Re: Viggo Brun, MOS, etc.

I found the two papers mentioned in my earlier post:

Selmer, Ernst S. 1961. On Flerdimensjonal Kjedebroek.
Nordisk Matematisk Tidskrift 9: 37-43.
This is the a-c algorithm.
(The oe stands for o-slash, which I can't
do in ascii).

Pipping, Nils. 1967. Approximation zweien realen Zahlen
durch rationale Zahlen mit gemeinsam Nenner. Acta Academiae
Aboensis, Math et Phys. 21(1):3-17.

(This is the ramified system. Start with a-b then immediately do an a-c
step. Henceforth, whenever a < b+c do a-b followed immediately by a-c.
When a > b+c, do a-c, then retest.)

I found an old paper by J. B. Rosser which starts where
J. Murray Barbour finished in the application of ternary
continued fractions to tuning theory.

Rosser, J. B. 1950. Generalized Ternary Continued Fractions.
The American Mathematical Monthly 57(8): 528-535.

The ref to Barbour's paper is
Barbour, J. M. 1948. Music and Ternary Continued Fractions.
American Mathematical Monthly 55(9): 545-555.

These new papers should be added to the microtonal bibliography if they
are not already listed. (Manuel are you on these lists still?

I've experimented a bit in the past with 4 or more factors and have
observed that the a-b form of the algorithm seems to converge quite
well, the a-n (were n is d or e, etc. for 4 or 5 factors,etc.) does not.

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Thanks for the references!
Do you know if any of these papers had an observation as to the placing the intervals in a 'correct' order?
While one can usually intuit how they are inserted, possibly someone has noticed a further 'pattern'. if you remember offhand.

Related to this is applying this to rhythm, where the method is similar to equal divisions.
In this case though there is the freedom in choosing either of the two possibilities.

John Chalmers wrote:
>
> I found the two papers mentioned in my earlier post:
>
> Selmer, Ernst S. 1961. On Flerdimensjonal Kjedebroek.
> Nordisk Matematisk Tidskrift 9: 37-43.
> This is the a-c algorithm.
> (The oe stands for o-slash, which I can't
> do in ascii).
>
> Pipping, Nils. 1967. Approximation zweien realen Zahlen
> durch rationale Zahlen mit gemeinsam Nenner. Acta Academiae
> Aboensis, Math et Phys. 21(1):3-17.
>
> (This is the ramified system. Start with a-b then immediately do an a-c
> step. Henceforth, whenever a < b+c do a-b followed immediately by a-c.
> When a > b+c, do a-c, then retest.)
>
> I found an old paper by J. B. Rosser which starts where
> J. Murray Barbour finished in the application of ternary
> continued fractions to tuning theory.
>
> Rosser, J. B. 1950. Generalized Ternary Continued Fractions.
> The American Mathematical Monthly 57(8): 528-535.
>
> The ref to Barbour's paper is
> Barbour, J. M. 1948. Music and Ternary Continued Fractions.
> American Mathematical Monthly 55(9): 545-555.
>
> These new papers should be added to the microtonal bibliography if they
> are not already listed. (Manuel are you on these lists still?
>
> I've experimented a bit in the past with 4 or more factors and have
> observed that the a-b form of the algorithm seems to converge quite
> well, the a-n (were n is d or e, etc. for 4 or 5 factors,etc.) does not.
>
> -- > No virus found in this outgoing message.
> Checked by AVG Free Edition.
> Version: 7.5.516 / Virus Database: 269.20.2/1271 - Release Date: > 2/11/08 8:16 AM
>
> -- Kraig Grady
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