There are many different ways to tune a piano, but the end goal should always be the same: to tune the piano so that it sounds as good as it can. One way to describe tuning methods is to consider the way that the piano technician decides to tune each note. Technicians employ one or both of the following approaches: (1) tuning aurally, that is, purely by listening, and (2) using an electronic tuning device, generally referred to as an ETD. I think it's fair to conclude that today, most technicians rely mainly on ETDs to tune pianos. I come to that conclusion after having communicated with a wide variety of piano technicians. Let's talk about these two approaches to tuning a piano.
Aural tuning, as you might expect, is the original way that pianos were tuned, and was the only way until the invention and refinement of ETDs in the latter half of the twentieth century. Moreover, aural tuning fundamentals are the bedrock upon which all ETDs are programmed.
I spent a year at North Bennet Street School, in Boston, learning basic piano technology. The lion's share of my time, and the school's emphasis, was on learning and developing aural tuning. Learning aural tuning was an enormous challenge for me. It was much more difficult than I had anticipated! About a month into the school year at North Bennet, I realized that I would need to spend time after school most days, and lots of time most weekends, practicing aural tuning for many hours to learn the skills. I'm proud to say that at the end of the school year at North Bennet, I passed the exams developed by the Piano Technicians Guild, and I earned the designation of Registered Piano Technician.
Aural tuning involves an understanding of some of the physics involved in the production of sound from vibrating strings, and an understanding of how different pitches combine to form different musical intervals, such as thirds, fourths, fifths, sixths, sevenths, octaves, tenths, twelfths, and even larger intervals. As aural tuners, we make use of all of these different intervals in tuning a piano. We start by tuning the A above middle C to an external pitch source. Perhaps that external pitch source is a tuning fork, or a tone generator from our phones, or some other way of getting an external reference for that initial A. The most commonly used "standard" for that A above middle C is 440 Hertz, which means that the sound wave generated by the string should vibrate at 440 cycles per second.
Once we have that A where we want it, we generally then tune the A one octave below, and from there we proceed to tune something we call the "temperament." The temperament is a series of 13 consecutive notes on the piano keyboard, generally right in the middle of the piano keyboard. Most technicians, including myself, tune a temperament from F3 to F4. We strive to get that set of 13 notes as accurate as possible, because that temperament octave then becomes the template for all of the notes on the piano, both above the temperament octave and below it. Any errors that are present in the temperament octave would be repeated in the rest of the piano keyboard, because the temperament is the "master plan" for the overall tuning.
There are many, many building blocks to learn to create a good temperament, and then to use that temperament to tune the rest of the piano. I won't go into the details here. I'll just briefly describe one important characteristic of pianos that makes them a challenge to tune, and that's a concept called "inharmonicity." The sound created by a piano string, like virtually all sounds that we encounter, is actually a series of different tones that are all combined in our listening experience. When we play A4 on the piano (the A above middle C), we hear A4, but we also hear A5 (the A one octave above A4), and we also hear E6 (which is the E a fifth above A5), and A6, and C#7, and E7, and on and on. This set of overtones is referred to as the "harmonic series."
Except on a piano those overtones are not "harmonic." In fact, they are "inharmonic." Each of the overtones produced by piano strings is sharp (higher in pitch) than would be pure. For this reason, piano technicians refer to piano string overtones as "partials" (rather than "harmonics", because they're not exactly "harmonic"!). Even more fun: the higher up you go on these partials of a string, the sharper the overtones are. So, going back to the example of the A4 string: the A5 overtone is a bit sharp of what A5 should actually be. E6 is even more sharp of what E6 should actually be. A6 is even more sharp of what A6 should be, and so on.
Whether we are aware of the overtones or not, rest assured that the piano is producing them in abundance. If you listen closely to individual piano notes, you'll hear them. Take a listen to a piano bass note, and you'll hear quite a lot of things going on. Because of the piano's inharmonicity, it can be quite a challenge to tune the piano so that many different important musical intervals (such as octaves, fifths, and fourths) sound good together.
For reasons that are best left to another day, it's impossible to "perfectly" tune a piano, because tuning involves compromises. If I make one interval sound "perfect," then another interval will sound a bit less perfect. Play any interval of a major third on the piano (for example, C4 together with E4), and listen to the "beating" of those two notes. It sure doesn't sound "perfect," does it? But beating thirds is one of the compromises we live with in order to have octaves and fifths that sound good. We've generally decided that octaves and fifths are more important than other intervals, at least in terms of tuning "purity."
Now, how does an ETD work? Electronic tuning devices "listen" to a sample of the piano's notes to determine the levels of inharmonicity (how sharp those partials are in relation to the fundamental tone of each note), and they then calculate the pitch of each note in the piano, again with the goal of making everything sound good. A skilled piano technician knows the ins and outs of the ETD and knows how to adjust the ETD's settings to get the best possible tuning for each piano. But at the end of the day, ETDs are programmed by humans who (we hope!) understand how pianos work, and how we perceive sound, and how best to set the pitch of each note so that the piano sounds good when notes are played together.
I enjoy tuning pianos using both aural techniques and an ETD. I am grateful to North Bennet Street School for giving me the hard-won skills to aurally tune a piano. Whenever I choose to use an ETD, I use it with the knowledge of how the device was programmed, and I know how to adjust the ETD to use it to make the piano sound good. I usually find that I enjoy tuning more when I do it aurally. Aural tuning is a fascinating puzzle, and it's great fun to put the puzzle pieces together and make a piano sound great. But either way - using aural techniques, using the ETD, or using them both in combination, the goal is the same - to make the piano sound good when many important musical intervals are played together in various combinations.