More on areas

I said a little about areas and wedge products on the big list, in
connection with wedge products of 5-limit commas, and how these relate
to Fokker blocks. What about counting lattice points in connection
with vals and wedge products of vals?

If the vals are mappings for period and generator, it is
straightforward; we are counting points in a rectangle of from 0 to
under an octave in one direction, and 0 less than generators in
another; in other words, we have the rectalgle with corners (0,0),
(p,0), (0,g) and (p,g), where a prime is represented by p periods and
g generators, and we count lattice points excluding the (p,0) to (p,g)
and (0,g) to (p,g) sides, giving p*g lattice points.

We could instead exlude the (0,0) to (p,0) and (0,0) to (0,g) sides,
and since the null val is boring this is what I want to do when
talking about et vals. Suppose, for example, I take 12 and 19 in the
5-limit,
so that we are looking at <12 19 28| and <19 30 44|. If we look just
at the primes 2 and 3, we get an area of one and only a single lattice
point, <31 49| which we can extend out to <31 49 72|, the 31-et. More
interesting are 2 and 5 and 3 and 5.

The area for 2 and 5 is 4, corresponding to the exponent of 3 in
81/80. The four lattice points we get correspond to linear
combinations a*<12 19 28| + b*<19 30 44| such that a and b are
positive rational numbers less than or equal to 1, and such that the
12*a+19*b and 28*a+44*b are integers. This gives us

(1/4)*<12 19 28| + <19 30 44| = <22 139/4 51|,
(1/2)*<12 19 28| + <19 30 44| = <25 79/2 58|,
(3/4)*<12 19 28| + <19 30 44| = <28 177/4 65|,
<12 19 28| + <19 30 44| = <31 49 72|.

Clearing denominators gives us 88, 50, 112, and 31 equal.

Similarly, if second and third coordinates must be integers, we get
area four once again (corresponding to the exponent of 2 in 81/80) and
four lattice points

(1/2)*<12 19 28| + (1/4)*<19 30 44| = <43/4 17 25|,
(1/2)*<12 19 28| + (3/4)*<19 30 44| = <81/4 32 47|,
<12 19 28| + (1/2)*<19 30 44| = <43/2 34 50|,
<12 19 28| + <19 30 44| = <31 49 72|

We finally end up with 31, 43, 50, 81, and 112 from these various blocks.

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

> We finally end up with 31, 43, 50, 81, and 112 from these various
blocks.

This should be 31, 43, 50, 81, 88 and 112. We might call these the
5-limit block vals for 12 and 19.

hi Gene,

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

> Suppose, for example, I take 12 and 19 in the 5-limit,
> so that we are looking at <12 19 28| and <19 30 44|.
> If we look just at the primes 2 and 3, we get an area
> of one and only a single lattice point, <31 49|

i understand how you got the area of 1, but can
you explain how you derive those lattice coordinates?

exactly what do these lattice-points represent?
this is apparently not the same lattice as the one
with the prime-factor exponents.

-monz

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

> i understand how you got the area of 1, but can
> you explain how you derive those lattice coordinates?

I derive them by treating them as you might treat monzos when looking
for Fokker blocks.

For example, suppose I choose 128/125 and 81/80 to define a block, and
look at all products (128/125)^a (81/80)^b where a and b are positive
rational numbers less than or equal to 1, and such that the expoents
of 3 and 5 are integers. The results are 12 numbers which are roots of
commas; if I clear the denominator of the exponent of 2 by taking the
appropriate power, I get the 12 commas. This is not ordinary what we
do; rather we ignore the 2 part, take just the 3 and 5 part, reduce
that to an octave and the result is a Fokker block.

I did formally the exact same thing with vals, but since I don't know
a use for that I did the equivalent of turning them into commsa, which
turns them into (in that case, meantone) vals.

The twelve fractional monzos I got I give below; since you like such
things maybe you can figure out a use for this. Ignore the fractional
exponent for 2 and you have a 12-note Fokker block; if you like you
can transpose it so that it includes a 1/1.

|-5/3 4 -2> |3 4 -4> |2/3 4 -3> |-1/12 3 -2>
|3/2 2 -2> |-5/6 2 -1> |9/4 3 -3> |23/6 2 -3>
|-29/12 3 -1> |37/12 1 -2> |65/12 1 -3> |3/4 1 -1>

> exactly what do these lattice-points represent?

What use there is to this, if any, is a question. When I looked at 12
and 19, it shone a light on equal temperaments between 19 and 31 which
in part resembeled a meantone. For instance, 22 is not a meantone
temperament since the 3 is sharp; but it has the same major third as
88, which *is* a meantone temperament. If you take the TOP tuning for
88 in log2 units and multiply it by 22, you get

<22.038 34.809 51.087|

which can be seen as a tuning for 22 notes.