"Hindoo" scale

Hindoo scale:

sa = 1

ri = 256/243, 16/15 = (81/80) x (256/243)

Ri = 10/9, 9/8 = (81/80) x (10/9)

ga = 32/27, 6/5 = (81/80) x (32/27)

Ga = 5/4, 81/64 = (81/80) x (5/4)

ma = 4/3, 20/27 = (81/80) x (4/3)

Ma = 64/45 , 45/32, 729/512 = (81/80) x (45/32)

pa = 40/27, 3/2 = (81/80) x (40/27)

dha = 128/81, 8/5 = (81/80) x (128/81)

Dha = 5/3, 27/16 = (81/80) x (5/3)

ni = 16/9, 9/5 = (81/80) x (16/9)

Ni = 15/8, 243/128 = (81/80) x (15/8)

Now for some reason they drop Ma and replace with tritone:

22 = 1 + 21 (<=> 256/243 x 243/128 = 2)

= 2 + 20 (<=> 16/15 x 15/8 = 2)

= 3 + 19 (<=> 10/9 x 9/5 = 2)

= 4 + 18 (<=> 9/8 x 16/9 = 2)

= 5 + 17 (<=> 32/27 x 27/16 = 2)

= 6 + 16 (<=> 6/5 x 5/3 = 2)

= 7 + 15 (<=> 5/4 x 8/5 = 2)

= 8 + 14 (<=> 81/64 x 128/81 = 2)

= 9 + 13 (<=> 4/3 x 3/2 = 2).

= 10 + 12 (<=> 27/20 x 40/27 = 2)

= 11 + 11 (<=> 2^(1/2) x 2^(1/2) =

So, it's true, the steps are way off from 22-tET. However, I notice
that there are three step sizes:

a=81/80
b=25/24
c=256/243

Also the difference between c/b is 2048/2025, the diaschisma.

So you get this scale: 1,c,a,b,a,c,a,b,a,c,a, and b,a together for
the 11th step, replaced by 2^1/2. Which for me is no big deal, since
the "tritone" isn't one of the Affine group intervals. (I call 11 of
22 the tritone). The Affine transform permutes the odd intervals,
except for 11. However, 11 is used in the "isomeric relation"

So I thought, cool, cuz I am concerned with odd intervals, the evens
merely adjust the odds by the syntonic comma and the odds are related
by the diaschisma!

The musical-set theory part of 22-tET is covered in my
paper "Isomeric Sets" Files - Paul Hj's Stuff

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> Hindoo scale:
>
> sa = 1
>
> ri = 256/243, 16/15 = (81/80) x (256/243)
>
> Ri = 10/9, 9/8 = (81/80) x (10/9)
>
> ga = 32/27, 6/5 = (81/80) x (32/27)
>
> Ga = 5/4, 81/64 = (81/80) x (5/4)
>
> ma = 4/3, 20/27 = (81/80) x (4/3)
>
> Ma = 64/45 , 45/32, 729/512 = (81/80) x (45/32)
>
> pa = 40/27, 3/2 = (81/80) x (40/27)
>
> dha = 128/81, 8/5 = (81/80) x (128/81)
>
> Dha = 5/3, 27/16 = (81/80) x (5/3)
>
> ni = 16/9, 9/5 = (81/80) x (16/9)
>
> Ni = 15/8, 243/128 = (81/80) x (15/8)
>
> Now for some reason they drop Ma and replace with tritone:
>
> 22 = 1 + 21 (<=> 256/243 x 243/128 = 2)
>
> = 2 + 20 (<=> 16/15 x 15/8 = 2)
>
> = 3 + 19 (<=> 10/9 x 9/5 = 2)
>
> = 4 + 18 (<=> 9/8 x 16/9 = 2)
>
> = 5 + 17 (<=> 32/27 x 27/16 = 2)
>
> = 6 + 16 (<=> 6/5 x 5/3 = 2)
>
> = 7 + 15 (<=> 5/4 x 8/5 = 2)
>
> = 8 + 14 (<=> 81/64 x 128/81 = 2)
>
> = 9 + 13 (<=> 4/3 x 3/2 = 2).
>
> = 10 + 12 (<=> 27/20 x 40/27 = 2)
>
> = 11 + 11 (<=> 2^(1/2) x 2^(1/2) =
>
> So, it's true, the steps are way off from 22-tET. However, I notice
> that there are three step sizes:
>
> a=81/80
> b=25/24
> c=256/243
>
> Also the difference between c/b is 2048/2025, the diaschisma.
>
> So you get this scale: 1,c,a,b,a,c,a,b,a,c,a, and b,a together for
> the 11th step, replaced by 2^1/2. Which for me is no big deal, since
> the "tritone" isn't one of the Affine group intervals. (I call 11 of
> 22 the tritone). The Affine transform permutes the odd intervals,
> except for 11. However, 11 is used in the "isomeric relation"
>
> So I thought, cool, cuz I am concerned with odd intervals, the evens
> merely adjust the odds by the syntonic comma and the odds are
related
> by the diaschisma!
>
> The musical-set theory part of 22-tET is covered in my
> paper "Isomeric Sets" Files - Paul Hj's Stuff
>
> PGH
>
Well, actually the increments of the evens over the odds are related
by the diaschisma since you have c and b overlapping.

Now, if one looks at the multipliers for the affine group, and
actually multiply the original sruti steps, and compare them
back on themselves you get this table. Question: Is it valid
to use the group theory normally applied to ET systems, and use
that for JI?

M1 M3 M5 M7 M9
1 Unison SP Unison SP Unison
3 Unison Porc D-Porc T-Magic T-Porc
5 Unison SP SP Unison Unison
7 Unison Porc Magic T-Magic D-Magic
9 Unison Unison Unison SP SP

I've only multiplied the odd intervals, except 11, since that is not
just, but 2^1/2. The evens aren't too interesting because they
just add the syntonic comma to each entry.

Key: SP=Superpythagorean D=Double T=Triple Porc=Porcupine

PS I need to sit down now and read the entire "All in the genes"
chain.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > Hindoo scale:
> >
> > sa = 1
> >
> > ri = 256/243, 16/15 = (81/80) x (256/243)
> >
> > Ri = 10/9, 9/8 = (81/80) x (10/9)
> >
> > ga = 32/27, 6/5 = (81/80) x (32/27)
> >
> > Ga = 5/4, 81/64 = (81/80) x (5/4)
> >
> > ma = 4/3, 20/27 = (81/80) x (4/3)
> >
> > Ma = 64/45 , 45/32, 729/512 = (81/80) x (45/32)
> >
> > pa = 40/27, 3/2 = (81/80) x (40/27)
> >
> > dha = 128/81, 8/5 = (81/80) x (128/81)
> >
> > Dha = 5/3, 27/16 = (81/80) x (5/3)
> >
> > ni = 16/9, 9/5 = (81/80) x (16/9)
> >
> > Ni = 15/8, 243/128 = (81/80) x (15/8)
> >
> > Now for some reason they drop Ma and replace with tritone:
> >
> > 22 = 1 + 21 (<=> 256/243 x 243/128 = 2)
> >
> > = 2 + 20 (<=> 16/15 x 15/8 = 2)
> >
> > = 3 + 19 (<=> 10/9 x 9/5 = 2)
> >
> > = 4 + 18 (<=> 9/8 x 16/9 = 2)
> >
> > = 5 + 17 (<=> 32/27 x 27/16 = 2)
> >
> > = 6 + 16 (<=> 6/5 x 5/3 = 2)
> >
> > = 7 + 15 (<=> 5/4 x 8/5 = 2)
> >
> > = 8 + 14 (<=> 81/64 x 128/81 = 2)
> >
> > = 9 + 13 (<=> 4/3 x 3/2 = 2).
> >
> > = 10 + 12 (<=> 27/20 x 40/27 = 2)
> >
> > = 11 + 11 (<=> 2^(1/2) x 2^(1/2) =
> >
> > So, it's true, the steps are way off from 22-tET. However, I
notice
> > that there are three step sizes:
> >
> > a=81/80
> > b=25/24
> > c=256/243
> >
> > Also the difference between c/b is 2048/2025, the diaschisma.
> >
> > So you get this scale: 1,c,a,b,a,c,a,b,a,c,a, and b,a together for
> > the 11th step, replaced by 2^1/2. Which for me is no big deal,
since
> > the "tritone" isn't one of the Affine group intervals. (I call 11
of
> > 22 the tritone). The Affine transform permutes the odd intervals,
> > except for 11. However, 11 is used in the "isomeric relation"
> >
> > So I thought, cool, cuz I am concerned with odd intervals, the
evens
> > merely adjust the odds by the syntonic comma and the odds are
> related
> > by the diaschisma!
> >
> > The musical-set theory part of 22-tET is covered in my
> > paper "Isomeric Sets" Files - Paul Hj's Stuff
> >
> > PGH
> >
> Well, actually the increments of the evens over the odds are
related
> by the diaschisma since you have c and b overlapping.

Correction: The increments of the odds over the evens are related by
the diaschisma since you have c and b alternating.
>
> Now, if one looks at the multipliers for the affine group, and
> actually multiply the original sruti steps, and compare them
> back on themselves you get this table. Question: Is it valid
> to use the group theory normally applied to ET systems, and use
> that for JI?
>
> M1 M3 M5 M7 M9
> 1 Unison SP Unison SP Unison
> 3 Unison Porc D-Porc T-Magic T-Porc
> 5 Unison SP SP Unison Unison
> 7 Unison Porc Magic T-Magic D-Magic
> 9 Unison Unison Unison SP SP
>
> I've only multiplied the odd intervals, except 11, since that is not
> just, but 2^1/2. The evens aren't too interesting because they
> just add the syntonic comma to each entry.
>
> Key: SP=Superpythagorean D=Double T=Triple Porc=Porcupine
>
> PS I need to sit down now and read the entire "All in the genes"
> chain.
>
> PGH
>