source file: mills2.txt Date: Tue, 4 Mar 1997 16:57:00 -0800 Subject: 5/4 vs 81/64 etc From: rtomes@kcbbs.gen.nz (Ray Tomes) The subject of 81/64 tuning vs 5/4 has appeared in several posts in the short time that I have been here. There are several examples of very near matches in harmonic ratios. They centre around the near equality of the numbers 81 and 80, 64 and 63 and some other lesser ones. These numbers are all important in my view because they are all numbers with many ways to be factorised and so have many harmonic relationships. Changes in the relative importance of notes come into play when we shift around in the multidimensional world of tonality. The 81/80 shift is the easiest one to get to happen and is no doubt familiar to all, but I repeat it just to set the scene. If we have a Just Intonation scale of 24 27 30 32 36 40 45 48 and move by a fifth then we multiply all these by 3/2 and get 36 40.5 45 48 54 67.5 72 which when brought back into the original range is 24 27 33.75 36 40.5 45 48. So we dropped 30 and added 33.75 which is our sharp coming in, and replaced 40 by 40.5 or an 81/80 shift. Of course if we repeat this 12 times they will all shift and we get Pythagoras comma problem. If we accept the ratios 28 and 42 as being the minor 3rd and minor 6th (please excuse my lack of proper musical jargon) then we can see that these will become 42 and 63 (or 31.5) when we transpose by a fifth. This 63 introduces another near miss with a difference of 63/64. There are other more subtle near misses. It is also possible to shift key by a major third and so we can multiply the JI scale by 5/4 to get 30 33.75 37.5 40 45 50 56.25 60 which in the original range is 25 28.125 30 33.75 37.5 40 45 and so lots of notes have gone wandering. The 28.125 value is however very close to 28 and the ratio of 225/224 is one of the many lesser near misses. When I used my little BASIC program to calculate all the strong harmonics according to the factorisations of each number (which is a measure of their total number of musical relationships) then I found a couple of interesting things happening as I went to higher numbers. Sometimes as I compared the pattern as I went up octaves one harmonic would gradually fade and a nearby one would come in. This happens with both 81/80 and 64/63 ratios. In fact, if we consider a tonic as 1 then although 6 octaves above is 64 alright, after a few more octaves the most dominant note is 63*2^n not 64*2^n. There is evidence of this type of thing in the Indian musical scale although I don't know enough about the music to say whether it happens in the music itself. Does anyone else have thoughts on whether there might reasonably be such a funny scale used which had the tonic tuned at successive octaves to say 16 32 64 126 252? This would be most likely to occur when the .. oh bugger, my musical knowledge doesn't allow me to express myself (HELP) let me go back a bit. Say we are in C major and we have Cs at frequency 16 32 64 126 252 with this funny discontinuity between 64 and 126. In parctice 63 wants to exist also and 128, but we don't have enough keys. ... anyway, to resume, this is likely to happen when we are also using Eb and Bb in our music because these have the ratios 28 and 42 already, which want the 63 and 126 rather than 64 and 128. Does this make sense? I am suggesting that the emphasis might change between the low and the high octaves. Here is a grand chord that demonstrates the simultaneous use of such "out of tune" octave notes: 8 12 16 20 24 28 32 42 56 84 126 Similar examples can be found that want both a 5/4 and 81/80 tuning in different octaves. Is this something that has ever been suggested before? Of course you realise that there is a secret plot. I have come here to drive you all mad HA HA HEE HEE ... -- 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; Wed, 5 Mar 1997 02:12 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA06191; Wed, 5 Mar 1997 02:12:16 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA06250 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id RAA05992; Tue, 4 Mar 1997 17:09:23 -0800 Date: Tue, 4 Mar 1997 17:09:23 -0800 Message-Id: <199703050100.UAA26357@sound.music.mcgill.ca> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu