reverse tempering and shuffling linear temperaments

Something I had looked at a while back, was changing the sequential
position of the 3/2 & the 4/3 on a "circle of fifths" by tempering
(i.e. dividing) the 3/2 & the 4/3... generally speaking this could
also be written as ((LOG(N)-LOG(D))*(1200/LOG(2^n)) where n>0 (in
other words, if n=1 and N/D=3/2, you would just have endlessly
extending Pythagorean columns of 3/2s & 4/3s). What I was initially
interested in here was trying to 'soften up' 8e in a particular piece
that I was working on... so for example, the 1 (i.e., 1/8th of an
octave) would put the fifth in the third position of the fourths
column in 8e:

0
1050 150
900 300
750 450
600 600

so I tried taking the N/D & n of ((LOG(N)-LOG(D))*(1200/LOG(2^n)) as
4/3 & 3, and got:

166 166 166 166 38 166 166 166
0 166 332 498 664 702 868 1034 1200

or:

166 166 166 38 166 166 166 166
0 166 332 498 536 702 868 1034 1200

when taken as 7 (the inversion at 1050�) where N/D=6/1 & n=3... Of
course this could also be seen as the fifth in the third position of
the fifths column where N/D=3/2 & n=3:

234 234 30 204 30 204 30 234
0 234 468 498 702 732 936 966 1200

...And so on and forth I went, on and on through a bunch of different
variations on this process, and eventually I did end up using:

3:8 / 2
204 147 147 57 147 147 204 147
0 204 351 498 555 702 849 1053 1200

and

1:6 / 7
129 130 184 130 129 184 130 184
0 129 259 443 573 702 886 1016 1200

But the process hadn't worked out in quite the way I had originally
envisioned it, so I put it aside for another day (as I did realize
that there seemed to be an awful lot of potential uses for the general
processes I was looking out).

And while I suppose that you could look at this as some sort of
reverse tempering -- as 12-tone equal temperament could be seen as
((LOG(N)-LOG(D))*(x/LOG(2^y)) where N/D=3/2, x=12, and y=1 -- it is
also just another way to render an endless variety of meantone like
tunings (e.g., ((LOG(5)-LOG(1))*(1200/LOG(2^4)) and
((LOG(3)-LOG(1))*(1200/LOG(2^6)) being synonymous with 1/4 comma and
1/6 kleisma, etc.).

So my question is -- as I'm starting to look at this process again --
is to what extent others here might have looked at this way of
shuffling linear temperaments... and what sorts of results they might
have gotten...

Dan

Dan Stearns wrote,

>generally speaking this could
>also be written as ((LOG(N)-LOG(D))*(1200/LOG(2^n))

You seem to have an extra left parenthesis, but anyway, this expression is
equivalent to

(log(n/d)/log(2))/n*1200

which is easier for me to understand.

>So my question is -- as I'm starting to look at this process again --
>is to what extent others here might have looked at this way of
>shuffling linear temperaments... and what sorts of results they might
>have gotten...

One thing your example reminds me of is Andrzej Gawel's 19-of-36-tET scale.
Gawel ingeniously took the 7-of-12-tET diatonic scale and divided each of
the six instances of the generator, 7/12 oct. = 19/12 oct., into a chain of
three sub-generators, 19/36 oct., allowing all six of the ordinary diatonic
triads to be completed as 7-limit tetrads, and in fact the scale has 14
7-limit tetrads.

>What I was initially interested in here was trying to 'soften up' 8e in a
>particular piece that I was working on... so for example, the 1 (i.e.,
1/8th >of an octave) would put the fifth in the third position of the fourths
>column in 8e:

Huh?

>So my question is -- as I'm starting to look at this process again --
>is to what extent others here might have looked at this way of
>shuffling linear temperaments... and what sorts of results they might
>have gotten...

Sorry, Dan, but I couldn't follow your post. What is it you were doing?
Trying to make an 8-tone scale with tetrachordal structure?

>One thing your example reminds me of is Andrzej Gawel's 19-of-36-tET scale.
>Gawel ingeniously took the 7-of-12-tET diatonic scale and divided each of
>the six instances of the generator, 7/12 oct. = 19/12 oct., into a chain of
>three sub-generators, 19/36 oct., allowing all six of the ordinary diatonic
>triads to be completed as 7-limit tetrads, and in fact the scale has 14
>7-limit tetrads.

Wow. Paul, is this right?
0 4 5 6 7 11 12 13 14 18 19 20 21 26 27 28 33 34 35

Only contains a single diatonic scale, and it's too big to be a generalized
diatonic in its own right. How do the tetrads look on the lattice?

-Carl

[me:]
>the 1 (i.e., 1/8th >of an octave) would put the fifth in the third
position of the fourths column in 8e:

[Carl Lumma:]
> Huh?

Well I guess another way to look at it is as a linear line of 1/8ths
centered on zero (i.e. 4 5 6 7 0 1 2 3 4) were your changing the
generator from 150� into a ~166� (thereby changing 5/8 into a 3/2).

Dan

[Paul Erlich:]
>and in fact the scale has 14 7-limit tetrads.

[Carl Lumma:]
>Wow. Paul, is this right? 0 4 5 6 7 11 12 13 14 18 19 20 21 26 27 28
33 34 35

[Paul:]
>No, it's

0 2 4 6 8 10 12 14 16 18 19 21
23 25 27 29 31 33 35

[Carl:]
>Only contains a single diatonic scale, and it's too big to be a
generalized diatonic in its own right. How do the tetrads look on the
lattice?

[Paul:]
>I'll leave that as an exercise for the reader.

I've probably got something skewed here... but when you first gave
this I thought there was also a minor tetrad on the 25/36 (making
fifteen), which I guess I'd illustrate as something like this:

14----------35
/ \ / \
/ \ / \
/ \ / \
10.-----/--31.--\--/--16. \
`. / / \`. \/ / \`. \
`02--/---\-`23--/---\-`08
/ \ / \
27.-----/--12.--\--/--33.--\-----18.
`. / / \`. \/ / \`. \ / \`.
`19--/---\-`04--/---\-`25--/---\-`10
/ \ / \ / \
08.-----/--29.--\--/--14.--\--/--35. \
`. / `. \/ `. \/ `. \
`00--------`21--------`06--------`27

Dan