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I get the feeling that the actual explanation is a lot simpler than what the manuals and textbooks offer. I've been trying to make sense of it all, and now I find myself thoroughly confused.

Here goes:

  1. When you play an octave on the piano (say, a C and the C following it), it sounds perfect. It's not jarring at all. There's no conflict between the two notes. To one's ear, it is essentially the same note. If you then play the same C and any other C, it'll still sound perfect.

  2. However, when you divide each octave into 12 semi-tones, and tune it mechanically, there's going to be a problem with the fifths. Some of the fifths will be off. (I'm not sure: which fifths? Like, the tonic of the second octave and the fifth of the third octave? Or is it the fifths in the sense that you play a note and then go +5, and then again +5, and so forth?)

  3. To combat this problem (i.e. to get the octaves and fifths to agree with one another), a piano tuner lengthens the ... what, exactly? The distance between the notes in the lower half of the piano?

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    The question is a bit ambiguous and thus the answers address various aspects. One question is how equal temperament works, second question is what temperament is used for pianos, and third question: what the piano tuner does to achieve it. Feb 11 at 21:34
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This is not nearly as complicated as a lot of people would have you believe. From a lot of the written descriptions you see out there, you'd imagine that (1) every interval on a piano was stretched by some fixed percentage, and also that (2) piano tuning has to proceed through a circle of fifths, tuning by ear using beats. I believed these two claims until I actually started tuning my own piano.

The reality is actually a lot simpler. The stretching of intervals is only desirable in the very high treble and very low bass, because those are the strings that have a significant amount of inharmonicity in their overtones. Over almost the entire keyboard, you can actually do fine just by tuning to an electronic tuner.

I've seen claims that each unison should actually be slightly detuned, and that this affects the sustain of the tone. This may be true, but the practical reality for my cheap baby grand is that the backlash in the tuning pegs and the stability of each string's pitch doesn't allow such tiny adjustments to stay stable.

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  • Might it be that stretching isn't as big as an issue for you because you have a grand? Uprights have shorter (but thicker) strings, inharmonicity might be more of a problem there (this is just guess work on my side, I don't have data or knowledge on this). Feb 12 at 9:07
  • @Quantumwhisp: What I'm pretty sure is true for all pianos is that the amount of stretching is not constant, and is negligible in the middle range. I'm sure you're right in comparative terms.
    – user9480
    Feb 14 at 22:20
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Let's get some terminology straight. In equal temperament, octaves aren't merely perfect; they are "just" or "pure". "Just" and "pure" are synonyms while "perfect" has a different technical meaning in music. Nevertheless, "just", "pure", and "perfect" happen to be the same when applied to octaves. So your point #1 is correct.

When we speak of just/pure intervals, we are talking about the relationship between frequencies. Ascending octaves are always twice the frequency of the starting note. Descending octaves are always half the frequency.

When we speak of fifths, however, just/pure fifths and perfect fifths are not necessarily the same. A just/pure fifth is always higher than the starting frequency by a ratio of 3:2. So, if you start at 110 Hz, for example, the just/pure fifth is 165 Hz. 110 Hz is A2, so 165 Hz must be E3, right? In "just intonation", yes, it would be. But in equal temperament, it's about 164.813778 Hz. Either pitch would be the perfect 5th of A2, but only 165 Hz is the just/pure fifth.

Now that we have that basis, let's unpack your point #2.

However, when you divide each octave into 12 semi-tones, and tune it mechanically, there's going to be a problem with the fifths. Some of the fifths will be off.

That's correct because when you add up just/pure octaves and you separately add up just/pure fifths, they will never meet. (I asked for a proof of this on the mathematics stack exchange and got several great answers.) But it so happens that there are a few points where a stack of fifths and a stack of octaves come pretty close to one another. One such point is seven octaves and twelve fifths. Nevertheless, there is a slight difference. The twelve fifths are a bit higher. So we've come up with many systems to fudge the difference. These systems are called temperaments. Equal temperament is just one example. In this system, the same amount is taken away from all the fifths--hence it is called equal--so that the 12 fifths line up exactly with 7 octaves. Other temperaments handle this differently. Some fifths are (or are closer to) just/pure fifths while others are further apart. That is why, in other temperaments, each key has its own flavor whereas, in equal temperament, all the keys are identical when you compare them relatively.

I'm not sure: which fifths?

All of them, by the same amount. Instead of a fifth being 3:2 (a frequency times 1.5), a fifth is defined as the twelfth root of 2 to the seventh power. Since octaves are double, or a factor of 2, and an octave is comprised of 12 half-steps/semitones, we need to take the twelfth root of 2 to get an equally tempered semitone. Then, we multiply that by itself 7 times (because there are seven semitones in a fifth) to get the ratio of an equally tempered fifth. That gives us a ratio of about 1.498307 instead of 1.5. (You can do this on a calculator by raising 2 to the power of 7/12.)

To combat this problem (i.e. to get the octaves and fifths to agree with one another), a piano tuner lengthens the ... what, exactly? The distance between the notes in the lower half of the piano?

No, the fifths are all reduced as described above. You might be conflating equal temperament with stretch tuning, which is still based on equal temperament, but makes the lower notes slightly lower and the higher notes slightly higher.

If you're interested in more of the theory, I recommend reading this question. It goes into greater detail about the math, science, and the music theory. But you wanted a simpler explanation so that is what I've tried to give you.

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    One big point that's missed here: sure in most octave-based tunings the octave is theoretically just. In real instruments, particularly wide-ranged ones like the piano, inharmonicity of the strings causes the second harmonics to be wider than 2:1. A tuner would need to stretch adjacent octaves to make them sound in tune, and again a mismatch ensues between widely separated octaves, since the inharmonicity doesn't play nice.
    – obscurans
    Feb 11 at 5:56
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    obscurans Stretch tuning was given a nod in the second to last paragraph with a link to learn more about it. But it didn’t seem like the focus of the question.
    – trw
    Feb 11 at 14:58
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You can choose which fifths you want to sound "off", you just can't avoid all of them.

Say you want to support the C major scale as well as possible. You tune all your C's to a given frequency. You then tune every G to exactly 3/2 of the C below it. So far, all your C's are perfectly in tune, and all C-G fifths are perfect fifths. (The G-C intervals are also perfect fourths, because 3/2 * 4/3 == 2 exactly.)

Now for the second note of the scale: the D. To form a perfect G-D fifth, the D would have to sound at 9/4 of the C below the G below it. No problem, you tune it like that and it doesn't really sound bad. You continue.

The fifth above the D is an A. Logically it should sound at 27/8 of the C below the D below the G below it. If you continue like this, eventually you will tune an F to (hold your breath) 177,147/2048, and to form a perfect fifth with the next C, that C should sound at 531,441/4092 of the first C.

And now the trouble begins: that C is supposed to sound at 128 times the frequency of the first C, because it is the seventh octave above that C, and 2^7 = 128. But 531,441/4092 != 128. In fact, it is more like 129. In other words, the circle of fifths and the ladder of octaves don't really line up. There is no reason why they should, really; the laws of physics precede humans, and they don't change just because we've invented a neat-sounding system and would like it to be perfect.

All this means that to be able to use a scale with 7 (or even 12) tones and have all intervals sound not too far off, compromises have to be made, and most people like the result better if you distribute the compromises so that all intervals are slightly off except the octaves. It's possible, of course, to tune a keyboard instrument so that more intervals are exact for a particular key, but that means that other keys sound even worse. And keyboard instruments tend to be expensive, so you don't really want one that can do only one key well. This is the major reason why equal temperament was invented in connectino with, and is primarily used, by pianos, harps, organs etc.

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Piano tuning is a bit more complicated than you might think. Tuning an old-fashioned electronic organ, such as a Conn or Allen from the 1960's, is easier. You have to temper the fifths, but the harmonics produced by each oscillator are just about perfect. A 1000 Hz tone has harmonics at 2000, 3000, 4000 etc.

But a piano string is not an electronic oscillator. The harmonics are close but not exact multiples of the fundamental tone. Every piano is different in this way. The third harmonic of one string may clash with the fourth harmonic of another string; if tuned not to clash, it may then clash with the sixth harmonic of a different string. A poorly-made piano can be very difficult to get to sound good.

Setting the temperament is a start, but a really good tuner adjusts each string to take full advantage of the piano's quirks. Here is an interesting memoir. The author bought a piano that sounded wonderful in the showroom but just ordinary in her home. Spoiler: after a quest of many years, she found a tuner who could recreate that sound.

https://www.amazon.com/Grand-Obsession-Odyssey-Perri-Knize/dp/0743276396

Grand Obsession: A Piano Odyssey, by Perri Knize.

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As an old-timey tuner, let me give you my take on your original questions:

  1. Yes, octaves and unisons should sound 'perfect'. What this really means is that the dominant harmonics in each note (or for a unison, for each string) are so close in frequency that no 'beats' are distinguished.

