The EQUALLY TEMPERED SCALE

Today, we have modern crystal-controlled electronic tuning devices which sound a very accurate series of pitches which we can use as standards or references to tune an instrument. Prior to the development of these devices, however, tuners would develop an equally tempered scale by ear, by sounding notes in pairs and listening for the slight amount of detuning necessary to get the desired result. Although this sounds quite formidable, instrument technicians could learn, in some cases fairly easily, what the various intervals should sound like and once having established an accurate C or an accurate A, could then easily get the remaining eleven notes. Even today many tuners still prefer to work by ear. This is, by the way, especially useful in piano tuning, because the upper harmonics of piano strings are not always exact, whole number integral multiples of the fundamental pitch, and an accurate "by ear" tuning can actually sound better then a tuning based on an electronic standard with perfect 1:2 octaves.

The actual creation of an equally tempered scale is done by sounding notes in progressive fifths and fourths. In the equally tempered scale, all fifths are slightly narrowed so that they are all off from a true 1:1.5 ratio by 0.112%. Likewise the fourths need to be widened so that they are off from a true 3:4 ratio by .114%. A very typical procedure used by many instrument tuners calls for working in the middle octave of a piano, for example and first accurately tuning Middle C as closely as possible to a standard. If A = 440 Hz, then Middle C, to be correctly tuned must have a frequency of 261.63 Hz.

Once the tuner has accurately tuned C, he next sounds the G which is a fifth above. The easiest way to tune this fifth accurately is first to tune it as close to exact as possible, so that it does indeed have a 1:1.5 relationship to C. Then, he proceeds to lower G very slightly. If G is not exactly equal to 1.5 times the pitch of C, or in this case (1.5)(261.63) = 392.45 Hz, then a slow beat will develop between the third harmonic of middle C and the second harmonic of Middle G. With practice, you can learn to hear this beat. As soon as the tuner slightly lowers the pitch of G, he will hear the beat between the aforementioned harmonics. When Middle G is tuned correctly in equal temperament with C, this beat rate will be about 0.9 beats per second. If we take C at 261.63 and multiply it by 1.5, we will get 392.45 Hz. But the tempered scale pitch should be 392 Hz which leaves a difference of 0.45, which we must multiply by two since we're dealing with the second harmonic of G. That's where we get 0.9 beats.

Once he has done that, the tuner then goes down a fourth to Middle D. Here, he first tunes Middle D exactly to the G that he just tuned, and then proceeds to lower D until he once again hears the beat, this time between the fourth harmonic of D and the third harmonic of G. To do this, he first tunes D dead on with G so that it comes in at 294 Hz. This is a precise 3:4 ratio between D and G. Then he lowers the D slightly until it is at 293.67 Hz. This will result in a beat between D's fourth harmonic and G's third harmonic of 1.32 beats per second.

All of the stuff in the above three paragraphs is strictly academic. In the middle octave of a keyboard instrument, it works out that the beat rates are very close to 10 beats in ten seconds for fifths, and 14 beats in ten seconds for fourths. (Actually each fifth and fourth is slightly different but who cares?) Believe it or not, you can secure a very nice-sounding, accurate equally tempered scale by using this ten-fourteen rule in the middle octave. Fifths beat ten beats in ten seconds; fourths beat 14 beats in ten seconds.

On a practical matter, it is not worth the time to tune each fourth and fifth to the exact, calculated beat rates, because a piano or organ tuner has to make a living and cannot afford to spend a whole day setting a theoretically perfect temperament correct to 3 decimal places, when the normal tolerance of the instrument he is tuning is not that close to begin with! With practice, it becomes after a while, very easy to recognize ten beats in ten seconds and fourteen beats in ten seconds, and then a tuner can secure a good temperament in about ten minutes or so. The discrepancy between a theoretically perfect equal temperament and that which you obtain with the ten-fourteen rule amounts to errors around a tenth of a percent, and as I just mentioned, there is no piano or pipe organ which does not over the course of a few minutes drift by more than that amount anyhow, just from normal temperature variations, and in the case of pipe organs, normal slight variations in the air pressure in the instrument and also atmospheric pressure as well as ambient temperature.