Re: [tuning] moving on to "scales"

Paul H. Erlich wrote,

> Wow. This is a very strong justification of the usual pentatonic and
diatonic scales (in particular, pseudo-meantone tunings), of 12-tET,
and of a few scales I've never seen before (let's start playing with
them!), purely on grounds of diadic concordance.

Very interesting. I've used a bunch of methods very similar to your
"relaxed ET" premise, but I never had as sharp a focus as your
saturated, or total dyadic concordance as the overriding emphasis, and
therefore the results were a lot more comparatively sprawling... right
off the bat I want to jump in and check out the 8 and the 10... (I'm
somewhat puzzled as to why 12 came out precisely as 12-tET though!
Also wouldn't some of the larger relaxed ETs with the rounded unisons
still round out to different total groups than just the ones you
posted; in other words isn't there a rounded 9, 11, 13, etc.,
somewhere up the line?)

Dan

Dan wrote,

(I'm
somewhat puzzled as to why 12 came out precisely as 12-tET though!)

Are you puzzled as to why 3 came out precisely as 3-tET? It's an analogous
situation, though much easier to fully grasp. Essentially, though, any small
deviations from 12-tET, no matter how many notes are involved in the
deviation, will increase discordance in some intervals more than they
decrease discordance in other intervals. Not a big surprise if you've ever
tried to tune a piano. More on relaxing 12-tone scales coming right up . . .

Also wouldn't some of the larger relaxed ETs with the rounded unisons
still round out to different total groups than just the ones you
posted; in other words isn't there a rounded 9, 11, 13, etc.,
somewhere up the line?)

Well, yes, but then the "notes" represented by the unisons are special in
that the intervals they form with all the other note are weighted doubly, or
triply, or whatever; and the intervals they form with one another are
weighted 4 times, or 9 times, etc., more than the "normal" intervals. So
such scales don't really have the same status.

Paul H. Erlich wrote,

> Are you puzzled as to why 3 came out precisely as 3-tET?

Hmm... 0 389 811 looks more like a 'meantonesque' 1/1 5/4 8/5 to me?

> Essentially, though, any small deviations from 12-tET, no matter how
many notes are involved in the deviation, will increase discordance in
some intervals more than they decrease discordance in other intervals.
Not a big surprise if you've ever tried to tune a piano.

Sorry, in this case I'm afraid this is still a surprise to me!

> Well, yes, but then the "notes" represented by the unisons are
special in that the intervals they form with all the other note are
weighted doubly, or triply, or whatever; and the intervals they form
with one another are weighted 4 times, or 9 times, etc., more than the
"normal" intervals. So such scales don't really have the same status.

Yes, I understand this. If you have it handy and you don't mind, what
would be the first multiply weighted, or rounded, nine and eleven?

thanks,
Dan

Dan wrote,

>Hmm... 0 389 811 looks more like a 'meantonesque' 1/1 5/4 8/5 to me?

That was with the error in the program. Did you see the corrected results? I
presumed you did, because you mentioned that you were going to play with the
10-tone scale . . .

>> Essentially, though, any small deviations from 12-tET, no matter how
>>many notes are involved in the deviation, will increase discordance in
>>some intervals more than they decrease discordance in other intervals.
>>Not a big surprise if you've ever tried to tune a piano.

>Sorry, in this case I'm afraid this is still a surprise to me!

Let's say fifths were all that mattered (not a bad approximation). Then
you've got 12 harmonic entropy curves, and for 12-tET, you're at a point
about 2 cents to the left of the minimum on all 12 curves. Now in any other
12-note tuning, you will move to the left on some and move to the right on
others, the total amount of movement being zero. Since the curve is roughly
parabolic around the 3:2, even all the way out to wolf-sized fifths, any
movement to the left will increase your entropy more than the same amount of
movement to the right decreases it. I can be more mathematical if you like.

Now if you don't ignore thirds, you might have a situation where, if you
start out with an uneven scale, a coordinated deviation to the left of
adjacent fifths gives you enough good thirds that you're in a local minimum
of discordance. That may be what was happening with that well-temperament --
though it's not stable enough to survive the theoretically innocuous
operation of using different modes. I suspect what's happening is that the
basin around that well-temperament is so small that the program, which is
really just looking for _any_ local minimum, will, in its search, sometimes
jump out of that basin and into the deeper 12-tET one. I'll see if I use
other optimization methods to gain insight into this issue.

>Yes, I understand this. If you have it handy and you don't mind, what
>would be the first multiply weighted, or rounded, nine and eleven?

OK -- here's what I get when I start with the other ETs:

9: 0 109 189 422 499 608 812 812 997 (8 notes)
11: 0 124 213 298 417 610 609 712 917 917 1109 (9 notes)
13: 0 102.5 198 306 390 506 596 698 698 894 1007 1007 1091 (11 notes)

There you go!

Paul Hahn wrote,

>> 3: 0 400 800
>> 4: 0 314 574 888

>Remember when John DeLaubenfels was talking about the augmented triad
>and the diminished seventh being best left in their 12TET forms, and I
>said I agreed about the Aug but not the d7?

>Heh.

Well, now you can say "I told you so" (though remember, when I was looking
at triads recently, and I wasn't using octave-equivalence, I found three
local minima near the augmented triad, none of which were 12-tET.

>> 8: 0 116 312 428 620 813 929 1124

>This looks (after you mess around a bit to find the right mode) like a
>pseudomeantone diatonic (major) scale, with a minor sixth added
>in.

Let's see...

(929)--699--(428)--696--(1124)--696--(620)--696--(116)--697--(813)--699--(31
2)

and the 0 would be the added note (actually an augmented fifth if the above
is made into a major scale, though a minor sixth, 40 cents sharper, would
work just as well . . .

>> 6: 0 197 388 620 811 1008

>Wholetone, with one tone almost exactly a 8/7 and the rest (roughly)
>meantonish.

I see four major thirds, three 5:7s, and a 7:4. In the usual lattice,
.
.
811
|
|
197
/
/
/
/
1008
/|
/ |
/ 388
/,'/
620 /
| /
|/
0
/
/
/
/
811
|
|
197
/
/
/
/
1008
/|
/ |
/ 388
/,'/
620 /
| /
|/
0
/
/
/
/
811
|
|
197
.
.

A pretty magical whole-tone scale, if you ask me!

On Thu, 24 Aug 2000, Paul H. Erlich wrote:
>>> 10: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3
>
>> This has one nearly-just 9/8;
>
> where?

Sorry--I mistyped a digit in my spreadsheet. Never mind.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

"> 8: 0 116 312 428 620 813 929 1124

This looks (after you mess around a bit to find the right mode) like a
pseudomeantone diatonic (major) scale, with a minor sixth added
in."

No, it's natural and harmonic minor superimposed. :) This also gives a
pseudo-17-limit dominant 7 (minor 9), or diminished 7, very widely used.

"> 10: 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3

This has one nearly-just 9/8; if you make put it between your 3/2 and
4/3 it has pretty good tetrachordality (one 4/3 contains alternating
diatonic- and chromatic-seeming semitones, and the other would too
except for one missing pitch)."

This is just #8 with the missing fifths put in to make a chain of 10.

Keenan P.