More on Fortuin & Peck at Sonic Circuits

A few days ago Kris Peck wrote:

>>Also, Harold Fortuin and I played a couple of numbers earlier in the
evening, including Harold's twisted 22ET arrangement of "Bali Hai" and
his
new arabic-inspired piece using a fractional ET of
27-point-something-notes-per-octave.

To clarify:

Kris & I, on the Nov. 5 concert in the lovely Landmark Center (a turn of
the century court building that
looks like a castle), played the "Bali Hi" transcription (including some
anticapitalist verses and interludes of my own creation). I also
presented my latest piece "A Voice from Iraq" which I prefaced with my
view that the sanctions on Iraq only punish the Iraqi people, not Saddam
Hussein. You might also be interested that I sang its ululated melody
(which highlights 5/4, 11/9, 6/5, and 7/6 representatives), and doubled
it heterophonically (almost in unison, actually) on sampled cello from
the Clavette playing a Kurzweil K2000.

"A Voice from Iraq" is in 27.35 ET, which, especially on a cent-accurate
K2000, is the nearly the same as 44-CET.
I arrived at this division by:

--taking the logs of 7/6, 6/5, 11/9, 5/4, and 9/7 (the five qualities of
thirds)
--seeking a "near" greatest common factor (GCF) for these logs
heuristically using a spreadsheet, that resulted in a
reasonably small non-octave ET

(I know Erv Wilson doesn't approve of terms like "non-octave", but this
ET doesn't precisely divide ANY
ratio that I'm concerned with. Perhaps a "factored logarithm ET" is a
better term.)

Interestingly, the above interval matches in 27.35 ET are alternately 3
and 8 cents off from the ratio values--
can any of the math professors or other mathematical wizards on this
list explain why? (I suspect this is not
merely a strange and wonderful accident.)

I'd also be interested in knowing about more systematic approaches to
the GCF approach, explained in as plain English as any of you can muster
(you can presume a basic understanding of logs, trig, and algebra).

Harold Fortuin wrote,

>I'd also be interested in knowing about more systematic approaches to
>the GCF approach, explained in as plain English as any of you can muster
>(you can presume a basic understanding of logs, trig, and algebra).

Personally, with modern computing power, I'd simply search all ETs from
0.01-tET to some pre-determined limit, say 69.31-tET (=log(2)/1%), eliminate
the inconsistent ones, and use the one with the lowest sum-of-squares error.
If you don't want to use a brute-force search, and don't care about
consistency (which shouldn't be an issue since really good approximations
are bound to be consistent), John Chalmers addressed the issue you're
interested in in TD 294.7:

"The Brun algorithm is easy to use when it is desired to approximate more
than two numbers simultaneously (e.g., find a temperament with good
3/2's, 5/4's, 7/4's, etc.). The corresponding ternary or higher-order
CF's tend to converge to rapidly, yielding only temperaments with very
large numbers of tones. J. M. Barbour developed a version which
converges slower, but Brun's algorithm is still easier to apply"

He subsequently listed the following references:

Barbour, James Murray. "Music and Ternary Continued Fractions", American
Mathematical Monthly vol. 55, 1948, p. 545.

Brun, Viggo. "En generalisation av kjedebr�ken", Norske Videnskabers
Selskapets Skrifter vol. 1-2, 1919-20, Oslo. (avec des r�sum�s en
Fran�ais)

Brun, Viggo. "Music and Ternary Continued Fractions", Det Kongelige
Norske Videnskabers Selskab, Forhandlinger Bd. 23, 1950, p. 38.

Brun, Viggo. "Algorithmes Euclidiens pour trois et quatre nombres",
Sartryck ur Trettonde Skandinavska Matematikerkongressen XIII, aug.
1957, p. 45.

Brun, Viggo. "Mehrdimensionale Algorithmen, welche die Eulersche
Kettenbruchentwicklung der Zahle verallgemeinern", Festschrift Leonard
Euler zum 250. Geburtstag, Akademie-Verlag, Berlin, 1959.

Brun, Viggo. "Musikk og Euklidske algoritmer", Nordisk matematisk
tidskrift vol. 9, Oslo, 1961, pp. 29-36.

Brun, Viggo. "Euclidean algorithms and music theory", L'enseignement
math�matique revue internationale tome X, fasc. 1-2, 1964, pp. 125-137.

Fokker, A.D. "Multiple antanairesis", Proceedings of the Koninklijke
Nederlandse Akademie van Wetenschappen, Series A, vol. 66, Amsterdam,
1963, pp. 1-6.

Olds. C. D. Continued Fractions. New Mathematical Library Monograph 9,
Random House, 1961? (Great introductory book!)

Pipping, Nils. Approximation Zweien Reelen Zahlen Durch Rationale Zahlen
Mit Gemeinsamer Nenner. Acta Academiae Aboensis (Math et Phys) 21(1)
1957, pp. 3-17

Selmer, Ernst S. Om Flerdimensjonal Kjedebro/K. Nordisk matematisk
tidskrift 9, Oslo, 1961, pp. 37-44.

Thanks, John!

>"A Voice from Iraq" is in 27.35 ET, which, especially on a cent-accurate
>K2000, is the nearly the same as 44-CET.

Every other step of which is Gary Morrison's 88CET. Which is not surprising,
since his tuning turned up in a similar search. These tunings join with 41ET
in a series dividing the 3/2 into 8, 16, and 24 equal parts.

Regarding the K2000 -- does the error occur before or after the stacking? If
before, you can still get closer to 27.35ET than 44CET.

>Interestingly, the above interval matches in 27.35 ET are alternately 3 and
>8 cents off from the ratio values--

Notice that in normal ET's, an interval and it's inversion about the octave
have equal errors in opposite directions. Since your tuning divides the 3/2
almost exactly, pairs of ratios which sum to 3/2 ought to have this same
property. Why 11/9 and 27/22 share the same error as 9/7 and 7/6 (and indeed
7/4!), I don't know. Probably these are all using the same comma...

This tuning (and 27ET) are interesting in that they distinguish the 225/224,
a comma of 8 cents, as one step while the 64/63 of 27 cents vanishes.

-Carl