reverse tempering

This is another type of a "reverse tempering" scheme that I've just
starting to put to actual use (primarily with low order EDOs)... And
while the basic premise involves working an equally divided octave
into a (loosely speaking) just intonation; I think that these (at
least in the low order EDOs that I've been looking at) work best as
they are - in other words, I see the multitudes of "commas" as a
decided plus (theoretically much like a well temperament).

If "e" is taken as an equal division of the octave and "n" is taken as
the fraction of "e," then D:N=(y-n):(y+n) where y=(x*e) and x=3...
Increasing and decreasing the cents value of 1/e only so much as to
not under or overshoot the rounded octave will alter sequence of
terraced overtone, and in these (2-13e) examples the upper ratios are
the "stretched octave" substitutes, and the lower ratios are the
"compressed octave" substitutes...

3/2
2) 1/1 7/5 2/1
4/3

5/3
3) 1/1 5/4 11/7 2/1
3/2

6/5 7/4
4) 1/1 13/11 7/5 5/3 2/1
7/6

4/3 9/5
5) 1/1 8/7 17/13 3/2 19/11 2/1
9/7 5/3

9/8 10/7 11/6
6) 1/1 19/17 5/4 7/5 11/7 23/13 2/1
10/9 12/7

11/9 3/2 5/3 13/7
7) 1/1 11/10 23/19 4/3 25/17 13/8 9/5 2/1
6/5 13/9

12/11 13/10 14/9 15/8
8) 1/1 25/23 13/11 9/7 7/5 29/19 5/3 31/17 2/1
13/12 3/2 16/9

7/6 15/11 8/5 17/9
9) 1/1 14/13 29/25 5/4 31/23 16/11 11/7 17/10 35/19 2/1
15/13 4/3 9/5

15/14 16/13 17/12 18/11 19/10
10) 1/1 31/29 8/7 11/9 17/13 7/5 3/2 37/23 19/11 13/7
16/15 19/12

... 2/1

17/15 9/7 19/13 5/3
11) 1/1 17/16 35/31 6/5 37/29 19/14 13/9 20/13 41/25 7/4
9/8 19/15 21/13

21/11
... 43/23 2/1
11/6

18/17 19/16 4/3 3/2 22/13
12) 1/1 37/35 19/17 13/11 5/4 41/31 7/5 43/29 11/7 5/3
19/18 21/16 22/15

9/5 23/12
... 23/13 47/25 2/1
24/13

10/9 21/17 11/8 23/15
13) 1/1 20/19 41/37 7/6 43/35 22/17 15/11 23/16 47/31 8/5
21/19 11/9 3/2

12/7 25/13
... 49/29 25/14 17/9 2/1
5/3

(etc.)

Although I've yet to give these a real (musical) setting, I'd still
have to say that I'm finding them to be some of the most pleasing
"reverse tempering" results I've come up with to date - in fact there
are so many that I've liked on just a cursory once-over, that I've
stopped listening to any new ones, and have instead just set my sights
on trying to assimilate a handful of the ones I already like...

Dan

I wrote,

> Increasing and decreasing the cents value of 1/e only so much as to
not under or overshoot the rounded octave will alter sequence of
terraced overtone,

I (somewhat confusingly) skewed this bit... it should have read:
"Increasing and decreasing the cents value of 1/e only so much as to
not under or overshoot the rounded octave will alter the sequence of
terraced overtones,"

Hmm, not really sure if that's actually any clearer(!), but at least
it's what I meant to write...

Dan

I've used a method of altering equal divisions of the octave by taking
the mean of say x, y, and z where

"x" = (LOG(2)-LOG(1))*(n/LOG(2))
"y" = round (LOG(3)-LOG(2))*(n/LOG(2))
"z" = round (LOG(4)-LOG(3))*(n/LOG(2))

and "n" is the specific EDO that you'd be altering. You could also
look at this mean as a subset of an EDO that is (x*y*z)*3, and where
the often extremely large size of these equal divisions of the octave
will be trivial as it's only the specific subset that's relevant.

Here's some examples using 22 and 19e (the left hand column is the
original EDO and the right hand column is the x y z mean):

22e (7722e)
-------------
0 0
55 49
109 112
164 161
218 223
273 272
327 321
382 384
436 433
491 495
545 544
600 593, 607
655 656
709 705
764 767
818 816
873 879
927 928
982 977
1036 1039
1091 1088
1145 1151
1200 1200

19e (5016e)
-------------
0 0
63 57
126 128
189 186
253 257
316 314
379 385
442 443
505 500
568 571
632 629
695 700
758 757
821 815
884 886
947 943
1011 1014
1074 1072
1137 1143
1200 1200

These are a few examples of some lower numbered EDOs that I like the
results of here as well:

5e
0 240 480 720 960 1200
(90e)
0 213 493 707 987 1200

7e
0 171 343 514 686 857 1029 1200
(252e)
0 157 348 505 695 852 1043 1200

8e (360e)
------------
0 0
150 130
300 313
450 443
600 627, 573
750 757
900 887
1050 1070
1200 1200

10e
0 120 240 360 480 600 720 840 960 1080 1200
(720e)
0 107 247 353 493 600 707 847 953 1093 1200

Dan