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# Harmonically Pure Equal Temperament

Access the infinite tones at the high end of the harmonic series by choosing a sufficiently low fundamental frequency
 (+2, -3) [vote for, against]

Just to be clear. This is precisely what the title says: Harmonically pure, as well as equally tempered. How? How is it both harmonically pure and equally tempered? Rather than simply tempering each note equally to its instrumental nearest neighbor according by the twelfth root of two (or whatever acoustically meaningless ratio modern tuning actually uses), we are going one step further, and tempering it to its *harmonic* nearest neighbor by 1/1. Every note is equally tempered toward its harmonic ideal counterpart by the same ratio AND those harmonic counterparts are all part of the same series. Thus, it is both harmonically pure AND equally tempered. The deviation from the archaic meaning of "equal temperament" is necessary. We use traditional equal temperament to get us close, and then we use real equal temperament to get us the rest of the way.

Next time your piano tuner comes over, tell her you want only harmonically pure intervals on your piano. When she protests, take her to the basement, and show her the 30-foot piano string you have strung against a dead cedar tree in the underground tunnel beneath your house.

Strike the monochord. An inaudibly low fundamental sounds out, the rich overtone bouquet fills the air, hinting at the root frequency. She thinks she knows what you're getting at. "It's not going to work" she says smugly.

You calmly walk to the other end of the chord and touch it lightly where there is a drawing of a piano keyboard on the wood. The overtones vanish from the air,and a single tone rings out. "Hear that?" you ask "That is 440 Hertz." You move to the next tone along the painted keyboard. "strike it". She does so, with a purposefully bored look on her face. "Hear that?" you ask, touching the point, trying to remain calm as you anticipate her insolent response. She wastes little time. "B Flat 466.164 Hertz. Can I tune your piano now? I have to ge..."

"WRONG! B Flat is 4663/10 Hertz !"

She reaches for her tuner, but she already knows the truth. Equal temperament, the self-reinforcing triumph of enlightenment positivism in music, the pinnacle of all temperament, the standard against which all tunings are judged, has been dethroned. She weakly offers "But the difference is too small to matter", but she already knows the problem with that logic. If there is any point to her tuning instrument having three decimal places, it is because precision matters. And the truncated, unabashedly irrational, non-repeating decimals in equal temperament can never be as precise as a rational interval.

"But why did you choose THAT harmonic? aren't there at least ten others to choose from that far into the series?" she asks. She's right, there are many other overtones in the neighborhood, but only one is closest to the equal temperament B flat, and that is 4663/10.

Every single note on the keyboard will point back to the same, very low 'A' (11/10 Hertz)

In short, this tuning compromises neither harmonic purity nor chromatic ability, while reconciling the baked in flaw that is Equal Temperament with harmony.

 — fishboner, Jan 03 2013

Well Temperament http://en.wikipedia...ll-Tempered_Clavier
A good read [csea, Jan 03 2013]

Harmonic_20Piano_20Strings [sqeaketh the wheel, Jan 04 2013]

Fishboner, for your eyes only https://fbcdn-sphot...09_1354189609_n.jpg
umm not for the faint of heart [Brian the Painter, Jan 05 2013]

 Either it is harmonically pure, or it is equally tempered. Not both.

 You have described a very distant just intonation which is pretty close to but not identical to equal temperament.

 For tuning a piano, the inharmonicity of the strings is much more important. That is to say, the partials are not true harmonics but are sharper than expected. That is what gives the piano its characteristic tone; it forces the tuner to make the higher octaves sharper than the lower octaves; and it also is what allows the use of equal temperament in the first place. Having tuned a harpsichord and experimented with different tunings, I can assure you that tuning equal temperament on an instrument with (thinner, longer) less inharmonic strings just sounds plain nasty.

Basically, that characteristic inharmonicity of piano strings means that your suggested tuning won't work. Those high harmonics you think you are tuning to are themselves out-of-tune.
 — pocmloc, Jan 03 2013

 Thanks for your thoughtful annotation. I know i should put quotes around "equal" temperament, since, as you point out, my intonation by definition cannot be guilty of the same crime as ET.

What could two tones in the harmonic series possibly out of tune with?
 — fishboner, Jan 03 2013

// the 30-foot piano string you have strung against a dead cedar tree in the underground tunnel beneath your house // gets my bun. Anyone willing to go to such effort simply to prove an esoteric point to an expert who doesn't give a flying fuck is a true 'baker.
 — Alterother, Jan 03 2013

 //What could two tones in the harmonic series possibly out of tune with?//

 Imagine your basement string. It is made of unobtanium, and is infinitessimally thin and totally flexible. The fundamental is 1.1Hz. Sound the 1st harmonic, i.e. the 2nd partial. It should sound 2.2Hz. All well and good.

 Now you have run out of unobtanium and all you have to hand is 3 inch girder. Never mind, string it up in the basement and tension it to sound a fundamental of 1.1Hz. “Clang”. My guess is that the 2nd partial (which a good acoustician would decline to call a harmonic) would be something like 3.7Hz (number pulled out of my arse to illustrate a theoretical point).

