source file: mills2.txt Date: Tue, 24 Jun 1997 09:05:44 +0200 Subject: Re: Octaves (Paul E) From: rtomes@kcbbs.gen.nz (Ray Tomes) On Mon, 23 Jun 1997, Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) wrote: >The fact that octaves are often not equal to 2, but slightly more than 2, >has nothing to do with harmonic series or inharmonic partials. The >effect is most pronounced when sine waves are used; I believe a large >experiment found that the average subject deemed two sine waves >equivalent when they differed by 1209 cents. The typical 2nd harmonic of >a piano string is more like 1203 cents. I don't know about the structure of the ear, but that is an interesting idea. However I do believe that in nature there are definite frequencies at ratios near to but not exactly equal to 2. One particularly interesting example is found in the periods of revolution of 3 of the 4 Galilean satellites of Jupiter. They are all being repelled by the action of tidal forces, with the strongest action on the innermost. However as they get near to exact 1:2 ratios it sets up a strong resonance which forces the next outer one away from an exact 1:2 resonance. The result is as follows: Jupiter Satellite I II III Period (days) 1.76913780 3.55118108 7.15455312 Frequency (nHz) 6542.21173 3259.21822 1617.72145 I - 2xII II - 2xIII Beat frequency (nHz) 23.77530 23.77531 Ratio II/beat 137.0842 137.0841 The interesting thing is that not only are the beats between each adjacent pair of satellites the same, but the ratio of these to the middle satellite period of 137.0841 is very similar to a number found in atomic physics called the fine structure constant which is 1/137.0360. The two ratios II/I and III/II are 1206.3 and 1212.7 cents respectively. It rather seems that nature likes to operate at a little distance from exact resonance. -- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 24 Jun 1997 12:55 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA23415; Tue, 24 Jun 1997 12:55:34 +0200 Date: Tue, 24 Jun 1997 12:55:34 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA23429 Received: (qmail 26704 invoked from network); 24 Jun 1997 10:55:12 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 24 Jun 1997 10:55:12 -0000 Message-Id: <199706240652_MC2-191F-7480@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu