Brink's pi-phi tuning

Hello all,

Brink showed me a tuning matrix, and i would like to
create a Tonescape file of it. I'm interested in knowing
what properties any of you might see in it.

The "ratio" of each note is calculated as a power
of pi multiplied by a power of phi.

The lattice has a horizontal axis where the steps
alternate, first pi^-5 * phi^12, then pi^8 * phi^-19,
going in the positive direction along the axis,
and with the signs reversed going in the negative
direction.

The vertical axis simply increments the power of phi
by 1 in the positive direction, and by -1 in the negative
direction.

Thus, around the origin, the first few notes look like this:

pi^5*phi^-11 ... pi^0*phi^1 ... pi^-5*phi^13 ... pi^3*phi^-6

pi^5*phi^-12 ... pi^0*phi^0 ... pi^-5*phi^12 ... pi^3*phi^-7

pi^5*phi^-13 ... pi^0*phi^-1 .. pi^-5*phi^11 ... pi^3*phi^-8

It's easier to see the exponents if i just write them
as pi,phi-monzos:

[5 -11> ... [0 1> ... [-5 13> ... [3 -6>
[5 -12> ... [0 0> ... [-5 12> ... [3 -7>
[5 -13> ... [0 -1> .. [-5 11> ... [3 -8>

Has anyone ever seen a tuning like this before?
Does it resemble any other tuning anyone knows about?
What are some of its properties?

-monz
http://tonalsoft.com
Tonescape microtonal music software

It's been a whole week and no response yet
... the proverbial lead balloon?

-monz

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

> Hello all,
>
>
> Brink showed me a tuning matrix, and i would like to
> create a Tonescape file of it. I'm interested in knowing
> what properties any of you might see in it.
>
> The "ratio" of each note is calculated as a power
> of pi multiplied by a power of phi.
>
> The lattice has a horizontal axis where the steps
> alternate, first pi^-5 * phi^12, then pi^8 * phi^-19,
> going in the positive direction along the axis,
> and with the signs reversed going in the negative
> direction.
>
> The vertical axis simply increments the power of phi
> by 1 in the positive direction, and by -1 in the negative
> direction.
>
> Thus, around the origin, the first few notes look like this:
>
>
> pi^5*phi^-11 ... pi^0*phi^1 ... pi^-5*phi^13 ... pi^3*phi^-6
>
> pi^5*phi^-12 ... pi^0*phi^0 ... pi^-5*phi^12 ... pi^3*phi^-7
>
> pi^5*phi^-13 ... pi^0*phi^-1 .. pi^-5*phi^11 ... pi^3*phi^-8
>
>
> It's easier to see the exponents if i just write them
> as pi,phi-monzos:
>
> [5 -11> ... [0 1> ... [-5 13> ... [3 -6>
> [5 -12> ... [0 0> ... [-5 12> ... [3 -7>
> [5 -13> ... [0 -1> .. [-5 11> ... [3 -8>
>
>
> Has anyone ever seen a tuning like this before?
> Does it resemble any other tuning anyone knows about?
> What are some of its properties?
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

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

> It's been a whole week and no response yet
> ... the proverbial lead balloon?
>
>
> -monz

I hit on the idea of making Brink's 2-D pi-phi
tuning matrix a 3-D lattice, with the following
generators:

base . . . . . . . . . cents . . . . . axis
-------------------------------------------------

2. . . . . . . . . . . 1200 . . . . . invisible

pi^3 * phi^-7 . . . . . 113.754 . . . x

phi^1 . . . . . . . . . 833.090 . . . y

pi^-5 * phi^12. . . . . 88.107. . . . z

-monz

>
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Hello all,
> >
> >
> > Brink showed me a tuning matrix, and i would like to
> > create a Tonescape file of it. I'm interested in knowing
> > what properties any of you might see in it.
> >
> > The "ratio" of each note is calculated as a power
> > of pi multiplied by a power of phi.
> >
> > The lattice has a horizontal axis where the steps
> > alternate, first pi^-5 * phi^12, then pi^8 * phi^-19,
> > going in the positive direction along the axis,
> > and with the signs reversed going in the negative
> > direction.
> >
> > The vertical axis simply increments the power of phi
> > by 1 in the positive direction, and by -1 in the negative
> > direction.
> >
> > Thus, around the origin, the first few notes look like this:
> >
> >
> > pi^5*phi^-11 ... pi^0*phi^1 ... pi^-5*phi^13 ... pi^3*phi^-6
> >
> > pi^5*phi^-12 ... pi^0*phi^0 ... pi^-5*phi^12 ... pi^3*phi^-7
> >
> > pi^5*phi^-13 ... pi^0*phi^-1 .. pi^-5*phi^11 ... pi^3*phi^-8
> >
> >
> > It's easier to see the exponents if i just write them
> > as pi,phi-monzos:
> >
> > [5 -11> ... [0 1> ... [-5 13> ... [3 -6>
> > [5 -12> ... [0 0> ... [-5 12> ... [3 -7>
> > [5 -13> ... [0 -1> .. [-5 11> ... [3 -8>
> >
> >
> > Has anyone ever seen a tuning like this before?
> > Does it resemble any other tuning anyone knows about?
> > What are some of its properties?
> >
> >
> >
> > -monz
> > http://tonalsoft.com
> > Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> It's been a whole week and no response yet
> ... the proverbial lead balloon?

The idea lacks any obvious rational basis, so it might generate more
enthusiasm if you explained *why* phi and pi.

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > It's been a whole week and no response yet
> > ... the proverbial lead balloon?
> >
> >
> > -monz
>
>
>
>
> I hit on the idea of making Brink's 2-D pi-phi
> tuning matrix a 3-D lattice, with the following
> generators:
>
>
> base . . . . . . . . . cents . . . . . axis
> -------------------------------------------------
>
> 2. . . . . . . . . . . 1200 . . . . . invisible

Why? It doesn't seem Brink is using any factors of 2 at all,
according to what you wrote in the previous message below.

> pi^3 * phi^-7 . . . . . 113.754 . . . x
>
> phi^1 . . . . . . . . . 833.090 . . . y
>
> pi^-5 * phi^12. . . . . 88.107. . . . z

In this lattice, you'd have an infinite number of places to put any
note in Brink's tuning. One of the axes would be redundant, since
(pi^-5 * phi^12)^3 * (pi^3 * phi^-7)^5 = phi.

In other words, 3*88.107 + 5*113.754 = 833.09, or 3*z + 5*x = y.

So it seems you would really want to eliminate one of these axes,
either x, y, or z, in order to be able to depict a Brink scale on it
with each note appearing in one and only one place.

>
>
>
>
> -monz
>
>
>
>
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > Hello all,
> > >
> > >
> > > Brink showed me a tuning matrix, and i would like to
> > > create a Tonescape file of it. I'm interested in knowing
> > > what properties any of you might see in it.
> > >
> > > The "ratio" of each note is calculated as a power
> > > of pi multiplied by a power of phi.
> > >
> > > The lattice has a horizontal axis where the steps
> > > alternate, first pi^-5 * phi^12, then pi^8 * phi^-19,
> > > going in the positive direction along the axis,
> > > and with the signs reversed going in the negative
> > > direction.
> > >
> > > The vertical axis simply increments the power of phi
> > > by 1 in the positive direction, and by -1 in the negative
> > > direction.
> > >
> > > Thus, around the origin, the first few notes look like this:
> > >
> > >
> > > pi^5*phi^-11 ... pi^0*phi^1 ... pi^-5*phi^13 ... pi^3*phi^-6
> > >
> > > pi^5*phi^-12 ... pi^0*phi^0 ... pi^-5*phi^12 ... pi^3*phi^-7
> > >
> > > pi^5*phi^-13 ... pi^0*phi^-1 .. pi^-5*phi^11 ... pi^3*phi^-8
> > >
> > >
> > > It's easier to see the exponents if i just write them
> > > as pi,phi-monzos:
> > >
> > > [5 -11> ... [0 1> ... [-5 13> ... [3 -6>
> > > [5 -12> ... [0 0> ... [-5 12> ... [3 -7>
> > > [5 -13> ... [0 -1> .. [-5 11> ... [3 -8>
> > >
> > >
> > > Has anyone ever seen a tuning like this before?
> > > Does it resemble any other tuning anyone knows about?
> > > What are some of its properties?
> > >
> > >
> > >
> > > -monz
> > > http://tonalsoft.com
> > > Tonescape microtonal music software

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

> > I hit on the idea of making Brink's 2-D pi-phi
> > tuning matrix a 3-D lattice, with the following
> > generators:
> >
> >
> > base . . . . . . . . . cents . . . . . axis
> > -------------------------------------------------
> >
> > 2. . . . . . . . . . . 1200 . . . . . invisible
>
> Why? It doesn't seem Brink is using any factors of 2 at all,
> according to what you wrote in the previous message below.

Oops ... i asked Brink about that and you're right,
2 never entered into this tuning as a factor at all.

That's not a problem, because Tonescape can create
tunings which have no equivalence-interval ... or
actually, i think Brink intends for phi to be the
equivalence-interval.

> > pi^3 * phi^-7 . . . . . 113.754 . . . x
> >
> > phi^1 . . . . . . . . . 833.090 . . . y
> >
> > pi^-5 * phi^12. . . . . 88.107. . . . z
>
>
> In this lattice, you'd have an infinite number of places to
> put any note in Brink's tuning. One of the axes would be
> redundant, since
> (pi^-5 * phi^12)^3 * (pi^3 * phi^-7)^5 = phi.
>
> In other words, 3*88.107 + 5*113.754 = 833.09, or 3*z + 5*x = y.
>
> So it seems you would really want to eliminate one of
> these axes, either x, y, or z, in order to be able to depict
> a Brink scale on it with each note appearing in one and only
> one place.

Thanks, Paul ... yes, i see it now. I tried to create
a cubic section of the 3-D Lattice, only to find that
Tonescape wouldn't put in some of the notes, because they
were redundant copies of notes that were already in the
Lattice.

So then a simpler 2-D Lattice is all that's needed to
provide all the notes of Brink's tuning?

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > I hit on the idea of making Brink's 2-D pi-phi
> > > tuning matrix a 3-D lattice, with the following
> > > generators:
> > >
> > >
> > > base . . . . . . . . . cents . . . . . axis
> > > -------------------------------------------------
> > >
> > > 2. . . . . . . . . . . 1200 . . . . . invisible
> >
> > Why? It doesn't seem Brink is using any factors of 2 at all,
> > according to what you wrote in the previous message below.
>
>
> Oops ... i asked Brink about that and you're right,
> 2 never entered into this tuning as a factor at all.
>
> That's not a problem, because Tonescape can create
> tunings which have no equivalence-interval ... or
> actually, i think Brink intends for phi to be the
> equivalence-interval.
>
>
>
> > > pi^3 * phi^-7 . . . . . 113.754 . . . x
> > >
> > > phi^1 . . . . . . . . . 833.090 . . . y
> > >
> > > pi^-5 * phi^12. . . . . 88.107. . . . z
> >
> >
> > In this lattice, you'd have an infinite number of places to
> > put any note in Brink's tuning. One of the axes would be
> > redundant, since
> > (pi^-5 * phi^12)^3 * (pi^3 * phi^-7)^5 = phi.
> >
> > In other words, 3*88.107 + 5*113.754 = 833.09, or 3*z + 5*x = y.
> >
> > So it seems you would really want to eliminate one of
> > these axes, either x, y, or z, in order to be able to depict
> > a Brink scale on it with each note appearing in one and only
> > one place.
>
>
> Thanks, Paul ... yes, i see it now. I tried to create
> a cubic section of the 3-D Lattice, only to find that
> Tonescape wouldn't put in some of the notes, because they
> were redundant copies of notes that were already in the
> Lattice.
>
>
> So then a simpler 2-D Lattice is all that's needed to
> provide all the notes of Brink's tuning?

It would seem so. Any of the (x,y), (x,z), (y,z) pairs provide a sufficient basis, correct?