Basically the thickness and stiffness of the string makes all of the partials a bit sharper than they “should” be. On a normal musical string it might be only one or two percent, but go high enough up the scale and it becomes noticeable. Piano strings are thicker and stiffer than e.g. harpsichord strings or lute strings, so piano partials are sharper than harpsichord partials. So the piano sounds more “fuzzy” or “warm” than a harpsichord, and so you can pull the tuning off of pure and it sounds fine. i.e. equal temperament becomes a “sensible” proposition.
 — pocmloc, Jan 03 2013

I wish I could understand this. But if I could, I would not be a janitor in an asylum for the criminally uninhibited.
 — Kansan101, Jan 03 2013

 The Fundamental problem (so to speak) is that 12 fifths do not equal 5 twelfths. If you choose to keep overtones truly harmonic in one key, you will be sacrificing those of another.

Please note (also so to speak) that JS Bach is often credited with developing equal temperament, but his famed preludes and fugues claim only to be "well" tempered. I'll see if I can dig up a reference.
 — csea, Jan 03 2013

 Okay, so take say B flat 7. Equal temperament dictates that it is 3729.31. It is harmonically bounded by 3729 and (37301/10).

 When the tuner sits down to tune, does she not turn the key until her hertz meter says 3729.31 ?

 If so, could she not simply choose the nearest harmonic neighbor instead?

I'm neither married to piano nor strings, btw. I actually tried this on my synth using sine waves. Any difference would register on a marginally conscious level, but I *think* I noticed a difference with large chords. It certainly sounds different when I press all the notes at the same time. I wonder to what extent the difference from ET is obliterated by error in the headphone drivers in a process similar to what you describe.
 — fishboner, Jan 03 2013

 //JS Bach is often credited with developing equal temperament, but his famed preludes and fugues claim only to be "well" tempered. I'll see if I can dig up a reference//

 Yes. This is true. Well Temperament is an irregular temperament, though pretty chromatically able. It is also contains more harmonically pure intervals than ET.

 // If you choose to keep overtones truly harmonic in one key, you will be sacrificing those of another.//

I realize that is the problem, though I have never heard it put so succinctly. Still, I don't see how the described arrangement does not solve it.
 — fishboner, Jan 03 2013

 //Either it is harmonically pure, or it is equally tempered. Not both.//

I still think its both. Clarification added to main section.
 — fishboner, Jan 03 2013

Reading this gave me a powerful desire to be sick.
 — WcW, Jan 03 2013

Well the piano tuners I have watched don’t use electric machines, but do it all in their ears by counting the ‘beats’.
 — pocmloc, Jan 03 2013

Shirley the solution is a dynamically retuned piano such that, whenever a certain combination of notes is played (as a chord or in rapid succession) it makes minor tweaks to the tuning to achieve the best compromise.
 — MaxwellBuchanan, Jan 03 2013

I thought when she wouldn't tune your piano that you were going to hang her from the piano string in the tree. I was about to give you a bun, when I re- read it. Please hang her so I can change my vote.
 — Brian the Painter, Jan 04 2013

 Either it is "Access the infinite tones at the high end of the harmonic series by choosing a sufficiently low fundamental frequency", or it is

 // Just to be clear //

But not both.
 — phundug, Jan 04 2013

Much of this has been beaten to death in prior posts, such as Harmonic_Piano_Strings [link]
 — sqeaketh the wheel, Jan 04 2013

yes [sqeaketh the wheel] but in the past post were piano tuners being treated this badly? And was the story this well written? I should think not! I still like the part where the tuning lady got hung by a piano wire in a tree in the basement. Some Ideas are good enough to warrant a fresh beating. Although I have it on good authority (David Klaver) that this is not possible. 'nuff said cuz Daves a genius
 — Brian the Painter, Jan 05 2013

 squeaketh,

 There are really two problems with the way pianos are tuned. You and pocmoc identified inharmonicity correction, but I'm more concerned with "equal" temperament. You solved one problem, I solved the other. To me, inharmonicity seems like music's way of telling us it wants to be quieter.

 maxwellbuchanan said: //Shirley the solution is a dynamically retuned piano such that, whenever a certain combination of notes is played (as a chord or in rapid succession) it makes minor tweaks to the tuning to achieve the best compromise.// This is the ideal. When I realized that this is what violinists do, I realized what a flawed instrument the piano is. But like so much in our culture, it's loudness and playability trumps not only its own harmonic quality, but that of much of modern music.

 There was a software that tried this called Justonic. I can't find a copy anywhere. But what I never understood is how it could know what fundamental you wanted.

I think a piano with something like bass pedals which, instead of playing a bass note, simply re-tunes all tones to the fundamental corresponding to the pressed pedal would be best.
 — fishboner, Jan 05 2013

You could make an instrument with 12 pedals to select the fundamental: one pedal if the note was in the previous chord, two pedals if it isn't (one to indicate the note that is, the other the new fundamental based off that note), and a "blank" pedal to indicate a return to the initial tuning frequencies. [edit: or I coulda just read the previous anno's last paragraph]
 — FlyingToaster, Jan 05 2013

The trouble with this retuning-on-the-fly system of chromatic just harmonies is that certain chord progressions cause embarrassing pitch shift. I believe acapella choirs such as barber shop quartets sometimes get this.
 — pocmloc, Jan 05 2013

yeah, you can really paint yourself into a corner. Most (well written) pieces can be returned to the original pitching though by deciding in advance which part to tune to where, and the ones that can't usually give breathing room between phrases to repitch.
 — FlyingToaster, Jan 05 2013

 [annotate]

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