In several places in the articles on various organs, I have made, or will make reference to the equally tempered scale. Exactly what is this?
There is much on the Internet about the many different musical temperaments that have been in use for centuries and of their advantages and shortcomings. A temperament is a tuning system by which the pitches or notes of the scale are tuned so that polyphonic music is possible in more than one key. You see, if you tune all the notes of the scale absolutely accurately so that you can play a song perfectly in the key of C, for example, it would not work if you want to play the same song in F. You'd need to create another scale and tune it perfectly for F. Obviously, if you have a keyboard instrument with a separate scale for each key, it would become pretty complicated and unwieldy. So, even as early as many centuries ago, musicians and instrument makers began to experiment with various ways of tuning so that things would sound good in several different keys and yet the instruments would not get too complicated.
The ultimate result of this is our modern equally-tempered musical scale. It takes the twelve tones of a complete musical scale and adds a slight amount of deliberate error to each pitch so that the error is spread out evenly across all twelve notes and no one error is too large. Then you can play a song in any key you choose and it will still sound good.
Modern pianos and organs would not be possible unless some type of temperament for tuning was available. The equally-tempered scale is not the only temperament available, but it's the best for keyboard instruments. Other temperaments were in use earlier which attempted to distribute tuning errors differently, making several keys sound really good and leaving certain intervals in other keys that were way off and were, interestingly enough, referred to as "wolves" probably because the glaring out-of-tune sounds that resulted from playing these so-called wolf intervals suggested the howling of a wolf.
The equally tempered scale takes any particular note's frequency and multiplies it by the twelfth root of two, which then gives the frequency for the next higher note on the scale. In modern music, It is essentially agreed that the pitch of Middle A will be 440 Hz. I understand that this was largely brought about by the J.C. Deagan company in the early 20th century as they were the leading manufacturer in the USA of tuned steel bars and tuning forks and other related items. Regardless of exactly who was responsible, we need to have a pitch standard if we're going to have more than one instrument playing simultaneously. A band or for that matter a symphony orchestra would not be possible unless everybody agreed on a standard pitch and everybody's As and Cs and F#s were all the same.
Anyhow, if we take the note Middle A which by standard convention has a frequency of 440 Hz and we multiply that by the 12th root of two which is 1.05946, we will get a frequency of 466.16376151. That's many more decimal places than we really need and is of academic interest only. For all practical purposes we can round that off to two decimal places and say that the frequency is 466.16 Hz. So, if Middle A is 440, then Middle A♯ (or B♭) will be 466.16 Hz. Even if we rounded it off to the nearest whole number or 466 Hz, it would still be sufficiently accurate to sound right. Modern electronic tuning devices are usually accurate only to two decimal places, and theoretically perfectly accurate tuning is a physical impossibility anyhow. If in the mid-range of an instrument, the tuning is within ± 0.5 or a half cycle per second, it will be very good indeed.
With the exception of octaves, there are no other exact pitch intervals in the equally tempered scale. Octavely related pitches have a 1:2 frequency ratio. True fifth-related pitches have a 1:1.5 ratio. If we were to play Middle A and the next higher E, for example, that musical interval is known as a fifth. It should have an exact 1:1.5 frequency ratio, so that if A is 440 Hz, then E should be (1.5)(440) = 660 Hz.
If, however, we are working with an equally tempered scale, A will of course be 440Hz, but E will be 659.26 Hz. The difference in pitch between the true E, and the equally tempered scale E in this case is only 0.74 Hz which is actually a very small error and is not really noticeable. But that slight discrepancy still lets that E work well with A, and it also works with any other key that we might wish to sound it with; thus the equally tempered scale lets us play in any key and still sound good. We trade off absolute perfection in any particular key for acceptable results in all keys and we also keep our keyboard instruments manageable, easy to play and affordable.
If we consider the normal musical scale and count the black as well as the white keys, we have 12 different tones or pitches in an octave, such as C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. The next note of course would be the next C which has exactly twice the frequency of the first C. Therefore, the way to divide things evenly so that each successive pitch of the twelve is higher by the same ratio is to take the 12th root of two and use that number as the constant factor by which to multiply the frequency of any particular note or tone to get the next higher one.