  2. dividing each octave into twelve equal intervals results in the 'equal temperament' tuning scheme in common use for well over a century. This mathematical division produces intervals whose dominant harmonics cannot be at the same frequency, and this means they will have a discernable beat. This means that ALL fifths are 'off' in that they cannot be perfectly harmonically aligned. And you don't want them to be - it would sound awful in context. Every interval on the keyboard - thirds, fourths, sixths, all of them - have a beat that is predictable based on their fundamental and harmonic frequencies.

  3. You're saying this is a 'problem' to be solved, but it really isn't. What you're leaving out of the equation is the inherent inharmonicity of a piano - a construction of wood, felt, leather and dissimilar metals, stretched across a cast iron frame, with tons of tension trying to fold the piano in half. Most strings on a piano can't be made to their natural speaking length because of size constraints (bass strings would need to be 20-30 feet long), so they are designed as steel cores with copper overwrappings. This creates tones with many subharmonics that come into play when listening to simple two-note intervals.

What good tuners do is compensate for a piano's natural tendency to deviate from perfect harmonicity by stretching certain intervals, shrinking others, all very subtly, in the context of the piano itself. Without this kind of stretching, a piano that is PERFECTLY tuned to EXACT harmonic intervals actually sounds 'wrong', sometimes even terrible.

In my early tuning days (in the seventies), I was fortunate to know a local tuner who was a physicist by education and trade. He was fascinated with the physics of artfully tuning a fine piano, and actually wrote a very well-respected book on the subject ('The calculating technician', by Dave Roberts). Unfortunately I believe it's out of print now, but you can google for more information if you want to plumb the depths of compensating for inharmonicity in piano construction.

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  • Thank you ever so much for your honest and highly informative answer. I'll find that book. Since you're a practicing tuner, here's a tricky (and silly) question for you: the electric Yamaha, 88 weighted keys and all: they tell me that in order to give it proper tuning, the engineers pick a well-tuned grand piano and run with it. How true is this?
    – Ricky
    Feb 12 at 20:40
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    I've had a few very good 'keyboards' and yes, you can bet the engineers very carefully create sophisticated sound profiles that mimic the waveforms generated by acoustic grand pianos. Feb 12 at 23:30
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    I'm not allowed to comment on other answers but my own, but many of the above answers, especially the ones that speculate about what tuners do, are either completely wrong or have some very mistaken ideas. But - this topic is not really about the complexities of piano tuning. Feb 13 at 1:21
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    Welcome. I hope that once your answer has gained enough upvotes for you to be able to comment everywhere that you will point out some of these mistaken ideas.
    – phoog
    Feb 13 at 4:33
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The answers explain what the different temperaments are . If you want to know what a professional piano tuner actually does to tune a piano correctly, it gets a bit more interesting. (Personally, I'd start with a bunch of tuning forks tuned to exactly the "equal temperament" frequencies but I'm not even pretending to be able to tune a piano).

Remember that most keys on the piano activate 2 or 3 strings. These strings are detuned relative to each other by some very tiny amount because that both increases the sustain and produces a more "pleasing" sound to most listeners. Professional tuners have incredibly sensitive pitch identification, and in the final setup have not only all octaves in perfect tune but have all keys set to the pitches they "desire."

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  • "These strings are detuned relative to each other by some very tiny amount because that...increases the sustain": do you have a reference for that? Physical theory suggests the opposite.
    – phoog
    Feb 12 at 1:00
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    The three strings of a piano note (well of those notes that have three strings) are tuned to be as close in pitch as is humanly possible. In spite of that, there is a chaotic component in their vibration due to the way they influence each other: it's a "three body problem" so to speak. That's the basis for the piano sound.
    – Kaz
    Feb 12 at 19:20
  • @Kaz as far as I can tell the beating that creates the piano sound is between the overtones of different pitches, thanks to equal temperament, the most prominent example being the major third. If you play a single key on a well tuned piano there is no beating or wavering.
    – phoog
    Feb 13 at 4:37
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The way a piano tuner tunes to equal temperament is by listening to beats or the sound of the interval getting louder and softer in a steady rhythm. Playing an octave should have no beats.

When playing a fifth, a piano tuner is listening for a beat approximately every second. Piano tuners will also listen to major thirds which will have a faster rate of about 4 or 5 beats per second.

To connect the sound back to the math, if one has a fundamental at 440 hz, a just fifth above that would be 660 hz whereas an equally tempered fifth is a little flat at 659.25 hz. The result is that the fifth gets behind one oscillation every 1.3 seconds which is the interval at which you'll hear a beat. A just major third above 440 hz would be 550 hz whereas an equally tempered major third will be a little sharp at 554.36 hz. The result is that the equally tempered third will get ahead 4.36 oscillations every second which means the tuner will hear 4+ beats per second.

