Introduction

The procedures described below are for tuning equal temperament. Equal temperament does not sound well with historical instruments such as fortepiano, harpsichord etc. They were tuned with historical temperaments. Equal temperament came into the forefront with the development of the modern instrument, starting around late 19th century.  The method for tuning equal temperament was devised around that time and is called the old method below.

Towards the latter part of the 20th century more emphasis was placed on tuning thirds and sixths. The temperament octave is anchored by contiguous thirds. With this foundation various procedures were devised. The procedure presented after the old method I will call the new method.

Nomenclature

The piano keyboard consists of 7 1/4 octaves. C marks the beginning of an octave. Below the lowest C are three notes we will relegate to octave 0. The lowest note on the piano is A and is called A0, the next notes are A#0 and B0. Then comes C1, C#1, D1, D#1 etc. The next octave is C2, C#2, D2 etc. so on and so forth until C8. Just so happens C8 is the topmost note.

Beats: when two strings of different frequencies are struck together, beats are produced. It can be heard as an audible pulse.
False Beats: if only a single string is struck and it produces
an audible pulse, then it is a false beat.

Intervals: Given any two notes, if there are three notes in between them, the interval is a third. For four notes in between, it is a
fourth. For six notes in between, it is a fifth. For eight notes in between, it is a sixth. For eleven notes in between, it is an octave.

Sections of the piano: The bass section is the section that is angled and crosses over the higher notes. After the bass section then comes the tenor section which goes all the way to the break. The break is where the strings are separated by a rib in the cast iron plate. Then comes the treble section. For larger pianos the treble section also has a break in it.




The Old Method

Tune A4 to a 440
tuning fork.
Tune A3 to A4 a pure octave.
Tune E4 to A3 a narrow fifth (tune perfect then lower E4 a tad such that there is about half beat per
second).
Tune F3 to A3 such that it beats 7 beats per second.
Tune F4 to F3 a perfect octave.
Tune A#3 such that F3 A#3 is a wide fourth and A#4 F4 is a narrow fifth. A wide fourth beats about 1 beat per
second, while a narrow fifth beats at about half beat per second.
Tune C4 to F3 a narrow fifth. Test: C4 E4 third should beat 10 beats per second.
Tune G3 to C4 a wide fourth. Test: G3 E4 sixth should beat 8.9 beats per second.
Tune D4 to G3 a narrow fifth. Test: F3 D4 sixth should beat 8 beats per second.
Tune B3 to E4 a wide fourth. Test: G3 B4 third should beat 7.8 beats per second.
Tune F#3 to B3 a wide fourth. Test: F# A# third should beat 7.3 beats per second.
Tune C#4 to F#3 a narrow fifth. Test: A3 C#4 third should beat at 8.7 beats per second.
Tune G#3 to C#4 a wide fourth. Test: G#3 C4 third should beat at 8.3 beats per second.
Tune D#4 to G#3 a narrow fifth: Test: F#3 D#4 sixth should beat at 8.4 beats per second.
Beginning at F3 A3 third play the thirds progressively up the keyboard, ending at C#4 F4 third. The beat rate should go progressively faster, from 7 bps for F3 A3 to 11 bps for C#4 F4. If this checks out reasonably (rarely if ever will check out perfectly), then the temperament is done.

There are many variations to this method, mainly just the sequencing of the notes and different tests. But basically it is tuning by fourths and fifths base on the foundation third F3 A3 which beats at 7 beats per second.

The problem is the tuner has to remember all these beat rates. Most don’t and just remember the foundation third F3A3 at 7 bps. Even if the tuner gets the 7 bps perfectly it doesn’t mean it is optimum for the particular piano. The harmonics are different for different pianos. For some the optimum is slower than 7 bps while for others it is faster.

The margin of error is big tuning by fourths and fifths but most tuners will leave at that and proceed to tune the rest of the piano. Another problem of the old method is there are no indications for how wide to tune the F3 F4 octave. Most instructions just say tune slightly wide.

Having done the temperament the tuner will proceed to tune the rest of the piano. It doesn’t matter if you tune bass first or treble first. Tuning the bass is simple, just tune perfect octaves, but the lower you go the harder it is to hear perfect octaves, so it comes down to the
experience of the tuner.

Tuning the tenor section is just plain perfect octaves until about F5. Afterwards tune the octaves ever so slightly wide. How wide will depend on the experience and preference of the tuner.

The New Method

Since the temperament octave is from F3 to F4, we will start
tuning at F4 and dispense with starting from A4 with a tuning fork.
Tune F4 to your electronic device (such as chromatic tuner) or smart phone app.
Tune F3 to F4 a perfect octave.
Listen to the harmonic at C6, it should have a slow beat.
Lower F3 until the C6 harmonic come to almost a standstill.
Play the C#3F3 third and then C#3F4 tenth. C#3F4 tenth should beat ever so slightly faster than C#3F3 third. Adjust F3 till it is so. Note: C#3 need not be tuned.
Tune A3 to F3 such that it beats at about 7 beats per second (no need to be exact). Tune C#4 to A3 to about 8.7 beats per second.Which is to say F3A3 and A3C#4 is in 4 to 5 ratio, for every 4 beats of F3A3, A3C#4 will have 5 beats.
Play A3C#4 third and C#4F4 third, they should also be in 4 to 5 ratio. If not fiddle around A3 and C#4 until the contiguous thirds F3A3 A3C#4 C#4F4 are in 4 to 5 ratio. It is important to get this right because it is the foundation of the tuning.

Next the tuning sequence is as follows: D4 (up a sixth from F3), A#3 (down a third), F3# (down a third), D#4 (up a sixth), B3 (down a third), G3 (down a third), E4 (up a sixth), C4 (down a third) and G#3 (down a third).

Tune D4 to A3 a wide fourth such that F3D4 sixth is beating in between F3A3 third and A3C#4 third. Which is: F3A3 7 bps, F3D4 7.8 bps, A3C#4 8.7bps.
Tune A#3 to F3 a wide fourth such that A#3D4 third beats slightly faster than A3C#4 third.
Now we tune a note out of sequence – G#3, which is the last note in the sequence to anchor the end. Tune G#3 to C#4 a wide fourth such that G#3F4 sixth beats the same as A#3D4.
Tune F#3 to C#4 a narrow fifth such that F#3A#3 third is slightly faster than F3A3 third.
Tune D#4 to A#4 a wide fourth such that F3#D#4 sixth beats slightly faster than F3D4 sixth.
Tune B3 to F#3 a wide fourth such that B3D#4 third beats slightly faster than A#3D4 third.
Tune G3 to D4 a narrow fifth such that G3B3 third beats slightly faster than F#3A#3 third.
Tune E4 to B3 a wide fourth such that G3E4 sixth beats slightly faster than F#3D#4 sixth.
Check that G3E4 sixth is beating slower than G#3F4 sixth.
Tune C4 to G3 a wide fourth such that G#3C4 third beats slightly faster than G3B3 third.

By now all notes have been tuned. We will check for the accuracy of the temperament. If it is not accurate than we will tune backwards through the sequence in reverse.
Check that the C4E4 third beats slightly faster than B3D#4 than and slower than C#4F4 third. If this is not so retune E4 so it is right. Then retune G3 to make it right, and the next note in the reverse sequence until the mistake is found. This is the value of G#3 as the end anchor, it allow us to work backwards.

Tuning the bass. Starting with E3, tune an octave with E4. Check the fifth E3B3, it should not beat over 1 beat per second, if not lower E3 to “quiet down” the fifth. The E3E4 octave will be wide but should not be objectionable to the ear, sometimes a compromise will have to be made between the fifth and the octave. Continue with D#3 tuning the same way until all the way to A0.

Tuning the treble. Starting at F#4, tune an octave with F#3. Check C#4F#4, should be a wide fourth and check B3F#4, should be a narrow fifth. Adjust F#4 so that you have a good octave, good fourth and a good fifth. Continue tuning octaves this way until D5. Beginning with D3F#4 tenth, play
tenths until A#3D5 tenth. The beats should progress evenly from slower to faster.

Tune D#5 an octave to D#4. Play the double octave D#3D#5, and play the octave and a fifth (12th) G#3D#5. Both of them should not beat, not perfectly beatless but only a hint of beating. If the 12th beats more than the double octave than D#5 is low. If the double octave beats more than the 12th than D#5 is high. Adjust D#5 until both is almost beatless. Continue tuning this way until A5.

Check the progression of tenths A#3D5, B3D#5, C4E5 and C#4F5. From here on the tenths will be beating to fast to hear properly, so we will start using 17ths. Starting from C#3F5 play the 17ths until F3A5. Once again the beats should progress evenly.

Tune A#5 an octave to A#4. Now we have three intervals to check – the double octave, the octave plus a fifth and the double octave plus a fifth. In this case it will be A#3A#5 (double octave), D#4A#5 (octave + fifth) and D#3A#5 (double octave + fifth). Once again all three intervals should be close to beatless. Tune this way until F6. Also check the progression of the 17ths.

After F6 the fifths will be harder to hear because of the false beats that are present in this area in most pianos. So we will just be tuning octaves and use 17ths to check. After C7 just tune pure octaves.

Now all the notes are tuned, if you are strip muting then just pull in the unisons. If you tuning mute as you go then the piano is done.




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