some continued experiments with limited transposability

This is an experiment in recasting proportions and truncating one- and
two-dimensional planes by a given periodicity within Olivier
Messiaen's "concept of limited transposability".

Threading the "P" (i.e., any given cents amount) in
(LOG(N)-LOG(D))*(P/LOG(2)) with Messiaen's concept of limited
transposability -- which assumes a repeating structure inside of a
larger periodicity (which in Messiaen's case was the octave) -- would
give a P = 1:2^(1/n) range of possibilities within some arbitrary
scope of useful distinctions.

Running these fractional octave generators out onto one and two
dimensions will give a proportional recasting, and a
linear/logarithmic hybrid, of the familiar "Pythagorean" and "Zarlino"
arrangements.

The first of these P = 1:2^(1/n) possibilities, P = 1:2^(1/1), would
of course just result in the familiar one- and two-dimensional
arrangements...

1200 0
498 702
996 204
294 906
792 408
90 1110
588 612
1086 114
384 816
882 318
180 1020
678 522
1177 23

182----884---386---1088
\ / \ / \ / \
\ / \ / \ / \
498----0----702---204

Fractional periodicity at the half-octave would result in following
one- and two-dimensional arrangements...

P = 1:2^(1/2)

600 0
249 351
498 102
147 453
396 204
45 555
294 306
543 57
192 408
441 159
90 510
339 261
588 12

91----442---193---544
\ / \ / \ / \
\ / \ / \ / \
249----0----351---102

Here's some of the resulting scales with a concept of limited
transposability.

1/1 2/1 4/1

0 600 1200

1/1 3/2 2/1 3/1 4/1

0 351 600 951 1200
0 249 600 849 1200

1/1 9/8 3/2 2/1 9/4 3/1 4/1

0 102 351 600 702 951 1200
0 249 498 600 849 1098 1200
0 249 351 600 849 951 1200

5/4
/ \
/ \
1/1---3/2

0 193 351 600 793 951 1200
0 158 407 600 758 1007 1200
0 249 442 600 849 1042 1200

1/1 9/8 3/2 27/16 2/1 9/4 3/1 27/8 4/1

0 102 351 453 600 702 951 1053 1200
0 249 351 498 600 849 951 1098 1200
0 102 249 351 600 702 849 951 1200
0 147 249 498 600 747 849 1098 1200

5/3---5/4
\ / \
\ / \
1/1---3/2

0 193 351 442 600 793 951 1042 1200
0 158 249 407 600 758 849 1007 1200
0 91 249 442 600 691 849 1042 1200
0 158 351 509 600 758 951 1109 1200

1/1 9/8 81/64 3/2 27/16 2/1 9/4 81/32 3/1 27/8 4/1

0 102 204 351 453 600 702 804 951 1053 1200
0 102 249 351 498 600 702 849 951 1098 1200
0 147 249 396 498 600 747 849 996 1098 1200
0 102 249 351 453 600 702 849 951 1053 1200
0 147 249 351 498 600 747 849 951 1098 1200

5/3---5/4
/ \ / \
/ \ / \
4/3---1/1---3/2

0 193 249 351 442 600 793 849 951 1042 1200
0 56 158 249 407 600 656 758 849 1007 1200
0 102 193 351 544 600 702 793 951 1144 1200
0 91 249 442 498 600 691 849 1042 1098 1200
0 158 351 407 509 600 758 951 1007 1109 1200

Fractional periodicity at the third of an octave would result in
following one- and two-dimensional arrangements...

P = 1:2^(1/3)

400 0
166 234
332 68
98 302
264 136
30 370
196 204
362 38
128 272
294 106
60 340
226 174
392 8

61----295---129---362
\ / \ / \ / \
\ / \ / \ / \
166----0----234---68

Here's some of the resulting scales with a concept of limited
transposability.

1/1 2/1 4/1 8/1

0 400 800 1200

1/1 3/2 2/1 3/1 4/1 6/1 8/1

0 234 400 634 800 1034 1200
0 166 400 566 800 966 1200

1/1 9/8 3/2 2/1 9/4 3/1 4/1 9/2 6/1 8/1

0 68 234 400 468 634 800 868 1034 1200
0 166 332 400 566 732 800 966 1132 1200
0 166 234 400 566 634 800 966 1034 1200

5/4
/ \
/ \
1/1---3/2

0 129 234 400 529 634 800 929 1034 1200
0 105 271 400 505 671 800 905 1071 1200
0 166 295 400 566 695 800 966 1095 1200

1/1---9/8---3/2---27/16

0 68 234 302 400 468 634 702 800 868 1034 1102 1200
0 166 234 332 400 566 634 732 800 966 1034 1132 1200
0 68 166 234 400 468 566 634 800 868 966 1034 1200
0 98 166 332 400 498 566 732 800 898 966 1132 1200

5/3---5/4
\ / \
\ / \
1/1---3/2

0 129 234 295 400 529 634 695 800 929 1034 1095 1200
0 105 166 271 400 505 566 671 800 905 966 1071 1200
0 61 166 295 400 461 566 695 800 861 966 1095 1200
0 105 234 339 400 505 634 739 800 905 1034 1139 1200

(etc.)

This general process could be altered in many ways, fractional
periodicity of larger non-octave groupings for instance, the
possibilities are indeed many... Whether the results must always
kowtow to psychoacoustic parameters, or whether there is an ability to
meaningfully discern and cognize such symmetries is something I have
yet to decide for myself.

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

http://www.egroups.com/message/tuning/15753

> This is an experiment in recasting proportions and truncating one-
and two-dimensional planes by a given periodicity within Olivier
> Messiaen's "concept of limited transposability".
>

Congrats, Dan, on your "updating" of some of Messiaen's concepts to
the xenharmonic realm. It will be interesting to hear what music you
come up with using these scales...
_______ ____ __ _ _
Joseph Pehrson

Joseph Pehrson wrote,

> Congrats, Dan, on your "updating" of some of Messiaen's concepts to
the xenharmonic realm. It will be interesting to hear what music you
come up with using these scales...

Boy o' boy, those scales are really REALLY strange. Their construction
is kind of like n amount of days lived in the space of 1 as seen
through dual reflecting funny mirrors... Not for the squeamish!

I think that the Gold and Silver "Apical" scales of limited
transposability that I posted a month or so ago would probably be a
less vertigo inducing leap if one wanted to try and recast some
Messiaen like actual musical usage's of these scales... in fact, I
think it's probably safe to say that this is one of the only Kornerup
like -- i.e., non-ET non-JI -- ways to frame *all* these types of
scales.

--Dan Stearns