Minimal rectangles for 7-limit planar temperaments

Given a 7-limit planar temperament in the strict sense, where 2 is a
generator, we can choose two other generators so that the tetrad or
pentad is contained in a generator rectangle of minimal area. Or we
could use diamonds instead, but this just doubles the sides of the
rectangles.

Below I give generators, normalized so that each generator g satisfies
1<g<sqrt(2), and the corresponding area for both tetrads and pentads.
The format is a list with members [gen1, gen2, area]. I list in order
of area for pentads, and then within that, area of tetrads, and then
most accurate temperaments first.

First on the list is marvel, 225/224, with its pentad square. Both
1728/1715 and 126/125 get the area of the tetrad down to 8, and breed
(2401/2400 planar) puts the tetrad into an area of 9, the same as
marvel, but with much greater accuracy. George Secor's generators of
35/32 and 48/35 for 6144/6125 giving an area of 16 for the pentad is
listed, but note another pair gives better results for tetrads.
Moreover, not only does 2401/2400 put the pentad into an area of 15,
4375/4374 puts it into another square for a 16, both with audibly just
or whatever the phrase is tuning.

This article really is a follow-up to
/tuning/topicId_67559.html#67622

Minimal area generators

225/224
tetrads [[15/14, 4/3, 9], [15/14, 5/4, 9], [5/4, 4/3, 9]]
pentads [[5/4, 4/3, 9]]

1728/1715
tetrads [[7/6, 6/5, 8], [8/7, 7/6, 8], [36/35, 7/6, 8]]
pentads [[8/7, 7/6, 12]]

126/125
tetrads [[6/5, 4/3, 8], [6/5, 5/4, 8], [8/7, 6/5, 8], [21/20, 6/5, 8]]
pentads [[6/5, 4/3, 12]]

245/243
tetrads [[7/6, 9/7, 12], [9/7, 4/3, 12]]
pentads [[7/6, 9/7, 12], [9/7, 4/3, 12]]

5120/5103
tetrads [[4/3, 27/20, 14], [10/9, 4/3, 14],
[21/16, 4/3, 14], [6/5, 4/3, 14], [64/63, 4/3, 14]]
pentads [[4/3, 27/20, 14], [10/9, 4/3, 14],
[21/16, 4/3, 14], [64/63, 4/3, 14]]

2401/2400
tetrads [[49/40, 7/5, 9]]
pentads [[7/6, 49/40, 15], [49/40, 7/5, 15]]

6144/6125
tetrads [[35/32, 5/4, 9]]
pentads [[35/32, 48/35, 16]]

1029/1024
tetrads [[21/20, 8/7, 10], [35/32, 8/7, 10],
[8/7, 7/5, 10], [8/7, 5/4, 10], [8/7, 6/5, 10]]
pentads [[21/20, 8/7, 16], [35/32, 8/7, 16],
[8/7, 7/5, 16], [8/7, 5/4, 16], [8/7, 6/5, 16]]

4375/4374
tetrads [[10/9, 6/5, 16], [27/25, 6/5, 16], [27/25, 10/9, 16]]
pentads [[10/9, 6/5, 16]]

3136/3125
tetrads [[25/24, 28/25, 12], [15/14, 28/25, 12], [28/25, 75/56, 12],
[28/25, 7/6, 12], [28/25, 4/3, 12], [28/25, 6/5, 12]]
pentads [[28/25, 4/3, 18], [28/25, 6/5, 18], [28/25, 75/56, 18]]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Given a 7-limit planar temperament in the strict sense, where 2 is a
> generator, we can choose two other generators so that the tetrad or
> pentad is contained in a generator rectangle of minimal area. ...
>
> Below I give generators, normalized so that each generator g satisfies
> 1<g<sqrt(2), and the corresponding area for both tetrads and pentads.
> The format is a list with members [gen1, gen2, area]. I list in order
> of area for pentads, and then within that, area of tetrads, and then
> most accurate temperaments first. ...

The above is from:
/tuning-math/message/15785

Gene, this is really nice. Thanks!

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Given a 7-limit planar temperament in the strict sense, where 2 is a
> generator, we can choose two other generators so that the tetrad or
> pentad is contained in a generator rectangle of minimal area. Or we
> could use diamonds instead, but this just doubles the sides of the
> rectangles.

Please remind me what a generator rectangle is, someone. Thanks!
I take it a pentad is (1, 6/5, 7/5, 8/5, 9/5, 2) ?

> Below I give generators, normalized so that each generator g
satisfies
> 1<g<sqrt(2), and the corresponding area for both tetrads and
pentads.
> The format is a list with members [gen1, gen2, area]. I list in
order
> of area for pentads, and then within that, area of tetrads, and then
> most accurate temperaments first.
>
> First on the list is marvel, 225/224, with its pentad square. Both
> 1728/1715 and 126/125 get the area of the tetrad down to 8, and
breed
> (2401/2400 planar) puts the tetrad into an area of 9, the same as
> marvel, but with much greater accuracy. George Secor's generators of
> 35/32 and 48/35 for 6144/6125 giving an area of 16 for the pentad is
> listed, but note another pair gives better results for tetrads.
> Moreover, not only does 2401/2400 put the pentad into an area of 15,
> 4375/4374 puts it into another square for a 16, both with audibly
just
> or whatever the phrase is tuning.
>
> This article really is a follow-up to
> /tuning/topicId_67559.html#67622
>
> Minimal area generators
>
> 225/224
> tetrads [[15/14, 4/3, 9], [15/14, 5/4, 9], [5/4, 4/3, 9]]
> pentads [[5/4, 4/3, 9]]
>
> 1728/1715
> tetrads [[7/6, 6/5, 8], [8/7, 7/6, 8], [36/35, 7/6, 8]]
> pentads [[8/7, 7/6, 12]]
>
> 126/125
> tetrads [[6/5, 4/3, 8], [6/5, 5/4, 8], [8/7, 6/5, 8], [21/20, 6/5,
8]]
> pentads [[6/5, 4/3, 12]]
>
> 245/243
> tetrads [[7/6, 9/7, 12], [9/7, 4/3, 12]]
> pentads [[7/6, 9/7, 12], [9/7, 4/3, 12]]
>
> 5120/5103
> tetrads [[4/3, 27/20, 14], [10/9, 4/3, 14],
> [21/16, 4/3, 14], [6/5, 4/3, 14], [64/63, 4/3, 14]]
> pentads [[4/3, 27/20, 14], [10/9, 4/3, 14],
> [21/16, 4/3, 14], [64/63, 4/3, 14]]
>
> 2401/2400
> tetrads [[49/40, 7/5, 9]]
> pentads [[7/6, 49/40, 15], [49/40, 7/5, 15]]
>
> 6144/6125
> tetrads [[35/32, 5/4, 9]]
> pentads [[35/32, 48/35, 16]]
>
> 1029/1024
> tetrads [[21/20, 8/7, 10], [35/32, 8/7, 10],
> [8/7, 7/5, 10], [8/7, 5/4, 10], [8/7, 6/5, 10]]
> pentads [[21/20, 8/7, 16], [35/32, 8/7, 16],
> [8/7, 7/5, 16], [8/7, 5/4, 16], [8/7, 6/5, 16]]
>
> 4375/4374
> tetrads [[10/9, 6/5, 16], [27/25, 6/5, 16], [27/25, 10/9, 16]]
> pentads [[10/9, 6/5, 16]]
>
> 3136/3125
> tetrads [[25/24, 28/25, 12], [15/14, 28/25, 12], [28/25, 75/56,
12],
> [28/25, 7/6, 12], [28/25, 4/3, 12], [28/25, 6/5, 12]]
> pentads [[28/25, 4/3, 18], [28/25, 6/5, 18], [28/25, 75/56, 18]]
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Please remind me what a generator rectangle is, someone. Thanks!

It's a generalized chain-of-generators scale. If we have the two
generators of a planar temperament, it is the set of interval classes
(scale) defined by {g1^i * g2^j|a1 <= i <= a2, b1 <= j <= b2}. I give
the generators in terms of rational numbers which temper to the
correct generators if you pick a tuning.

> I take it a pentad is (1, 6/5, 7/5, 8/5, 9/5, 2) ?

Yes, but in root position this would be 1, 9/8, 5/4, 3/2, 7/4.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> Gene, this is really nice. Thanks!

Thanks. Do you use rectangular arragements for your purposes?

Some points that can be made: we can generalize the idea of Graham
complexity to each of the generators separately. This isn't as
interesting as in the case of linear temperaments, because there only
chains of generators give convexity, but it still might be of
interest. If h1 is the partial complexity of generator g1 (with
respect to tetrads or pentads), and h2 of g2, then the area I've been
computing is (h1+1)*(h2+1). If we consider an AxB rectangle then the
number of otonal tetrads (resp, pentads) is (A-h1)*(B-h2).

We can find h1 and h2 as follows: take the matrix defined by
[2, g1, g2, c], where g1 and g2 are given as rational numbers, and c
is the comma. Upon inversion, the second and third columns give vals
defining the exponents for g1 and g2 respectively, for any 7-limit
interval. Now consider each range independently on tetrads (pentads.)

For example, take 8/7 and 7/6 as generators and 1728/1715 as the
comma. Then [2, 8/7, 7/6, 1728/1715] gives the matrix with rows
[|1, 0, 0, 0>, |3, 0, 0, -1>, |-1, -1, 0, 1>, |6, 3, -1, -3>].
Inverting this unimodular matrix gives
[<1, 2, 3, 3|, <0, -1, 0, -1|, <0, -1, -3, 0|, <0, 0, -1, 0|]
where the vals in this list now represent columns.

Applying the second and third vals to [1, 3, 5, 7] gives the list
[[0, 0], [-1, -1], [0, -3], [-1, 0]]. The g1 partial complexity is 1,
and the g2 partial complexity is 3, and the area is (1+1)*(3+1)=8.
The number of tetrads in the scale with g1 running from -1 to 1 and g2
running from -2 to 2 will be (3-1)*(5-3) = 4. The area of the
rectangle will be (2+1)*(4+1) = 15, so we have a 15-note scale with
four otonal tetrads, four untonal tetrads (and also two each of pentads.)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > Gene, this is really nice. Thanks!
>
> Thanks. Do you use rectangular arragements for your purposes?

Only if linear ones aren't efficient enough, which seems to be the case
for the 13 limit in 130-ET. For 130 I was able to find a relatively
compact planar configuration for the 9 limit, but then 11 and 13 were
off in opposite directions.

--George