# What rotation does the Bjorklund algorithm produce - and why is it special?

[EDIT - I added "... is it special" to the title. If you are using some rotated Euclid in your patterns - how did you choose the rotation? Or, why did you choose to not rotate?]

I was looking into properties of Euclidean patterns, and possible alternate implementations
( euclidOff \neq euclidOffBool )

The important property is "even-ness", and this is defined, and unique, only up to rotation. My question is - what rotation do we actually want? And what rotation is produced by Bjorklund's algorithm?

The papers I find (e.g., Erik D. Demaine, Francisco Gomez-Martin, Henk Meijer, David Rappaport, Perouz Taslakian, Godfried T. Toussaint, Terry Winograd, and David R. Wood: The Distance Geometry of Music) are silent on this.

Example: all rotations for `bjorklund (5, 13)` are (in lexicographically decreasing order)

``````([1,0,1,0,0,1,0,1,0,0,1,0,0],-5,False)
([1,0,1,0,0,1,0,0,1,0,1,0,0],-4,False)
([1,0,0,1,0,1,0,0,1,0,1,0,0],-3,True)
([1,0,0,1,0,1,0,0,1,0,0,1,0],-2,False)
([1,0,0,1,0,0,1,0,1,0,0,1,0],-1,False)
``````

the second component is also the start value of a simple algorithm that produces the sequence, and the third component says `True` to indicate the sequence returned by tidal's actual implementation. That index goes all over the place, e.g.,

``````([1,1,0,1,0,1,1,0,1,0,1,0],-7,False)
([1,1,0,1,0,1,0,1,1,0,1,0],-6,False)
([1,0,1,1,0,1,0,1,1,0,1,0],-5,True)
([1,0,1,1,0,1,0,1,0,1,1,0],-4,False)
([1,0,1,0,1,1,0,1,0,1,1,0],-3,False)
([1,0,1,0,1,1,0,1,0,1,0,1],-2,False)
([1,0,1,0,1,0,1,1,0,1,0,1],-1,False)

([1,0,0,1,0,0,0,1,0,0,0],-3,False)
([1,0,0,0,1,0,0,1,0,0,0],-2,False)
([1,0,0,0,1,0,0,0,1,0,0],-1,True)

([1,0,1,0,1,0,1,0,1,0,0],-5,True)
([1,0,1,0,1,0,1,0,0,1,0],-4,False)
([1,0,1,0,1,0,0,1,0,1,0],-3,False)
([1,0,1,0,0,1,0,1,0,1,0],-2,False)
([1,0,0,1,0,1,0,1,0,1,0],-1,False)
``````

we could also discuss the "proper" definition of "even-ness" but that's easier since several are equivalent (see cited paper). The definition I like best is that the sets of factors (of equal length) are balanced (cf. lothaire chapter 3 - Sturmian words, p. 42)

1 Like

I haven't looked at your references yet but if you haven't seen it, I think you should find something around syncopation in Toussaint's book "the geometry of rhythm".

But I like Kofi Agawu's criticism of this as being only about 'paper rhythms' -- that when we rotate rhythms they look like the same rhythm on paper.. But actually what you end up with is a completely different rhythm, you don't feel the rotation, it doesn't exist in practice.

Thanks for the references. I will browse Toussaint's book (I have it, but not right in front of me now).

Where is "Kofi Agawu's criticism" published? (I don't see it here https://www.gc.cuny.edu/people/kofi-agawu)

NB - I was thinking about euclid rotations exactly three years ago, when I wrote this: euclid Â· master Â· waldmann / computer-mu Â· GitLab - and I completely forgot about that .. so, apologies for double posting. Serves to show that my memory is bad, but my consistency is still good? (contents and references are identical) ...

I think it was "Structural Analysis or Cultural Analysis? Competing Perspectives on the â€śStandard Patternâ€ť of West African Rhythm"

OK, from Agawu's paper:

Skepticism about the relevance of â€śrotationâ€ť and other such terms
is therefore not only not inappropriate, but potentially illuminating ...

(the exercise is to remove double negations ...) but yes, I think it means that rotation should not be considered as an equivalence operation on rhythms.

But then (my conclusion) choosing the starting point of an Euclidean rhythm becomes even more important, and it's not reasonable to just copy the rotation produced by Bjorklund's algorithm, which I think is - arbitrary. (I.e., can only be explained by performing the algorithm, and not by a property of its output)

I was re-reading Toussaint Chapter 21, which presents Bjorklund's algorithm
(I assume - it is not named as such) but does not answer my question (I think.)

I will try to read some more writings by Agawu. Thanks again for the reference.