Planar lattice of 7-limit tetrads

I've expounded on a number of occasions on the cubic space lattice of
7-limit tetrads. Here is a planar lattice of tetrads, based on breed
tempering. It therefore gives chord relationships which work for
Keenan JI in the 7-limit.

This lattice consists of square grid lattices points (a, b), with
metric defined by the norm

|| (a, b) || = sqrt(8a^2 + b^2)

If b is an even number, then the lattice point represents the otonal
tetrad with root (49/40)^a (10/7)^(b/2), which may be reduced by
octaves and 2401/2400. If b is an odd number, then it is the utonal
tetrad (of form 1-6/5-3/2-12/7) with root (49/40)^a (10/7)^(b+3),
which also may be reduced. Two lattice points represent tetrads with
an interval in common if, and only if, they are at a distance of 3
from each other, and these lattice points therefore surround the
starting one in a nice hexagonal pattern, rather than being the six
verticies of an octahedron as in the strict 7-limit chord relationship.

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

> If b is an even number, then the lattice point represents the otonal
> tetrad with root (49/40)^a (10/7)^(b/2), which may be reduced by
> octaves and 2401/2400. If b is an odd number, then it is the utonal
> tetrad (of form 1-6/5-3/2-12/7) with root (49/40)^a (10/7)^(b+3),
> which also may be reduced.

Should be (49/40)^a (10/7)^((b+3)/2)

If [a, b, c] is a tetrad in the cubic lattice of 7-limit tettrads, then
the corresponding breed chord lattice tetrad is [b+c, 3a-b+c]. If
[a, b] is a tetrad in the breed chord lattice, then the corresponding
cubic lattice tetrads are [a+b, 2a+b, -a-b] modulo [2, 3, -3]; the
modulo means that [a+b+2n, 2a+b+3n, -a-b-3n] is a corresponding tetrad
for any integer n. You can check that [a+b+2n, 2a+b+3n, -a-b-3n] goes
to [a, b] under the cubic to planar transformation I gave first. This
can serve as a definition for the breed chord lattice in place of the
one I first gave if you prefer. The deal with [2, 3, -3] is that it
represents the otonal tetrad with root 2401/2400.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've expounded on a number of occasions on the cubic space lattice of
> 7-limit tetrads. Here is a planar lattice of tetrads, based on breed
> tempering. It therefore gives chord relationships which work for
> Keenan JI in the 7-limit.

I'd prefer if you referred to it as "sensory JI" or similar.

But if you wanted to say "sensory JI (as promoted/defined by Dave
Keenan)" that's fine with me.

what, you are taking this over here now, you sneaky dog you!
what about sensory ET then :)
IF JI is a mathematical construct , then ET is even more so.
Merely observe what people call equal around the world and look at the measurements

Dave Keenan wrote:

>--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
>wrote:
> >
>>I've expounded on a number of occasions on the cubic space lattice of
>>7-limit tetrads. Here is a planar lattice of tetrads, based on breed
>>tempering. It therefore gives chord relationships which work for
>>Keenan JI in the 7-limit.
>> >>
>
>I'd prefer if you referred to it as "sensory JI" or similar. >
>But if you wanted to say "sensory JI (as promoted/defined by Dave
>Keenan)" that's fine with me.
>
>
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> >Yahoo! Groups Links
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> >
>
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> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> what, you are taking this over here now, you sneaky dog you!
> what about sensory ET then :)
> IF JI is a mathematical construct , then ET is even more so.

Pure rational intonation and precise equal tempering are both equally
mathematical. They are idealizations, and as is often the case,
working with the idealizations is actually easier.

> Merely observe what people call equal around the world and look at the
> measurements

Scary, huh?

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> what, you are taking this over here now, you sneaky dog you!

Hi Kraig,

I was a sneaky dog, pushing you to go public like that, wasn't I. I
hope you didn't mind too much. I hope Paul was sufficiently gracious
in response.

I can't really afford time to continue the discussion anywhere at the
moment. I just search on my name now and then to see how I'm being
misrepresented ;-). So I saw Gene was calling it "Keenan JI". I don't
mean to imply that was misrepresentation, far from it, but I didn't
really like it since the idea seriously pre-dates me, being first used
that way in English nearly 200 years ago.

Having though some more about it, I think "perceptible JI" would be
better since people might think that "sensory JI" was more
specifically related to Bill Sethares "sensory dissonance" than it is.

> what about sensory ET then :)

It doesn't exist. ET is a purely mathematical idea, as is is rational
tuning (your kind of JI).

> IF JI is a mathematical construct , then ET is even more so.

Yes. But it's your meaning of "JI" that is mathematical, not mine. In
fact that's one way of characterising yours: "mathematical JI" (as
opposed to "perceptible JI").

> Merely observe what people call equal around the world and look at the
> measurements

I'm not sure I have understood your point here.

-- Dave Keenan

for instance what the chopi call equal, you end up with Mavila which was considered the best of all their scales, or slendro for that matter. the Mavila ensembles do no longer exist BTW.
but i won't belabor this discussion here

> >
>>Merely observe what people call equal around the world and look at the >>measurements
>> >>
>
>I'm not sure I have understood your point here.
>
>-- Dave Keenan
>
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>
>
>
> >Yahoo! Groups Links
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>
>
> >
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> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles