source file: mills2.txt Date: Tue, 4 Mar 1997 13:14:15 -0800 Subject: Re: New Paper From: rtomes@kcbbs.gen.nz (Ray Tomes) "Asmussen, Robert" erote: >I have recently posted a paper that explores tuning by ratios. Included >in this paper are illustrations, programs, WAV files, Csound examples >and a retuned two-part invention by J.S. Bach. >http://www.terraworld.net/users/r/robert/default.htm Robert, on arriving at your pages I had a feeling of deju vu. Your three coloured waves logo with periods in the ratio 2:3:5 is quite similar to one on my pages. The problem with which you are dealing, that of working out the exact correct frequency of every note in any composition, is the same one that I address in my AJI pages at http://www.kcbbs.gen.nz/users/rtomes/aji-main.htm Robert, congratulations on a nice clear presentation. In your pages you say: >Another point to note is that most pieces of traditional tonal music, >even short and seemingly simple pieces by Mozart and Haydn, cannot be >translated without modification from traditional notation into integer >ratios. Chromatic passages are especially difficult to recast into this >new framework. Right! I have considered a number of options for this and believe that in general the best solution is one based on a sort of information theory idea. The following may be a bit heavy, but once grasped can be seen as a powerful tool. To explain this idea, consider that each chord played has notes related by ratios to the tonic. Suppose that these ratios for a particular chord are 5/4, 15/8, 3/1. Then it is necessary to find the LCM (lowest common multiple) of the numerators and divisors which are 15 and 8. My system then consists of counting demerit points for the factors of these two numbers. We have 2^3 and 3^1*5^1 and so must score demerits by weighting the value for the indices here. Generally ratios of 2 are most acceptable and then 3 and then 5 etc. My preferred formula for demerits is in proportion to p*ln(p) so that if 2 is set as unity each 3 and 5 will score 2.4 and 5.8 respectively. In that case 8 and 15 will score 3 and 2.4+5.8 for a total of 11.2 demerits. If there is a better way to "read" the chord then it will score less. Some intervals such as a minor sixth may be interpreted as 7/4, 16/9 or 9/8 depending on circumstances. The one that minimises the demerits is to be considered better because it is more simply related to the other notes present or even previously played. BTW, there is a missing graphic on one of your pages. Unfortunately I just pressed the wrong key and lost everything including the reference. -- 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, 4 Mar 1997 22:43 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05980; Tue, 4 Mar 1997 22:43:44 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA05965 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id NAA19501; Tue, 4 Mar 1997 13:41:30 -0800 Date: Tue, 4 Mar 1997 13:41:30 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu