43edo 7-limit periodicity-block

i've just added some 7-limit lattices to my
Tuning Dictionary "meride" entry, showing the
"closest to 1/1" 7-limit periodicity-block
for 43edo.

http://sonic-arts.org/dict/meride.htm

(at the bottom of the page)

just above the lattice, i refer to Gene's
"7-limit MT reduced bases for 43edo". but
i find that on these lattices, 225:224 is closer
than 126:125. is that because i'm using the
rectangular rather than triangular/hexagonal
taxicab metric?

so anyway, the bases i see are 81:80 and 225:224.
what's the third one?

here's a list of [3,5,7] vectors for the ratios
in my periodicity-block; asterisks indicate pitches
which occur twice (**) or 3 times (***) equally far
away from 1/1, with the 43edo-degree number -- they're
shown in darker shades of grey on the 5-limit "sheets"
lattices:

[ 0 5 0] ***27
[ 0 4 0] **13
[ 0 3 0] **42
[-1 2 0] **3
[ 0 2 0]
[-1 1 0]
[ 0 1 0]
[ 1 1 0]
[-2 0 0]
[-1 0 0]
[ 0 0 0]
[ 1 0 0]
[ 2 0 0]
[-1 -1 0]
[ 0 -1 0]
[ 1 -1 0]
[ 0 -2 0]
[ 1 -2 0] **40
[ 0 -3 0] **1
[ 0 -4 0] **30
[ 0 -5 0] **16
[ 0 3 1]
[-1 2 1]
[ 0 2 1]
[ 1 2 1]
[ 2 2 1] ***27
[-1 1 1]
[ 0 1 1]
[ 1 1 1]
[ 2 1 1] **13
[-1 0 1]
[ 0 0 1]
[ 1 0 1]
[ 2 0 1] **42
[ 0 -1 1]
[ 1 -1 1] **3
[-1 1 -1] **40
[ 0 1 -1]
[-2 0 -1] **1
[-1 0 -1]
[ 0 0 -1]
[ 1 0 -1]
[-2 -1 -1] **30
[-1 -1 -1]
[ 0 -1 -1]
[ 1 -1 -1]
[-2 -2 -1] **16
[-1 -2 -1]
[ 0 -2 -1]
[ 1 -2 -1]
[-1 -3 -1] ***27
[ 0 -3 -1]

-monz
"all roads lead to n^0"

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> i've just added some 7-limit lattices to my
> Tuning Dictionary "meride" entry, showing the
> "closest to 1/1" 7-limit periodicity-block
> for 43edo.
>
> http://sonic-arts.org/dict/meride.htm
>
> (at the bottom of the page)
>
> just above the lattice, i refer to Gene's
> "7-limit MT reduced bases for 43edo". but
> i find that on these lattices, 225:224 is closer
> than 126:125. is that because i'm using the
> rectangular rather than triangular/hexagonal
> taxicab metric?

partially, yes. really, the "T" in MT refers to the Tenney Harmonic
Distance function, in which the ratio with smaller numbers is always
represented by a shorter distance than a ratio with larger numbers.
geometrically, it *is* a rectangular lattice, but it (crucially)
includes 2 as a factor, and the length of each rung along the axis
for prime p is log(p). kees van prooijen's page, which i just
referred you to in a private e-mail, presents an impressive attempt
to incorporate the smaller-numbers-ratio->smaller-distance idea onto
an octave-equivalent lattice (at least for the small, comma-like
intervals), the octave-equivalence being necessary for representing
periodicity blocks with a finite number of points. i tried very hard
to get members of this list interested in kees' idea, and to help
figure out what was going on with this metric, but i found it akin to
beating my head against a wall.

> so anyway, the bases i see are 81:80 and 225:224.

these are not bases: a basis for an et in the 7-limit would have to
consist of three unison vectors.

> what's the third one?

there are of course an infinite number of possibilities. note that
225:224 * 126:125 = 81:80. so these three are not linearly
independent. any two imply the third. since gene's list was

81:80
126:125
12288:12005

we therefore know that 12288:12005 is one possible choice for forming
a complete basis for 43 along with 81:80 and 225:224.

Paul wrote:

>kees van prooijen's page, which i just
>referred you to in a private e-mail, presents an impressive attempt
>to incorporate the smaller-numbers-ratio->smaller-distance idea onto
>an octave-equivalent lattice (at least for the small, comma-like
>intervals), the octave-equivalence being necessary for representing
>periodicity blocks with a finite number of points. i tried very hard
>to get members of this list interested in kees' idea, and to help
>figure out what was going on with this metric, but i found it akin to
>beating my head against a wall.

I haven't told yet that this metric is implemented in Scala:
SET ATTRIBUTE PROOIJEN
and there's a little text in tips.par.

>we therefore know that 12288:12005 is one possible choice for forming
>a complete basis for 43 along with 81:80 and 225:224.

This PB doesn't look like what Joe plotted on his page.
Is it really a PB? I'm not sure.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Paul wrote:
>
> >kees van prooijen's page, which i just
> >referred you to in a private e-mail, presents an impressive attempt
> >to incorporate the smaller-numbers-ratio->smaller-distance idea
onto
> >an octave-equivalent lattice (at least for the small, comma-like
> >intervals), the octave-equivalence being necessary for representing
> >periodicity blocks with a finite number of points. i tried very
hard
> >to get members of this list interested in kees' idea, and to help
> >figure out what was going on with this metric, but i found it akin
to
> >beating my head against a wall.
>
> I haven't told yet that this metric is implemented in Scala:
> SET ATTRIBUTE PROOIJEN
> and there's a little text in tips.par.
>
> >we therefore know that 12288:12005 is one possible choice for
forming
> >a complete basis for 43 along with 81:80 and 225:224.
>
> This PB doesn't look like what Joe plotted on his page.

i said a complete basis, not necessarily a set of edges for a fokker
parallelepiped.

> Is it really a PB? I'm not sure.

is what really a PB? Joe's block? you have to resolve the ambiguous
positions he indicated for different occurences of the same note at
the same distance from 1/1, but once you've done that, it most
certainly is a PB, since it contains one and only one instance of
each of the 43 tones of 43-equal.

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, November 15, 2002 11:47 AM
> Subject: [tuning-math] Re: 43edo 7-limit periodicity-block
>
>
> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > Paul wrote:
> >
> > > we therefore know that 12288:12005 is one possible
> > > choice for forming a complete basis for 43 along
> > > with 81:80 and 225:224.
> >
> > This PB doesn't look like what Joe plotted on his page.
>
> i said a complete basis, not necessarily a set of edges
> for a fokker parallelepiped.

i don't know the difference between a "complete basis" and
"a set of edges for a fokker parallelepiped", and would
greatly welcome an explanation.

> > Is it really a PB? I'm not sure.
>
> is what really a PB? Joe's block? you have to resolve
> the ambiguous positions he indicated for different
> occurences of the same note at the same distance from
> 1/1, but once you've done that, it most certainly is
> a PB, since it contains one and only one instance of
> each of the 43 tones of 43-equal.

paul's explanation here is exactly right. i left in the
doubled and tripled instances of pitches which are an
equal-number of taxicab steps away from 1/1 in any of
the six directions (+/- 3/5/7), and connected them with
lines showing their equivalence.

but in any case, Manuel is absolutely correct that
12288:12005 is *not* one of the unison-vectors defining
the periodicity-block in my graphic.

the matrix i published on the webpage to represent
Gene's "7-limit MT reduced bases" for 43-edo is:

2 3 5 7

[-4 4 -1 0] = 81/80
[ 1 2 -3 1] = 126/125
[12 1 -1 -4] = 12288/12005

the periodicity-block in my graphic only uses 7^(-1,0,+1).

so my question still has not been answered: in addition
to 81:80 and 225:224, which i can easily see in my
diagram, what is the third necessary unison-vector which
defines my periodicity-block? and how does one figure
that out?

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning-math@y...>
> > Sent: Friday, November 15, 2002 11:47 AM
> > Subject: [tuning-math] Re: 43edo 7-limit periodicity-block
> >
> >
> > --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > > Paul wrote:
> > >
> > > > we therefore know that 12288:12005 is one possible
> > > > choice for forming a complete basis for 43 along
> > > > with 81:80 and 225:224.
> > >
> > > This PB doesn't look like what Joe plotted on his page.
> >
> > i said a complete basis, not necessarily a set of edges
> > for a fokker parallelepiped.
>
>
> i don't know the difference between a "complete basis" and
> "a set of edges for a fokker parallelepiped", and would
> greatly welcome an explanation.

any set of unison vectors which get you a 43-tone periodicity
block, which can be reasonably identified with 43-equal (in the
ways you've been doing), are a complete basis for 43. the
"edges of the fokker parallelepiped" refers to the method of
creating periodicity blocks discussed in the gentle introduction
(except the "excursion"). for example, you can see that (say, by
comparing ramos' tuning vs. other 12-note tunings) there are
different 12-tone periodicity blocks with different unison vectors
serving as their edges -- but any of these pairs of unison vectors
suffices as a basis for 12.

if this still isn't making sense, look at the "excursion". note that
*any* of the unison vectors:

syntonic comma
chromatic semitone
greater limma

describes how the first hexagonal periodicity block repeats itself
in the lattice. any *two* of these unison vectors will form a basis
for a 7-tone universe. similarly, *any* of the unison vectors

syntonic comma
diesis
diaschisma

describes how the second hexagonal periodicity block repeats
itself in the lattice. any *two* of these unison vectors will form a
basis for a 12-tone universe.

in neither case, though, are two of the unison vectors
representing the edges of a parallelogram -- because here, we
have a hexagon instead of a parallelogram!

> > > Is it really a PB? I'm not sure.
> >
> > is what really a PB? Joe's block? you have to resolve
> > the ambiguous positions he indicated for different
> > occurences of the same note at the same distance from
> > 1/1, but once you've done that, it most certainly is
> > a PB, since it contains one and only one instance of
> > each of the 43 tones of 43-equal.
>
>
> paul's explanation here is exactly right. i left in the
> doubled and tripled instances of pitches which are an
> equal-number of taxicab steps away from 1/1 in any of
> the six directions (+/- 3/5/7), and connected them with
> lines showing their equivalence.
>
>
> but in any case, Manuel is absolutely correct that
> 12288:12005 is *not* one of the unison-vectors defining
> the periodicity-block in my graphic.
>
> the matrix i published on the webpage to represent
> Gene's "7-limit MT reduced bases" for 43-edo is:
>
> 2 3 5 7
>
> [-4 4 -1 0] = 81/80
> [ 1 2 -3 1] = 126/125
> [12 1 -1 -4] = 12288/12005
>
>
> the periodicity-block in my graphic only uses 7^(-1,0,+1).
>
>
> so my question still has not been answered: in addition
> to 81:80 and 225:224, which i can easily see in my
> diagram, what is the third necessary unison-vector which
> defines my periodicity-block?

first of all, how are you "seeing" that 81:80 and 225:224 are two
of the unison vectors defining your periodicity block?

> and how does one figure
> that out?

just draw up a nice big block of the 7-limit lattice, divvy it up
according to how your PB tiles it, and observe the vectors at
which the tile repeats itself. simple! (there will be more than one
answer, depending on which direction in the lattice you look.)

i would have an easier time doing this by eye if you were to
resolve the ambiguous positions one way or the other . . .

anyway, 12288:12005 is certainly going to work for this purpose.
i don't see why you're denying it above. if you're worried that just
because its 7s exponent is -4, while you're only using 3 different
levels along the 7 direction in your plot, it isn't going to work,
worry no further. it will work just fine. if you prefer to keep the 7s
exponent within the plus-or-minus 3 range, you can always add
or subtract (the exponents of ) any of the other unison vectors. for
example, if you add the exponents of 126:125 ([2 -3 1]) to those
of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about
adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6 -2] .
. . all of these are valid choices for the third unison vector as well.

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, November 17, 2002 12:19 AM
> Subject: [tuning-math] Re: 43edo 7-limit periodicity-block
>
>
> anyway, 12288:12005 is certainly going to work for this purpose.
> i don't see why you're denying it above. if you're worried that just
> because its 7s exponent is -4, while you're only using 3 different
> levels along the 7 direction in your plot, it isn't going to work,
> worry no further. it will work just fine. if you prefer to keep the 7s
> exponent within the plus-or-minus 3 range, you can always add
> or subtract (the exponents of ) any of the other unison vectors. for
> example, if you add the exponents of 126:125 ([2 -3 1]) to those
> of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about
> adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6 -2] .
> . . all of these are valid choices for the third unison vector as well.

ah, yes -- of course! duh, my bad. i just didn't see it at first.
got it.

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning-math@y...>
> > Sent: Sunday, November 17, 2002 12:19 AM
> > Subject: [tuning-math] Re: 43edo 7-limit periodicity-block
> >
> >
> > anyway, 12288:12005 is certainly going to work for this
purpose.
> > i don't see why you're denying it above. if you're worried that
just
> > because its 7s exponent is -4, while you're only using 3
different
> > levels along the 7 direction in your plot, it isn't going to work,
> > worry no further. it will work just fine. if you prefer to keep the
7s
> > exponent within the plus-or-minus 3 range, you can always
add
> > or subtract (the exponents of ) any of the other unison
vectors. for
> > example, if you add the exponents of 126:125 ([2 -3 1]) to
those
> > of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about
> > adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6
-2] .
> > . . all of these are valid choices for the third unison vector as
well.
>
>
> ah, yes -- of course! duh, my bad. i just didn't see it at first.
> got it.

[1 -6 2] can actually be seen on your chart, separating two pairs
of "0"s, each having one member on the 7^1 plane and one
member on the 7^(-1) plane . . . see it?

in any case, it's clear that the third unison vector, no matter what
you decide to make it, is going to be a lot longer than the other
two . . . so in a sense, it's the other two unison vectors that will
determine what scales in 43 allow one to exploit its abundance
of 7-limit harmony . . . and as we know from this list, the pair
<81:80, 126:125>, or equivalently <81:80, 225:224>, defines the
septimal meantone system, whose generator is the perfect fifth .
. . thus, the most efficient 7-limit scales in 43 will tend to lie along
the perfect fifth direction in the plot.