# [seqfan] Re: Into subtleties of musical information

Antti Karttunen antti.karttunen at gmail.com
Tue May 26 11:39:10 CEST 2015

```Hmm, just realized that there shouldn't be any a priori reason that
floor(tan(n)) could not have also other fixed points than just 0 and
1.
For example, we have A000503(37362253) = 37754921, and 37754921 -
37362253 = 392668 (just one percent difference between the argument
and the result).

So, I have to define some of these sequences more defensively,
allowing the possibility for other fixed points than 0 or 1 as well.

Best,

Antti

On Tue, May 26, 2015 at 8:47 AM, Veikko Pohjola <veikko at nordem.fi> wrote:
> 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.
>>>
>>> http://www.moz.ac.at/sem/lehre/lib/bib/software/cm/Notes_from_the_Metalevel/chaos.html
>>>
>>>
>>>
>>> Terveisin,
>>>
>>> Antti
>>>
>>>>
>>>> Best regards,
>>>> Veikko Pohjola
>>>>
>>
>
```