source file: mills3.txt Date: Thu, 18 Dec 1997 16:47:36 +0100 Subject: 19-tone Equal Octave Stretch From: Gregg Gibson Paul Ehrlich said: > >My calculations indicate that 2-3 cents is indeed the optimal amount of > >stretching for the 19-tone octave in order to maximize the overall consonance > >of intervals which are approximations of ratios using numbers no higher than > >6 (including the octave and double octave). Very good. No matter what one's particular bias, matters such as the optimal temperature of the octave are determinable using objective criteria. Your figures for relative consonance, in particular for those of the 22-tone equal, are however utterly incorrect, for, as I have tried to demonstrate to you - and this is by no means a difficult matter to understand - that temperament is noncyclic. You cannot blithely assign the closest values to all consonances when evaluating the consonance of a noncyclic temperament, for in such systems these consonances do not exist together either in diatonic music (which you have admitted) nor yet in chromatic or enharmonic music, which also require a modicum of consonance and tonal coherence. I formerly made the same error which you are making, but had no one to tell me so; I was forced to discover this for myself. In the present case, it is _not_ all the consonances that are _ultimately_ relevant to the determination of the amount by which the 19-tone equal octave can or should be tempered, even though I have considered them all, as a kind of courtesy to Fokker, and a demonstration that all of them have to be considered initially. For, finally, the fourth cannot be tempered by much more than 8.4 cents and remain satisfactory, i.e no more deteriorated than the more sensitive fifth at 5.7 cents deviation, and no less consonant than 5:3 or 5:4. This puts a limit on the permissible temperature of the octave, as does the fifteenth, I need not tell you why. You should be very chary of reducing congruence with consonances to percentages in the facile manner you have done. This is a habit of thought that too often leads one to overlook many important factors. A rigorous mathematical calculation of all the important factors is certainly beyond you; it is also as yet beyond anyone. But it is unnecessary to spend 300 years with the equations to know a good or bad temperament when one hears and sees one. SMTPOriginator: tuning@eartha.mills.edu From: William Sethares Subject: melodic lemons PostedDate: 18-12-97 16:53:59 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 18-12-97 16:51:57-18-12-97 16:51:58,18-12-97 16:51:34-18-12-97 16:51:35 DeliveredDate: 18-12-97 16:51:35 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256571.00572414; Thu, 18 Dec 1997 16:53:45 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA29663; Thu, 18 Dec 1997 16:53:59 +0100 Date: Thu, 18 Dec 1997 16:53:59 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA29656 Received: (qmail 23243 invoked from network); 18 Dec 1997 07:53:55 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 18 Dec 1997 07:53:55 -0800 Message-Id: <199712181551.AA24698@eceserv0.ece.wisc.edu> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu