source file: mills3.txt Date: Thu, 23 Oct 1997 19:20:01 +0200 Subject: The limits of the ear From: Carl Lumma I hope I haven't generated any harsh feelings with these flaming posts. I just enjoy 'em. Mr. Erlich can rest assured that I have nothing but the highest respect for his work. His was one of my favorite cuts on the tape swap and has got me wanting more. He also may or may not wish to know that one of the first tunings I'm going to map to my generalized keyboard (when I get it) will be 22 equal. >Today is my last day as a full-time employee of an investment company. I >already spend almost all my free time playing music... So how is that working out? Are you able to resist the temptation of playing with your 12 tone friends enough to go home and be microtonal? Lastly, I will say again that altough I used a posting of Mr. Erlich's as an example, I was addressing the community at large with an issue at large. Anyway, here's my response... >Now these intervals are getting closer and closer to 3/2, and after just >a few of them even the best ear can't tell the difference between them >and 3/2. Yet according to Carl, 3/2 is more consonant than any of these >ratios. Thus we must conclude that Carl's definition of consonance has >little to do with anything you can actually hear. The idea of defining Consonance as a product of the sizes of the numbers in a ratio wasn't Just something I dreamt up. It is littered all over the literature, especially the JI Primer. But there's always something vague like "strong implied preference" involved. It is told straight in Denny Genovese's thesis "The Harmonic Series as a Practical Approach to Just Intonation", in chapter 2, "Quantification of Dissonance and an Acoustic Definition of Consonance". To paraphrase: If you've got two frequencies represented by a ratio in lowest terms whose decimal value is between 1 and 2, then the period of the composite waveform of these two frequencies is the product of the two numbers in the ratio. I believe this makes a more useful definition of Consonance than any other I've ever heard. If somebody's got one they think I havn't heard, please share. This definition is independent of what a person may or may not hear, and it is independent of how they might like or dislike what they hear. For those of you who do not view these as assets, don't use this definition. The ear has a limit of resolution, just like anything else. That the one example given by Mr. Erlich is beyond the ear's resolution does not mean that this definition has no practical application. Clorox is poison but lots of folk find it useful to chlorinate their water. I think the practical usefulness of the definition is obvious. While I can't hear the difference between a 3/2 and a 300001/200001, I can hear the difference between a 19/16 and a 19723/16565. I'm listening to it right now. But I've been told that an error of 2 cents is not significant when comparing tempered intervals to just ones. Working within the practical limits of human hearing, the perception of subtle mis-tuning is very sensitive to the timbres used, the voicing of the intervals, how high the identities are, and how they are used in combination with other intervals. If your timbres have a high degree of inharmonicity, you'll loose resolution. Don't use bowed strings for high resolution work. If your interval is down in the lowest octave of the piano, don't expect to hear a 10 cent difference. You're much less likely to notice a 3/2 2 cents off than a 11/7 mistuned by 2 cents. You can't hear the difference between a 3/2 and 700 cents in a melody, but in an otherwise just triad, it sticks out like a sore thumb. So my original point was: When comparing an equal-step tuning to a just tuning on a broad, theoretical level, define Consonance on a broad theoretical level. My other original point was: Equal step tunings have nothing to apologize for. I don't view them as imitating just tunings. They're a different breed of cat. The kind of music that makes sense in an equal temperament doesn't make sense in JI and vice versa. It's like apples and oranges. I used to think that an equal step tuning was like having all the pitches that you needed to modulate in every key of a just tuning, except you used a single average pitch instead of two or three or four just ones. So you could make music on this huge matrix of a just tuning with less notes and worse sound. But the difference is much deeper than that. I can't explain it. The Pauls (Erlich and Hahn) turned me on to this, and their ideas on this matter are much farther along than mine. And I never said dissonance was bad, Carl Aline Surman writes... >I always enjoy the discussions about ratios and eq temps. It is true >that being "in tune" is a mathematical phenomenon of frequencies matching >up perfectly...I agree that when we say eq temps approximate a pure >ratio, it doesn't correlate in a way because as soon as you are off of >the ratio, the new note is NOT the ratio, so it IS something else. Thanks for the support! I know writing these posts is beneficial for me. It's nice to know they are of interest to others as well. SMTPOriginator: tuning@eartha.mills.edu From: gbreed@cix.compulink.co.uk (Graham Breed) Subject: fourth based keyboard mapping PostedDate: 23-10-97 19:23:27 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: 23-10-97 19:22:38-23-10-97 19:22:38,23-10-97 18:23:18-23-10-97 18:23:18 DeliveredDate: 23-10-97 18:23:18 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 C1256539.005F7132; Thu, 23 Oct 1997 19:22:29 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA25485; Thu, 23 Oct 1997 19:23:27 +0200 Date: Thu, 23 Oct 1997 19:23:27 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA25473 Received: (qmail 25782 invoked from network); 23 Oct 1997 10:23:22 -0700 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Oct 1997 10:23:22 -0700 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu