UVs for 46-ET 11-limit PB

i came up with the following pseudo-periodicity-block
for 46-ET, by making a 4-dimensional bingo-card lattice
and selecting notes by eye.

i call it a "pseudo" periodicity-block because it
still includes notes at the edges of the block which
are duplicates or triplicates.

here it is, as a list of [3,5,7,11]-monzos,
with the associated 46-ET mapping:

(if viewing on the Yahoo web interface, please forward
a copy to your email account to see the proper formatting)

3 5 7 11 46-ET

single notes:

[ 0 3 0 0] 45
[ 0 2 0 0] 30
[-2 1 0 0] 7
[-1 1 0 0] 34
[ 0 1 0 0] 15
[ 1 1 0 0] 42
[-2 0 0 0] 38
[-1 0 0 0] 19
[ 0 0 0 0] 0
[ 1 0 0 0] 27
[ 2 0 0 0] 8
[-1 -1 0 0] 4
[ 0 -1 0 0] 31
[ 1 -1 0 0] 12
[ 2 -1 0 0] 39
[ 0 -2 0 0] 16
[ 0 -3 0 0] 1
[ 0 0 1 0] 37
[ 1 0 1 0] 18
[-1 -1 1 0] 41
[ 0 -1 1 0] 22
[ 0 1 -1 0] 24
[ 1 1 -1 0] 5
[-1 0 -1 0] 28
[ 0 0 -1 0] 9
[ 0 0 0 1] 21
[ 1 0 0 1] 2
[-1 0 0 -1] 44
[ 0 0 0 -1] 25

all of the following pairs or triples
of notes are the same number of steps
away from the origin:

the following 5 pairs are separated by
a diaschisma:

[-3 1 0 0] 26
[ 1 3 0 0] 26

[-3 0 0 0] 11
[ 1 2 0 0] 11

[-2 -1 0 0] 23
[ 2 1 0 0] 23

[-1 -2 0 0] 35
[ 3 0 0 0] 35

[-1 -3 0 0] 20
[ 3 -1 0 0] 20

these are the other pairs:

[ 0 4 0 0] 14
[-2 -1 1 0] 14

[-1 2 0 0] 3
[ 1 -1 1 0] 3

[ 0 -4 0 0] 32
[ 2 1 -1 0] 32

[ 1 -2 0 0] 43
[-1 1 -1 0] 43

[-2 0 1 0] 29
[-1 -1 0 -1] 29

[-1 0 1 0] 10
[ 0 -1 0 -1] 10

[ 1 0 -1 0] 36
[ 0 1 0 1] 36

[ 2 0 -1 0] 17
[ 1 1 0 1] 17

the following are the triples:

[ 0 1 1 0] 6
[ 0 -1 0 1] 6
[ 1 0 0 -1] 6

[ 1 1 1 0] 33
[ 1 -1 0 1] 33
[ 2 0 0 -1] 33

[-1 -1 -1 0] 13
[-2 0 0 1] 13
[-1 1 0 -1] 13

[ 0 -1 -1 0] 40
[-1 0 0 1] 40
[ 0 1 0 -1] 40

=================

i tried to derive a 46-note periodicity-block
which is a subset of this list, using our software,
with the following unison-vectors:

[ 3 -6 1 0]
[-4 -2 0 0] (diaschisma)
[ 2 -3 1 0] (small septimal comma)
[ 3 -4 0 -1]

and the software gave me a nice 46-note
periodicity-block, but it was entirely
in the [3,5]-plane.

as can be seen from the first list of monzos
at the top of this post (the single notes),
the block i want has at least 8 notes with
7 as a prime-factor, and at least 4 notes
with 11 as a factor ... and there could be
more, depending on how the duplicates and
triplicates are filtered out.

so my question is: what unison-vectors do i
need, to produce the 46-tone periodicity-block
in which every note is part of a subset of
monzos i listed above?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i tried to derive a 46-note periodicity-block
> which is a subset of this list, using our software,
> with the following unison-vectors:
>
> [ 3 -6 1 0]
> [-4 -2 0 0] (diaschisma)
> [ 2 -3 1 0] (small septimal comma)
> [ 3 -4 0 -1]

First Dave, and now you. Commas and any other interval, as opposed to
an octave class, should ALWAYS be given with the correct value for 2.
Anything else simply will not do. If I was referee for Dave's
article, it would bounce like a rubber ball until he fixed that
problem.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i tried to derive a 46-note periodicity-block
> > which is a subset of this list, using our software,
> > with the following unison-vectors:
> >
> > [ 3 -6 1 0]
> > [-4 -2 0 0] (diaschisma)
> > [ 2 -3 1 0] (small septimal comma)
> > [ 3 -4 0 -1]
>
> First Dave, and now you. Commas and any other interval, as opposed
to
> an octave class, should ALWAYS be given with the correct value for
2.
> Anything else simply will not do. If I was referee for Dave's
> article, it would bounce like a rubber ball until he fixed that
> problem.

good think you weren't a referree for fokker's original papers.

>> First Dave, and now you. Commas and any other interval, as opposed
>> to an octave class, should ALWAYS be given with the correct value for
>> 2. Anything else simply will not do. If I was referee for Dave's
>> article, it would bounce like a rubber ball until he fixed that
>> problem.
>
>good think you weren't a referree for fokker's original papers.

Why? Sounds like Fokker could have used the feedback too.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> good think you weren't a referree for fokker's original papers.

Why? Referees are always insisting on one change or another,
generally for the better.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i tried to derive a 46-note periodicity-block
> which is a subset of this list,

of course, *any* 46-note subset which has each of the 46-equal
degrees exactly once is already a periodicity block . . . (by the
way, i don't agree with your reckoning of "equally close" . . .)

> using our software,
> with the following unison-vectors:
>
> [ 3 -6 1 0]
> [-4 -2 0 0] (diaschisma)
> [ 2 -3 1 0] (small septimal comma)
> [ 3 -4 0 -1]
>
>
> and the software gave me a nice 46-note
> periodicity-block, but it was entirely
> in the [3,5]-plane.

the determinant of this matrix is 14, so i'm not sure how you're
getting a 46-note periodicity block out of it!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i tried to derive a 46-note periodicity-block
> > which is a subset of this list,
>
> of course, *any* 46-note subset which has each of the 46-equal
> degrees exactly once is already a periodicity block . . . (by the
> way, i don't agree with your reckoning of "equally close" . . .)
>
> > using our software,
> > with the following unison-vectors:
> >
> > [ 3 -6 1 0]
> > [-4 -2 0 0] (diaschisma)
> > [ 2 -3 1 0] (small septimal comma)
> > [ 3 -4 0 -1]
> >
> >
> > and the software gave me a nice 46-note
> > periodicity-block, but it was entirely
> > in the [3,5]-plane.
>
> the determinant of this matrix is 14, so i'm not sure how you're
> getting a 46-note periodicity block out of it!

i grouped a few quartets of 11-limit commas together at random, and
after a couple of 72s, found that 9801:9800, 3025:3024, 441:440 and
176:175 together give 46. then i tried multiplying and dividing pairs
of these to get simpler ratios (being sure to keep 4 linearly
independent ones at each stage); one possibility is 896:891, 385:384,
125:126, and 176:175. the matrix of these:

-4 0 1 -1
-1 1 1 1
-2 3 -1 0
0 -2 -1 1

now i looked at the periodicity block defined by the unit hypercube
lying between 0 and 1 (instead of the usual -.5 and .5) along each of
the four transformed coordinate axes using the matrix above:

numerator denominator
55 54
33 32
25 24
16 15
275 256
11 10
10 9
9 8
55 48
7 6
33 28
6 5
175 144
99 80
5 4
32 25
165 128
33 25
4 3
27 20
11 8
7 5
99 70
275 192
35 24
165 112
3 2
55 36
99 64
25 16
8 5
825 512
33 20
5 3
297 175
55 32
7 4
99 56
9 5
11 6
297 160
15 8
48 25
495 256
99 50
2 1

this has plenty of ratios with factors of 7 and 11 -- hopefully it's
close to what you need!

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i tried to derive a 46-note periodicity-block
> > which is a subset of this list, using our software,
> > with the following unison-vectors:
> >
> > [ 3 -6 1 0]
> > [-4 -2 0 0] (diaschisma)
> > [ 2 -3 1 0] (small septimal comma)
> > [ 3 -4 0 -1]
>
> First Dave, and now you. Commas and any
> other interval, as opposed to an octave class,
> should ALWAYS be given with the correct value for 2.
> Anything else simply will not do. If I was referee
> for Dave's article, it would bounce like a rubber
> ball until he fixed that problem.

OK, thanks for pointing that out ... but it's no big deal.

here is the matrix of unison-vectors with the exponents
of 2 included.

... and also note that the sign for the exponent of 7
in the first comma was incorrect, which is probably
why paul got a determinant of 14 rather than 46.

2 3 5 7 11

[ 12 3 -6 -1 0]
[ 11 -4 -2 0 0] (diaschisma)
[ 1 2 -3 1 0] (small septimal comma)
[ 8 3 -4 0 -1]

-monz

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > i tried to derive a 46-note periodicity-block
> > > which is a subset of this list,
> >
> > of course, *any* 46-note subset which has each of
> > the 46-equal degrees exactly once is already a
> > periodicity block . . .

right, i know that ... but i deliberately left in
the duplicates and triplicates to see what you, Gene,
et al would come up with.

> > (by the way, i don't agree with your reckoning of
> > "equally close" . . .)

i know ... you use the hexagonal reckoning rather than
the rectangular one i used. i considered doing that
from the start, but it was just easier for me to do
the one i did since i'm still using Excel for this
kind of stuff.

> >
> > > using our software,
> > > with the following unison-vectors:
> > >
> > > [ 3 -6 1 0]
> > > [-4 -2 0 0] (diaschisma)
> > > [ 2 -3 1 0] (small septimal comma)
> > > [ 3 -4 0 -1]
> > >
> > >
> > > and the software gave me a nice 46-note
> > > periodicity-block, but it was entirely
> > > in the [3,5]-plane.
> >
> > the determinant of this matrix is 14, so i'm not sure how you're
> > getting a 46-note periodicity block out of it!

a sign in the first comma was reversed. its complete
[2,3,5,7,11]-monzo should be [12 3 -6 -1 0] .
see my post to Gene.

> i grouped a few quartets of 11-limit commas together
> at random, and after a couple of 72s, found that
> 9801:9800, 3025:3024, 441:440 and 176:175 together
> give 46. then i tried multiplying and dividing pairs
> of these to get simpler ratios (being sure to keep
> 4 linearly independent ones at each stage); one
> possibility is 896:891, 385:384, 125:126, and 176:175.
> the matrix of these:
>
> -4 0 1 -1
> -1 1 1 1
> -2 3 -1 0
> 0 -2 -1 1
>
> now i looked at the periodicity block defined by the
> unit hypercube lying between 0 and 1 (instead of the
> usual -.5 and .5) along each of the four transformed
> coordinate axes using the matrix above:
>
> numerator denominator
> 55 54
> 33 32
> 25 24
> 16 15
> 275 256
> <etc., snip>
>
> this has plenty of ratios with factors of 7 and 11 --
> hopefully it's close to what you need!

thanks ... but it would be easier for me to tell if
instead of ratios the notes had already been factored
into monzos. if anyone else cares to do it for me,
that would be nice! ;-)

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > > i tried to derive a 46-note periodicity-block
> > > > which is a subset of this list,
> > >
> > > of course, *any* 46-note subset which has each of
> > > the 46-equal degrees exactly once is already a
> > > periodicity block . . .
>
>
> right, i know that ... but i deliberately left in
> the duplicates and triplicates to see what you, Gene,
> et al would come up with.
>
>
>
> > > (by the way, i don't agree with your reckoning of
> > > "equally close" . . .)
>
>
> i know ... you use the hexagonal reckoning rather than
> the rectangular one i used. i considered doing that
> from the start, but it was just easier for me to do
> the one i did since i'm still using Excel for this
> kind of stuff.
>
>
>
> > >
> > > > using our software,
> > > > with the following unison-vectors:
> > > >
> > > > [ 3 -6 1 0]
> > > > [-4 -2 0 0] (diaschisma)
> > > > [ 2 -3 1 0] (small septimal comma)
> > > > [ 3 -4 0 -1]
> > > >
> > > >
> > > > and the software gave me a nice 46-note
> > > > periodicity-block, but it was entirely
> > > > in the [3,5]-plane.
> > >
> > > the determinant of this matrix is 14, so i'm not sure how
you're
> > > getting a 46-note periodicity block out of it!
>
>
>
> a sign in the first comma was reversed. its complete
> [2,3,5,7,11]-monzo should be [12 3 -6 -1 0] .
> see my post to Gene.
>
>
>
>
> > i grouped a few quartets of 11-limit commas together
> > at random, and after a couple of 72s, found that
> > 9801:9800, 3025:3024, 441:440 and 176:175 together
> > give 46. then i tried multiplying and dividing pairs
> > of these to get simpler ratios (being sure to keep
> > 4 linearly independent ones at each stage); one
> > possibility is 896:891, 385:384, 125:126, and 176:175.
> > the matrix of these:
> >
> > -4 0 1 -1
> > -1 1 1 1
> > -2 3 -1 0
> > 0 -2 -1 1
> >
> > now i looked at the periodicity block defined by the
> > unit hypercube lying between 0 and 1 (instead of the
> > usual -.5 and .5) along each of the four transformed
> > coordinate axes using the matrix above:
> >
> > numerator denominator
> > 55 54
> > 33 32
> > 25 24
> > 16 15
> > 275 256
> > <etc., snip>
> >
> > this has plenty of ratios with factors of 7 and 11 --
> > hopefully it's close to what you need!
>
>
>
> thanks ... but it would be easier for me to tell if
> instead of ratios the notes had already been factored
> into monzos. if anyone else cares to do it for me,
> that would be nice! ;-)
>
>
>
>
> -monz

here are the factorizations: first column is 3, second column is 5,
third column is 7, fourth column is 11:

-3 1 0 1
1 0 0 1
-1 2 0 0
-1 -1 0 0
0 2 0 1
0 -1 0 1
-2 1 0 0
2 0 0 0
-1 1 0 1
-1 0 1 0
1 0 -1 1
1 -1 0 0
-2 2 1 0
2 -1 0 1
0 1 0 0
0 -2 0 0
1 1 0 1
1 -2 0 1
-1 0 0 0
3 -1 0 0
0 0 0 1
0 -1 1 0
2 -1 -1 1
-1 2 0 1
-1 1 1 0
1 1 -1 1
1 0 0 0
-2 1 0 1
2 0 0 1
0 2 0 0
0 -1 0 0
1 2 0 1
1 -1 0 1
-1 1 0 0
3 -2 -1 1
0 1 0 1
0 0 1 0
2 0 -1 1
2 -1 0 0
-1 0 0 1
3 -1 0 1
1 1 0 0
1 -2 0 0
2 1 0 1
2 -2 0 1
0 0 0 0

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> i grouped a few quartets of 11-limit commas together at random, and
> after a couple of 72s, found that 9801:9800, 3025:3024, 441:440 and
> 176:175 together give 46. then i tried multiplying and dividing pairs
> of these to get simpler ratios (being sure to keep 4 linearly
> independent ones at each stage); one possibility is 896:891, 385:384,
> 125:126, and 176:175.

