[seqfan] Re: XronoMorph - Overlaid polygons and music

[Bob Selcoe]

Hi  Charles and Seqfans,

I wrote: >>>we have (2^16-1)*(2^15-1)*(2^14-1) ~ 35 trillion combinations.

I should have been paying more attention!  I think it really ought to be
(2^16-1)*(2^16-2)*(2^16-3) applying circular shifts.  Charles, aren't there
exactly 16 circular shifts here, one for each starting beat (represented by
each of the 16 vertices)?  That would give us ~ 17.6 trillion.

Cheers,
Bob

--------------------------------------------------
From: "Charles Greathouse" <charles.greathouse at case.edu>
Sent: Monday, July 18, 2016 11:33 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] XronoMorph - Overlaid polygons and music

> I found an interesting article:
> http://theconversation.com/how-a-little-mathematics-can-help-create-some-beautiful-music-61812
>
> It discusses music generated from fairly simple mathematical structures:
> (small) subsets of the vertices of a polygon. Each subset is displayed as 
> a
> polygon of a different color.
>
> From the article: "There are more than 17 trillion different rhythms, and
> that is only counting rhythms with three levels where every beat occurs at
> one of 16 distinct time locations (16 being a very common temporal
> subdivision in music)."
>
> I'm trying to figure out how this number is computed. I've tried a few
> different ways but none are close enough. binomial(2^16, 3) is too big, 
> but
> reducing it by circular shift makes it a bit more than 1/16th the size
> which is too small. Any ideas? (I assume that by "more than 17 trillion"
> they mean between 17e12 and 18e12.)
>
> Charles Greathouse
> Case Western Reserve University
>
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