Viggo Brun algorithm

Thanks for the suggestions regarding the Brun algorithm.

I was happy to stumble across quite a good explanation of it in my copy
of M. Joel Mandelbaum's 1962 dissertation (p. 370), although it is only
applied to 3 ratios at a time here (and it's not clear how to
extrapolate this to more--I couldn't find any of Paul Ehrlich's quote of
Chalmers-suggested articles in the Minneapolis libraries other than the
Barbour one in the 1948 American Mathematical Monthly).

I certainly can see its value for the JI community in particular, or for
those who'd like to create precise subdivisions (ET or not) of some
'substitute for the octave' ratio.

However, my effort, which was to find a close match to several ratios
(without necessarily any precise match), seems still to be most easily
accomplished by seeking out common factors to values near the logs of
the ratios by experimentation with a spreadsheet.

I suspect that a spline algorithm might also be a more formal way to
solve this kind of problem. Any ideas along those lines?

The most useful to me so far is 27.35 ET, nearly the same as 44-CET,
which matches 7/6, 6/5, 11/9, 5/4, and 9/7 within either 3 or 8 cents.

I'm still looking for one or more which would match the inversion of
these ratios (meaning each of the above * 4/3) with similar accuracy,
and also have a reasonably useable fifth.

>However, my effort, which was to find a close match to several ratios
>(without necessarily any precise match), seems still to be most easily
>accomplished by seeking out common factors to values near the logs of
>the ratios by experimentation with a spreadsheet.

Harold, you may want to check out the method Gary Morrison used to find
88CET. I think it's a graphic version of the GCF method. The neat thing
about it was it could be done by hand, with paper and pencil, and you can
see exactly the compromise you're making. I have a sheet on it I could dig
up, or maybe Gary could explain it. Did I see it on the web somewhere, Gary?

-Carl

>I certainly can see its value for the JI community in particular, or for
>those who'd like to create precise subdivisions (ET or not) of some
>'substitute for the octave' ratio.

These days I see no practical reason to spend too much time thinking about
this, since a brute-force computing approach can find the best answer in a
very short period of time.

>The most useful to me so far is 27.35 ET, nearly the same as 44-CET,
>which matches 7/6, 6/5, 11/9, 5/4, and 9/7 within either 3 or 8 cents.

The errors in those intervals actually round to 4 or 9 cents.

>I'm still looking for one or more which would match the inversion of
>these ratios (meaning each of the above * 4/3) with similar accuracy,
>and also have a reasonably useable fifth.

I'm not sure why you call that the "inversion", but here goes. I did a
brute-force search on all ETs from 0.01 ET to 99.99 ET (the computer did
this essentially instantaneously). The local minima of maximum error over
your original five intervals and those five * 4/3, which are smaller than
all previous local minima, come out as:

21.70 ET max. error 16.16�
27.00 ET max. error 13.69�
31.00 ET max. error 9.28�
41.00 ET max. error 6.31�
68.00 ET max. error 6.09�
72.09 ET max. error 3.49�
99.21 ET max. error 3.36�

Paul!
As someone who has used this algorithm, I have found one can use it in a
variety of creative ways. Like as one moves down in layers, I have made
alterations to the formula to introduce new limit ratios. I have found out
that Wilson has done the same. One need not always stick to the obvious to
get good results and sometimes these little excursions had a real artistic
flair to your tuning.

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> >I certainly can see its value for the JI community in particular, or for
> >those who'd like to create precise subdivisions (ET or not) of some
> >'substitute for the octave' ratio.
>
> These days I see no practical reason to spend too much time thinking about
> this, since a brute-force computing approach can find the best answer in a
> very short period of time.

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

[Harold Fortuin:]
> However, my effort, which was to find a close match to several
ratios (without necessarily any precise match) . . . The most useful
to me so far is 27.35 ET, nearly the same as 44-CET, which matches
7/6, 6/5, 11/9, 5/4, and 9/7 within either 3 or 8 cents. I'm still
looking for one or more which would match the inversion of these
ratios (meaning each of the above * 4/3) with similar accuracy, and
also have a reasonably useable fifth.

Hi Harold,

As I'm starting to find it difficult to keep up with all the posts,
I'm not sure if anyone might've already mentioned this or not (and if
so, sorry for the repeat), but have you considered looking at an equal
division of the 4/3 as opposed to an equal division of the 3/2 here
(as I'd look at the above as a 3/2 divided into 16 equal parts)? A 4/3
divided into 13 equal parts (~31.3 ET) would give the "inversion of
these ratios (meaning each of the above * 4/3)" with matching
(rounded) errors of -1, 9, 3, 3, and 14� for instance (this would also
give a fifth of ~690�, which I think would still fall into your
"reasonably useable" range)...

Dan

Hi Mike~

i think it is extremely useful. and can be a bit fun too

you can find a discussion of it in Mandelbaums book on starting on page 370
http://anaphoria.com/mandelbaum.html
in chapter 13 i think.

It is worth looking here too
http://anaphoria.com/viggo3.PDF
to see an example of how it answers produce things in between.
also it shows how you can seed the algorithm with more than one term.
In fact you could put in there anything you wanted from rationals to irrationals if there was some of those one liked

The other articles in
http://anaphoria.com/wilson.html
on viggo brun show how it can also be applied to scale numbers too

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Aug 10, 2010 at 7:54 AM, Kraig Grady <kraiggrady@...> wrote:

>
> What is the Viggo Brun algorithm?
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:

> also it shows how you can seed the algorithm with more than
> one term.
> In fact you could put in there anything you wanted from
> rationals to irrationals if there was some of those one liked

If you are using computer algebra and not a hand computation you can make things easier on yourself if you use only rational numbers and forget about the part where you keep track of how many of each interval make up your original list of intervals. This is because that sort of bookkeeping is easily recoverable from the results by using matrix inversion.

/tuning-math/message/8999

It's not clear to me that in the computer age it has serious musical applications, but it works very well for finding tiny commas and the associated temperaments.

Thanks Gene.
I bookmarked this and will have to spend some time with it.

The reason i like Viggo Brun is that it tends to catch some interesting material in between that i don't seem popping up otherwise.
I also like that it is quite flexible in that i can treat it as a certain limit up to a certain point and then introduce another limit after a few steps or anywhere i like. I have not found it useful for rhythms though

I find i like doing things by hand often because i can watch how a scale develops and can be aware of the different direction it might take. Which is also the case when i pursue constant structures. When i was trying to solve the problem with the dekanies, finding the least number of tones i needed to add there were often alternatives that i am glad i could choose from instead of just getting an answer.

I meant to mention how the 22 scale of India might be one of the earliest examples of a microtemperment where the chain of 22 3/2s forms skismatic equivalents of 5/4 after 8 steps down.

With MOS and what i think we might call SubMOS patterns at this time i find that i sometimes can go to a third layer. the MOS of an MOS of an MOS. Past that point i have not once made it work once that i can think of. With Rhythm too as it gives quite a bit of variation in duration

At this point it is almost like a Xenakis Sieve method but surely applied to something with potentially a more acoustical basis .

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
>
> > also it shows how you can seed the algorithm with more than
> > one term.
> > In fact you could put in there anything you wanted from
> > rationals to irrationals if there was some of those one liked
>
> If you are using computer algebra and not a hand computation you can make things easier on yourself if you use only rational numbers and forget about the part where you keep track of how many of each interval make up your original list of intervals. This is because that sort of bookkeeping is easily recoverable from the results by using matrix inversion.
>
> /tuning-math/message/8999
>
> It's not clear to me that in the computer age it has serious musical applications, but it works very well for finding tiny commas and the associated temperaments.
>