source file: mills2.txt Date: Fri, 7 Feb 1997 10:32:58 -0800 Subject: Re: TUNING digest 979 From: James Kukula > The traditional mathematical model of musical tuning and > acoustics states that these "ghost tones" (for sake of a better > term) are found at frequencies which are small integer ratios > to the fundamental or open string frequency. > This is the assumption upon which Just Intonation is based. In physics I think they're called modes of oscillation. The fundamental mode is the one with lowest frequency. Maybe the others are just called non-fundamental modes. A while back someone described a delightful experiment with prepared guitar, verifying that these non-fundamental modes need not have frequencies which are small integer ratios of the fundamental. Under ideal conditions - uniform strings, and I hear one would need massless frictionless pivots to hold the ends of the string - ideal conditions would yield a spectrum of frequencies all integer multiples of the fundamental. But this is not the only place that integers come up. I just read an amazing thing, I think in IEEE SPECTRUM, an engineering magazine. There's this new kind of loudspeaker that's been invented. It's based on the fact that air is a non-linear acoustic medium. Non-linear means that air isn't a perfect spring, as you squeeze air it doesn't resist in exact proportion to how much you squeeze. So the new loudspeaker takes advantage of this. It's composed of two ultrasonic loudspeakers that each emit sound up around 200 KHz. Dogs etc. can't hear way up there either. For all I know these may be the best way to eliminate cockroaches! Anyway, the two ultrasonic speakers actually play at somewhat different frequencies - for example, suppose one played at 200 KHz and the other at 201 KHz. What happens, when two oscillating forces hit a nonlinear medium, is that new oscillations get induced at the sum and difference frequencies. So new oscillations appear at 401 KHz and 1 KHz. Of course 401 KHz is inaudible. But the 1 KHz... hey, where'd that sound come from! The IEEE Spectrum mentioned as I recall that Carver is involved in producing a product based on this new invention. Anyway, if you start with two frequencies and start taking sums and differences: a, b, a+b, a-b, a+2b, 2a-b, 2a+2b, 2a, 2b, 3a+b, etc. somehow amazingly integers pop up. Now I'm not particularly enamoured of the creationist view of the world, but I'll follow Haverstick at least to the point that it looks like the integers are just somehow the way things are. The ear itself is undoubtedly non-linear. I imagine that's why we can hear "beat frequencies". Anyway, my whole point is that just intonation need not be founded on the frequencies of modes of oscillation of a string. Personally I don't think you can find secure foundations for any conceptual system, though even insecure foundations can be quite useful. The best foundation for just intonation, seems to me, is just the way various pitch combinations sound to us humans. Of course, our perception is also shaped by the conceptual systems that have been used to construct our environment. But anyway if you look for why humans might like simple ratios of frequencies, there are more answers around than just strings. Jim Kukula Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 8 Feb 1997 00:22 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA29580; Sat, 8 Feb 1997 00:22:51 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA29479 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id PAA00099; Fri, 7 Feb 1997 15:20:54 -0800 Date: Fri, 7 Feb 1997 15:20:54 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu