[microsound] Piano Paradox

[Andrew Salch]

On Tue, 16 Feb 2010, Adam Davis wrote:
> Imagine a piano. The length of one string of this piano, in whatever unit or
> order of magnitude, is equal to the sum total of all the numbers that
> comprise the aleph-0 set.


This sum diverges (to infinity), which I guess is your point; this means 
that the wave in the string produced by striking the string will never 
reach the far end of the string, so the string has no harmonic frequency; 
the sound produced by striking the string is essentially the same as if 
the string is completely damped.


> Similarly, the next string is, in whatever
> measurement, equal in length to the sum total of the figures that make the
> aleph-1 set.


You would have to be more precise about what you mean about "figures that 
make the aleph-1 set" (do you mean countable ordinal numbers? How do you 
take the sum of all such things?) but in any case, even if such a thing 
were possible (which is extremely dubious), you would have another string 
whose sound, when struck, is essentially the same as that of a completely 
damped string.



> The string after that, aleph-2, and so on. When the piano is
> played, will the strings sound different pitches, if at all?


No, for the reasons stated above.


> How could the
> tension of the strings be kept if the ends could not be reached? Will there
> even be other ends for the keys, hammers and other mechanisms to situate? If
> the piano has an infinite amount of keys with an infinite amount of
> respective strings, are a highest starting note, aleph-0, even possible?


If this last question is supposed to ask whether there exists an infinite 
cardinal number less than aleph_0, the answer is no--it is not hard to 
prove (at least with the axiom of choice) that any infinite set admits an 
injection of the natural numbers into it, meaning that aleph_0 is less 
than or equal to every other infinite cardinal.


It might be more interesting to consider the following: suppose that we 
start playback of a digital signal whose sample rate is X hz. So 
immediately on beginning playback, the first sample is "played," i.e., 
some voltage is sent to the speakers; then, 1/X seconds after playback 
begins, the next sample is played, i.e., the voltage sent to the speakers 
is changed; then, 2/X seconds after playback begins, the next sample is 
played, and so on. In other words, when playing back a digital sample of 
sample rate X hz, the voltage sent to the speakers is changed N/X seconds 
after playback begins, where N is any nonnegative integer. So, even if the 
signal is left to continue playing forever, only a countable number of 
samples are ever played.

It is typical to think of an analog signal as one which is able to produce 
a change in the voltage sent to the speakers at any moment of time, or in 
other words, an analog signal may change voltage after r seconds, where r 
is any real number. (Representing the set of all moments in time as the 
continuum, or the real numbers, is common, and probably about as justified 
as representing space as locally looking like a product of copies of the 
real numbers: it's a good model and works well for most purposes in 
physics and engineering but it is perhaps not how space, or time, really 
work, e.g. because of the Planck length in the case of space. But for most 
purposes it is probably appropriate to think of time as a continuum.)

Suppose that one is able to produce a signal which can change voltage at 
any moment in time that any digital signal can change voltage. In other 
words, this signal can change voltage r/s seconds after playback begins, 
where r and s are integers, and it can ONLY change voltage at moments in 
time which are r/s seconds after playback begins, where r and s are 
integers. (This means that our hypothetical signal can change voltage 1/2 
a second after playback begins, or 1.8 seconds after playback begins, for 
example, but it CANNOT change voltage exactly sqrt(2) seconds after 
playback begins, or exactly pi seconds after playback begins, etc.) Let's 
call this hypothetical signal the "perfect digital signal" since it is 
able to record at any sampling rate that any digital signal can record at. 
The set of numbers of the form r/s, where r and s are integers, are called 
the rational numbers.

The rational numbers are countable, i.e., their cardinality is that of the 
cardinal number aleph_0. However, the real numbers are uncountable--their 
cardinality is strictly greater than the cardinal number aleph_0--and this 
means that, if an analog signal is played for any nonzero length of time 
at all, the voltage is able to change an uncountable number of times; but 
no matter how long the perfect digital signal is allowed to play for, it 
can only change voltage a countable (even if infinite) number of times. In 
some sense this perfect digital signal still carries less information than 
an analog signal, even though both signals carry infinitely many samples.

It is extremely hard to believe that any human ear could hear this 
difference in information carried, i.e., that a human ear could hear the 
difference between the perfect digital signal and an analog signal. Maybe 
it is still an interesting idea. For what it's worth, the "generalized 
continuum hypothesis" (you can Google this if you want to read about it) 
is a statement about the possible cardinal numbers strictly between 
aleph_0 and the cardinality of the real numbers; if the GCH is true, then 
there are essentially no possible signals which carry more information 
than the perfect digital signal, but less information than an analog 
signal; but if the GCH is false then such signals do exist. The GCH was 
established in the 1960s to be undecidable in the standard axioms for 
mathematics; in other words, the standard logical axioms of mathematics 
are not enough to prove the existence OR the non-existence of signals 
carrying more information than the perfect digital signal but less 
information than an analog signal.

Maybe some of this is interesting to some people on this list? I hope so.