9-EDO and some ideas

http://xenharmonic.wikispaces.com/9edo

(Pelogic, Negri, Augmented, Beep)

7-limit focus is more on Pelogic and Negri.

This requires 3 dimensions. This is a great 7-limit tuning.

M9 and Affine(3) plane come into play here, a S(2,3,9) Steiner system.

123
456
789

Kind of like Sadoku, or Rubik's face.

Steiner blocks:

123,456,789,147,258,369,159,348,267,357,168,249

This covers all pairs, conveniently, these are intervals in the tuning system. Four intervals, 01,02,03,04 (05,06,07,08)

The blocks are lines, Horizontal, Vertical, Left and Right Diagonals.

So we have four types of blocks

012,036,015,024, based on generators 1,3,5 and 2

and four types of intervals (01,02,03,04) which are really the same as the generators (1,2,3,4=5)

7-limit tuning for each interval type is

<3,-2,0> (1)
<-1,0,1> (2)
<2,-2,1> (3)
<-2,0,2> (4)

In terms of p=3,5,7 with 2-equivalence.

Now, with the twelve blocks, all 9C2 intervals are represented, so that is 36 intervals or 9 transpositions of each type. Playing around with the fractions of course produces every combination you could have based on the 9-EDO article in xenharmonic wiki. So it really comes down to the lines (Steiner blocks). Here we have 3 transpositions each of each type of block. So let's tune the blocks:

I suppose just use 1 and 2 for the first block, 3 and 6 (3 again) for the second block, 5 (4) and 1 for the third block, and 2 and 4 for the fourth block. So this is

1,2
3,3
4,1
2,4

So two again of each type, either way (the other being)

1,1
3,3
4,4
2,2

Is this stupid or is this fun? It really is the "kernel" of the Steiner idea (and my study of M11, M12, M22, M23 and M24 w.r.t. tuning); it cannot be reduced down further, but 9-EDO is a cool 7-limit tuning, (almost exact as wiki says), so now the challenge is to work this into a projective space of 3 dimensions.

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
> http://xenharmonic.wikispaces.com/9edo
>
> (Pelogic, Negri, Augmented, Beep)
>
> 7-limit focus is more on Pelogic and Negri.
>
> This requires 3 dimensions. This is a great 7-limit tuning.
>
> M9 and Affine(3) plane come into play here, a S(2,3,9) Steiner system.
>
> 123
> 456
> 789
>
> Kind of like Sadoku, or Rubik's face.
>
> Steiner blocks:
>
> 123,456,789,147,258,369,159,348,267,357,168,249
>
> This covers all pairs, conveniently, these are intervals in the tuning system. Four intervals, 01,02,03,04 (05,06,07,08)
>
> The blocks are lines, Horizontal, Vertical, Left and Right Diagonals.
>
> So we have four types of blocks
>
> 012,036,015,024, based on generators 1,3,5 and 2
>
> and four types of intervals (01,02,03,04) which are really the same as the generators (1,2,3,4=5)
>
> 7-limit tuning for each interval type is
>
> <3,-2,0> (1)
> <-1,0,1> (2)
> <2,-2,1> (3)
> <-2,0,2> (4)
>
> In terms of p=3,5,7 with 2-equivalence.
>
> Now, with the twelve blocks, all 9C2 intervals are represented, so that is 36 intervals or 9 transpositions of each type. Playing around with the fractions of course produces every combination you could have based on the 9-EDO article in xenharmonic wiki. So it really comes down to the lines (Steiner blocks). Here we have 3 transpositions each of each type of block. So let's tune the blocks:
>
> I suppose just use 1 and 2 for the first block, 3 and 6 (3 again) for the second block, 5 (4) and 1 for the third block, and 2 and 4 for the fourth block. So this is
>
> 1,2
> 3,3
> 4,1
> 2,4
>
> So two again of each type, either way (the other being)
>
> 1,1
> 3,3
> 4,4
> 2,2
>
> Is this stupid or is this fun? It really is the "kernel" of the Steiner idea (and my study of M11, M12, M22, M23 and M24 w.r.t. tuning); it cannot be reduced down further, but 9-EDO is a cool 7-limit tuning, (almost exact as wiki says), so now the challenge is to work this into a projective space of 3 dimensions.
>
> PGH
>
Quite simply, the primes 3,5,7 correspond with 5/9 steps, 3/9 steps, and 7/9 steps so this is absolute 2,3, and 4 steps. Now using a block
like 0,1,5 we cover 1 step; so use this one for p=3; and perhaps 024 and 036 to cover primes 7 and 5 respectively. I like these macrotemperments because they are so Escher-like (11-EDO is also)

To be nice I would also throw in say 0,1,2 so H,V,DL, DR are all used.
This last one would have to be derived from 0,5,1 and 0,2,4

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> >
> > http://xenharmonic.wikispaces.com/9edo
> >
> > (Pelogic, Negri, Augmented, Beep)
> >
> > 7-limit focus is more on Pelogic and Negri.
> >
> > This requires 3 dimensions. This is a great 7-limit tuning.
> >
> > M9 and Affine(3) plane come into play here, a S(2,3,9) Steiner system.
> >
> > 123
> > 456
> > 789
> >
> > Kind of like Sadoku, or Rubik's face.
> >
> > Steiner blocks:
> >
> > 123,456,789,147,258,369,159,348,267,357,168,249
> >
> > This covers all pairs, conveniently, these are intervals in the tuning system. Four intervals, 01,02,03,04 (05,06,07,08)
> >
> > The blocks are lines, Horizontal, Vertical, Left and Right Diagonals.
> >
> > So we have four types of blocks
> >
> > 012,036,015,024, based on generators 1,3,5 and 2
> >
> > and four types of intervals (01,02,03,04) which are really the same as the generators (1,2,3,4=5)
> >
> > 7-limit tuning for each interval type is
> >
> > <3,-2,0> (1)
> > <-1,0,1> (2)
> > <2,-2,1> (3)
> > <-2,0,2> (4)
> >
> > In terms of p=3,5,7 with 2-equivalence.
> >
> > Now, with the twelve blocks, all 9C2 intervals are represented, so that is 36 intervals or 9 transpositions of each type. Playing around with the fractions of course produces every combination you could have based on the 9-EDO article in xenharmonic wiki. So it really comes down to the lines (Steiner blocks). Here we have 3 transpositions each of each type of block. So let's tune the blocks:
> >
> > I suppose just use 1 and 2 for the first block, 3 and 6 (3 again) for the second block, 5 (4) and 1 for the third block, and 2 and 4 for the fourth block. So this is
> >
> > 1,2
> > 3,3
> > 4,1
> > 2,4
> >
> > So two again of each type, either way (the other being)
> >
> > 1,1
> > 3,3
> > 4,4
> > 2,2
> >
> > Is this stupid or is this fun? It really is the "kernel" of the Steiner idea (and my study of M11, M12, M22, M23 and M24 w.r.t. tuning); it cannot be reduced down further, but 9-EDO is a cool 7-limit tuning, (almost exact as wiki says), so now the challenge is to work this into a projective space of 3 dimensions.
> >
> > PGH
> >
> Quite simply, the primes 3,5,7 correspond with 5/9 steps, 3/9 steps, and 7/9 steps so this is absolute 2,3, and 4 steps. Now using a block
> like 0,1,5 we cover 1 step; so use this one for p=3; and perhaps 024 and 036 to cover primes 7 and 5 respectively. I like these macrotemperments because they are so Escher-like (11-EDO is also)
>
> To be nice I would also throw in say 0,1,2 so H,V,DL, DR are all used.
> This last one would have to be derived from 0,5,1 and 0,2,4
>
> PGH
>
Without checking whether this is Pelogic, Negri, etc. I will just use the maps in the article for the four fundamental steps of 9-EDO

<3,-2,0> (1)
<-1,0,1> (2)
<2,-2,1> (3)
<-2,0,2> (4)

And proceed to find a nice way to incorporate this into the Steiner blocks (Lines here in Aff(3)). Anyway, so, taking the generators for
3,5, and 7 we get

5,1,6
3,6,0
7,5,3

Which are transpositions of (or actual) Steiner blocks.

123,456,789,147,258,360,150,340,267,357,168,240 (9 is sure nice 'cuz no double-digits)

Anyway, based on the tuning maps above we have | | for steps

<|6,3,0|> tuning 1s (low by one step)
<|4,0,7|> tuning 2s (works)
<|1,3,7|> tuning 3s (low by one step)
<|8,0,5|> tuning 4s (works)

So the evens work, the odds don't. Obviously, p=5 is a problem. This is 3 steps, so I guess we look at tunings sans-5. Very strange since 5 is tuned the same as in 12-EDO but presents a challenge in conjunction with 3 and 7. Luckily, this is an odd tuning, so using just 2 and 4 will cover everything (2,4,6,8,1,3,5,7,0 will generate the scale.) Tying 2s and 4s togethers shows that they are really the same, 4+4=8 and 7+7=5 is not too interesting. Actually none of this is too interesting. Oh Well. So 7/6 is 2s, or the same as 8/7. So 49/48 is tempered out, that is about all one can take from this.

2,4,6 would work, in that 7^3 = 5 (343/320); 4,8,3 is the same as 1,5,6 which tunes p=3, doesn't tie in p=7, but <|8,0,5|> does, in a sense, in the sense that two "fourths (4/3)" and two "sevenths (7/4)"
equals a "fourth" (4/3)^2 * (7/4)^2 = 4/3 or (7/3)^2 = 4/3 or really
49/48 is tempered out again. 3,6,0 leads nowhere, in the above maps, so this leaves 1,2,3 which merely says there is a unit interval between 8/7 and 5/4, which in itself means nothing. 35/32 is a unit interval then. Which has nothing to do with <3,-2,0> or <|6,3,0|> map.

So there really, is nothing, here to say, not really, anything, at all. I could start a religion like Mormonism and get more followers.

PGH