On 2002-07-24 08:40, Feadan wrote:
Although I understand the concept of Just vs. Equal, I don’t get what > real > difference it would make in such a small range as 2 octaves. Anyone care to spell it out? What would the difference in cents really be between the two scales in the range of a whistle?
Cheers,
David
I don’t know if I remember this stuff all very clearly, but fwiw:
The issue is not melodic range, how many octaves up or down you play. The issue is harmonic range, that is how many keys you stray from the main or home key. Picture a circle of fifth* - the further you stray from your whistle’s key the more the difference between just and equal temprament matters. (There are many other tempraments, btw).
The way it works is that if you think about a the pitch of a note in Hz, like A=440, you can go up an octave by doubling that number, and you go down an octave by halving it: A’=880 etc. If you went up 12 fifth from a starting note, say C, you would arrive at a C again. (C → G → D → A → E → B → F# → G#/Ab → Eb → Bb → F → C) If you took that new C and divided the Hz in half as many times as you needed to get back to your original C (7 times, in fact), you would find that you would not have your original C, but something close to it. This divergance is called the Pythagorean Comma and it has puzzled philosophers, physicians and musicians throughout the ages.
In the eighteenth century somebody took the Pythagorian Comma and divided it evenly over all the fifth, making each one of them unpure, if you will. The effect was that the fifths would be in sync with the octaves, and that a piano tuned in this way could play in any key it wanted, as far removed from C (the white keys) as it wanted. That was called Equal Temprament and was popularized by JS Bach (who was into this sort of thing) in his Well-Tempered Clavier, which as you remember has a prelude and fuge in each of the 12 different keys, major and minor. Our ears have become so accustomed to Equal temprament that we don’t hear it as “dirty” or “off” anymore. Quite the reverse.
If you don’t do the equal temprament trick, you can’t really equate G# with Ab, and D# with E etc. So if you went up the ladder of fifth, you would end up at “B#”, and if you went down, you’d get a “D-flatflat”, both of which would note quite be C. The further you go from the key you started in, that is, the more accidentals (sharps or flats) you add, the more pronounced the “error” becomes (and that’s what I mean by harmonic distance).
For chromatic insturments, the only way to go is equal temprament nowadays (except for fiddles, which are fretless, and can mess with the temprament, as good Irish musicians very often will do). For diatonic instruments or instruments with drones, like whistles or pipes, there is less need for equal temprament and maybe even a problem with it. (Brass instruments, btw, notice this effect because they are built on the series of overtones–which is pure and not “equal”-- and therefore the players will have to compensate for equal temprament.)
I have never been able to figure out whether tin whistles were originally made in tempraments other than equal, such as the just temprament, or if their tuning has simply never been considered all that important.
Finally, I think that with mass popularity the trend in Irish Trad is to move away from the older, pipes-based tempraments toward mainstream equal temprament, which today sounds more “in tune” to most listeners. (I am sure piano accordions are no help.) That is part of why the high-end makers go for equal temprament, I’d guess.
HTH
*) The circle of fifths, is just the fifth written in a circle, usually C at the 12 o’clock position. Notice that you need equal temprament to make the circle of fith really a circle, meaning that the twelfth fifth will equal the starting tone, seven octaves up.
/bloomfield
[ This Message was edited by: Bloomfield on 2002-07-24 10:55 ]