source file: mills2.txt Date: Sat, 1 Mar 1997 15:24:17 -0800 Subject: Re: natural music From: rtomes@kcbbs.gen.nz (Ray Tomes) Paul Hahn wrote: "Ray: have you read "The Year of the Jackpot" by Heinlein? If not, I strongly recommend it." I have read a few Heinlein but not that one. I will have a look for it thanks. To continue the story about nature's music, after visiting the Cycles Foundation and finding that others before me had also find that cycles in many things seemed to have periods that were harmonically related I gradually developed a theory which would explain this. It also turned out to make a lot of valid predictions about the universe at large. On my WWW site (at the URL in sig block) I explain the harmonics theory which lead to explaining these observations. I will give a brief outline here which is oriented towards music and which will tell us something about tuning systems. Imagine an instrument which when it is plucked (or struck or blown or whatever) produces some fundamental frequency plus all of its harmonics. The higher order harmonics will have less energy in them. Now imagine that over time each harmonic loses energy to its harmonics. For the curious, this is caused by a non-linearity in the wave equation and only happens in a 3D wave so will not happen in a string for example). So what goes on is this, energy is transferred as follows: Frequency --> harmonics 1 --> 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... 2 --> 4 6 8 10 12 14 ... 3 --> 6 9 12 15 ... 4 --> 8 12 ... 5 --> 10 15 ... 6 --> 12 ... 7 --> 14 ... .. etc etc So what is happening here is that some harmonics are receiving energy from multiple different paths, in particular 12 gets lots of energy, while other harmonics such as 11 and 13 receive very little. The resulting energy pattern depends totally on how many ways each harmonic number can be factorised. When calculations are carried out to higher numbered harmonics an interesting pattern develops. These calculations can be carried out by a very simple BASIC program: n = (highest harmonic number) dim x(n): x(1)=1 for i=1 to n/2 for j=i*2 to n x(j)=x(j)+x(i) next j next i and the results are now in the x array and may be printed. The accurate details of the patterns generated can be seen at: http://www.kcbbs.gen.nz/users/rtomes/rt-ha-no.gif Harmonics to 1000000 http://www.kcbbs.gen.nz/users/rtomes/rt-hanox.gif Harmonics 2 to 288 http://www.kcbbs.gen.nz/users/rtomes/rt-hanoy.gif Harmonics 288 to 69120 but for those who don't have WWW access here is a rough ascii reconstruction of the relative harmonic energy for the strong harmonics between 48 and 96: --------------------------------------------- 48 ---- --------------------- 54 -------------------- 56 ------------------------------------ 60 -------------------------- 64 --- ------------------------------------------ 72 --- --------------------------- 80 ------------------------- 84 ----------------------- 90 ------------------------------------------- 96 Now a tuning person will instantly recognise the relationships present. The 4 strongest harmonics are 48-60-72-90 or a major chord with ratios 4:5:6:8 and the just intonation scale is present as double the traditional 24:27:30:32:36:40:45:48 ratios. However there are two additional values which are quite strong and these 56 and 84 which put the ratios 28 and 42 in the just scale so as to allow a couple of good dominant 7th chords with ratios 4:5:6:7:8. There are additional weaker harmonics between these strong ones and I have investigated indian music which has extra notes and found that these extra notes are where they would be expected by the harmonics theory. At higher harmonics this pattern is repeated with variations. I was going to write "minor variations" but realised that "minor" would have a different connotation. There are also examples of minor keys to be found, such as the range 1440 to 2880: ------------------------------------- 1440 ---- ---- -------------------------------- 1728 --- --------------------------- 1920 ----------------------- 2016 ------------------------------ 2160 ---------------------------- 2304 ---- ----------------------- 2688 --- ----------------------------------------- 2880 Here the strongest harmonics are 1440-1728-2160-2880 which is a 10:12:15:20 tuning. The predicted pattern of harmonics by this very simple rule is found to very rich and to have many interesting musical meanings present. The pattern is characterised by many ratios of 2, lots of ratios of 3 and lesser still of 5 and 7. Although the other primes do appear as ratios they do so very seldom. Also, as we move up the harmonics scale we find that the key keeps changing. After every 2 or 3 ratios of 2 (2.38 on average) the key shifts by a ratio of 3 and so that means that every about every 4 octaves (including the 2.38 and 1.585 from the 3) there is a shift to the dominant. At still larger intervals there is a key shift by a major third. It is at these points that minors appear. The above patterns also match closely to the pattern of cycles found by Dewey. From this pattern I have calculated what periods should exist for longer and shorter cycles and found that I get the right answers even for long geological cycles of ~600 million years. I am quite convinced that the universe is a giant musical instrument oscillating in the pattern that I described. -- 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; Sun, 2 Mar 1997 11:11 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01875; Sun, 2 Mar 1997 11:11:16 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA01825 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id CAA13929; Sun, 2 Mar 1997 02:09:28 -0800 Date: Sun, 2 Mar 1997 02:09:28 -0800 Message-Id: <33194CE1.70FA@cavehill.dnet.co.uk> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu