val: new definition

hi Robert, Gene, and everyone,

i've written an entirely new "newbie" definition
for val, and placed it at the top of my Encyclopaedia
entry, above Gene's more technical definitions.

http://tonalsoft.com/enc/val.htm

(thanks for prodding me into this, Robert!)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Robert, Gene, and everyone,
>
>
> i've written an entirely new "newbie" definition
> for val, and placed it at the top of my Encyclopaedia
> entry, above Gene's more technical definitions.
>
> http://tonalsoft.com/enc/val.htm

Pretty good! I'd say "list of numbers" or "list of integers", not "set
of numbers". Also, a val is not restricted to giving the mapping in a
tuning system. <12 19 28| gives the mapping of the 5-limit to 12-equal
and also the number of scale steps in certain perodicity blocks or other
JI scales which temper to 12 equal, but other vals do other things.

hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Robert, Gene, and everyone,
> >
> >
> > i've written an entirely new "newbie" definition
> > for val, and placed it at the top of my Encyclopaedia
> > entry, above Gene's more technical definitions.
> >
> > http://tonalsoft.com/enc/val.htm
>
> Pretty good!

thanks!

i'm trying hard to grasp this stuff, and then
present it in a way that lots of other readers
will understand it.

> I'd say "list of numbers" or "list of integers", not
> "set of numbers".

hmm, interesting ... my first inclination was to call
it a "list" ... i'll change it.

> Also, a val is not restricted to giving the mapping
> in a tuning system. <12 19 28| gives the mapping of
> the 5-limit to 12-equal and also the number of scale
> steps in certain perodicity blocks or other JI scales
> which temper to 12 equal, but other vals do other things.

please tell us more!

-monz

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

> > Also, a val is not restricted to giving the mapping
> > in a tuning system. <12 19 28| gives the mapping of
> > the 5-limit to 12-equal and also the number of scale
> > steps in certain perodicity blocks or other JI scales
> > which temper to 12 equal, but other vals do other things.
>
>
> please tell us more!

<1 *| gives the exponent of 2; this is the "2-adic valuation" of
number theory; <0 1 *| is the 3-adic valuation and so forth. <0 1 4|
gives the number of meantone fifths needed to get to 3 and 5 in
meantone, and
<1 1 0| the corresponding number of octaves; in other words as a pair
this is the octave-generator mapping for 5-limit meantone. Equal
temperaments are associated to a single val, linear temperaments to a
pair of vals, and so forth.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > Also, a val is not restricted to giving the mapping
> > > in a tuning system. <12 19 28| gives the mapping of
> > > the 5-limit to 12-equal and also the number of scale
> > > steps in certain perodicity blocks or other JI scales
> > > which temper to 12 equal, but other vals do other things.
> >
> >
> > please tell us more!
>
> <1 *| gives the exponent of 2; this is the "2-adic
> valuation" of number theory; <0 1 *| is the 3-adic valuation
> and so forth. <0 1 4| gives the number of meantone fifths
> needed to get to 3 and 5 in meantone,

and i see that [-4 4, -1> /\ [1 0, 0> = <0 1 4| .

> and <1 1 0| the corresponding number of octaves;

and i see that [-4 4, -1> /\ [0 0, 1> = <1 1 0| .

is that how you got that val?

> in other words as a pair this is the octave-generator
> mapping for 5-limit meantone. Equal temperaments are
> associated to a single val, linear temperaments to a
> pair of vals, and so forth.

more examples of how to calculate these "other vals"
(those other than the ones which show the prime-mapping)
would be most welcome.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > > Also, a val is not restricted to giving the mapping
> > > > in a tuning system. <12 19 28| gives the mapping of
> > > > the 5-limit to 12-equal and also the number of scale
> > > > steps in certain perodicity blocks or other JI scales
> > > > which temper to 12 equal, but other vals do other things.
> > >
> > >
> > > please tell us more!
> >
> > <1 *| gives the exponent of 2; this is the "2-adic
> > valuation" of number theory; <0 1 *| is the 3-adic valuation
> > and so forth. <0 1 4| gives the number of meantone fifths
> > needed to get to 3 and 5 in meantone,
>
>
> and i see that [-4 4, -1> /\ [1 0, 0> = <0 1 4| .
>
>
>
> > and <1 1 0| the corresponding number of octaves;
>
>
> and i see that [-4 4, -1> /\ [0 0, 1> = <1 1 0| .
>
> is that how you got that val?
>
>
> > in other words as a pair this is the octave-generator
> > mapping for 5-limit meantone.

and [-4 4, -1> /\ [0 1, 0> = <1 0 -4] .

is there some significance to that one?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > > > Also, a val is not restricted to giving the mapping
> > > > > in a tuning system. <12 19 28| gives the mapping of
> > > > > the 5-limit to 12-equal and also the number of scale
> > > > > steps in certain perodicity blocks or other JI scales
> > > > > which temper to 12 equal, but other vals do other things.
> > > >
> > > >
> > > > please tell us more!
> > >
> > > <1 *| gives the exponent of 2; this is the "2-adic
> > > valuation" of number theory; <0 1 *| is the 3-adic valuation
> > > and so forth. <0 1 4| gives the number of meantone fifths
> > > needed to get to 3 and 5 in meantone,
> >
> >
> > and i see that [-4 4, -1> /\ [1 0, 0> = <0 1 4| .
> >
> >
> >
> > > and <1 1 0| the corresponding number of octaves;
> >
> >
> > and i see that [-4 4, -1> /\ [0 0, 1> = <1 1 0| .
> >
> > is that how you got that val?
> >
> >
> > > in other words as a pair this is the octave-generator
> > > mapping for 5-limit meantone.
>
>
>
> and [-4 4, -1> /\ [0 1, 0> = <1 0 -4] .
>
> is there some significance to that one?

i made an error thru-out this post.
wedging together two monzos gives a bimonzo,
not a val. a val can be easily calculated
from the bimonzo, but my use of the equal sign
is simply incorrect.

anyway, i've worked this all out.
i was using prime-mappings when i should
have been using monzos for the generator itself,
i.e., i should use [-1 1, 0> instead of [0 1, 0> .

so .....

in 5-limit meantone:

[-4 4, -1> /\ [1 0, 0> => <0 1 4]
wedging the vum and the period produces the generator-mapping.

[-4 4, -1> /\ [-1 1, 0> => <1 1 0]
wedging the vum and the generator produces the period-mapping.

[1 0, 0> /\ [-1 1, 0> => <0 0 1]
wedging the period and the generator produces .... ?

-monz

monz wrote:

> in 5-limit meantone:
> > [-4 4, -1> /\ [1 0, 0> => <0 1 4]
> wedging the vum and the period produces the generator-mapping.

Yes, this is true in general.

> [-4 4, -1> /\ [-1 1, 0> => <1 1 0]
> wedging the vum and the generator produces the period-mapping.

Oh wow, so it does. And in magic (19&22) as well. Of course, it depends on knowing a ratio for the generator.

> [1 0, 0> /\ [-1 1, 0> => <0 0 1]
> wedging the period and the generator produces .... ?

The octave-equivalent mapping for a linear temperament that tempers out 3:2.

Try wedging the period-mapping and generator-mapping as vals. Then you get the wedgie for the linear temperament.

Graham

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

> i made an error thru-out this post.
> wedging together two monzos gives a bimonzo,
> not a val. a val can be easily calculated
> from the bimonzo, but my use of the equal sign
> is simply incorrect.

In the 5-limit; in the 7-limit the bimonzo converts to a bival, etc.

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

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > i made an error thru-out this post.
> > wedging together two monzos gives a bimonzo,
> > not a val. a val can be easily calculated
> > from the bimonzo, but my use of the equal sign
> > is simply incorrect.
>
> In the 5-limit; in the 7-limit the bimonzo converts
> to a bival, etc.

right, i forgot to spell that out. thanks for doing so.

-monz