[5-11]-prime-space p-block and 13edo

playing around with my software, i noticed that
13edo, which does not have anything resembling a
3:2 "perfect-5th", gives a not-really-horrible
approximation of 5:4 (just a little worse than
that of 12edo, but in the opposite direction),
and a very good approximation of 11:8 :

2^(4/13) = ~369.2307692 cents, ~17.08294463 less than 5:4

2^(6/13) = ~553.8461538 cents, ~2.528211481 more than 11:8

so i set up a prime-space consisting of the
euler-genus 5^(-1...+5) * 11^(-2...+3) .

and found these two unison-vectors:

~cents [2,5,11]-monzo ratio

56.30525686 [ 0 3 -2] 125/121

26.58125482 [-15 2 3] 33275/32768

... and voila, out popped 13edo.

the 13-tone JI periodicity-block given by these
unison-vectors is shown on this lattice:

1/1 is the reference note, the unison-vectors
are shown by "X", and notes not included in the
periodicity-block are shown as "()"

()------()--------X-----------()----------()
| | | | |
()------()----3025/2048---15125/8192------()
| | | | |
()-----55/32---275/256-----1375/1024---6875/4096
| | | | |
1/1-----5/4-----25/16-------125/64------625/512
| | | | |
()------()------25/22-------125/88--------()
| | | | |
()------()-------()------------X----------()

i'd be interested in any feedback anyone has
on this.

-monz

I will add this scale to the archive.

Speaking of lattices, I have improved the lattice player
in Scala. Now it will also display nonrational intervals,
so one can use it for the "bingocard" type.
Go to Analyse:Lattice and player, click the 2D play button,
tick the "Nearest in emtpy positions" checkbox and set an
amount of rows and columns.

Manuel

hi Manuel,

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

>
> I will add this scale to the archive.

thanks! i can easily create tons more if you want them.

> Speaking of lattices, I have improved the lattice player
> in Scala. Now it will also display nonrational intervals,
> so one can use it for the "bingocard" type.
> Go to Analyse:Lattice and player, click the 2D play button,
> tick the "Nearest in emtpy positions" checkbox and set an
> amount of rows and columns.

cool.

-monz

>thanks! i can easily create tons more if you want them.

Me too, so thanks but not really. But if you have ones
you think are nice or special they can of course be sent.
Cheers,

Manuel

since the ASCII lattice i posted below got messed up by
Yahoo (even "Expand Messages" didn't work for me this time),
here's a good quality image of the lattice:

/tuning-math/files/monz/[5-11]-
primespace-13edo.gif

or

http://tinyurl.com/rc2c

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> so i set up a prime-space consisting of the
> euler-genus 5^(-1...+5) * 11^(-2...+3) .
>
>
> and found these two unison-vectors:
>
>
> ~cents [2,5,11]-monzo ratio
>
> 56.30525686 [ 0 3 -2] 125/121
>
> 26.58125482 [-15 2 3] 33275/32768
>
>
> ... and voila, out popped 13edo.
>
>
> the 13-tone JI periodicity-block given by these
> unison-vectors is shown on this lattice:
>
> 1/1 is the reference note, the unison-vectors
> are shown by "X", and notes not included in the
> periodicity-block are shown as "()"
>
>
> ()------()--------X-----------()----------()
> | | | | |
> ()------()----3025/2048---15125/8192------()
> | | | | |
> ()-----55/32---275/256-----1375/1024---6875/4096
> | | | | |
> 1/1-----5/4-----25/16-------125/64------625/512
> | | | | |
> ()------()------25/22-------125/88--------()
> | | | | |
> ()------()-------()------------X----------()

-monz