It is frustrating to have just set up and perfectly tune and intonate a guitar to have the owner immediately start messing with the machine heads. It is equally frustrating to try to explain Equal Temperment to those people who are so stubbornly proud of their "Golden Ears". In my recording studio I insist everyone use my Peterson V-Sam Tuner, there is always someone who tunes by ear or will start messing with the tuning after we start the session. I hope Equal Temperment will become more of a topic in the future.
20 years ago I was confused by the science of tuning and stummbled on Equal Temperment in an Encyclopedia (remember those?) I didnt want to accept that music was not so exact and pure, but I got the message. I attempted to see what would happen if a plotted the "Circle Of Fifths" in a chart. I took an arbitrary value of 100 to start. multipied that by 3. I then divided by 2 to get 150 and plotted that value in its proper place in the 12 note interval I was plotting. I then multiplied 150 by 3 and then divided by 2 and plotted that value. (I would have to divide by 2 more than once to ensure my final value would end up between 100 and 200) I did this until all 12 spaces where full. I then compared the ratio of any 2 intervals. There were 2 ratios that repeated. I forgot exactly what they were, I have the whole experiment written down someplace. The most interesting part of all this was that I recognised a pattern of the 2 different ratios, It was the same pattern as the black and white keys of the piano, which is of course the Major Scale. I felt relieved that there seemed to be some order to this after all. This system is called "Just Intonation". The problem is you are stuck to play in the one key. Equal Temperment takes an "average" of these 2 ratios and makes one repeated interval. The ratio of Equal Temperment is 2 to the exponent 1/12 or 1.059463094. If you are curious you can take any number and multiply it by this ratio 12 times and you will end up with exactly an octave of your original number. A guitar tuned to 12ET with an accurate tuner is pure bliss!
VP