source file: mills2.txt Date: Thu, 2 Nov 1995 08:38:00 -0800 From: "John H. Chalmers" From: mclaren Subject: The Gelfond-Schneider Theorem and non-just non-equal-tempered scales --- It occurs to me that n-j n-e-t scales undoubtedly seem more mysterious than either just or equal tempered tunings because the mathematical basis of n-j n-e-t scales is not as obvious. This post attempts to clarify that mathematical basis. First, a word about traditional tunings: If we describe the scale steps of any given tuning as a number between 0 and 1, the mathematical basis of just tunings is simple and straightforward: k[n] = A/B where both A and B are integers. The only ambiguity here is the question of whether or not A and B are "small." This is clearly a matter of personal taste. John Chalmers and Your Humble E-Mail Correspondent consider many of Erv Wilson's CPS tuning to be examples of just intonation, and thus made up of the ratio of "small" integers: however, many of those integers are not small in the usual sense defined by Partch, Doty, Johnston, et al. For example, the [1,7, 19, 37] hexany consists of ratios of numbers to the generator 7*19*37 = 4921. This number is "small" compared to a googol, or to the number of atoms in the Local Cluster of galaxies: but it is large compared to, say, 31 or 43. Thus, some tuning theorists would consider this Wilson CPS tuning *not* to be a just tuning, while others *would.* It is a matter of taste. For equal-tempered tunings, the scale- step is described with equal simplicity: k[n] = 2^[n/M] In this case every scale-step of every equal-tempered tuning is described by an irrational number. So far, so good. Just tunings have scale steps described by ratios of integers: equal-tempered tunings have scale steps described by irrational numbers. An irrational number is a real solution of an algebraic equation. An integer is the real solution of an ordinal arithmetic equation. Thus tunings can be defined by the class of equations to which the ratios which define the scale-steps of the tuning form solutions. For example: Algebraic equations involve a finite number of integers raised to integer powers: say, X = A + By^2 + Cy^5. Ordinal arithmetic equations involve integers: X = A/C + D or X = A*B, etc. (You might have noticed that my definition of integers is circular. This is because the question of what an integer is happens to be a very deep one. It is indeed very, *very* difficult to define an integer in abstract terms without using an equation which involves integers. If memory serves, the Bourbaki collective devoted an entire volume to the definition of an integer.) This gives us a handle on what is meant by a "non-just non-equal-tempered scale." Clearly, if the scale-steps of a just scale are *integers* and if the scale-steps of an equal-tempered scale are *irrationals,* then the scale-steps of a non-just non- equal-tempered scale must be given by transcendental numbers. What is a transcendental number? How is such a number defined? Is there a general mathematical procedure for obtaining transcendental numbers? This, as it happens, is also a deep question. In fact it is often *VERY* difficult to *prove* mathematically that a given number is transcendental. For instance, e^pi is known to be transcendental--but pi^e has never been proven transcendental (though most mathematicians believe it to be). In fact the number pi itself was not proven transcendental until 1882. (F. Lindemann, in the paper "Ueber die Zahl Pi," took that honor.) One of the perplexities which attend transcendental numbers is the fact that while there is a general criterion for determining whether a number is an integer (Does it satisfy an ordinal arithmetic equation?) and for determining whether a number is irrational (Does it *NOT* satisfy an ordinary arithmetic equation and is it a real root ofan algebraic equation?) there appears to be NO general criterion for determining whether a number is transcendental. Consequently, many different (non-obvious, counterintuitive) equations generate transcendental numbers. For example, the number i^i is transcendental--in fact it is equal to e^[-pi/2] = 0.2078795. ("i" here refers to the square root of -1.) This sounds absolutely insane, but it happens to be true...and provable! Like quantum mechanics, many of the results of mathematics follow the rule: "If it makes intuitive sense to you, then you don't really understand it." The real part of log(i) is also transcendental: log(i) = i*pi/2. Thus, while there's no general formula or method for generating transcenedental numbers, quite a few transcendental numbers have been discovered over the last few milennia...mainly by chance. Here are few: Liouville numbers have been proven transcendental. They were discovered in 1851 (much later than pi or e) and are given by the formula: Sum from k = 1 to infinity over a[k]*r^[-k!] where "!" means "factorial" and a[k] is an integer twixt 0 and r. There are infinitely many Liouville numbers. For instance, if all a[k] = 1 and base r = 10, we get 1/10 + 1/(10^[1*2]) + 1/(10^[1*2*3]) + ... = 0.1100010000000000000000001000... Depending on how the a[k] are chosen, many different Liouville numbers arise. One might pick the a[k] as the fractional part of the decimal expansion of e, or pi, or of an irrational number such as 2^[1/3], etc. Euler's constant gamma is transcendental. It's given by the limit for n = - infinity of the series 1 + 1/2 + 1/3 + 1/4 + ...+ 1/n - ln(n) Gamma = 0.577215... Catlan's constant is another lesser-known number. It has not yet been proven transcendental, but mathematicians widely believe it to be. It's given by the formula G = sum of (-1)^k/(2k+ 1)^2 = 1 - 1/9 + 1/25 - 1/49... Chapernowne's number is also generally believed transcendental. It is constructed by concatenating the digits of the positive integers: C = 0.1234567891011121314151617181920... As mentioned in my series of posts on generating non-just non-equal-tempered scales last year, the zeta function also yields transcendental numbers. However, my post did not specify that the zeta function must be evaluated at rational points: zeta(K) can be either real or imaginary, since K can be either real or imaginary. For K not equal to a real integer, zeta(K) is in general not transcendental. However this still leaves us with zeta(2), zeta(3), zeta(5), etc., all trascendental. Of particular interest in the construction of non-just non-equal-tempered scales is the Gelfond-Schneider theorem. According to this theorem, any number of the form a^b is trascendental where a and b are algebraic (a <> 0, a <> 1) and b is not a rational number. This formula spews out an infinite number of transcendental numbers, since (for example) Hilbert's number, 2^[sqrt(2)] is clearly transcendental, ditto 2^[sqrt(5)], 3^[fifth root of 7], etc. Feigenbaum numbers are also transcendental. These numbers arise from chaos theory and are related to properties of dynamical systems which exhibit period-doubling and other chaotic behaviour. The Feigenbaum number is 4.66920160910299067185320382046620161725... Alas, equations involving transcendental numbers do not necessarily produce solutions which are transcendental. e^[i*pi] = 1, an integer, while (e^[i*pi]) + 2*phi = sqrt(5), an irrational number. Clearly phi, the Golden Ratio, is not transcendental since it is the solution of an algebraic equation: phi = [sqrt(5) - 1]/2 = (5^[1/2] - 1)/2 One of my own amateur mathematical discoveries (I've not seen it published elsewhere, at any rate) is an infinite number series given by the iterated absolute log of K, where K is an integer. I believe (but cannot prove) that the scale-steps given by the terms of this series form a non-just non-equal-tempered scale. This is a peculiar and interesting series of numbers since the terms oscillate between 0 and 1. The first term is transcendental, but I have not been able to prove that the succeeding terms are (or are not) transcendental. For instance, the first 10 terms of the iterated abs log of 2 are: i[1] = abs(log(2)) = 0.30103... i[2] = abs(log(i[1])) = 0.5213902... i[3] = abs(log(i[2])) = 0.2828372... i[4] = abs(log(i[3])) = 0.5484636... i[5] = abs(log(i[4])) = 0.2608521... i[6] = abs(log(i[5])) = 0.2338806... i[7] = abs(log(i[6])) = 0.6310057... i[8] = abs(log(i[7])) = 0.1999666... i[9] = abs(log(i[8])) = 0.6990424... i[10] = abs(log(i[9])) = 0.1554964... and so on. There does not appear to be an obvious pattern to the terms. If one were so inclined, one might call this the McLaren series: this is surely the first post on this tuning forum to feature an original mathematical discovery, albeit a trivial one. As can be seen, all of the above methods for generating transcendental numbers can produce an infinite variety of 'em. Depending on the pattern of generators (the numbers you plug into the various equations to produce transcendental numbers), you get an endless variety of non-just non-equal-tempered scales. The choice of whether to terminate the series with a 2/1 or not is a matter of taste. (One might call it "terminating the series with extreme prejudice.") In that case, one obtains a non-just non-equal-tempered scale which repeats at the octave. Choosing another termination integer (or irrational) would produce a non-just non-equal-tempered scale without octaves. As can readily be seen, these are the obverse of the equal tempered class of octave and non-octave scales. My limited experiments with non-just non-equal-tempered octave and n-j n-e-t non-octave scales appear to show that there is a marked difference in "sound" between the two classes of tunings. As Gary Morrison so aptly put it, "non-octave scales sound like rich thick chocolate milk shakes." They are very exotic and harmonically rich, and seem to have a sonic colouration which lends an eldritch quality to the compositions one produces in such scales. For n-j n-e-t non-octave scales, the same appears to be true, only more so. They are among the most exotic and sonically luxuriant of all tunings in my experience, and an extraordinary realm for new music exploration. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 2 Nov 1995 18:40 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA29513; Thu, 2 Nov 1995 08:40:23 -0800 Date: Thu, 2 Nov 1995 08:40:23 -0800 Message-Id: <9511020838.aa10245@cyber.cyber.net> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu