source file: mills2.txt Date: Wed, 14 May 1997 19:19:20 +0200 Subject: RE: Why x-ET? From: gbreed@cix.compulink.co.uk (Graham Breed) Reply to David Finnamore: >Thanks for joining the fray! This is good stuff. Great sense of humor, too! Hmmm, worrying. Are you detecting my finely tuned wit, or did I make some more laughable errors in my post? Hopefully the former. The problem with, as you put it, "joining the fray," is that I handle my e-mail offline so I end up replying to the digest before the one I'm picking up. Anyway, here goes . . . >In any event, it seems evident >to me that the math of roots and exponents is more complex than that of >whole-number ratios. If you can solve for the nth root of x in your head, >and multiply it by the factor of the previous note of the scale in your head, >you really do have some mathematical skills to show off! And if I'm not >mistaken, that math is necessary for ETs whether or not you're trying to >approximate a JI tuning. If you're tuning a digital synth to an ET scale, all you need to do is one division, and from then on it's just addition. This is easier than all that multiplying of fractions you need to work with JI -- unless you use ratio space axes. My point is that there's no reason why you can't do all your work in a logarithmic scale, and then an ET is the simplest scale to define. Most of the work in microtonality now is done with electronic synthesis rather than dividiing up strings. Actually, I probably could calculate an nth root of 2 to a few of decimal places (which is all you need for practical tuning) using the Taylor series for (1+x)**n A first approx to the nth root of two is 1+0.7/n for 10>I consider small number ETs to be the less valuable >>because the natural order, which can be made audible by >>playing equal intervals in series, sounds pretty boring. > >This sounds like interesting analysis but I'm not sure what it means. Do you >mean that the larger the number of equal divisions of a given interval, the >more musically useful the resulting scale? If so, why? If not, how small is >small? And why should a scale played in order be boring? "Joy to the world" >by G. F. Handel is a good example of musically interesting and effective use >of scale tones in their "natural" order. I must be missing your point. By the limited experience I've had of playing in alternative tunings, I have found that ETs do have a common characteristic. My ear can definitely tell that all the intervals are the same. Melodically, this makes ET to me more natural than JI. But, the effect is to make the scales sound "linear". Diatonic scales fitted to 19tet sound great but, once I play all the notes within a third, in any order, it starts to sound bland. This means that the equal tempered nature of a scale is audible, but should be hidden. Handel may have got away with it, but he was a better composer than me! >It is true that human perception of audio >phenomena, pitch included, is essentially logarithmic. But why does it make >musical sense to simply quantize this logarithmic space? I'm not saying it >doesn't. But I don't yet know of any reason that it should, either >theoretically or experientially. Once you define a space, the simplest thing you can do is divide it into equal steps. If that doesn't work, you can try something more complicated, but there's no point in adding complexity for the sake of it. JI comes from the harmonic series, which is a quantization of linear frequency space. >>>nature is the best basis of judging any tuning >>>system. > >> doesn't hold unless you consider ETs to be less natural >than JIs. > >I do. :-) Sort of. . . . To correct my own grammar, I suppose something's either natural or it isn't. My point is that ET scales _are_ natural. Of course, the word "natural" could mean a lot of things. In this case, in relation to mathematics, acoustics, human instinct or patterns found in the natural world. I think it's the last one you mean, which is fair enough. More importantly, though, I disagree with your original statement which, as you did say, is largely a question of personal taste. A tuning should be judged primarily by its subjective musical interest. By this criterion, just scales score very well. But, with most music, they can be approximated quite adequately to, say, 31tet. This is the usual, but not sole, reason for using ETs. If you want to exploit JI with hair splitting accuracy, I expect you'd have to work with slow, harmonic music with perfectly harmonic timbres and no vibrato. Otherwise, go with whatever's simplest. On my Wave Blaster, major triads actually sound better in 31tet than JI. This may be because of sample looping (could someone explain this, please?), phase shifting vibrato or a bug in my program. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 15 May 1997 02:28 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02266; Thu, 15 May 1997 02:28:51 +0200 Date: Thu, 15 May 1997 02:28:51 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA02264 Received: (qmail 26358 invoked from network); 15 May 1997 00:22:02 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 15 May 1997 00:22:02 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu