[seqfan] Re: Musical information about a sequence

[Antti Karttunen]

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
>>