source file: mills2.txt Date: Thu, 2 Jan 1997 14:19:31 -0800 Subject: A Letter from David Feldman From: Daniel Wolf <106232.3266@compuserve.com> I received the following mail from composer and mathematician David Feldman - one of our lurkers - who has allowed me to forward it to the list: I had a tuning idea the other day which led me to some computations which I would be happy to share with you. Here is the idea. Suppose that you are primarily interested in two intervals. Classically you might think of r11 and r22, but any pair will serve. Then you might think of dividing one of these intevals into equal parts so as to produce a good approximation of the second. To decide how many parts leads you to the irrational quantity qog r1)/(log r2) which you must then approximate by a rational number. Continued fractions provide the best tool. All that is well known of course. The new (?) idea which currently intrigues me is working with four intervals, r1, r2, r3 and r4 with (log r1)/(log r2) (log r3)/(log r4) (approximately) With such a foursome you could transform a musical structure which depended on the first two intervals into one which depended on the last two. For example log (2/1)/log (3/2) 1.70951... while log (15/8)/log (13/9) 1.70945... so up to a point (5, 12, 41, 53, ...) subdivisions of the octave that produce good fifths corresponds to subdivisions of a 15/8 that produce good 13/9 's. I call the pairs (2/1,3/2) and (15/8,13/9) (mutually) isotempered. The table I used to learn this fact (and many others like it) I made as follows. I computed qg(r1)/log(r2) for every r1 and r2 (in lowest terms) with denominator up to 10 and numerator not more then 10 greater then denominator. Then I sorted the table by the magnitude of q. Finally I computed the difference between consecutive lines in order to look for good isotempered pairs. The table includes the continued fraction partial quotients of each q. I made it all with a simple maple program. The format is very wide (it all fits on my new wide monitor screen) which may be inconvenient for you. Let me know if you need it reformated. I will send the table under separate cover. David Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 2 Jan 1997 23:29 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00913; Thu, 2 Jan 1997 23:32:28 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA00911 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id OAA13342; Thu, 2 Jan 1997 14:32:26 -0800 Date: Thu, 2 Jan 1997 14:32:26 -0800 Message-Id: <32CC376F.6E1B@sprynet.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu