source file: mills2.txt Date: Mon, 18 Dec 1995 18:19:28 -0800 From: "John H. Chalmers" From: mclaren Subject: transcendental numbers, the entropy of musical scales, and Shannon's theorems applied to tunings --- Some time ago, Gary Morrison very unjustly impugned his own mathematical abilities (which are considerable) in the process of asking the question: What is a transcendental number? Looking back at my previous posts on the subject, the source of the confusion is easy to spot. My statement: "In general, it is extremely difficult to determine whether a given number is transcendental or not" could be interpreted in several ways. One could take the statement to mean that the Turing halting problem for a general algorithm that sieves the field of reals and always produces all transcendental numbers has not yet been solved. Or one could take the statement to mean: "It's impossible to define the term 'transcendental number.'" This is surely untrue. There are many ways to define a transcendental number. One way is: a transcendental number is the solution of a transcendental equation--that is, an equation involving logarithms or antilogarithms. This is not always true, since the equation e^[i*pi] + X = 0 has the solution 1. However, perhaps one could say that a transcendental number is one which arises *only* as the solution of an equation involving logarithms or antilogarithms. (Manuel and John Fitch may jump on me for this one; it may not be 100% true all the time. Can you think of a counter- example?) Another way of defining a transcendental number is: It's a number which is not the solution of an ordinal arithmetic equation or an algebraic equation with rational-fraction coefficients and exponents and a finite number of terms. Algebraic and arithmetic equations can, of course, also involve an infinite number of terms: pi and e both arise as a result of many different infinite series, the most spectacular of which were discovered by Srinivasas Ramanujan. Pi and e also arise from equations involving logarithms and antilogarithms: X^[i*pi] + 1 = 0 defines e, while e^[i*X] + 1 = 0 defines pi. This latter definition is close to Grolier's, although strictly speaking Grolier's definition is incorrect since it appears to leave out the requirement that a transcendental number cannot be the solution of an arithmetic or algebraic equation with a *finite* number of terms. (Many of Ramanujan's series involve algebraic products & quotients but an infinite number of terms.) Transcendental numbers also tend to arise when the exponents of an algebraic equation are imaginary. Another way of defining a transcendental number is by the amount of information required to describe it. How complex an algorithm is needed to generate the number? How long does it take to run? Claude Shannon proved elegantly that the amount of information required to generate (or parse) a message is proportional to the log to the base 2 of the number of bits in the message. By this standard, an integer requires very little information to parse (or generate). Even if very long, any finite integer can be entirely written down. Once the last digit is written, your're finished. A simple "copy this array" algorithm suffices. A rational fraction requires somewhat more information to parse (or generate). However, the algorithm required to describe the digits in the number 1/9 (for example) is still simple: "Keep writing ones." I.e., 1/9 = .11111... The information contained in an algebraic irrational is somewhat greater. (There are two kinds of irrational reals: algebraic irrationals and transcendental irrationals. Algebraic irrationals are those numbers which arise as real roots of equations involving rational coefficients and rational exponents. Transcendental irrationals arise when the algebraic exponent, for example, is itself an algebraic irrational--as in Hilbert's number, 2^[sqrt(2)]. ) In the case of an algebraic irrational, the algorithm required to generate the number is lengthier than that required to generate the decimal expansion of a rational fraction--thus the algebraic irrational contains more information than does a finite rational fraction. However, transcendental numbers appear to require the most information of all. To my knowledge, there is as yet no known algorithm by which the entire field of reals may be sieved and by which all transcendental numbers will always be found. Yet, since we live in a Goedelian universe, such an algorithm might well exist-- and worse still, it might be very simple. Another way of stating this proposition is that the simplest description of a transcendental number appears to be...itself. If true, this makes transcendental number unique. It has been speculated that the digits in the decimal expansion of transcendental numbers never repeat. One subscriber has even stated as much on this forum. However, this proposition has never been proven mathematically. (Most mathematicians believe this supposition to be true, but belief is *not* the same thing as proof.) Thus it is not yet known how random the digits of (say) pi or e really are. Many functions, graphs and plots seem random from one perspective, but when rotated they reveal hidden patterns. The same might be true of pi or e. So it's entirely conceivable that out beyond a googol decimal places, all the digits of pi might turn to 1's, for example. A disturbing thought...yet one which cannot be dismissed until a proof of the true randomness of pi's digits is found. In view of this possibility, the randomness of a transcendental number's digits cannot be defined. If an infinite number of pi's digits are, say, 1, or 3, or 9, or what-have-you, after a given point, then clearly the number is hardly random at all. However, in order to determine this by brute force we would have to calculate an *infinite* number of digits in pi's decimal expansion. For no matter how far we go, there's always the possibility that at the next digit, the digits fall into a predictable and eternally repeating pattern. Thus the devilish undecidability in so many cases of the question: "Is a given number transcendental?" For example: the number 1 + a googol zeroes + 1 + an infinite number of zeroes might be transcendental. Is it? I don't know. You don't know. It's impossible to calculate. No proof exists that this number is transcendental (or not transcendental). Thus we will likely never know. Mathematicians widely believe the digits in pi to be completely random. If so, this provides another way of defining transcendental numbers: by the power spectral density of their decimal expansion. By taking the Fourier transform of a number's decimal expansion, its spectrum can be determined. The spectrum will contain Dirac delta functions at those frequencies which define a periodicity. Thus the number 1.212121212.... will have a sharp spike in its spectrum at 2, since the decimal expansion has a periodicity of 2 digits. There will be little energy anywhere else. Integers have an FFT which forms a narrow or broad Gaussian, depending on the length of the integer. Rational fractions have broader spectra: algebraic irrationals have spectra which are broader still. Transcendental numbers (if the mathematicians are right) likely have flat spectra: that is, their digits never repeat. (This has not been proven, but is universally believed.) Since Parseval's Theorem tells us that the Fourier Transform of the autocorrelation function is the power spectral density, this is only as we would expect--it is, after all, merely another way of saying that the digits of transcendental numbers appear to exhibit no correlation with one another. There is no pattern hidden in the decimal expansion. Returning to the question of musical scales, this gives us another way of defining tunings: by their entropy. Since statistical mechanics teaches us that entropy is a measure of the total number of available states in a system, clearly the number of states available in a number's decimal expansion is greatest for transcendental numbers and least for integers. The next digit in a transcendental number could be anything: the number has maximum extropy. Thus the entropy of a musical scale and be defined by summing the entropies of the numbers which comprise it. The harmonic series clearly has least entropy; next come JI scales made up of rational fractions, next equal tempered scales made up of algebraic irrationals, and finally non-just non-equal-tempered scales made up of transcendental numbers. In this sense, transcendental numbers can be thought of as micro-universes, containing an infinite possible number of available states, and requiring an infinite amount of energy to parse. Although the full workings of the ear/brain system are not yet completely understood, it is safe to assume that however it operates, the human auditory system can be modelled as some sort of state-space machine. In this case, we have a possible explanation for the response of the ear to the octave as well as Enrique Moreno's extended chroma phenomenon. Since information is logarithmically proportional to energy (another of Shannon's theorems), it requires the least amount of information/energy to parse a musical interval which is an integer ratio of another interval. Next most energy is required to parse JI scales, still more to parse scales involving algebraic irrationals (ET scales), and the most energy/information is required when parsing non-just non-equal-tempered scales. This accords well with Gary Morrison's and my own findings about non-octave scales. JI scales sound more "bland" than equal tempered scales--or one might prefer to put it the other way around and say that Nth root of 2 tunings sound muddier and more turbulent than JI tunings. Non-octave scales sound "like thick rich chocolate milk shakes," as Gary has pointed out, and my own experience with non-just non-equal- tempered scales indicates that these tunings sound most complex and sonically luxuriant of all. However, this hypothesis is not supported by the psychoacoustic data which demonstrate clearly that listeners universally hear intervals about 15 cents > the octave as "pure octaves" and intervals of 2.0 as "too flat" and "out of tune." Moreover, this assumes that the human auditory system can be modelled as a Turing machine which obeys linearity and the superposition principle. However, the data appears to indicate that many parts of the human auditory system are non-linear and do not obey the superposition principle. In this case, the analogy with finite automata may not be apt. Lastly, if one wanted to go completely over the edge, one could describe numbers in terms of their dB signal-to-noise ratio by comparing the normalizing power spectral density of the integer 1 to the psd of the target number. In this case transcendentals would exhibit a zero dB signal-to-noise ratio, while integers would exhibit no noise whatever and thus an infinite signal-to-noise ratio. (Describing numbers in terms of their signal-to-noise ratio sounds absolutely insane until you realize that this explains the extreme noisiness of digital reverb systems; the input signal becomes progressively degraded by roundoff error during its trip through the recirulating delay lines of the reverb algorithm, and thus each sample of the input suffers a progressive randomization of its bits and thus a progressive decrease in its singal-to-noise ratio.) Incidentally, Gary's purported "mathematical idiocy" pales before my own. Alert readers will still be guffawing at my statement that "i is the square root of -1." Obviously untrue, since -i is also the square root of -1... As Manuel op de Coul so delicately pointed out during our meeting across the street from Disneyland (an apt venue for wild-eyed microtonalists). Moreover, e^[i*pi] = -1, not 1. No duh dude, as Gauss would doubtless say. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 19 Dec 1995 15:20 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id GAA28265; Tue, 19 Dec 1995 06:20:47 -0800 Date: Tue, 19 Dec 1995 06:20:47 -0800 Message-Id: <199512191519.QAA09965@elevator.source.co.at> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu