From: mclaren
Subject: transcendental numbers
---
Some weeks back, 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,
it's clear that they are not
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 agebraic
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 of that a transcendental number
cannot be the solution of an arithmetic or
algebraic equation with a *finite* number of
terms. (Many of Ramaujan's series involve
algebraic products & quotients with an infinite
number of terms.)
Many functions, graphs and plots seem random from
one perspective, but when rotated they reveal hidden
patterns. The same might be true of pir or e.
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 relatively little
information to parse (or generate). Even if very
long, any finite integer can be entirely written
down.
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 after the decimal point."
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 fraction exponents.
Transendental irrationals arise when the
algebraic exponent, for example, is itself
an algebraic irrational--as in Hilbert's number,
2^[sqrt(2)]. )
Transcendental numbers also tend to arise when
the exponents of an algebraic equation are
imaginary.
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.
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, there might well be such an algorithm--
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
certainly makes transcendental number unique.
It has been speculated that the numbers in the
decimal expansion of transcndental numbers
never repeat. One subscriber has even stated as
much on this forum. However, this proposition
has never been proven mathematically. Thus it
is not yet known how random the digits of (say)
pi or e really are. It's always 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 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 pit'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. Thus we will likely never
know.
Mathematicians widely believe the digits
in pi to be completely random. If so, this
gives another way of defining transcendental
numbers: by the power sepctral density of
the order of their digits.
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 that's the periodicity
of the number. 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 irrations have
spectra which are broader still.
Transcendental numbers (if the
mathematicians are right) 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--since it is
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 transcndental 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.
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 finite-state 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. Yet much
data indicates taht erposition 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 (and apt venue for wild-eyed
microtonalists).
Moreover, e^[i*pi] = -1, not 1.
No duh dude, as Gauss would doubtless say.
--mclaren
