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OK, so what *is* a temperament, anyway?

  1. temperament -- (n.) any method of distributing the tones of a musical octave. In most Western music the octave is divided into twelve tones, and the most common Western temperament is equal temperament (ET), in which the twelve tones are evenly spaced. Unequal temperaments, which include most baroque and world music scales, result in differences in the harmonics of different keys, so music written for a given key will sound somewhat different if transposed to another key.
  2. Physics 101 -- a refresher:
    About sound -- when something vibrates in a fluid (air, water, whatever) it makes the fluid around it vibrate (pushes/pulls it) in waves that cycle (crest/trough) at the same rate, (ok, maybe a little slower, not relevant to this discussion). This rate is expressed in cycles per second or Hertz (abbreviated Hz) -- same thing. If this rate falls within a certain range of frequencies ('audible range' = approx. 20 to 20,000 Hz) and is sufficiently energetic ('loud') these vibrations can perceived by the human ear, if there is one around. We call that sound.
  3. Some useful behaviours of pieces of string -- when a piece of string (or a reed or a column of air, for that matter) is made to vibrate -- by shaking, plucking, bowing or blowing, whatever -- it vibrates at a rate that is a function of its length. Yes, there are other factors, such as tension, diameter, material, but we are only concerned with the length just now. Generally speaking, if you shake (or pluck or bow, etc.) a long string it will vibrate more slowly ('lower pitch') than a shorter one ('higher pitch') and the great beauty of it is that a string *exactly* half as long will vibrate *exactly* twice as fast, all other things being equal*. We call that an octave. This, BTW, is a good science project.
  4. So, making music with octaves is pretty limited, but we can use lengths of string in between, of course. On harpsichords, pianos, xylophones, harmonicas and such the individual strings or whatever are cut to the correct length, so you have a s many notes on the instrument as you have strings. On violins, guitars, clavichords, trombones and such the player can change the sounding length of the string, which accomplishes the same thing, and allows you to get lots of notes with a few strings (or whatever).
  5. Some interesting behaviours of waves -- if two waves are occuring at the same time they will behave as if they were added together.
  6. Some interesting behaviours of the human brain -- If two sounds are sufficiently close, we tend to hear them as the same note. If two sounds are slightly different we hear tend to perceive that as unpleasant. If two sounds are in frequencies that are simply related to one another -- doubles, halves, thirds, quarters, etc. -- we find that pleasant, it is called harmony.
  7. If you want to know about historic temperaments for harpsichords or most any European instruments, Da Man is Dr. Bradley Lehman of University of Michigan. He's done extensive research on historic temperaments; he discovered, or perhaps deduced is the correct word, the Bach (Lehman) Wohltemperirte temperament, full story here http://www.larips.com/ (it is fascinating). His website is a treasure trove of temperament discussion, theory and practice -- well worth reading whether you want to get into the weeds or just want to know how he figured it out.

* There are a *lot* of 'other things', but we will ignore them for now.







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