The Well-Tempered Tune

Devoted to perfecting the sound of pianos and harpsichords
7D2_3551The Well-Tempered Tune
Tidbits of Information (and hopefully wisdom)


The equal tempered scale is designed so that an octave is composed of 12 equally spaced intervals or semitones. An octave results from the doubling of a frequency: f2 = f1 * 2 (e.g. A3 is at 220 Hz, and A4 is at 220 * 2 or 440 Hz). To determine the size of a semitone, we need to determine the factor that we need to multiply a frequency by to increase it one semitone such that if we were we were to continue this multiplication 12 times we will double the original frequency. Note that this is not given by 1/12 the difference between the two limits. That is, A#3 is not at 220 + 1/12 of 440-220. In other words, it is important to realize that the scale is not linear.

The size of a semitone is 2 raised to the power (1/12), which corresponds to a change in frequency by the 12th root of two. Multiplying this factor by itself 12 times equals 2. After making 12 steps by this amount we double the frequency. The 12th root of 2 is 1.05946. You can verify for yourself that if you multiply 1.05946 by itself 12 times, you obtain 2.

The cent is defined such that there are 1200 cents in an octave, or 100 cents in a semitone interval. Therefore, if we increase by 1200 cents, we double the frequency, or

b/a = 2^(n/1200)

where n is the size of the interval in cents, and a and b are the lower and upper frequencies, respectively. Taking the log to base two, we can obtain the interval in cents between two frequencies using:

n = 1200 Log2 (b/a)

Or to calculate a frequency given the size of an interval in cents:

b = a 2^(n/1200)

Tuning Procedure

An excellent tuning can be obtained using the Veritune software without understanding how it works. However, a basic understanding of stretch and inharmonicity can be useful if one wants to move beyond the basic styles provided. In the beginning one can use the standard stretch (style) files provided in the software along with equal temperament.

Start a new tune with the Verituner in Coarse mode. Be careful to only allow the Verituner to "hear" single strings. Determine the inharmonicity of A4 using the center single string (muting the two outer strings with rubber mutes) by playing A4 until the I is filled. Do not tune any string while the I is being filled. Tune the remaining strings on A4 individually to the Verituner. Similarly, measure the inharmonicity of A3 using the center string. Fill the I and then tune that string and the other two strings individually. Move up the scale in a similar manner from A#3 to C8, filling the I for each note using the most stable single string for each, and then down from A3 to A0.

Switch to FIne mode. This locks the targets (notice the lock close on the display). Check the tuning of each note/string and touch up as needed. Leave the inharmonicity turned on to allow the program to update the dataset if better measurements are obtained. That is, let the Verituner listen and learn while you tune. Turn off the Verituner if you are going to play intervals or unisons.

When returning to the piano at a later date, pull up the most recent tuning file for that piano, and do a recalculation to apply the most recent inharmonicity data to the target calculation (this is assuming that the same style and temperament are going to be used).

Reproducibility - Checking your Tuning Technique

In order for an electronic tuning device to be useful, it must provide an excellent tune that is reproducible. The quality of the tune obtained by the Veritune software is outstanding, and it is highly reproducible. The latter can be demonstrated by performing repetitive complete tunings, starting each time with a fresh tuning file with identical settings (stretch and temperament).

Shown below are the results of two tunings of a Steinway B, the second done 2 weeks after the first. The SK stretch (given below) with equal temperament (ET) was used for both, and the tuning procedure was as described above. Both tunings are complete tunings, with filling of the I's for every note on the piano in Coarse mode, then touching up the tuning as needed in Fine mode with Inharmonicity turned on. Upon reloading the tuning files to measure the difference, a recalculation was allowed. The results for the temperament octave (A3-A4, highlighted in grey) and all the A's are shown. The positions of the lowest partial displayed in Fine mode after each tuning are given in the 3rd and 4th columns. The differences between these values are shown in the fifth column (in cents). See below for a definition of cents.

Since the software achieves a tune using a number of partials and not a single value (certainly not the fundamental), a better indication of the reproducibility of the tune is indicated by the targets, i.e. the deviation indicated by the strobe. The deviation of the targets was therefore derived from the strobe after retuning the piano using the first tuning file (i.e. tuning to set the strobe to 0 after loading the initial file with recalculating), then reloading the second tuning file (with recalculating) and checking the strobe difference for each note. It can be seen that the deviation between the two tunings is negligible, typically less than 0.5 cents.

This is also an excellent way to check your tuning technique.
Stacks Image 15
(High resolution image of the above table can be found here.)

Inharmonicity on a Poorly Tuned Piano

The inharmonicity of a string depends slightly on its pitch. Thus if a piano is far out of tune at the start, the inharmonicity is not going to be accurate. One must therefore proceed with a coarse tune, but when finished discard the file and start over. This should place the strings close to the final pitch so that the inharmonicities are accurate the second time through.

Steve Brady’s Forearm Smash

How to make sure that you have a “bullet-proof” tuning for a concert performance: Steve Brady’s Forearm Smash