source file: mills2.txt Date: Sat, 8 Mar 1997 05:11:16 -0800 Subject: Re: New Paper From: rtomes@kcbbs.gen.nz (Ray Tomes) Robert Asmussen wrote: >As to some specific details, I believe you are incorporating intervals >in your Mozart example that are not in keeping with the classical style. >For example, you use octave equivalents of 7/4 as some of your frequency >ratios. I challenge you or anyone to create a dominant seventh chord, >using the number seven as a multiple in the denominator of the chord’s >seventh, that would not sound out of place in a piece of chamber music >by Mozart. And yet if you look at the derived fundamental frequency (defined below) for the three bars I think that the "correct" interpretation is undoubtedly F for the 1st and 3rd bars and C for the second. This fits quite naturally with the 7 ratios but not with your interpretation. See http://www.kcbbs.gen.nz/users/rtomes/aji-xmpl.htm for this Mozart example and the Beethoven example mentioned below. >I would suggest giving it the acid test, which would be to make a WAV >sound file of your Mozart score using Csound. With the tempo nice and >slow, try 7/4, then 16/9 as the relative frequency for the seventh of >your dominant chord. I certainly don't know enough about music history to argue on that score and anyway accept that 16/9 is sometimes the correct value for that note. However I would argue on the basis of logic. If you have a dominant 7th chord and the other notes have frequency ratios of 4:5:6:8 why should the extra note be 64/9 in that ratio scheme when 63/9 would cancel down nicely to 7 and make an elegant 4:5:6:7:8? Whatever was actually played historically I still feel that the intention or meaning of such a chord is 4:5:6:7:8. >You state, ”Some intervals such as a minor sixth may be interpreted as >7/4, 16/9 or 9/8 depending on circumstances.” >Without going into the potential merits of your statistical approach to >pitch selection, I must point out that 7/4 is approximately a minor >seventh; 16/9 is an in-tune minor seventh; and 9/8 is a major second. >Certainly none of these ratios is a minor sixth. I assume this is just a >grammatical error. Oops, I really went to sleep on that one didn't I? Two typos. What I meant to say was that a minor 7th could be 7/4, 16/9 or 9/5. >When I stated in my paper that pieces of traditional tonal music could >not be translated without modification from traditional notation into >integer ratios, I could have elaborated further with the following two >points: >1. Often, such as when employing chromatic scales, the composer is >simply filling in the space from point A to point B with notes. Such >passages do not need to be translated, nor often can they be, into >ratios. One might even argue that such passages are not really music, >but rather more like dust on a mirror. >2. Composers of the past wrote their music using limited instruments and >tuning systems that were available at the time. We should not expect to >translate their imperfect creations into a purely mathematical >framework; in fact, we should be surprised when it is possible to do so. I have to agree at the surprise, but believe that the results often tell us more than we expected to find. Have a look at my second example, Beethoven's "Romance". The melody goes b c d g g g b g a b b g b c d but the fundamental frequency b c d g g g b c d g g g b c d is calculated. (I define fundamental frequency as the frequency which has ratio 1 when all the ratios are reduced to integers, so it divides all the played frequencies). To me it seems certain that the part with melody b g a has been correctly interpreted because the fundamental frequency echoes the other parts that go b c d. Notice that the ratio 7 occurs in this short passage several times and is necessary to give this beautiful structure. Actually I cheated a wee bit, because one of those b's is actually an incidental b/a. Thanks for the information about Csound. I will access this. >Regarding the problems you are having with the graphics in my paper, >would you please contact me if you experience additional difficulties? I >have reloaded all ten HTML files off my server quite easily using only 8 >megs of RAM while running a minimum of applications and windows. I accessed all your pages and there was only one missing graphic. Unfortunately when I went to gets its name I hit the wrong key and lost it all. You already saw above how butterfingered I was that night from my typing :-) If I find this again I will let you know. Robert, thanks for the interesting comments, Ray. -- 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; Sat, 8 Mar 1997 14:14 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA09568; Sat, 8 Mar 1997 14:14:21 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA09520 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id FAA04164; Sat, 8 Mar 1997 05:10:04 -0800 Date: Sat, 8 Mar 1997 05:10:04 -0800 Message-Id: <33374f5e.672998538@kcbbs.gen.nz> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu