Re: wendy carlos scales

>> Actually, I believe it is defined as the 11th root of 1.5.
>
>Nope, it's defined strictly as a local minimum of the root-mean-square
>error function that Wendy set up with 3/2, 5/4, 6/5, 7/4, and 11/8 as
>target intervals.

The article I have says something like (from memory) "11 steps to a perfect
(no kidding) fifth"... The article does mention the computer search, but
doesn't give any sort of detailed report.

C.

On Mon, 22 Mar 1999, Carl Lumma wrote:
>> Nope, [Beta]'s defined strictly as a local minimum of the root-mean-square
>> error function that Wendy set up with 3/2, 5/4, 6/5, 7/4, and 11/8 as
>> target intervals.
>
> The article I have says something like (from memory) "11 steps to a perfect
> (no kidding) fifth"...

Well, I'm sure she expresses it various ways at various times--it's
certainly a lot easier to say "11 steps to a perfect fifth" than "okay,
I set up this error function yada yada yada local minimum at a stepsize
of 63.8 cents", especially to most readers who don't know or
particularly care to know about root-mean-square or cents notation.
Heck, in the liner notes to _Beauty in the Beast_ she describes Beta as
having eight steps to the perfect fourth, when in fact Beta's fourth
(510.4 cents) isn't all that perfect.

(OTOH, 11 * 63.8 = 701.8, which is a pretty damn good approximation to
3/2.)

> The article does mention the computer search, but
> doesn't give any sort of detailed report.

The article I quoted is, as far as I know, the earliest publication
(Computer Music Journal, Spring 1987) of her Alpha-Beta-Gamma scales,
and as "original" a definition of their derivation as I think we can
get.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

NOTE: dehyphenate node to remove spamblock. <*>

>> The article I have says something like (from memory) "11 steps to a
>> perfect (no kidding) fifth"...
>
>Well, I'm sure she expresses it various ways at various times--it's
>certainly a lot easier to say "11 steps to a perfect fifth" than "okay,
>I set up this error function yada yada yada local minimum at a stepsize
>of 63.8 cents"...

Right, makes sense.

>Heck, in the liner notes to _Beauty in the Beast_ she describes Beta as
>having eight steps to the perfect fourth, when in fact Beta's fourth
>(510.4 cents) isn't all that perfect.

But did she call it a perfect no kidding fourth? :)

>> The article does mention the computer search, but doesn't give any
>> sort of detailed report.
>
>The article I quoted is, as far as I know, the earliest publication
>(Computer Music Journal, Spring 1987) of her Alpha-Beta-Gamma scales,
>and as "original" a definition of their derivation as I think we can
>get.

The article I have is from her web site, entitled "Three Asymmetric
Divisions of the Octave." I'll have to check out CMJ. Is that the same
article where she shows a diagram of her generalized keyboard design?

Carl

On Tue, 23 Mar 1999, Carl Lumma wrote:
> I'll have to check out CMJ. Is that the same
> article where she shows a diagram of her generalized keyboard design?

Yes, it is.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

NOTE: dehyphenate node to remove spamblock. <*>

--Paul Hahn--
> The citation is: Wendy Carlos, "Tuning: at the crossroads", in the Spring
> 1987 issue of _Computer Music Journal_, pp. 29-43.
>
>[-snip-]
> Yes, it is.

Check! (thanks!)

--Joe Monzo--
> Also, any further technical info on the Carlos scales would be appreciated

Did you catch my two posts on non-octave scales in early Feb?

Alpha, 78.0 cents/step

15tET (x1) = 80.0
31tET (x2) = 77.1
46tET (x3) = 78.3
77tET (x5) = 77.9

Beta, 63.8 cents/step

19tET (x1) = 63.2
56tET (x3) = 64.3
75tET (x4) = 64.0

Gamma, 35.1 cents/step

34tET (x1) = 35.3
103tET (x3) = 35.0

88CET, 88.0 cents/step

14tET (x1) = 85.7
27tET (x2) = 88.9
41tET (x3) = 87.8

Equalized Bohlen-Pierce, 146.3 cents/step

8tET (x1) = 150.0
25tET (x3) = 144.0
33tET (x4) = 145.6

Just as the 7th root of 3/2 is related to 12tET, so are Alpha and Beta
related to 15 and 19tET. Gamma can be considered two interlaced 10th root
of 3/2 scales, much as 34tET can be considered two interlaced 17tET scales.

The idea is that when tempering an MOS, the generator usually takes the
punishment. But why not give the interval of equivalence a taste of the
medicine? The difference between the versions will be proportional to
D/(G+I) where "D" is the size difference between the two chains, "G" is the
number of members in the chain of generators, and "I" is the number of
members in the chain of intervals of equivalence.

Of course any tuning can be explained as a retempering of any other tuning,
and so much the better if it helps us think about them.

88CET is then the "other" version of the 11-tone 7/4 vs. 2/1 MOS. 27 and
41 are both higher MOS's of this interval pair. I don't know how to
classify BP, which I find to be a scale of limited (I should say specific)
usefulness.

For sufficiently low values of the above formula I think various temperings
of an MOS ought to be quite similar conceptually, differing mainly in
"mood". My experience with 12tET vs. the 7th root of 3/2 backs this up.

C.

At 03:26 PM 3/22/99 -0600, Paul Hahn wrote:

>Heck, in the liner notes to _Beauty in the Beast_ she describes Beta as
>having eight steps to the perfect fourth, when in fact Beta's fourth
>(510.4 cents) isn't all that perfect.

Although the fourth is not just, I would call it perfect rather than
augmented or diminished.

By the way, is a distance of 1902 cents known as a triapason? Or am I confused.

--Charlie Jordan <http://www.rev.net/~aloe/music>