source file: mills3.txt Date: Tue, 28 Oct 1997 16:15:45 +0100 Subject: psychoacoustic foundations From: William Sethares Paul, mind if I quibble a bit? Yesterday you wrote: >... the probability that two physical strings are tuned >to a rational inteval is zero, as Cantor showed... This is true only if the two strings vibrate independently. When two strings are mounted on the same instrument, then there may well be some kinds of interaction between them. They may, for instance, become "entrained" to each other, that is, locked into various "simple integer" modes of vibration. Besides strings, there are other examples of entrainment, for instance, the orbits of several moons are entrained to simple integer ratios about their planets. On pianos, it is well known that adjacent strings interact. Might they, under certain conditions, entrain? Then, in talking about consonance and dissonance, you said: >The only definition I care about is one with a psychoacoustic >correlative. The sensory consonance theory (including Helmholtz, Plomp and Levelt, Terhardt, your own work, and my own work) is not the only theory that has a "psychoacoustic correlate." Consider for a moment the slightly different question of how the ear determines pitch. There are two major theories, the "place" and the "periodicity" theories of pitch perception. In the place theory, pitch is determined by where on the basilar membrane the maximum excitation occurs. This classic model, expounded by Von Bekesy in the 30's was considered the final word for a long time, but careful measurements of JND's (Just Noticeable Differences) for pitch showed that the basilar membrane is not sensitive enough to account for all thats known about the ear's resolution powers. The ear can also resolve very small temporal differences (as used for instance, to locate sounds in the environment). The periodicity theorists suggest that the ear basically "counts up" how many maxima (or minima or zero crossings, depending on who you read) occur in a given time interval to determine pitch. There is currently no experimental evidence that definitely points to one of these theories over the other, and J Pierce has argued (convincingly, in my opinion) that the most likely scenario is that both mechanisms are at work. Now let's return to the "simple integer ratio" theory of consonance. One can easily turn the simple integer ratio version into a "length of waveform" argument. The periodicity pitch theorists already have a mechanism for determining length of period, so it would be easy to believe that the this same mechanism could also be used by the "dissonance determining algorithm" of the brain. The problem with irrational ratios then becomes the problem of determining the pitch of "almost periodic waveforms". This is then not a huge problem since presumably the brain will interpret ambiguous interval ratios in much the way as it interprets ambiguous pitches. Thus the "simple integer ratio" theory of consonance has (roughly) as firm a psychoacoustic foundation as the sensory consonance theory. SMTPOriginator: tuning@eartha.mills.edu From: "Fred Kohler" Subject: Starr Labs/Erv Wilson Keyboard PostedDate: 28-10-97 16:55:11 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: 28-10-97 16:54:18-28-10-97 16:54:19,28-10-97 15:54:53-28-10-97 15:54:53 DeliveredDate: 28-10-97 15:54:53 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 C125653E.00575A28; Tue, 28 Oct 1997 16:54:07 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA29461; Tue, 28 Oct 1997 16:55:11 +0100 Date: Tue, 28 Oct 1997 16:55:11 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA29767 Received: (qmail 4398 invoked from network); 28 Oct 1997 07:55:07 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 28 Oct 1997 07:55:07 -0800 Message-Id: <01bce3b9$51164480$4617c2cf@a1a05977> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu