source file: mills2.txt Date: Thu, 19 Oct 1995 21:11:46 -0700 Subject: RE: Militant JI'ism From: Mmcky@aol.com Denny says, > For those of you who think that I'm not being militant >enough on the side of just intonation, ... I certainly think it's better not to be militant about just intonation, or anything else, for that matter. >I would like to point out something that Easly Blackwood once >said: "There is no such thing as just intonation. All you >can do is approximate a ratio by some standard of precision. It >all depends on the tolorance that you are willing to accept". I >think that the same thing could be said about equal >temperaments, as well. That may well be true for mechanical instruments, but it does not seem to represent the situation for frequency dividers, and since all modern electronic musical instruments use some form of frequency division, it may be a distinction of some importance. Take for example the humble fifth, which is 3/2 in JI and 1.498307077 in 12ET. To produce the JI fifth to an almost unbelievable level of accuracy simply takes three flip-flops with no gating, assuming they are the right kind of flip-flops (JK or D). The circuit needs to be fed with a frequency that is 3 times the frequency of the lower tone. But to produce the 12ET fifth to the accuracy represented above is quite complex, and since it is a non-repeating decimal, to produce the 12ET fifth with the same accuracy with which we can easily produce the JI fifth is quite impossible. Since 498307077, and 1000000000 have no common factors, we would have to use two frequency dividers of 29 and 30 flip-flops respectively, and a substantial amount of gating logic. We would have to feed this circuit with a frequency 4.983070769E+17 times higher than the lowest tone we wished to produce. There is the alternative of using the "phase accumulator" method of generating the frequency, as in FM synthesis or Wavetable synthesis. That eliminates the need for the very high frequency, but introduces phase jitter and does little or nothing for circuit complexity. With electronics, we can easily produce a very accurate JI fifth, or we can spend thousands of times as much and get a much less accurate approximation of an ET12 fifth. That is a principle reason why I prefer exact JI. Another reason is that the ear/brain system looks a lot more like an electronic mechanism to me that a mechanical system. And then, of course, there is ease of analysis. Marion Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 20 Oct 1995 07:17 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id WAA28815; Thu, 19 Oct 1995 22:17:16 -0700 Date: Thu, 19 Oct 1995 22:17:16 -0700 Message-Id: <951020045912_71670.2576_HHB31-2@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu