Fun with secors

Joseph P, I sensed that perhaps you were as confused as I was when Paul
E recently said:

/tuning/topicId_29874.html#30170

[Paul:]
>Well, if the analogy is to varieties of meantone, then yes, you have
>31-tET Blackjack at one extreme, 41-tET Blackjack at another extreme,
>and 72-tET Blackjack in the middle.

I've not followed the Blackjack discussions in complete detail, and the
above drew a blank with me, so I started playing.

A Blackjack scale is by definition a chain of 21 notes separated by one
secor. A secor is precisely 116.7155940982074 cents, based upon a
calculated optimization for the chords that result. But, it can be
closely approximated in several ET's:

in 31-tET, 3 steps = 116.13 cents
in 72-tET, 7 steps = 116.67 cents
in 41-tET, 4 steps = 117.07 cents

Now the pattern that Paul is talking about becomes clearer. And perhaps
this is all clear to _you_, Joe, but I'd been unaware of some of these
details of the different ET's.

Something else confused the heck out of me: 31-tET is a meantone scale,
but the other two aren't. Since, in all three scales, the common
intervals are represented by the same number of secors, how is this
possible? More specifically, what happens to the "pump" sequence
C->A->D->G->C, where though I'm not showing accidentals, the implication
is to follow the JI ratios best represented by each scale; thus,
5/6 * 4/3 * 4/3 * 4/3 (* 1/2) = 80/81 . In one scale, this vanishes,
but in the other two it doesn't.

The answer is that the pump's intervals are:

5/6 -13 secors (+ 1 octave)
4/3 -6 secors (+ 1 octave)

Altogether, then the pump sequence traverses -31 secors (-13 -6 -6 -6),
plus some octaves. In 31-tET, that exactly closes the circle, thus
meantone; in 41-tET and 72-tET one comes up a microtone shy, thus not
meantone.

(Note that this "pump" sequence wouldn't fit onto the 21 notes of a
Blackjack subset).

Pretty kyool! Thanks for the catalyst, Joe & Paul.

JdL

> From: John A. deLaubenfels <jdl@adaptune.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, November 16, 2001 5:43 AM
> Subject: [tuning] Fun with secors
>
>
> Joseph P, I sensed that perhaps you were as confused as I was when Paul
> E recently said:
>
> /tuning/topicId_29874.html#30170
>
> [Paul:]
> >Well, if the analogy is to varieties of meantone, then yes, you have
> >31-tET Blackjack at one extreme, 41-tET Blackjack at another extreme,
> >and 72-tET Blackjack in the middle.
>
> I've not followed the Blackjack discussions in complete detail, and the
> above drew a blank with me, so I started playing.
>
> A Blackjack scale is by definition a chain of 21 notes separated by one
> secor. A secor is precisely 116.7155940982074 cents, based upon a
> calculated optimization for the chords that result. But, it can be
> closely approximated in several ET's:
>
> in 31-tET, 3 steps = 116.13 cents
> in 72-tET, 7 steps = 116.67 cents
> in 41-tET, 4 steps = 117.07 cents

Nice work, John! The part which followed, which I snipped,
is interesting!

Just thought I'd mention in passing, if someone hasn't
done it already, that the best low-integer rational
approximation to this optimum secor value is 46:43.
Don't know if that has any practical significance.

I was inspired by all of this to expand my Dictionary
entry for "secor", inlcuding a graph showing how various
EDOs converge on a close approximation of it:
http://www.ixpres.com/interval/dict/secor.htm

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

/tuning/topicId_30262.html#30262

> Joseph P, I sensed that perhaps you were as confused as I was when
Paul E recently said:
>
> /tuning/topicId_29874.html#30170
>
> [Paul:]
> >Well, if the analogy is to varieties of meantone, then yes, you
have 31-tET Blackjack at one extreme, 41-tET Blackjack at another
extreme, and 72-tET Blackjack in the middle.
>

Hi John!

This is *definitely* incorrect. I was *more* confused...

> I've not followed the Blackjack discussions in complete detail, and
the above drew a blank with me, so I started playing.
>
> A Blackjack scale is by definition a chain of 21 notes separated by
one secor. A secor is precisely 116.7155940982074 cents, based upon a
> calculated optimization for the chords that result. But, it can be
> closely approximated in several ET's:
>
> in 31-tET, 3 steps = 116.13 cents
> in 72-tET, 7 steps = 116.67 cents
> in 41-tET, 4 steps = 117.07 cents
>

I appreciate this reprise, John. Paul was hinting at this in some of
his smaller posts, but it's nice to see it all put together!

JP

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I was inspired by all of this to expand my Dictionary
> entry for "secor", inlcuding a graph showing how various
> EDOs converge on a close approximation of it:
> http://www.ixpres.com/interval/dict/secor.htm

Hi Monz.

You might want to change "calculated optimum value" to "calculated
minimax value" or "minimax optimum value" on that page. As we've
seen, there are many different ways to define the optimum, and many
slightly different values for the secor result.

The minimax value, by the way, which you give correct to many digits,
is exactly equal to (18/5)^(1/19). This satisfies the property that
Secor mentions in http://www.anaphoria.com/secor.PDF, that all the
primary 11-limit ratios (aka 11-limit consonances) will be
approximated with an error not greater than 3.32 cents. He
incorrectly gave the value 116.69 cents for the generator satisfying
this property, but he corrected this in the errata (according to
Margo).

Also, on your Blackjack page, you should make each instance of the
word "MIRACLE" link to the definition of MIRACLE. As it stands, it's
a

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, November 16, 2001 1:34 PM
> Subject: [tuning] Re: Fun with secors
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > I was inspired by all of this to expand my Dictionary
> > entry for "secor", inlcuding a graph showing how various
> > EDOs converge on a close approximation of it:
> > http://www.ixpres.com/interval/dict/secor.htm
>
> Hi Monz.
>
> You might want to change "calculated optimum value" to "calculated
> minimax value" or "minimax optimum value" on that page. As we've
> seen, there are many different ways to define the optimum, and many
> slightly different values for the secor result.

Done. I went with "minimax optimum".

> The minimax value, by the way, which you give correct to many digits,
> is exactly equal to (18/5)^(1/19). This satisfies the property that
> Secor mentions in http://www.anaphoria.com/secor.PDF, that all the
> primary 11-limit ratios (aka 11-limit consonances) will be
> approximated with an error not greater than 3.32 cents. He
> incorrectly gave the value 116.69 cents for the generator satisfying
> this property, but he corrected this in the errata (according to
> Margo).

I've also added most of this to my definition, with
credit given to you at the bottom. Thanks.

Can you point me to a good link explaining "minimax"?
I've searched, but don't understand enough about it
to know what I should be directing others to look at.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

[Monz wrote:]
>Can you point me to a good link explaining "minimax"?
>I've searched, but don't understand enough about it
>to know what I should be directing others to look at.

Well, I'm sure you'll get some more precise responses from our heavy
hitting math boys, but conceptually minimax is quite simple: you adjust
[whatever variable you're trying to solve for] until, when you look at
all the errors (in this case, the list of cents deviation for each
interval being considered), and pick the maximum (absolute) error, and
that's how bad you are off. That is, minimizing this worst error is
your goal. Here's an example:

System A System B
-------- --------
Error 1 +5.0 -5.5
Error 2 -4.0 +0.2
Error 3 -4.5 -1.3
------- -------- --------
RMS error 4.52 3.26

Which system is worse? In minimax, the -5.5, worse than +5.0, means
that System B is worse than System A. If, however, we use Root Mean
Square, System A is worse than System B. My own preference in general
would be for RMS calculation, but it is largely a matter of preference.

As Paul E has mentioned, minimax calculations are more difficult than
RMS, because the derivatives of the "pain" functions are discontinuous,
so it is not possible to set up a single polynomial to solve for.

HTH.

JdL

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> Can you point me to a good link explaining "minimax"?

Minimax simply means "minimizing the maximum error". The lowest value
of largest error is acheived there. For example, 1/4-comma meantone
is the "minimax" meantone with respect to the 5-limit, as all 5-limit
intervals are approximated with a maximum error of 1/4-comma, and no
other meantone can improve on that (see the "historical meantones"
chart at the bottom of your meantone definition page to see why).

You can add this as a footnote to the occurence of "minimax" in your
page.

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, November 17, 2001 5:39 PM
> Subject: [tuning] Re: Fun with secors
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Can you point me to a good link explaining "minimax"?
>
> Minimax simply means "minimizing the maximum error". The lowest value
> of largest error is acheived there. For example, 1/4-comma meantone
> is the "minimax" meantone with respect to the 5-limit, as all 5-limit
> intervals are approximated with a maximum error of 1/4-comma, and no
> other meantone can improve on that (see the "historical meantones"
> chart at the bottom of your meantone definition page to see why).
>
> You can add this as a footnote to the occurence of "minimax" in your
> page.

OK Paul, thanks. Done.
http://www.ixpres.com/interval/dict/secor.htm

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> http://www.ixpres.com/interval/dict/secor.htm

Further correction, Monz. George Secor is (as far as I know) not the
discoverer of the MIRACLE tunings, if you mean the 21-, 31-, or 41-
tone MOS scales we associate with that name. He did discover the
generator of the MIRACLE tunings, but the scale he proposed,
mimicking Partch's 43-tone scale, had a couple of holes in the chain
of generators, depriving it of many of the miraculous properties we
associate with the 21-, 31-, and 41-tone MOSs.

By the way, on the Blackjack page,
<http://www.ixpres.com/interval/monzo/blackjack/blackjack.htm>, I
really think you should make the word "MIRACLE" link to the
definition of "MIRACLE". Actually this page looks like it consists of
various bits thrown together that don't make sense as a whole -- it
may be time for a re-write.