6. Tuning Procedures and Temperament

Tuning consists of “setting the bearings” in an octave near middle C, and then “copying” this octave to all the other keys. You will need various harmonic tunings to set the bearings and only the middle string of each note in the “bearings octave” is initially tuned. The “copying” is performed by tuning in octaves. Once one string of each note is tuned in this way, the remaining string(s) of each note are tuned in unison.

In setting the bearings, we must choose which temperament to use. As explained in section 2 above, most pianos today are tuned to Equal temperament (ET), but the historical temperaments may be showing signs of gaining popularity, especially the Well temperaments (WT). Therefore, I have chosen ET and one WT, Kirnberger II (K-II), for this chapter. K-II is one of the easiest temperaments to tune; therefore, we will visit that first. Most people who are unfamiliar with the different temperaments may not notice any difference at first between ET and K-II; they will both sound terrific compared to a piano out of tune. Most pianists, on the other hand, should hear a distinct difference and be able to form an opinion or preference if certain pieces of music are played and the differences are pointed out to them. The easiest way to listen to the differences for the uninitiated is to use an electronic piano that has all these temperaments built into it, and to play the same piece, using each temperament. For an easy test piece, try Beethoven’s Moonlight Sonata, 1st movement; for a more difficult piece, try the 3rd movement of his Waldstein Sonata. Also, try some of your favorite Chopin pieces. My suggestion is for a beginner to learn K-II first so that you can get started without too much difficulty, and then learn ET when you can tackle more difficult stuff. One drawback of this scheme is that you may like K-II so much over ET that you may never decide to learn ET. Once you get used to K-II, ET will sound a little lacking, or “muddy”. However, you cannot really be considered a tuner unless you can tune ET. Also, there are many WTs that you may want to look into, that are superior to K-II in several respects (see c. Pythagorean, Equal, Meantone, and “Well” Temperaments).

You can start tuning ET anywhere, but most tuners use the A-440 fork to start, because orchestras generally tune to A-440. The objective in K-II is to have C major and as many “nearby” scales as possible to be just (have perfect chords), so the tuning is started from middle C (C4 = 261.6Hz, but most tuners will use a C-523.3 tuning fork to tune C4 partly because the higher harmonic gives twice the accuracy). Now, the A that results from K-II tuned from the correct C does not result in A-440. Therefore, you will need two tuning forks: A for ET and C for K-II. Alternatively, you can just start with only a C fork and start tuning ET from C. Having two tuning forks is an advantage because whether you start from C or from A, you can check your ET when you get to the other note.

a. Tuning the Piano to the Tuning Fork

One of the most difficult steps in the tuning process is tuning the piano to the tuning fork. This difficulty arises from two causes:

  1. The tuning fork has a different (usually shorter) sustain than the piano so that the fork dies off before you can make an accurate comparison.

  2. The fork puts out a pure sine wave, without the loud harmonics of the piano strings.

Therefore, you cannot use beats with higher harmonics to increase the accuracy of the tuning as you can with two piano strings. One advantage of electronic tuners is that they can be programmed to provide square wave reference tones that contain large numbers of high harmonics. These high harmonics (they create those sharp corners of square waves – you will need to know polynomial math or Fourier transforms to understand this) are useful for increasing the tuning accuracy. We must therefore solve these two problems in order to tune the piano accurately to the tuning fork.

Both difficulties can be solved if we can use the piano as the tuning fork and make this transfer from fork to piano using some high piano harmonic. To accomplish such a transfer, find any note within the muted notes that makes loud beats with the fork. If you can’t find any, use the note a half tone down or up; for example, for tuning fork A, use Ab or A# on the piano. If these beat frequencies are a bit too high, try these same notes an octave lower. Now tune the A on the piano so it makes the same frequency beats with these reference notes (Ab, A#, or any other note you had picked). The best way to hear the tuning fork is to press it against your ear lobe, as described above, 3. Tuning Tools, or to press it against any large, hard, flat surface.

b. Kirnberger II

  1. Mute all side strings from F3 to F4.

  2. Tune C4 (middle C) to the fork.

  3. Then use C4 to tune G3 (4th), E4 (3rd), F3 (5th), and F4 (4th), and G3 to tune D4 (5th) and B3 (3rd).

  4. Then use B3 to tune F#3 (4th),

  5. F#3 to tune Db4 (5th),

  6. F3 to tune Bb3 (4th),

  7. Bb3 to tune Eb4 (4th) and

  8. Eb4 to tune Ab3 (5th).

  9. All tunings up to here are just. Now tune A3 such that the F3-A3 and A3-D4 beat frequencies are the same.

You are done with setting the bearings!

Now tune up in just octaves to the highest notes, then tune down to the lowest notes, using the bearings octave as reference. In all these tunings, tune just one new octave string while muting the others, then tune the remaining one or two strings in unison to the newly tuned string.

This is one time you might break the “tune one string against one” rule. If your reference note is a (tuned) 3-string note, use it as it is. This will test the quality of your tuning. If you have a hard time using it to tune a new single string, then your unison tuning of the reference note may not have been sufficiently accurate and you should go back and clean it up. Of course, if after considerable effort, you cannot tune 3 against 1, you will have no choice but to mute two of the three in order to advance. When all the treble and bass notes are done, the only un-tuned strings left are the ones you muted for setting the bearings. Tune these in unison to their center strings, starting with the lowest note, by pulling the felt off one loop at a time.

c. Equal Temperament (ET)

I present here the simplest ET tuning scheme. More accurate algorithms can be found in the literature (Reblitz, Jorgensen). No self-respecting professional tuner would use this scheme; however, when you get good at it, you can produce a useable ET. For the beginner, the more complete and precise schemes will not necessarily give better results. With those complex methods, a beginner can quickly get confused without any idea of what he did wrong. With the method shown here, you can quickly develop the ability to find out what you did wrong.

Mute the side strings from G3 to C#5. Tune A4 to the A-440 fork. Tune A3 to A4. Then tune A3-E4 in a contracted 5th; by tuning E4 slightly flat until you hear a beat of about 1 Hz. The contracted 5th should beat a little under 1 Hz at the bottom of the muted range (A3) and about 1.5 Hz near the top. The beat frequencies of the 5ths should increase smoothly with increasing pitch. Keep tuning up in contracted 5ths until you cannot go up any more without leaving the muted range, then tune one octave down, and repeat this up-in-5ths and down-one-octave procedure until you get to A4. For example, you started with a contracted A3-E4. Then tune a contracted E4-B4. The next 5th will take you above the highest muted note, C#4, so tune one octave down, B4-B3. All octaves are, of course, just. To get the contracted 5th, start from just and tune flat in order to increase the beat frequency to the desired value and set the pin correctly at the same time. If you had done everything perfectly, the last D4-A4 should be a contracted 5th with a beat frequency of slightly over 1 Hz without any tuning. Then, you are done. You have just done a “circle of fifths”. The miracle of the circle of fifths is that it tunes every note once, without skipping any within the A3-A4 octave!

If the final D4-A4 is not correct, you made some errors somewhere. In that case, reverse the procedure, starting from A4, going down in contracted 5ths and up in octaves, until you reach A3, where the final A3-E4 should be a contracted 5th with a beat frequency slightly under 1 Hz. For going down in 5ths, you create a contracted 5th by tuning the lower note sharp from just. However, this tuning action will not set the pin. In order to set the pin correctly, you must first go too sharp, and then decrease the beat frequency to the desired value. Therefore, going down in 5ths is more difficult than going up in 5ths.

An alternative method is to start with A and tune to C by going up in 5ths, and checking this C with a tuning fork. If your C is too sharp, your 5ths were not sufficiently contracted, and vice versa. Another variation is to tune up in 5ths from A3 a little over half way, and then tune down from A4 to the last note that you tuned coming up.

Once the bearings are set, continue as described in the Kirnberger section above.