If you TM reduce this, you get the 11-limit TM basis for h46, namely
[121/120, 126/125, 176/175, 245/243]. A Fokker block for this can be
obtained from

q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46)
(176/175)^round(38i/46) (245/243)^round(31i/46)

This gives the following scale:

1 [0, 0, 0, 0, 0]
55/54 [-1, -3, 1, 0, 1]
36/35 [2, 2, -1, -1, 0]
25/24 [-3, -1, 2, 0, 0]
35/33 [0, -1, 1, 1, -1]
15/14 [-1, 1, 1, -1, 0]
11/10 [-1, 0, -1, 0, 1]
10/9 [1, -2, 1, 0, 0]
198/175 [1, 2, -2, -1, 1]
25/22 [-1, 0, 2, 0, -1]
7/6 [-1, -1, 0, 1, 0]
90/77 [1, 2, 1, -1, -1]
6/5 [1, 1, -1, 0, 0]
11/9 [0, -2, 0, 0, 1]
216/175 [3, 3, -2, -1, 0]
5/4 [-2, 0, 1, 0, 0]
14/11 [1, 0, 0, 1, -1]
9/7 [0, 2, 0, -1, 0]
33/25 [0, 1, -2, 0, 1]
4/3 [2, -1, 0, 0, 0]
1188/875 [2, 3, -3, -1, 1]
15/11 [0, 1, 1, 0, -1]
7/5 [0, 0, -1, 1, 0]
140/99 [2, -2, 1, 1, -1]
10/7 [1, 0, 1, -1, 0]
22/15 [1, -1, -1, 0, 1]
875/594 [-1, -3, 3, 1, -1]
3/2 [-1, 1, 0, 0, 0]
50/33 [1, -1, 2, 0, -1]
14/9 [1, -2, 0, 1, 0]
11/7 [0, 0, 0, -1, 1]
8/5 [3, 0, -1, 0, 0]
175/108 [-2, -3, 2, 1, 0]
18/11 [1, 2, 0, 0, -1]
5/3 [0, -1, 1, 0, 0]
77/45 [0, -2, -1, 1, 1]
12/7 [2, 1, 0, -1, 0]
44/25 [2, 0, -2, 0, 1]
175/99 [0, -2, 2, 1, -1]
9/5 [0, 2, -1, 0, 0]
20/11 [2, 0, 1, 0, -1]
28/15 [2, -1, -1, 1, 0]
66/35 [1, 1, -1, -1, 1]
48/25 [4, 1, -2, 0, 0]
35/18 [-1, -2, 1, 1, 0]
108/55 [2, 3, -1, 0, -1]

>If you TM reduce this, you get the 11-limit TM basis for h46, namely
>[121/120, 126/125, 176/175, 245/243]. A Fokker block for this can be
>obtained from
>
>q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46)
>(176/175)^round(38i/46) (245/243)^round(31i/46)

Whoa, the makings of an intelligible method, safe to try at home.

What's i?

And where has 81/80 come from? The TM basis has 4 commas, which
should be enough to enclose an 11-limit block.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >If you TM reduce this, you get the 11-limit TM basis for h46,
namely
> >[121/120, 126/125, 176/175, 245/243]. A Fokker block for this can
be
> >obtained from
> >
> >q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46)
> >(176/175)^round(38i/46) (245/243)^round(31i/46)
>
> Whoa, the makings of an intelligible method, safe to try at home.
>
> What's i?

Sorry. "i" is an integer which runs from 0 to 45.

> And where has 81/80 come from?

81/80 is one step in 46-et.

The TM basis has 4 commas, which
> should be enough to enclose an 11-limit block.

Right. I'm giving the block. If you toss in the 81/80, giving you a
unimodular matrix, and invert the matrix, you get the values I used
in the above computation: 46, -19, -24, 38, 31. They are the top row
of the inverted matrix.

>> And where has 81/80 come from?
>
>81/80 is one step in 46-et.

How are we supposed to know that?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> And where has 81/80 come from?
> >
> >81/80 is one step in 46-et.
>
> How are we supposed to know that?

Surely you can calculate that? What are you asking?

Gene wrote:
>namely
> >[121/120, 126/125, 176/175, 245/243]. A Fokker block for this can

I found this one too, but thought it was too uneven. Paul's block is
better in that regard. I wonder though if an 11-limit 46-note
PB in which the four primes occur and which is strictly proper exists.

Manuel

that does look more like what monz wanted, hope he takes a look.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > i grouped a few quartets of 11-limit commas together at random,
and
> > after a couple of 72s, found that 9801:9800, 3025:3024, 441:440
and
> > 176:175 together give 46. then i tried multiplying and dividing
pairs
> > of these to get simpler ratios (being sure to keep 4 linearly
> > independent ones at each stage); one possibility is 896:891,
385:384,
> > 125:126, and 176:175.
>
> If you TM reduce this, you get the 11-limit TM basis for h46, namely
> [121/120, 126/125, 176/175, 245/243]. A Fokker block for this can be
> obtained from
>
> q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46)
> (176/175)^round(38i/46) (245/243)^round(31i/46)
>
> This gives the following scale:
>
> 1 [0, 0, 0, 0, 0]
> 55/54 [-1, -3, 1, 0, 1]
> 36/35 [2, 2, -1, -1, 0]
> 25/24 [-3, -1, 2, 0, 0]
> 35/33 [0, -1, 1, 1, -1]
> 15/14 [-1, 1, 1, -1, 0]
> 11/10 [-1, 0, -1, 0, 1]
> 10/9 [1, -2, 1, 0, 0]
> 198/175 [1, 2, -2, -1, 1]
> 25/22 [-1, 0, 2, 0, -1]
> 7/6 [-1, -1, 0, 1, 0]
> 90/77 [1, 2, 1, -1, -1]
> 6/5 [1, 1, -1, 0, 0]
> 11/9 [0, -2, 0, 0, 1]
> 216/175 [3, 3, -2, -1, 0]
> 5/4 [-2, 0, 1, 0, 0]
> 14/11 [1, 0, 0, 1, -1]
> 9/7 [0, 2, 0, -1, 0]
> 33/25 [0, 1, -2, 0, 1]
> 4/3 [2, -1, 0, 0, 0]
> 1188/875 [2, 3, -3, -1, 1]
> 15/11 [0, 1, 1, 0, -1]
> 7/5 [0, 0, -1, 1, 0]
> 140/99 [2, -2, 1, 1, -1]
> 10/7 [1, 0, 1, -1, 0]
> 22/15 [1, -1, -1, 0, 1]
> 875/594 [-1, -3, 3, 1, -1]
> 3/2 [-1, 1, 0, 0, 0]
> 50/33 [1, -1, 2, 0, -1]
> 14/9 [1, -2, 0, 1, 0]
> 11/7 [0, 0, 0, -1, 1]
> 8/5 [3, 0, -1, 0, 0]
> 175/108 [-2, -3, 2, 1, 0]
> 18/11 [1, 2, 0, 0, -1]
> 5/3 [0, -1, 1, 0, 0]
> 77/45 [0, -2, -1, 1, 1]
> 12/7 [2, 1, 0, -1, 0]
> 44/25 [2, 0, -2, 0, 1]
> 175/99 [0, -2, 2, 1, -1]
> 9/5 [0, 2, -1, 0, 0]
> 20/11 [2, 0, 1, 0, -1]
> 28/15 [2, -1, -1, 1, 0]
> 66/35 [1, 1, -1, -1, 1]
> 48/25 [4, 1, -2, 0, 0]
> 35/18 [-1, -2, 1, 1, 0]
> 108/55 [2, 3, -1, 0, -1]

>> >> And where has 81/80 come from?
>> >
>> >81/80 is one step in 46-et.
>>
>> How are we supposed to know that?
>
>Surely you can calculate that? What are you asking?

Calculate it from what? One step of 46-et could be lots
of things. Why did you pick 81:80?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > i grouped a few quartets of 11-limit commas together at random,
and
> > after a couple of 72s, found that 9801:9800, 3025:3024, 441:440
and
> > 176:175 together give 46. then i tried multiplying and dividing
pairs
> > of these to get simpler ratios (being sure to keep 4 linearly
> > independent ones at each stage); one possibility is 896:891,
385:384,
> > 125:126, and 176:175.
>
> If you TM reduce this, you get the 11-limit TM basis for h46, namely
> [121/120, 126/125, 176/175, 245/243]. A Fokker block for this can be
> obtained from
>
> q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46)
> (176/175)^round(38i/46) (245/243)^round(31i/46)
>
> This gives the following scale:
>
> 1 [0, 0, 0, 0, 0]
> 55/54 [-1, -3, 1, 0, 1]
> 36/35 [2, 2, -1, -1, 0]
> 25/24 [-3, -1, 2, 0, 0]
> 35/33 [0, -1, 1, 1, -1]
> 15/14 [-1, 1, 1, -1, 0]
> 11/10 [-1, 0, -1, 0, 1]
> 10/9 [1, -2, 1, 0, 0]
> 198/175 [1, 2, -2, -1, 1]
> 25/22 [-1, 0, 2, 0, -1]
> 7/6 [-1, -1, 0, 1, 0]
> 90/77 [1, 2, 1, -1, -1]
> 6/5 [1, 1, -1, 0, 0]
> 11/9 [0, -2, 0, 0, 1]
> 216/175 [3, 3, -2, -1, 0]
> 5/4 [-2, 0, 1, 0, 0]
> 14/11 [1, 0, 0, 1, -1]
> 9/7 [0, 2, 0, -1, 0]
> 33/25 [0, 1, -2, 0, 1]
> 4/3 [2, -1, 0, 0, 0]
> 1188/875 [2, 3, -3, -1, 1]
> 15/11 [0, 1, 1, 0, -1]
> 7/5 [0, 0, -1, 1, 0]
> 140/99 [2, -2, 1, 1, -1]
> 10/7 [1, 0, 1, -1, 0]
> 22/15 [1, -1, -1, 0, 1]
> 875/594 [-1, -3, 3, 1, -1]
> 3/2 [-1, 1, 0, 0, 0]
> 50/33 [1, -1, 2, 0, -1]
> 14/9 [1, -2, 0, 1, 0]
> 11/7 [0, 0, 0, -1, 1]
> 8/5 [3, 0, -1, 0, 0]
> 175/108 [-2, -3, 2, 1, 0]
> 18/11 [1, 2, 0, 0, -1]
> 5/3 [0, -1, 1, 0, 0]
> 77/45 [0, -2, -1, 1, 1]
> 12/7 [2, 1, 0, -1, 0]
> 44/25 [2, 0, -2, 0, 1]
> 175/99 [0, -2, 2, 1, -1]
> 9/5 [0, 2, -1, 0, 0]
> 20/11 [2, 0, 1, 0, -1]
> 28/15 [2, -1, -1, 1, 0]
> 66/35 [1, 1, -1, -1, 1]
> 48/25 [4, 1, -2, 0, 0]
> 35/18 [-1, -2, 1, 1, 0]
> 108/55 [2, 3, -1, 0, -1]

my fokker block for [121/120, 126/125, 176/175, 245/243] is

ratio 3^ 5^ 7^ 11^
891/875 4 -3 -1 1
45/44 2 1 0 -1
21/20 1 -1 1 0
77/72 -2 0 1 1
15/14 1 1 -1 0
11/10 0 -1 0 1
875/792 -2 3 1 -1
9/8 2 0 0 0
25/22 0 2 0 -1
7/6 -1 0 1 0
33/28 1 0 -1 1
6/5 1 -1 0 0
175/144 -2 2 1 0
27/22 3 0 0 -1
5/4 0 1 0 0
77/60 -1 -1 1 1
9/7 2 0 -1 0
33/25 1 -2 0 1
175/132 -1 2 1 -1
27/20 3 -1 0 0
15/11 1 1 0 -1
7/5 0 -1 1 0
99/70 2 -1 -1 1
36/25 2 -2 0 0
35/24 -1 1 1 0
81/55 4 -1 0 -1
3/2 1 0 0 0
55/36 -2 1 0 1
54/35 3 -1 -1 0
25/16 0 2 0 0
35/22 0 1 1 -1
45/28 2 1 -1 0
33/20 1 -1 0 1
5/3 -1 1 0 0
297/175 3 -2 -1 1
75/44 1 2 0 -1
7/4 0 0 1 0
135/77 3 1 -1 -1
9/5 2 -1 0 0
11/6 -1 0 0 1
324/175 4 -2 -1 0
15/8 1 1 0 0
21/11 1 0 1 -1
27/14 3 0 -1 0
99/50 2 -2 0 1
2 0 0 0 0

i transformed the lattice by the inverse of the fokker matrix and
threw out anything with a coordinate >.50000001 or <-.49999999. it
can't be that we differ only because you kept some or all of the -
0.5s, because that would only affect #23. hmm . . .

sorry folks -- there was a mysterious extra line in my program. when
i removed it, i got the same block as gene, except 77/54 instead of
140/99 -- that's the point with a -.5 coordinate . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > i grouped a few quartets of 11-limit commas together at random,
> and
> > > after a couple of 72s, found that 9801:9800, 3025:3024, 441:440
> and
> > > 176:175 together give 46. then i tried multiplying and dividing
> pairs
> > > of these to get simpler ratios (being sure to keep 4 linearly
> > > independent ones at each stage); one possibility is 896:891,
> 385:384,
> > > 125:126, and 176:175.
> >
> > If you TM reduce this, you get the 11-limit TM basis for h46,
namely
> > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this can
be
> > obtained from
> >
> > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-24i/46)
> > (176/175)^round(38i/46) (245/243)^round(31i/46)
> >
> > This gives the following scale:
> >
> > 1 [0, 0, 0, 0, 0]
> > 55/54 [-1, -3, 1, 0, 1]
> > 36/35 [2, 2, -1, -1, 0]
> > 25/24 [-3, -1, 2, 0, 0]
> > 35/33 [0, -1, 1, 1, -1]
> > 15/14 [-1, 1, 1, -1, 0]
> > 11/10 [-1, 0, -1, 0, 1]
> > 10/9 [1, -2, 1, 0, 0]
> > 198/175 [1, 2, -2, -1, 1]
> > 25/22 [-1, 0, 2, 0, -1]
> > 7/6 [-1, -1, 0, 1, 0]
> > 90/77 [1, 2, 1, -1, -1]
> > 6/5 [1, 1, -1, 0, 0]
> > 11/9 [0, -2, 0, 0, 1]
> > 216/175 [3, 3, -2, -1, 0]
> > 5/4 [-2, 0, 1, 0, 0]
> > 14/11 [1, 0, 0, 1, -1]
> > 9/7 [0, 2, 0, -1, 0]
> > 33/25 [0, 1, -2, 0, 1]
> > 4/3 [2, -1, 0, 0, 0]
> > 1188/875 [2, 3, -3, -1, 1]
> > 15/11 [0, 1, 1, 0, -1]
> > 7/5 [0, 0, -1, 1, 0]
> > 140/99 [2, -2, 1, 1, -1]
> > 10/7 [1, 0, 1, -1, 0]
> > 22/15 [1, -1, -1, 0, 1]
> > 875/594 [-1, -3, 3, 1, -1]
> > 3/2 [-1, 1, 0, 0, 0]
> > 50/33 [1, -1, 2, 0, -1]
> > 14/9 [1, -2, 0, 1, 0]
> > 11/7 [0, 0, 0, -1, 1]
> > 8/5 [3, 0, -1, 0, 0]
> > 175/108 [-2, -3, 2, 1, 0]
> > 18/11 [1, 2, 0, 0, -1]
> > 5/3 [0, -1, 1, 0, 0]
> > 77/45 [0, -2, -1, 1, 1]
> > 12/7 [2, 1, 0, -1, 0]
> > 44/25 [2, 0, -2, 0, 1]
> > 175/99 [0, -2, 2, 1, -1]
> > 9/5 [0, 2, -1, 0, 0]
> > 20/11 [2, 0, 1, 0, -1]
> > 28/15 [2, -1, -1, 1, 0]
> > 66/35 [1, 1, -1, -1, 1]
> > 48/25 [4, 1, -2, 0, 0]
> > 35/18 [-1, -2, 1, 1, 0]
> > 108/55 [2, 3, -1, 0, -1]
>
> my fokker block for [121/120, 126/125, 176/175, 245/243] is
>
> ratio 3^ 5^ 7^ 11^
> 891/875 4 -3 -1 1
> 45/44 2 1 0 -1
> 21/20 1 -1 1 0
> 77/72 -2 0 1 1
> 15/14 1 1 -1 0
> 11/10 0 -1 0 1
> 875/792 -2 3 1 -1
> 9/8 2 0 0 0
> 25/22 0 2 0 -1
> 7/6 -1 0 1 0
> 33/28 1 0 -1 1
> 6/5 1 -1 0 0
> 175/144 -2 2 1 0
> 27/22 3 0 0 -1
> 5/4 0 1 0 0
> 77/60 -1 -1 1 1
> 9/7 2 0 -1 0
> 33/25 1 -2 0 1
> 175/132 -1 2 1 -1
> 27/20 3 -1 0 0
> 15/11 1 1 0 -1
> 7/5 0 -1 1 0
> 99/70 2 -1 -1 1
> 36/25 2 -2 0 0
> 35/24 -1 1 1 0
> 81/55 4 -1 0 -1
> 3/2 1 0 0 0
> 55/36 -2 1 0 1
> 54/35 3 -1 -1 0
> 25/16 0 2 0 0
> 35/22 0 1 1 -1
> 45/28 2 1 -1 0
> 33/20 1 -1 0 1
> 5/3 -1 1 0 0
> 297/175 3 -2 -1 1
> 75/44 1 2 0 -1
> 7/4 0 0 1 0
> 135/77 3 1 -1 -1
> 9/5 2 -1 0 0
> 11/6 -1 0 0 1
> 324/175 4 -2 -1 0
> 15/8 1 1 0 0
> 21/11 1 0 1 -1
> 27/14 3 0 -1 0
> 99/50 2 -2 0 1
> 2 0 0 0 0
>
> i transformed the lattice by the inverse of the fokker matrix and
> threw out anything with a coordinate >.50000001 or <-.49999999. it
> can't be that we differ only because you kept some or all of the -
> 0.5s, because that would only affect #23. hmm . . .

i performed the same calculation but this time the transformed
coordinates were between zero (inclusive) and one (exclusive). this
was to check monz who did the same thing offlist. most of the pitches
of this block agree with his, but not all (i even used the same signs
on the unison vectors as him):

ratio 3^ 5^ 7^ 11^
1 0 0 0 0
782/779 -3 -3 1 2
77/75 -1 -2 1 1
847/810 -4 -1 1 2
2156/2025 -4 -2 2 1
242/225 -2 -2 0 2
1232/1125 -2 -3 1 1
539/486 -5 0 2 1
847/750 -1 -3 1 2
154/135 -3 -1 1 1
999/853 -4 -3 2 2
88/75 -1 -2 0 1
2137/1774 -2 -4 1 2
599/491 -5 -1 2 2
154/125 0 -3 1 1
847/675 -3 -2 1 2
2393/1873 -3 -3 2 1
484/375 -1 -3 0 2
2464/1875 -1 -4 1 1
539/405 -4 -1 2 1
847/625 0 -4 1 2
308/225 -2 -2 1 1
2971/2114 -3 -4 2 2
176/125 0 -3 0 1
77/54 -3 0 1 1
467/319 -4 -2 2 2
22/15 -1 -1 0 1
1694/1125 -2 -3 1 2
616/405 -4 -1 1 1
968/625 0 -4 0 2
847/540 -3 -1 1 2
1078/675 -3 -2 2 1
121/75 -1 -2 0 2
616/375 -1 -3 1 1
3388/2025 -4 -2 1 2
1056/625 1 -4 0 1
77/45 -2 -1 1 1
2932/1669 -3 -3 2 2
44/25 0 -2 0 1
3388/1875 -1 -4 1 2
1232/675 -3 -2 1 1
749/403 1 -5 0 2
847/450 -2 -2 1 2
2156/1125 -2 -3 2 1
242/125 0 -3 0 2
1232/625 0 -4 1 1

> > > If you TM reduce this, you get the 11-limit TM basis for h46,
> namely
> > > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this
can
> be
> > > obtained from
> > >
> > > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-
24i/46)
> > > (176/175)^round(38i/46) (245/243)^round(31i/46)
> > >
> > > This gives the following scale:
> > >
> > > 1 [0, 0, 0, 0, 0]
> > > 55/54 [-1, -3, 1, 0, 1]
> > > 36/35 [2, 2, -1, -1, 0]
> > > 25/24 [-3, -1, 2, 0, 0]
> > > 35/33 [0, -1, 1, 1, -1]
> > > 15/14 [-1, 1, 1, -1, 0]
> > > 11/10 [-1, 0, -1, 0, 1]
> > > 10/9 [1, -2, 1, 0, 0]
> > > 198/175 [1, 2, -2, -1, 1]
> > > 25/22 [-1, 0, 2, 0, -1]
> > > 7/6 [-1, -1, 0, 1, 0]
> > > 90/77 [1, 2, 1, -1, -1]
> > > 6/5 [1, 1, -1, 0, 0]
> > > 11/9 [0, -2, 0, 0, 1]
> > > 216/175 [3, 3, -2, -1, 0]
> > > 5/4 [-2, 0, 1, 0, 0]
> > > 14/11 [1, 0, 0, 1, -1]
> > > 9/7 [0, 2, 0, -1, 0]
> > > 33/25 [0, 1, -2, 0, 1]
> > > 4/3 [2, -1, 0, 0, 0]
> > > 1188/875 [2, 3, -3, -1, 1]
> > > 15/11 [0, 1, 1, 0, -1]
> > > 7/5 [0, 0, -1, 1, 0]
> > > 140/99 [2, -2, 1, 1, -1]
> > > 10/7 [1, 0, 1, -1, 0]
> > > 22/15 [1, -1, -1, 0, 1]
> > > 875/594 [-1, -3, 3, 1, -1]
> > > 3/2 [-1, 1, 0, 0, 0]
> > > 50/33 [1, -1, 2, 0, -1]
> > > 14/9 [1, -2, 0, 1, 0]
> > > 11/7 [0, 0, 0, -1, 1]
> > > 8/5 [3, 0, -1, 0, 0]
> > > 175/108 [-2, -3, 2, 1, 0]
> > > 18/11 [1, 2, 0, 0, -1]
> > > 5/3 [0, -1, 1, 0, 0]
> > > 77/45 [0, -2, -1, 1, 1]
> > > 12/7 [2, 1, 0, -1, 0]
> > > 44/25 [2, 0, -2, 0, 1]
> > > 175/99 [0, -2, 2, 1, -1]
> > > 9/5 [0, 2, -1, 0, 0]
> > > 20/11 [2, 0, 1, 0, -1]
> > > 28/15 [2, -1, -1, 1, 0]
> > > 66/35 [1, 1, -1, -1, 1]
> > > 48/25 [4, 1, -2, 0, 0]
> > > 35/18 [-1, -2, 1, 1, 0]
> > > 108/55 [2, 3, -1, 0, -1]

d'oh! i was using format rat, which approximates ratios that are too
complex with simpler ratios! the real ratios are . . . well who
cares, they agree perfectly with monz's, and of course the monzos
below agree . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> i performed the same calculation but this time the transformed
> coordinates were between zero (inclusive) and one (exclusive). this
> was to check monz who did the same thing offlist. most of the
pitches
> of this block agree with his, but not all (i even used the same
signs
> on the unison vectors as him):
>
> ratio 3^ 5^ 7^ 11^
> 1 0 0 0 0
> 782/779 -3 -3 1 2
> 77/75 -1 -2 1 1
> 847/810 -4 -1 1 2
> 2156/2025 -4 -2 2 1
> 242/225 -2 -2 0 2
> 1232/1125 -2 -3 1 1
> 539/486 -5 0 2 1
> 847/750 -1 -3 1 2
> 154/135 -3 -1 1 1
> 999/853 -4 -3 2 2
> 88/75 -1 -2 0 1
> 2137/1774 -2 -4 1 2
> 599/491 -5 -1 2 2
> 154/125 0 -3 1 1
> 847/675 -3 -2 1 2
> 2393/1873 -3 -3 2 1
> 484/375 -1 -3 0 2
> 2464/1875 -1 -4 1 1
> 539/405 -4 -1 2 1
> 847/625 0 -4 1 2
> 308/225 -2 -2 1 1
> 2971/2114 -3 -4 2 2
> 176/125 0 -3 0 1
> 77/54 -3 0 1 1
> 467/319 -4 -2 2 2
> 22/15 -1 -1 0 1
> 1694/1125 -2 -3 1 2
> 616/405 -4 -1 1 1
> 968/625 0 -4 0 2
> 847/540 -3 -1 1 2
> 1078/675 -3 -2 2 1
> 121/75 -1 -2 0 2
> 616/375 -1 -3 1 1
> 3388/2025 -4 -2 1 2
> 1056/625 1 -4 0 1
> 77/45 -2 -1 1 1
> 2932/1669 -3 -3 2 2
> 44/25 0 -2 0 1
> 3388/1875 -1 -4 1 2
> 1232/675 -3 -2 1 1
> 749/403 1 -5 0 2
> 847/450 -2 -2 1 2
> 2156/1125 -2 -3 2 1
> 242/125 0 -3 0 2
> 1232/625 0 -4 1 1
>
> > > > If you TM reduce this, you get the 11-limit TM basis for h46,
> > namely
> > > > [121/120, 126/125, 176/175, 245/243]. A Fokker block for this
> can
> > be
> > > > obtained from
> > > >
> > > > q[i] = (81/80)^i (121/120)^round(-19i/46) (126/125)^round(-
> 24i/46)
> > > > (176/175)^round(38i/46) (245/243)^round(31i/46)
> > > >
> > > > This gives the following scale:
> > > >
> > > > 1 [0, 0, 0, 0, 0]
> > > > 55/54 [-1, -3, 1, 0, 1]
> > > > 36/35 [2, 2, -1, -1, 0]
> > > > 25/24 [-3, -1, 2, 0, 0]
> > > > 35/33 [0, -1, 1, 1, -1]
> > > > 15/14 [-1, 1, 1, -1, 0]
> > > > 11/10 [-1, 0, -1, 0, 1]
> > > > 10/9 [1, -2, 1, 0, 0]
> > > > 198/175 [1, 2, -2, -1, 1]
> > > > 25/22 [-1, 0, 2, 0, -1]
> > > > 7/6 [-1, -1, 0, 1, 0]
> > > > 90/77 [1, 2, 1, -1, -1]
> > > > 6/5 [1, 1, -1, 0, 0]
> > > > 11/9 [0, -2, 0, 0, 1]
> > > > 216/175 [3, 3, -2, -1, 0]
> > > > 5/4 [-2, 0, 1, 0, 0]
> > > > 14/11 [1, 0, 0, 1, -1]
> > > > 9/7 [0, 2, 0, -1, 0]
> > > > 33/25 [0, 1, -2, 0, 1]
> > > > 4/3 [2, -1, 0, 0, 0]
> > > > 1188/875 [2, 3, -3, -1, 1]
> > > > 15/11 [0, 1, 1, 0, -1]
> > > > 7/5 [0, 0, -1, 1, 0]
> > > > 140/99 [2, -2, 1, 1, -1]
> > > > 10/7 [1, 0, 1, -1, 0]
> > > > 22/15 [1, -1, -1, 0, 1]
> > > > 875/594 [-1, -3, 3, 1, -1]
> > > > 3/2 [-1, 1, 0, 0, 0]
> > > > 50/33 [1, -1, 2, 0, -1]
> > > > 14/9 [1, -2, 0, 1, 0]
> > > > 11/7 [0, 0, 0, -1, 1]
> > > > 8/5 [3, 0, -1, 0, 0]
> > > > 175/108 [-2, -3, 2, 1, 0]
> > > > 18/11 [1, 2, 0, 0, -1]
> > > > 5/3 [0, -1, 1, 0, 0]
> > > > 77/45 [0, -2, -1, 1, 1]
> > > > 12/7 [2, 1, 0, -1, 0]
> > > > 44/25 [2, 0, -2, 0, 1]
> > > > 175/99 [0, -2, 2, 1, -1]
> > > > 9/5 [0, 2, -1, 0, 0]
> > > > 20/11 [2, 0, 1, 0, -1]
> > > > 28/15 [2, -1, -1, 1, 0]
> > > > 66/35 [1, 1, -1, -1, 1]
> > > > 48/25 [4, 1, -2, 0, 0]
> > > > 35/18 [-1, -2, 1, 1, 0]
> > > > 108/55 [2, 3, -1, 0, -1]

when i get rid of that bad line in the program, this 896:891,
385:384, 125:126, and 176:175 block becomes:

ratio 3^ 5^ 7^ 11^
1 0 0 0 0
100/99 -2 2 0 -1
33/32 1 0 0 1
25/24 -1 2 0 0
16/15 -1 -1 0 0
15/14 1 1 -1 0
35/32 0 1 1 0
10/9 -2 1 0 0
9/8 2 0 0 0
8/7 0 0 -1 0
7/6 -1 0 1 0
33/28 1 0 -1 1
6/5 1 -1 0 0
40/33 -1 1 0 -1
99/80 2 -1 0 1
5/4 0 1 0 0
32/25 0 -2 0 0
9/7 2 0 -1 0
21/16 1 0 1 0
4/3 -1 0 0 0
27/20 3 -1 0 0
48/35 1 -1 -1 0
7/5 0 -1 1 0
64/45 -2 -1 0 0
10/7 0 1 -1 0
35/24 -1 1 1 0
40/27 -3 1 0 0
3/2 1 0 0 0
32/21 -1 0 -1 0
14/9 -2 0 1 0
25/16 0 2 0 0
8/5 0 -1 0 0
160/99 -2 1 0 -1
33/20 1 -1 0 1
5/3 -1 1 0 0
56/33 -1 0 1 -1
12/7 1 0 -1 0
7/4 0 0 1 0
16/9 -2 0 0 0
9/5 2 -1 0 0
64/35 0 -1 -1 0
28/15 -1 -1 1 0
15/8 1 1 0 0
48/25 1 -2 0 0
64/33 -1 0 0 -1
99/50 2 -2 0 1

i think this is the closest yet to fulfilling monz's original
requirements . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi paul,
> >
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > > wrote:
> > > > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > > >
> > > > > i tried to derive a 46-note periodicity-block
> > > > > which is a subset of this list,
> > > >
> > > > of course, *any* 46-note subset which has each of
> > > > the 46-equal degrees exactly once is already a
> > > > periodicity block . . .
> >
> >
> > right, i know that ... but i deliberately left in
> > the duplicates and triplicates to see what you, Gene,
> > et al would come up with.
> >
> >
> >
> > > > (by the way, i don't agree with your reckoning of
> > > > "equally close" . . .)
> >
> >
> > i know ... you use the hexagonal reckoning rather than
> > the rectangular one i used. i considered doing that
> > from the start, but it was just easier for me to do
> > the one i did since i'm still using Excel for this
> > kind of stuff.
> >
> >
> >
> > > >
> > > > > using our software,
> > > > > with the following unison-vectors:
> > > > >
> > > > > [ 3 -6 1 0]
> > > > > [-4 -2 0 0] (diaschisma)
> > > > > [ 2 -3 1 0] (small septimal comma)
> > > > > [ 3 -4 0 -1]
> > > > >
> > > > >
> > > > > and the software gave me a nice 46-note
> > > > > periodicity-block, but it was entirely
> > > > > in the [3,5]-plane.
> > > >
> > > > the determinant of this matrix is 14, so i'm not sure how
> you're
> > > > getting a 46-note periodicity block out of it!
> >
> >
> >
> > a sign in the first comma was reversed. its complete
> > [2,3,5,7,11]-monzo should be [12 3 -6 -1 0] .
> > see my post to Gene.
> >
> >
> >
> >
> > > i grouped a few quartets of 11-limit commas together
> > > at random, and after a couple of 72s, found that
> > > 9801:9800, 3025:3024, 441:440 and 176:175 together
> > > give 46. then i tried multiplying and dividing pairs
> > > of these to get simpler ratios (being sure to keep
> > > 4 linearly independent ones at each stage); one
> > > possibility is 896:891, 385:384, 125:126, and 176:175.
> > > the matrix of these:
> > >
> > > -4 0 1 -1
> > > -1 1 1 1
> > > -2 3 -1 0
> > > 0 -2 -1 1
> > >
> > > now i looked at the periodicity block defined by the
> > > unit hypercube lying between 0 and 1 (instead of the
> > > usual -.5 and .5) along each of the four transformed
> > > coordinate axes using the matrix above:
> > >
> > > numerator denominator
> > > 55 54
> > > 33 32
> > > 25 24
> > > 16 15
> > > 275 256
> > > <etc., snip>
> > >
> > > this has plenty of ratios with factors of 7 and 11 --
> > > hopefully it's close to what you need!
> >
> >
> >
> > thanks ... but it would be easier for me to tell if
> > instead of ratios the notes had already been factored
> > into monzos. if anyone else cares to do it for me,
> > that would be nice! ;-)
> >
> >
> >
> >
> > -monz
>
> here are the factorizations: first column is 3, second column is 5,
> third column is 7, fourth column is 11:
>
>
> -3 1 0 1
> 1 0 0 1
> -1 2 0 0
> -1 -1 0 0
> 0 2 0 1
> 0 -1 0 1
> -2 1 0 0
> 2 0 0 0
> -1 1 0 1
> -1 0 1 0
> 1 0 -1 1
> 1 -1 0 0
> -2 2 1 0
> 2 -1 0 1
> 0 1 0 0
> 0 -2 0 0
> 1 1 0 1
> 1 -2 0 1
> -1 0 0 0
> 3 -1 0 0
> 0 0 0 1
> 0 -1 1 0
> 2 -1 -1 1
> -1 2 0 1
> -1 1 1 0
> 1 1 -1 1
> 1 0 0 0
> -2 1 0 1
> 2 0 0 1
> 0 2 0 0
> 0 -1 0 0
> 1 2 0 1
> 1 -1 0 1
> -1 1 0 0
> 3 -2 -1 1
> 0 1 0 1
> 0 0 1 0
> 2 0 -1 1
> 2 -1 0 0
> -1 0 0 1
> 3 -1 0 1
> 1 1 0 0
> 1 -2 0 0
> 2 1 0 1
> 2 -2 0 1
> 0 0 0 0

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> And where has 81/80 come from?
> >> >
> >> >81/80 is one step in 46-et.
> >>
> >> How are we supposed to know that?
> >
> >Surely you can calculate that? What are you asking?
>
> Calculate it from what? One step of 46-et could be lots
> of things. Why did you pick 81:80?

The answer to "Why a duck?" is "Why not a duck?" I needed any 11-
limit interval which counted as one step of 46-equal, and it didn't
matter which one I picked.

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> d'oh! i was using format rat, which approximates ratios
> that are too complex with simpler ratios!

yeah, you have to watch out for that kind of thing!

i used to have lots of problems with that when using Excel,
until i finally just did everything using monzos instead
of ratios.

> the real ratios are . . . well who cares, they agree
> perfectly with monz's, and of course the monzos
> below agree . . .

OK, good! so then that means that the method being used
in our software is producing the same results as those used
by you and Gene, correct? that was the reason why i started
this thread in the first place, to check that we're going
about things the right way.

-monz

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> when i get rid of that bad line in the program, this 896:891,
> 385:384, 125:126, and 176:175 block becomes:
>
> ratio 3^ 5^ 7^ 11^
> 1 0 0 0 0
> 100/99 -2 2 0 -1
> 33/32 1 0 0 1
> 25/24 -1 2 0 0
> 16/15 -1 -1 0 0
> 15/14 1 1 -1 0
> 35/32 0 1 1 0
> 10/9 -2 1 0 0
> 9/8 2 0 0 0
> 8/7 0 0 -1 0
> 7/6 -1 0 1 0
> 33/28 1 0 -1 1
> 6/5 1 -1 0 0
> 40/33 -1 1 0 -1
> 99/80 2 -1 0 1
> 5/4 0 1 0 0
> 32/25 0 -2 0 0
> 9/7 2 0 -1 0
> 21/16 1 0 1 0
> 4/3 -1 0 0 0
> 27/20 3 -1 0 0
> 48/35 1 -1 -1 0
> 7/5 0 -1 1 0
> 64/45 -2 -1 0 0
> 10/7 0 1 -1 0
> 35/24 -1 1 1 0
> 40/27 -3 1 0 0
> 3/2 1 0 0 0
> 32/21 -1 0 -1 0
> 14/9 -2 0 1 0
> 25/16 0 2 0 0
> 8/5 0 -1 0 0
> 160/99 -2 1 0 -1
> 33/20 1 -1 0 1
> 5/3 -1 1 0 0
> 56/33 -1 0 1 -1
> 12/7 1 0 -1 0
> 7/4 0 0 1 0
> 16/9 -2 0 0 0
> 9/5 2 -1 0 0
> 64/35 0 -1 -1 0
> 28/15 -1 -1 1 0
> 15/8 1 1 0 0
> 48/25 1 -2 0 0
> 64/33 -1 0 0 -1
> 99/50 2 -2 0 1
>
> i think this is the closest yet to fulfilling monz's original
> requirements . . .

yes, indeed! -- this is *very* close to the original
pseudo-PB that i devised by eye, trying to keep all notes
as close as possible (according to the rectangular metric)
to the 1/1.

of course, the biggest difference is that your PB contains
only one instance of each note, whereas my pseudo-PB had
several duplicates and triplicates which were the same
number of steps from 1/1. i realize that i could use your
unison-vectors to find similar duplicates/triplicates in
your PB.

but brushing that aside (since i knew about it and expected
it from the beginning), the two big differences between
yours and mine are:

1) your first (after 1/1) and last notes contain both 7
and 11 as factors, whereas all notes in mine had either
7 *or* 11 *or* neither; and

2) your PB does *not* include 11/8 or 16/11, which i felt
should be included.

now, to continue the puzzle: can you or Gene (or another
adventurous tuning-math-er) find a PB which corrects those
two conditions?

-monz

i've uploaded a graphic to tuning_files, showing both my
original pseudo-PB and paul's latest PB, for 46-tone 11-limit:

/tuning-math/files/monz/compact_46-
et_pb.gif

or

http://tinyurl.com/t3as

i know that it's too small for the numbers and letters to
be legible, but the point is simply to see by the colors
which notes are in the PB and which are not.

in both diagrams, grey shading indicates notes which occur
only one time in the PB.

my original pseudo-PB, blue indicates duplicate notes and
green indicates triplicate, which are the same number of
(rectangular metric) steps away from 1/1. ... the brown
shading was only used to keep track of notes and can be
ignored.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> yes, indeed! -- this is *very* close to the original
> pseudo-PB that i devised by eye, trying to keep all notes
> as close as possible (according to the rectangular metric)
> to the 1/1.
>
>
> of course, the biggest difference is that your PB contains
> only one instance of each note, whereas my pseudo-PB had
> several duplicates and triplicates which were the same
> number of steps from 1/1. i realize that i could use your
> unison-vectors to find similar duplicates/triplicates in
> your PB.
>
>
> but brushing that aside (since i knew about it and expected
> it from the beginning), the two big differences between
> yours and mine are:
>
> 1) your first (after 1/1) and last notes contain both 7
> and 11 as factors, whereas all notes in mine had either
> 7 *or* 11 *or* neither; and
>
> 2) your PB does *not* include 11/8 or 16/11, which i felt
> should be included.
>
>
>
> now, to continue the puzzle: can you or Gene (or another
> adventurous tuning-math-er) find a PB which corrects those
> two conditions?
>
>
>
> -monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i've uploaded a graphic to tuning_files, showing both my
> original pseudo-PB and paul's latest PB, for 46-tone 11-limit:
>
>
> /tuning-math/files/monz/compact_46-
> et_pb.gif
>
> or
>
> http://tinyurl.com/t3as
>
>
> i know that it's too small for the numbers and letters to
> be legible, but the point is simply to see by the colors
> which notes are in the PB and which are not.
>
> in both diagrams, grey shading indicates notes which occur
> only one time in the PB.
>
> my original pseudo-PB, blue indicates duplicate notes and
> green indicates triplicate, which are the same number of
> (rectangular metric) steps away from 1/1. ... the brown
> shading was only used to keep track of notes and can be
> ignored.

sorry ... i was in a hurry when i posted that.
i should have added:

the lattice only show 2 dimensions at a time, those of
prime-factors 3 and 5. the horizontal axis is 3, the
vertical is 5.

the big diagrams on the left show only 3 and 5. the smaller
ones on the right show, from the top down respectively,
7^1, 7^-1, 11^1, and 11^-1.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> OK, good! so then that means that the method being used
> in our software is producing the same results as those used
> by you and Gene, correct?

right, but you'd get better matches to your 'closest-to-origin'
periodicity blocks if you used coordinate ranges of -.5 to .5 instead
of 0 to 1; i.e., putting 1/1 in the center of the block instead of a
corner.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > when i get rid of that bad line in the program, this 896:891,
> > 385:384, 125:126, and 176:175 block becomes:
> >
> > ratio 3^ 5^ 7^ 11^
> > 1 0 0 0 0
> > 100/99 -2 2 0 -1
> > 33/32 1 0 0 1
> > 25/24 -1 2 0 0
> > 16/15 -1 -1 0 0
> > 15/14 1 1 -1 0
> > 35/32 0 1 1 0
> > 10/9 -2 1 0 0
> > 9/8 2 0 0 0
> > 8/7 0 0 -1 0
> > 7/6 -1 0 1 0
> > 33/28 1 0 -1 1
> > 6/5 1 -1 0 0
> > 40/33 -1 1 0 -1
> > 99/80 2 -1 0 1
> > 5/4 0 1 0 0
> > 32/25 0 -2 0 0
> > 9/7 2 0 -1 0
> > 21/16 1 0 1 0
> > 4/3 -1 0 0 0
> > 27/20 3 -1 0 0
> > 48/35 1 -1 -1 0
> > 7/5 0 -1 1 0
> > 64/45 -2 -1 0 0
> > 10/7 0 1 -1 0
> > 35/24 -1 1 1 0
> > 40/27 -3 1 0 0
> > 3/2 1 0 0 0
> > 32/21 -1 0 -1 0
> > 14/9 -2 0 1 0
> > 25/16 0 2 0 0
> > 8/5 0 -1 0 0
> > 160/99 -2 1 0 -1
> > 33/20 1 -1 0 1
> > 5/3 -1 1 0 0
> > 56/33 -1 0 1 -1
> > 12/7 1 0 -1 0
> > 7/4 0 0 1 0
> > 16/9 -2 0 0 0
> > 9/5 2 -1 0 0
> > 64/35 0 -1 -1 0
> > 28/15 -1 -1 1 0
> > 15/8 1 1 0 0
> > 48/25 1 -2 0 0
> > 64/33 -1 0 0 -1
> > 99/50 2 -2 0 1
> >
> > i think this is the closest yet to fulfilling monz's original
> > requirements . . .
>
>
>
> yes, indeed! -- this is *very* close to the original
> pseudo-PB that i devised by eye, trying to keep all notes
> as close as possible (according to the rectangular metric)
> to the 1/1.
>
>
> of course, the biggest difference is that your PB contains
> only one instance of each note, whereas my pseudo-PB had
> several duplicates and triplicates which were the same
> number of steps from 1/1. i realize that i could use your
> unison-vectors to find similar duplicates/triplicates in
> your PB.
>
>
> but brushing that aside (since i knew about it and expected
> it from the beginning), the two big differences between
> yours and mine are:
>
> 1) your first (after 1/1) and last notes contain both 7
> and 11 as factors, whereas all notes in mine had either
> 7 *or* 11 *or* neither; and

the first note after 1/1 is

> > 100/99 -2 2 0 -1

and the last is

> > 99/50 2 -2 0 1

so i don't know whay you mean. the only notes i have with both 7s and
11s are

> > 33/28 1 0 -1 1

and

> > 56/33 -1 0 1 -1

but these have *opposite* signs on 7 and 11, so are no more complex
than having 11 by itself.

> now, to continue the puzzle: can you or Gene (or another
> adventurous tuning-math-er) find a PB which corrects those
> two conditions?

it might be impossible, but i'll keep trying.

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > the two big differences between yours [46-tone 11-limit PB]
> > and mine are:
> >
> > 1) your first (after 1/1) and last notes contain both 7
> > and 11 as factors, whereas all notes in mine had either
> > 7 *or* 11 *or* neither; and
>
> the first note after 1/1 is
>
> > > 100/99 -2 2 0 -1
>
> and the last is
>
> > > 99/50 2 -2 0 1
>
> so i don't know what you mean. the only notes i have with
> both 7s and 11s are
>
> > > 33/28 1 0 -1 1
>
> and
>
> > > 56/33 -1 0 1 -1
>
> but these have *opposite* signs on 7 and 11, so are no more complex
> than having 11 by itself.

oops ... my bad! i should have taken another look before
i typed that. yes, those are the two notes i was referring to.
in the graphic i posted last night, you can see them in grey
at the bottom of the left side of your PB.

> > now, to continue the puzzle: can you or Gene (or another
> > adventurous tuning-math-er) find a PB which corrects those
> > two conditions?
>
> it might be impossible, but i'll keep trying.

thanks!

... but at this point, with the advances we've made in the
software over the last day, i can have fun myself just
trying out different unison-vectors.

i'm beginning to gain an awful lot of respect for 46-ET.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> ... but at this point, with the advances we've made in the
> software over the last day, i can have fun myself just
> trying out different unison-vectors.

great!