differing lengths of each instrument.

• Donn Linton says:

In physics, when we double the frequency of a note we perceive it as an octave. For A-440 the next A will be 880,the next 1760 etc. and there will be no beat. On a Peterson strobe tuner, there 12 strobe displays, one each for C, C#,D, D# etc. Each display has concentric “rings” representing that note at each octave. The tuner will respond not only the fundamental frequency, but also to any harmonics. When we tune an organ, we tune octaves till there is no beat and we have no problem. On the tuner, all the rings stand still. But on a piano each harmonic is slightly sharp of what it should be. We can see this on the Peterson tuner where, when the correct octave ring is standing still, the ring for successive octave move progressively forward. Hence our problem; we can’t tune to both the fundamental and the harmonic of the octave below at the same time.

• David A Cole says:

This is a very enjoyable discussion thread. Thank you. It brings to mind a class I took 40 years ago at UC Santa Cruz, “The Acoustics and Psycho-Acoustics of Music.” It was wonderful. Here is another discussion related (but tangential) to this piano tuning thread: Have you ever wondered how/why you can listen to a small speaker (say, in your car, or nowadays on a phone), which speaker, due to its size, cannot physically create a sound wave long enough to produce a low bass note; yet, you hear the low fundamental bass note anyway? This is the result of a psycho-acoutic phenomenon called the “Missing Fundamental.” Here is a link to a Wikipedia article on the subject: https://en.wikipedia.org/wiki/Missing_fundamental

• John says:

Where did the 9/8 come from at 2:22 in the video? Is it because each string is a different length? It seems like that’s what this whole video is all about, but it was never said explicitly.

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