Why You Can Never Tune a Piano

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…

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.

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.

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

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.

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

Thanks a lot Bro. Easy expla­na­tion. God bless you…