question: partch scale as 41-tone periodicity block

in http://www.ixpres.com/interval/td/erlich/partchpblock.htm, i found
that partch's scale, seen as a 41-tone scale with wilson's two
auxillaries, is quite similar to the fokker (parallelepiped)
periodicity block with unison vectors 896/891, 441/440, 245/243, and
100/99 (the latter being the deviation of the auxillaries, hence a
sort of 'chromatic' unison vector).

questions:

1. is there a better choice, in terms of capturing more of Partch's
choices correctly?

2.

(a) if the answer to (1) is no, what is the linear temperament
defined by the unison vectors 896/891, 441/440 and 245/243?
(b) if the answer to (1) is yes, then . . .

3. is there a different convex shape, rather than a parallelepiped,
that can be specified in a general way (perhaps using a few extra,
*dependent* unison vectors) which captures Partch's choices even
better than any parallelepiped could?

4. peter piper picked a peck of pickled peppered parellelepipeds. how
much wood could a woodchuck chuck if a woodchuck could chuck wood?

In-Reply-To: <a3pkqi+cmvj@eGroups.com>
paulerlich wrote:

> 1. is there a better choice, in terms of capturing more of Partch's
> choices correctly?

One way to find out would be to list the step vectors (I did this once,
but can't find them) and then list all the intervals between pairs of
step vectors. Take the simplest, or something like that.

> 2.
>
> (a) if the answer to (1) is no, what is the linear temperament
> defined by the unison vectors 896/891, 441/440 and 245/243?
> (b) if the answer to (1) is yes, then . . .

I have a CGI for getting temperaments from unison vectors as well.

In this case, using your 100:99 as the chromatic unison vector, I get

8/41, 234.3 cent generator

basis:
(1.0, 0.19528912541079024)

mapping by period and generator:
[(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)]

mapping by steps:
[(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)]

highest interval width: 30
complexity measure: 30 (31 for smallest MOS)
highest error: 0.004454 (5.345 cents)
unique

which is interesting in that it's a 41 note scale with a 31 note MOS so it
looks suspiciously like Miracle. In fact it isn't because it isn't
consistent with h31. Meaning the mapping of 11-limit harmony to the
nearest intervals of 31-equal doesn't fit the mapping you get from this
temperament, which is

[31, 49, 71, 87, 108]

instead of

[31, 49, 72, 87, 107]

and that gives a g72 of

[72, 114, 166, 202, 250]

instead of

[72, 114, 167, 202, 249]

In fact, it's this part of the scale tree

.5 31
. 36
. 41 67
. 46 77 103 98

so g31&h41 has contorsion, and g72 doesn't figure.

Ah, so that is h41&h46 (taking the nearest-prime mapping of 46-equal).

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a3pkqi+cmvj@e...>
> paulerlich wrote:
>
> > 1. is there a better choice, in terms of capturing more of Partch's
> > choices correctly?
>
> One way to find out would be to list the step vectors (I did this once,
> but can't find them) and then list all the intervals between pairs of
> step vectors. Take the simplest, or something like that.

Interesting you should say this, because I just did this and was about to post about it.

Here are the step vectors and their differences:

[121/120, 100/99, 99/98, 81/80, 64/63, 56/55, 55/54, 50/49, 49/48, 45/44]

[9801/9800, 3025/3024, 2401/2400, 4000/3993, 540/539, 441/440, 5120/5103]

Here are the linear temperaments these give which appeared in the previous post:

Hemiennealimmal
wedgie [36, 54, 36, 18, 2, -44, -96, -68, -145, -74]

Octoid
wedgie [24, 32, 40, 24, -5, -4, -45, 3, -55, -71]

Miracle
wedgie [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]

Unidec
wedgie [12, 22, -4, -6, 7, -40, -51, -71, -90, -3]

wedgie [12, 34, 20, 30, 26, -2, 6, -49, -48, 15]
(Consistent with both 58 and 72--probably deserves a name.)

wedgie [2, -4, -16, -24, -11, -31, -45, -26, -42, -12]

Here are ones that did not appear--including Paul's parallel 29-et
system, I note:

wedgie [1, 33, 27, -18, 50, 40, -32, -30, -156, -144]

map [[0, -1, -33, -27, 18], [1, 2, 16, 14, -4]]

bad 285.4371097 rms 1.115729551 g 27.84651812

generators [.4144789553, 1] 497.3747464 1200.

ets

41
70
82
111
152
193
234
345

wedgie [42, 47, 34, 33, -23, -64, -93, -53, -86, -25]

map [[0, 42, 47, 34, 33], [1, -13, -14, -9, -8]]

bad 249.5656038 rms .5795196051 g 38.06010286

generators [.3472619033, 1] 416.7142840 1200.

ets

72
95
144
167
239
311
383
455
478
550
622
694
861
933

wedgie [6, 75, 39, 73, 105, 45, 95, -120, -90, 70]

map [[0, 6, 75, 39, 73], [1, -1, -30, -14, -28]]

bad 664.6829813 rms 1.006667259 g 49.18550890

generators [.4309653303, 1] 517.1583964 1200.

ets

58
181
239
420
478
601

wedgie [3, 17, -1, -13, 20, -10, -31, -50, -89, -33]

map [[0, 3, 17, -1, -13], [1, 1, -1, 3, 6]]

bad 253.7980928 rms 3.005389215 g 14.32031524

generators [.1953763085, 1] 234.4515702 1200.

ets

5
41
46
82
87
92
128
133
169
174

wedgie [4, 50, 26, -31, 70, 30, -63, -80, -245, -177]

map [[0, 4, 50, 26, -31], [1, 1, -5, -1, 8]]

bad 441.6305312 rms .9569121300 g 39.67456904

generators [.1464510525, 1] 175.7412630 1200.

ets

41
82
157
198
239
280
437
478

wedgie [18, 15, -6, 9, -18, -60, -48, -56, -31, 46]

map [[0, 6, 5, -2, 3], [3, 0, 3, 10, 8]]

bad 219.5497298 rms 1.357051968 g 21.15250745

generators [.2640465184, 1/3] 316.8558221 400.

ets

15
30
57
72
87
102
144
159
174
231

wedgie [6, 46, 10, 44, 59, -1, 49, -106, -57, 89]

map [[0, 3, 23, 5, 22], [2, 1, -12, 2, -9]]

bad 314.3980722 rms 1.127989485 g 29.31601221

generators [.3618387335, 1/2] 434.2064802 600.

ets

58
94
152
246
340

wedgie [2, 25, 13, 5, 35, 15, 1, -40, -75, -31]

map [[0, 2, 25, 13, 5], [1, 1, -5, -1, 2]]

bad 252.7955445 rms 3.221444274 g 13.70349278

generators [.2929392038, 1] 351.5270446 1200.

ets

41
58
82

wedgie [10, 26, -34, -28, 18, -82, -79, -152, -155, 39]

map [[0, 5, 13, -17, -14], [2, 1, -1, 13, 13]]

bad 352.7792183 rms .8819853677 g 36.41035959

generators [.2171375651, 1/2] 260.5650781 600.

ets

46
92
106
152
198
244
350

wedgie [0, 29, 29, 29, 46, 46, 46, -14, -33, -19]

map [[0, 0, 1, 1, 1], [29, 46, 58, 72, 91]]

bad 433.0072655 rms 2.285966134 g 23.25172805

generators [.3239193886, 1/29] 388.7032663 41.37931034

ets

29
58
87
145
174
232

wedgie [9, -7, -61, -10, -32, -122, -47, -122, 1, 183]

map [[0, -9, 7, 61, 10], [1, 2, 2, 0, 3]]

bad 401.1533812 rms .8397457939 g 40.50396806

generators [.4603708883e-1, 1] 55.24450660 1200.

ets

87
152
174
239
326
391
413
478
565

wedgie [4, 50, 26, 68, 70, 30, 94, -80, -15, 101]

map [[0, 2, 25, 13, 34], [2, 2, -10, -2, -13]]

bad 544.1352214 rms 1.140709497 g 40.46868277

generators [.2929368122, 1/2] 351.5241746 600.

ets

58
140
198
338
536

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a3pkqi+cmvj@e...>
> paulerlich wrote:
>
> > 1. is there a better choice, in terms of capturing more of
Partch's
> > choices correctly?
>
> One way to find out would be to list the step vectors (I did this
once,
> but can't find them) and then list all the intervals between pairs
of
> step vectors. Take the simplest, or something like that.

that doesn't do the trick. my original choice came from this approach
and i had to keep using multiples or quotients of the intervals
between the steps in order to capture more and more of partch's
choices.

>
> > 2.
> >
> > (a) if the answer to (1) is no, what is the linear temperament
> > defined by the unison vectors 896/891, 441/440 and 245/243?
> > (b) if the answer to (1) is yes, then . . .
>
> I have a CGI for getting temperaments from unison vectors as well.
>
> In this case, using your 100:99 as the chromatic unison vector, I
get
>
> 8/41, 234.3 cent generator
>
> basis:
> (1.0, 0.19528912541079024)
>
> mapping by period and generator:
> [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)]
>
> mapping by steps:
> [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)]
>
> highest interval width: 30
> complexity measure: 30 (31 for smallest MOS)
> highest error: 0.004454 (5.345 cents)
> unique

was this in gene's list of 35?

>
> which is interesting in that it's a 41 note scale with a 31 note
MOS so it
> looks suspiciously like Miracle. In fact it isn't because it isn't
> consistent with h31. Meaning the mapping of 11-limit harmony to
the
> nearest intervals of 31-equal doesn't fit the mapping you get from
this
> temperament, which is
>
> [31, 49, 71, 87, 108]
>
> instead of
>
> [31, 49, 72, 87, 107]
>
> and that gives a g72 of
>
> [72, 114, 166, 202, 250]
>
> instead of
>
> [72, 114, 167, 202, 249]
>
> In fact, it's this part of the scale tree
>
> .5 31
> . 36
> . 41 67
> . 46 77 103 98
>
> so g31&h41 has contorsion, and g72 doesn't figure.
>
> Ah, so that is h41&h46 (taking the nearest-prime mapping of 46-
equal).

one day, all of this is going to have to be written up with precise
definitions and the like. at least i hope so. otherwise, what a waste
of intellect.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > In-Reply-To: <a3pkqi+cmvj@e...>
> > paulerlich wrote:
> >
> > > 1. is there a better choice, in terms of capturing more of
Partch's
> > > choices correctly?
> >
> > One way to find out would be to list the step vectors (I did this
once,
> > but can't find them) and then list all the intervals between
pairs of
> > step vectors. Take the simplest, or something like that.
>
> Interesting you should say this, because I just did this and was
about to post about it.
>
> Here are the step vectors and their differences:
>
> [121/120, 100/99, 99/98, 81/80, 64/63, 56/55, 55/54, 50/49, 49/48,
45/44]
>
> [9801/9800, 3025/3024, 2401/2400, 4000/3993, 540/539, 441/440,
5120/5103]

note that this does not even produce the unison vectors i used to
make the block! 245:243 and 896:891 are missing, for example. and how
about my original question above? no one seems interested in my
questions :(

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> note that this does not even produce the unison vectors i used to
> make the block! 245:243 and 896:891 are missing, for example. and how
> about my original question above? no one seems interested in my
> questions :(

That's because it wasn't (and isn't) clear to me what the question is. It isn't finding temperaments of dimension 0 or 1, since we tried that; do you want to find a block containing Genesis?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)]
> >
> > mapping by steps:
> > [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)]
> >
> > highest interval width: 30
> > complexity measure: 30 (31 for smallest MOS)
> > highest error: 0.004454 (5.345 cents)
> > unique
>
> was this in gene's list of 35?

No, but it was on my second list of temperaments derived from Genesis itself. It's done quite well by 17/87 as a generator.

> > Ah, so that is h41&h46 (taking the nearest-prime mapping of 46-
> equal).

There you go--41+46=87.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > note that this does not even produce the unison vectors i used to
> > make the block! 245:243 and 896:891 are missing, for example. and how
> > about my original question above? no one seems interested in my
> > questions :(
>
> That's because it wasn't (and isn't) clear to me what the question >is. It isn't finding temperaments of dimension 0 or 1, since we tried >that;

actually, i was very enthusiastic about those answers so far. thanks to both of you.

>do you want to find a block containing Genesis?

that was one of the questions, yes.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > [(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13)]
> > >
> > > mapping by steps:
> > > [(36, 5), (57, 8), (83, 12), (101, 14), (125, 17)]
> > >
> > > highest interval width: 30
> > > complexity measure: 30 (31 for smallest MOS)
> > > highest error: 0.004454 (5.345 cents)
> > > unique
> >
> > was this in gene's list of 35?
>
> No, but it was on my second list of temperaments derived from Genesis itself. It's done quite well by 17/87 as a generator.
>
> > > Ah, so that is h41&h46 (taking the nearest-prime mapping of 46-
> > equal).
>
> There you go--41+46=87.

well, i'm officially behind you guys in understanding. i really want to get the whole clifford algebra thing down . . . maybe i should shut up until i do.