well-temperament wazoo

I wrote,

>I'm afraid I'll have to attribute that well-temperament that came out to
algorithmic >instabilities -- because when I [stick] different modes in, it
sometimes remains there, but it sometimes >"relaxes" to 12-tET.

Well, I thought I'd seed my "relaxer" with a bunch of 12-tone scales, each
of which is 12-tET with each note shifted by a normal random variable with
standard deviation 10 cents. These are the circles of fifths that come out
(you can safely ignore everything after the decimal place):

1.
703.02 699.65 698 698.02 698.43 698.5
698 698 698 699.71 703.89 706.79

2.
697.71 695.38 698.01 697.98 697.01 699.01
697.58 696 698.42 703 713.89 706

3.
[collapse to 11 notes]

4.
[within 1¢ of 12-tET]

5.
701.09 702.14 703 701.36 699.48 699
698.98 699 699 698.99 699 698.95

6.
700 699.22 699.28 699 699 700
699.2 700 699.99 700.99 701.84 701.47

7.
700.18 702.02 703 702.25 700 699.04
699 699.06 699 699.02 698.78 698.65

8.
701.94 705 704.44 701 698.89 698
698.12 698.95 698.99 698 698 698.68

9.
699 698.73 699 699 699 698.99
699.02 699.39 700.9 702.98 703 701

10.
[within 1¢ of 12-tET]

11.
699.09 701.01 703 703.67 702 699
698 699 699 698.49 699 698.74

12.
701 702.02 703.65 702 699.93 698.39
699 699 699 699 699 698

13.
[collapse to 11 notes]

14.
702.99 699 697.07 698 699 698.39
698.02 697.98 698 699.99 704.68 706.88

15.
697.99 698.85 699 698.99 698.01 699
699 701 703 704 702 699.15

16.
704 708 704.13 700 697.02 697.96
698 698.99 698.07 697.96 696.87 699

17.
703 707 704.48 700.02 698 698
698.35 698.17 698.78 698 697.2 699

18.
[collapse to 11 notes]

19.
698.02 698.1 698.98 698.92 698.66 698.42
698.02 700.14 703 705 702.75 700

20.
700 700.37 701.03 701.3 701 700
700 699 699.77 699.53 699 699

21.
701.75 698 697.93 698.03 698.3 698.71
698 698 698 701 706 706.28

22.
700.01 701 701.94 702 701 699.69
699.19 699.1 699.06 699 699 699

23.
699.12 699 699 699.9 699.47 699
699.02 700 700.99 701.98 701.52 701

24.
701.19 705.1 705 701 698.19 698
698.52 698.76 698.93 698.47 697.85 699

25.
701 702 701.95 701 699.99 699
699 699.35 699.37 699 699 699.35

26.
700.47 701 701 700 700 700
699.78 699.39 699.7 699.87 699.08 699.71

27.
705.01 704.8 701.01 698.48 698 698.51
698.49 699 698 698 699 701.7

28.
[collapse to 11 notes]

29.
701.77 699 698.86 698.14 699 699
699 698.64 698.36 700.52 703.71 704

30.
[collapse to 11 notes]

31.
700.1 698 698 699 698.16 698.93
698 698 699 702.57 706 704.23

32.
699.83 699.18 699 699.55 699.96 700
700 700.63 701 700.85 700.01 699.99

33.
703.6 700 698 697.97 699 698.6
699.03 698.07 698 699.93 702.8 705

34.
[collapse to 11 notes]

35.
[collapse to 11 notes]

36.
699 699.08 699.02 699.11 699.11 700
700.99 701.35 702 701 699.97 699.37

37.
697.33 698.2 698.54 698.41 698 698
699 702 705.79 704.98 701 698.75

38.
703.45 700 699 698 699 698.53
699 698.73 698 699.29 702 705

39.
703.62 707 705.23 700 697 697.65
699.02 699 698.47 698.01 697 698

40.
698.77 698.94 699 699.61 701.99 703.95
701.74 700 698.98 699 699.01 699.01

41.
701 699.01 699 699 699 699
699 699 699.06 701 702.93 703

42.
702 706.66 708.07 703 698.62 696.66
698 698 698 698 697 696

43.
[collapse to 11 notes]

44.
[within 1¢ of 12-tET]

45.
697.58 697 698.35 698.06 698.12 698.91
696.99 697 700.92 706 708 703.07

There's a pattern to all the well-temperaments that deviate significantly
from 12-tET -- can anyone spot it?

> [Paul Erlich]

> http://www.egroups.com/message/tuning/11813
>
> There's a pattern to all the well-temperaments that deviate
> significantly from 12-tET -- can anyone spot it?

> http://www.egroups.com/message/tuning/11846
>
> ... there may be some significance to the fact that all the
> well-temperaments significantly different from 12-tET that
> the optimizer got stuck at share a common pattern, beyond
> the obvious one you mentioned -- anyone see it?

Paul,

I looked at the first 16 of the scales that deviated significantly
from 12-tET, your numbers 1 thru 21, and stopped there because I
found a pattern; whether it's the one you mean or not I don't know.

The pattern I see is that, assuming the starting-note in the cycle
of '5ths' to be called 'C', the overall distribution of deviation
from 700 cents (the 12-tET '5th') is fairly wide for the first '5th',
C:G, then largest for the '5th' G:D, gradually getting smaller for
the next few (D:A, A:E, E:B), smallest (!) for the central '5ths'
B:F# and F:C#, then becoming wider again thru the next few (C#:G#,
G#:Eb = Ab:Eb, Eb:Bb), until quite wide again for Bb:F and F:C.

Is that it? I'm quite surprised that the '5ths' around the
tritone and semitone degrees of the scale are the ones showing
the narrowest spread of deviation: I would have expected it to
be exactly the opposite.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> Well, I thought I'd seed my "relaxer" with a bunch of 12-tone
scales, each of which is 12-tET with each note shifted by a normal
random variable with standard deviation 10 cents. These are the
circles of fifths that come out
> (you can safely ignore everything after the decimal place):

Well, this is a fun game... However, I'm almost certain I won't
come
up with anything. Just off the top, it looks as though some of the
sequences of fifths go "large-- small-- large" and others go
"small--large" 7-6-7, 7-6-7 or 6-7, 6-7, and since, apparently all
the
shifting was done totally randomly, I haven't the vaguest idea why it
would be?? Could it be the way the machine processes them in
sequence... Does that contribute to it getting "stuck??" I can't
even
figure out why the optimizer is getting stuck!"

It's those chirping birds again...

____________ _____ __ __ _
Joseph Pehrson

Well, since no one caught on, I'll tell you what I mean. In all the
ones which differed enough from 12-tET, the very smallest fifths were
not in the middle of the "good" keys, as one might expect. Instead,
they "frame" the large fifths of the "bad" keys, providing a buffer.
In other words, the central fifths of the good keys are narrow, but
more remote fifths are narrower still, before giving way to the few
greatly expanded fifths at the opposite side of the circle. This
seems to be a very clever way of increasing the number of "good"
thirds, since the narrowest fifths will cancel out the really large
ones in the construction of thirds that are pretty far from C-E along
the circle. Have any historical temperaments exploited this strategy?