When you hear a piano tuner at work, they loudly play these intervals over and over, listening for beats because they can't judge if they're getting exactly 4.36 beats per second but they can judge if they made too much of an adjustment one place, causing problems in another place on the scale. If all fifths were just instead of being a little flat, octaves would be sharp. So that's why they keep listening to octaves.

Piano tuners tune the equalized temperament in the middle of the keyboard and extend this to higher and lower octaves by tuning the octave. As a result, there is not a lot of variation in the rate of the beats for the subset of notes in the middle of the keyboard tuned by listening to beats.

Equal temperament has been achieved for 400+ years by listening to beats, not by devices unavailable until recently. See https://en.wikipedia.org/wiki/Piano_tuning under the heading of Temperament.

You should be able to hear beats on a tuned piano - even an electronic one. Compare playing an octave with a fifth and listen for a slow pulse to the fifth.

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    Am I right in thinking that the required number of beats per second is proportional to the frequency, so that higher notes will beat faster than lower ones?  If I recall, piano tuners historically used tables of beat frequencies, so they could look up the right beats-per-second for each interval they needed to tune that way.  (I believe that these days, they tend to use electronic tuners instead, which are obviously much more straightforward to use.)
    – gidds
    Feb 10 at 19:47
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    @gidds you are correct. The statement "when playing a fifth, a piano tuner is listening for a beat approximately every second" is incorrect. But electronic tuners (as far as I'm aware) don't adjust for the inharmonicity of the string, so they would not result in the "stretched" tuning that is required to make the piano sound best. (As I understand it, the degree of stretching depends on the specific properties of the strings in a given piano, so it will differ for example from longer to shorter instruments.) So I suspect that the best tuners rely as little as possible on electronics.
    – phoog
    Feb 10 at 21:10
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Although this can be a complex subject, there is a simple way to conceptualize it. It's a way to tweak notes, away from the "natural" tuning based on the overtones of a single root note, in order to make a wide range of key changes possible.

  • There is a way to derive a usable scale from simple mathematical ratios between the wavelengths of notes --which is easier to visualize if we translate it into the lengths of strings. An octave is half the string length of the original note. A fifth is two-thirds the length of the starting note. These are the places a string naturally resonates, generating what is called the "overtone series." By translating twelve of these tones into a single octave, we can derive something close to our familiar twelve-tone scale. These notes tend to harmonize well together because of that resonance.

  • You can't do much in the way of key changes in this tuning however, because it is tied to a specific root note, and the different "half steps" and "whole steps" are not of equal sizes. Thus, if you modulated from C to D on a keyboard tuned this way, the song would sound entirely different. There are many different "tempering" systems to adjust for this, but the only one in common modern use is "equal" tempering. In this tuning, only the octaves are tuned to the natural mathematical ratio. All other notes are spaced out equally between them, twelve equal-sized half-steps per octave. This makes each key sound the same as each other, relatively speaking. Some notes, like the fifths, are very close to what they "should" be, others, like the thirds, are pretty far distant. But they are all at least a little bit off from their natural versions, except the octave (and none of them, except the octave, is a simple mathematical ratio).

Interestingly enough, there is some reason to believe that, although most people, particularly without a trained ear, don't consciously perceive the difference between the tempered notes and the naturally tuned ones, there is some subconscious resistance to the artificiality of the tempered sound.

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    I didn't downvote, but (1) your usage of the wod "tempered" is not right. (2) It is very complicated. (3) There's no single "just" tuning. (4) It's really important to distinguish between "just" (basically works in one tonality/modality only, but this again is a simplification), "midtone" (works almost just in couple modes), "equal" (that's what you seem to mean when you write tempered) and "tempered" (that's e.g. what Bach meant when he composed The Well-Tempered Clavier). C'mon, if it were simple, my digital organ wouldn't offer me a dozen of choices.
    – yo'
    Feb 10 at 22:51
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This question mixes two distinct concepts that happen to overlay in the case of piano tuning.

(2) is an issue of the equal temperament, which will happen to anyone and anything (musical instrument maker, singer, or some synthetic tone generator) trying to divide the octave into 12 tones of equal distance.

(3) is a property of vibrating strings (inharmonicity) that occurs when the thickness of a string is not negligibly small compared to its length.

The piano tuner needs to respect both aspects when tuning the piano.

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