[seqfan] Into subtleties of musical information

[Veikko Pohjola]

It is worth mentioning that the following numbers seem to appear in the sequence of differences:
3, 6, 7, 9, 10, 12, 13, 16, 19.
Each of them has its own density and distribution exhibiting its own rhythmic pattern and thus resulting in polyrythmic music, a phenomenon studied by Robert Walker, for instance in the case of Fibonacci numbers (See reference in A000045).
Veikko

Antti Karttunen kirjoitti 25.5.2015 kello 5.42:

> Veikko wrote in http://list.seqfan.eu/pipermail/seqfan/2015-May/014904.html
> 
>> Well, I did not refer directly to the values given in A258024, which are the n’s which reduce to 1 instead of 0, but to their differences, which of course is a separate, though related, sequence
>> 3, 19, 3, 19, 3, 19, 3, 19, 3, 13, 6, 3, 7, 6, 6, 3, 7, 6, 6, 3, 13, 6, 3, 3, 10, 6, 3, 3, 10, 6, 3, 3, 16, 3, 3, 16, 3….
> 
> This is now in https://oeis.org/draft/A258200
> 
>> In the search of the pattern this latter is more informative as can be readily seen. It may well deserve a separate sequence, especially if its musicality either as such or due to the potential of drawing conclusions on its basis has some interest. I can submit it and then we’ll see.
> 
>> Then there is a possibility for a sequence of n’s which reduce to 0 instead of 1,
> 
> This is now https://oeis.org/draft/A258022
> 
>> and their differences…
> 
> It is:
> 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
> 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1,
> 1, 2, 1, 2
> but I didn't create that yet.
> 
> Also, I created
> https://oeis.org/draft/A258021
> "Eventual fixed point of map x -> floor(tan(x)) when starting the
> iteration with the initial value x = n."
> 
> and
> 
> https://oeis.org/draft/A258020
> "Number of steps to reach either 0 or 1 with map x -> floor(tan(x))
> when starting iteration with the initial value x = n. "
> 
> Now, because I don't trust that in MIT/GNU Scheme (floor->exact (tan
> n)) would not at some point be one off because of the loss of the
> precision, I didn't try to compute b-file for any of these sequences.
> Could somebody with real CAS (and the knowledge how tweaking various
> parameters affects that risk) do that, up to say a few thousands at
> least? (For A258020 and A258200 at least).
> 
> 
> Regards,
> 
> Antti
> 
> 
> 
> 
> 
> On Mon, May 25, 2015 at 12:56 AM, Antti Karttunen
> <antti.karttunen at gmail.com> wrote:
>> On Sun, May 24, 2015 at 12:00 AM,  <seqfan-request at list.seqfan.eu> wrote:
>> 
>>> ------------------------------
>>> 
>>> Message: 16
>>> Date: Sun, 24 May 2015 00:00:32 +0300
>>> From: Veikko Pohjola <veikko at nordem.fi>
>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> Subject: [seqfan] Musical information about a sequence
>>> Message-ID: <2C234C13-4D18-4D6C-AC08-E57A1A0DAFB0 at nordem.fi>
>>> Content-Type: text/plain; charset=windows-1252
>>> 
>>> Dear seqfans,
>>> 
>>> Applying floor(tan(n)) repeatedly, a limiting sequence results composed of 0’s and 1’s only. The proportion of 1’s is somewhat over 12% and they distribute interestingly forming a repeating pattern. Converting the sequence of distances between the positions of 1’s into music (say piano) turns it to fascinating music manifesting a steady beat with a theme and delicate variations. Counting the number of beats in a sequence of known length permits to assess the number of individual sounds (terms) within each measure to about 42.
>>> 
>>> I am wondering whether such a steady beat could be inherited from the periodic nature of the mother function (tan) and if so, should the length of the pattern thus be predicted.
>>> And in general, are this sort of musical findings regarded to belong to recreational domain and not at all to hard mathematics, not ...
>> 
>> Dear Veikko,
>> 
>> I don't care what other people think about what is "hard enough
>> mathematics" (some people have very restrictive biases), but think
>> that your find is very interesting.
>> 
>> I have been myself trying to find good examples of the general idea
>> behind Per Nørgård's "infinity sequence"
>> http://oeis.org/A004718
>> "invented in an attempt to unify in a perfect way repetition and variation".
>> (See also https://oeis.org/A056239 for another Danish comment in
>> another, not related sequence. Also https://oeis.org/A126759 )
>> 
>> In other words, anything on the sweet but rare region between (too
>> much) regularity (most base-sequences) and (too much) chaos. (Compare
>> also to some Wolfram's CA-classifications, although I'm not now
>> interested about Turing-capability. Also, it seems that human mind
>> cannot relish complete chaos until it is regularly repeated and thus
>> "amplified"?)
>> 
>> So far, my attempts have concentrated on "entanglement-permutations"
>> and "beanstalk-sequences" (my neologisms but not my inventions) both
>> of which mix together a repeating pattern with some "new material",
>> although in different ways. I haven't yet much experimented of
>> actually producing any sounds of these, except some random playing
>> with "Listen-button" which leaves much to be desired regarding the
>> actual mapping, not just to notes but to rhythm/dynamics as well (or
>> maybe I should learn to use its various options better?) In any case,
>> maybe it's better to leave their exact mapping to rhythm and sounds to
>> more musical talents, and for me to just keep on producing more
>> patterns and hope that some of them are mathematically interesting and
>> useful as well.
>> 
>> 
>>> ... even when some useful mathematical information could be obtained by listening.
>> 
>> For this, please see also:
>> http://www.moz.ac.at/sem/lehre/lib/bib/software/cm/Notes_from_the_Metalevel/chaos.html
>> 
>> 
>> 
>> Terveisin,
>> 
>> Antti
>> 
>>> 
>>> Best regards,
>>> Veikko Pohjola
>>> 
>