source file: mills2.txt Date: Fri, 20 Sep 1996 11:41:17 -0700 Subject: RE: RE: Consistency generalized From: PAULE >> Paul H., >> >> I believe your original definition leads to consistency for level >> N>1.5. For 1> will be consistent with one another. >Yes, I noted that in my "erratum" followup message. However, the >algorithm I just posted in response to John Chalmers is correct. Your new algorithm is not correct either -- at least it does not reproduce my definition of consistency. In particular, no tuning can fail to be level 1 consistent according to your algorithm, and some tunings would be level 2 consistent according to this algorithm that wouldn't be according to your original definition. I will give the correct consistency algorithm if you wish. John C., you are correct that "fractional consistency" is kind of meaningless; I was using it in the sense of Paul H.'s original definition with respect to the maximum error size. >> I don't think the concept of consistency has much to do with whether a >> particular JI interval is approximated well enough to be used as a harmonic >> consonance. >I don't think that's exactly what I meant, or even what I said. I was responding to your assertion that common-practice harmony did not find 7-tET sufficient. Clearly, 7-tET would not provide the desired harmonic effect when used for common-practice harmony. I believe the reasons lie not with higher-level consistency but with accuracy. But the actual historical reasons for not adopting 7-tET lie not with 5-limit, but even earlier with 3-limit harmony, as practiced in Medieval music. Although 7-tET may have a high level of consistency within the 3-limit, its accuracy in approximating 3-limit consonances is questionable. Medieval musicians tuned fifths and fourths with enough accuracy that is was clearly seen that the eighth note in a circle of fifths was different from the first. Thus the process of chromatic alteration began that ultimately led to 12-tET. In China, fifths and fourth were also the main harmonic consonances, and China developed 12-tET before the West did. In Thailand, however, the accuracy of the fifths and fourths was less important, since their role was more melodic than harmonic and because most Thai instruments do not have harmonic partials. Thus it is not surprising that Thailand did develop 7-tET. [bigsnip] >> What consistency offers is a supplement to considerations of how good >> the approximations to JI are. Essentially, composing with a consistent >> tuning will be no more difficult than composing in JI, while in inconsistent >> tunings, complications may arise if one attempts to always use the best >> approximations to JI intervals. >Here's where we part company, I'm afraid. I find level 1 consistency to >be far too low a standard. Very weird things can happen in level 1 >consistent tunings. A simple example: 5TET is consistent at the >5-limit. A 16/15, represented as a 4/3 less a 5/4, becomes 2-2=0 steps, >or a unison. However, a 25/24, represented as a 5/4 less a 6/5, becomes >2-1=1 step. In other words, a larger interval becomes a unison while a >smaller interval does not. Maybe this doesn't bother you, but for me it >causes cognitive dissonance. Well, the idea of using 5-tET for 5-limit harmony bothers me. Didn't you read my post? I explained why, even though 7-tET is consistent at the 5-limit, it is not a good tuning for 5-limit harmony. The same goes for 5-tET. If a tuning is both consistent and has good approximations to JI however, then I will admit it, even if the second-order intervals do not behave as in JI. I think 22-tET is marginally good enough for 9-limit harmony, if the texture doen't get too thin, but observe: 64/63, an 8/7 less a 9/8, is 4-4=0 steps, while 81/80, a 9/8 less a 10/9, is 4-3=1 step. "In other words, a larger interval becomes a unison while a smaller interval does not." This doesn't bother me in the slightest -- in fact I like it. The ear does not have a pre-defined map of JI; it does, however, perceive consonance, dissonance, and scale structures. Sometimes the most effective scale structures for consonance differ significantly from JI, but I think these differences can be just fine, even necessary, unless they violate simple consistency. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 20 Sep 1996 21:51 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA29365; Fri, 20 Sep 1996 21:53:15 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA31963 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id MAA13137; Fri, 20 Sep 1996 12:53:13 -0700 Date: Fri, 20 Sep 1996 12:53:13 -0700 Message-Id: <55960920195155/0005695065PK2EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu