source file: mills3.txt Date: Mon, 27 Oct 1997 21:51:25 +0100 Subject: Reply to Carl Lumma From: "Paul H. Erlich" I've posted at length on definitions of consonance and dissonance (as well as discussions of melody and modulation) in previous Tuning Digests. The issue is complicated but that is no reason to retreat to a simpler, less useful definition. Your definition makes no sense, because you'll never come up with an answer for irrational intervals. How do you propose to "measure [the mistuning of an equal temperament] in cents deviation _and_ consonance* deviation" when you can't define the consonance of an irrational interval? Presumably, you want to use rational approximations, but how do you decide whether to use 25/21, 44/37, 1785/1501, 10754/9043, or 19723/16585? You seem to like the latter, but if you really go to that level of accuracy, you face a very large danger that some of your so-called JI perfect fifths will turn out to be closer to 30001/20001, in which case you'll have to call the equal-tempered minor third more consonant that the just perfect fifth! Actually, if you want to get into silly mathematical discussions, the probability that two physical strings are tuned to a rational inteval is zero, as Cantor showed that the cardinality of the reals is greater than that of the rationals. The probability that the strings are tuned to an n-tone-equal-tempered interval is also zero, since the algebraic numbers have the same cardinality as the rationals (which have the same cardinality as the integers). The only definition I care about is one with a psychoacoustic correlative. Otherwise we're not talking about music, we're talking about abstract marks on a piece of paper or computer screen. My calculation of beat rates was an attempt to show that the one psychoacoustic correlate you did mention does not lead to a first-order qualitative difference between 19/16 and 2^(1/4). Personally I think 19/16 in a high register has a peculiar stability due to its denominator being octave-equivalent with the fundamental (the brain often fills in the missing fundamental with a "virtual pitch" and any combination tones will only be in agreement with the virtual pitch in the case of JI). Oh, I finally saw that cream cheese commercial. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: Inharmonicity (again!) PostedDate: 27-10-97 22:23:36 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: 27-10-97 22:22:38-27-10-97 22:22:39,27-10-97 21:23:14-27-10-97 21:23:14 DeliveredDate: 27-10-97 21:23:14 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 C125653D.00756B9B; Mon, 27 Oct 1997 22:22:32 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA28317; Mon, 27 Oct 1997 22:23:36 +0100 Date: Mon, 27 Oct 1997 22:23:36 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA29009 Received: (qmail 10540 invoked from network); 27 Oct 1997 13:23:27 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 27 Oct 1997 13:23:27 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu