source file: mills2.txt Date: Wed, 18 Sep 1996 07:50:09 -0700 Subject: RE: Consistency generalized From: Paul Hahn Thanks for your comments, Paul (E). After four days I was beginning to wonder if anybody gave a damn about my goofy ideas. On Tue, 17 Sep 1996, PAULE wrote: > The musical meaning of integer levels of > consistency higher than 1, namely, the importance of matching just > intonation in combinations of n-limit intervals, I find dubious. So level 2 > consistency seems like an arbitrary requirement, as level 2.1 consistency > and level 1.9 consistency would lead to results which are slightly different > but just as meaningful musically. This is a good question, and I'm not sure if I have a good answer for it. Probably the best answer I can give is this: the main thrust of my theoretical wanderings in xenharmony is to discover tunings that are analogous to common practice harmony, in the sense that they have similar structures according to various hypothesized perceptual models. Common practice 5-limit harmony did not find 5TET or 7TET sufficient for its purposes, despite the fact that they are level 1 consistent at the 5-limit; it settled on 12TET, which is level 3 consistent. Therefore, I tentatively conclude that there is some kind of advantage to higher levels of consistency. A possible explanation is this: an ET which is level 1 consistent at a given limit may err from a just interval within that limit by up to half a step. It seems to me that my ear, at least, would not readily accept such an ET as an approximation to that interval when almost any interval randomly chosen from the continuous gamut would be more closely approximated by that ET. Since the average error of random intervals is 1/4 step, level 2 consistency at the N-limit ensures that N-limit intervals will be better approximated than the average randomly chosen interval. Thought of in this way, one can justify choosing level 2 as a lower limit on consistency. Does that make any sense? --pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote) O /\ "Foul? What the hell for?" -\-\-- o "Because you are chalking your cue with the 3-ball." Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 18 Sep 1996 18:00 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07925; Wed, 18 Sep 1996 18:02:04 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA06765 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA06983; Wed, 18 Sep 1996 09:02:02 -0700 Date: Wed, 18 Sep 1996 09:02:02 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu