A piano tuner will explain that when tuning they are listening to "beats" - being able to recognize and correctly judge the speed of these is one of the key skills developed in the course of learning to tune. So I though it might be useful to put something on the blog about what these "beats" actually are, in physical terms, and why listening to these is an essential part of the process when tuning. To be a tuner you don't need the ability to hear low-flying aircraft, noises that only bats or dogs can normally hear, or strange mystical powers. What you do need to do over time is to train yourself to recognize particular sounds within the large number of apparently random noises coming out of a piano.
Firstly, very few tuners have perfect (or absolute) pitch, namely the ability to recognize the pitch of a sound ("that's an F#") instantly - this is a rare gift even amongst musicians. It's very rare amongst tuners as well, and not always an advantage!
For the rest of us (and that includes me), to get a piano to pitch, it is necessary to tune the first string either to a tuning fork (the traditional way) or to an electronic tuner. Personally, I use an electronic tuner for the pitch of the first string only - everything is then tuned by ear relative to that first string. For that reason, a tuner needs to be able to recognize different musical intervals and have a good relative sense of pitch. Normally I use middle C as the reference point, but many tuners will use A instead.
The next thing is to get the two unison strings of middle C in tune with the first string.This is where the tuner needs to be able to recognize "beats" in the sound of two strings together, when one is slightly out of tune with the other. It would help at this point to explain why "beats" occur, and what actually causes these sounds in practice.
This graph represents a sound wave as a sine curve:
This is a simplified model - of course in practice the sound isn't a perfect sine wave, and the curve is simply a representation of compressions and rarefactions in the air itself - but it does help to illustrate how two sound waves combine together. This second graph shows the effect of two sound waves perfectly in tune with each other - one is coloured green and the other yellow, but you can't see the yellow one because it's covered by the green one. The red curve (of double the amplitude) is the combination of the two.
As you can see, the two sound waves just reinforce each other.
Now watch what happens if the green and yellow sound waves are at a slightly different frequency:
In this particular illustration, the yellow sound wave is at a higher frequency than the green sound wave; the ratio is 16:15 - this isn't typical of the kind of ratio you would normally be listening to as a tuner, but will do very nicely to illustrate the principle. The combination of these two sound waves gives the following result (red line):
This graph shows the same pattern - this time I have increased the frequency of the green and yellow waves and the ratio is now 51:50. You can clearly see that the red wave - the result of addition of the green and yellow - shows a clear, regular "beating" pattern which is the consequence of wave interference. This is what a tuner is listening for when attempting to put the two strings in unison.
Here's the same graph except that I have now increased the frequency of the yellow wave again, so it's now in a ratio of 52:50 to the green one. As you can see, the "beat" is now faster - double what it was previously. This illustrates the important point that as two strings get further apart in frequency, the speed of beating increases until the strings are just obviously out of tune. The other thing to note is that the speed of beating tells the tuner how far away the two strings are from each other, but not whether the string being tuned is higher or lower than the reference string - that has to be worked out by listening to the speed of beats changing as the tuning pin is moved.
Obviously, the tuner's objective in this case is to try and get rid of the "beating" sound, which should put the two strings in tune. So this is fine for getting two strings perfectly into unison with each other, but that isn't good enough if we want to tune a whole piano - we need to be able to tune other intervals as well. In practice, a tuner will put in a "scale" or temperament octave - that is tuning every note in an octave around the middle of the piano and then work outwards in octave steps from there. I'm not going to deal with exactly how the scale octave is done now because it involves an understanding of musical temperament (in particular equal temperament, which is nearly always used in piano tuning) and in practice, this process involves putting in "beats" instead of just getting rid of them. I expect to return to these subjects in some future posts, but for now I'll deal with the simpler case of tuning an octave.
The first thing to understand is that two notes an octave apart should have a frequency ratio of 2:1, so the tuner will need to be able to hear the result if two notes are in a perfect octave or slightly out. For this reason, I've set up the graph to show the resulting pattern for an 81:40 ratio - i.e nearly an octave but not quite, to see what results. Although there is some kind of regular change in the wave pattern, there is no clear variation in amplitude (loudness) as with the case of the near-unison we looked at earlier. The bottom line, in fact, is that this does not produce any clearly discernible "beat" which is of use to the tuner; how, then, is it possible to hear when a octave is in tune?
Hearing the harmonics
What saves the day here is a property of a vibrating string fixed at both ends: as well as its "fundamental frequency" (i.e. the soundwave produced when it oscillates along its entire length), it also has separate vibrations through 1/2, 1/3, 1/4, 1/5 of its length and so on. These are called upper harmonics or partials of each note, and help to give the tone of an acoustic piano its richness.
This is called the harmonic series. What this means is that if we look at a particular note (for example the C two octaves blow middle C), it is producing all the following sounds:
The fundamental frequncy: C two octaves below middle C
2nd harmonic: C one octave below middle C
3rd harmonic: G below middle C
4th harmonic: Middle C
5th harmonic: E above middle C
6th harmonic: G above middle C
7th harmonic: Doesn't quite correspond exactly, but close to Bb above middle C
8th harmonic: C above middle C
Yes! Really, all these noises are coming out of your piano when you play just the one note. In fact, the series goes on ad infinitum, but in practice the harmonics above the 8th are not really important for piano tuning and above about the 12th they are very weak indeed, to the point of being virtually undetectable. Also, a bass note will have more audible harmonics than one in the treble - the additional partials get progressively weaker the higher the note.
If you're interested in a longer explanation, there's one here:
How does this help the tuner? Well, if two notes an octave apart are being tuned, the second harmonic of the lower note should be at the same frequency as the note above, and thus produce a recognizable beat, exactly the same as a unison, if the notes are slightly out of tune; thus, the tuner is trying simply to remove the "beat" in exactly the same way as before.
So that's how an octave is tuned - for completeness, I should briefly add that when an octave is tuned on a piano, it's actually slightly wider than a theoretically perfect one. The reason for this is that the strings on a piano aren't "ideal strings", i.e. perfectly flexible; rather they have stiffness and this results in the upper harmonics being slightly sharp of where they ought to be. The technical name for this is "inharmonicity".
But that's enough for now - tying up some of the loose ends should provide topics for future posts!