Quartaminorthirds and muggles

Here are quartaminorthirds[16] and muggles[16], more 36/35-chroma systems.

Quartaminorthirds[16]

[9, 5, -3, -13, -30, -21]
[1, 21/20, 35/32, 8/7, 6/5, 5/4, 21/16, 48/35, 10/7, 35/24, 32/21,
8/5, 5/3, 7/4, 64/35, 40/21]

[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 49/40, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]

Muggles[16]

[5, 1, -7, -10, -25, -19]
[1, 21/20, 16/15, 8/7, 6/5, 5/4, 21/16, 4/3, 10/7, 3/2, 25/16, 8/5,
5/3, 7/4, 15/8, 40/21]

[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 6/5, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 32/27, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here are quartaminorthirds[16] and muggles[16], more 36/35-chroma
systems.

could i put in a request -- when you look at these, try to find
omnitetrachordal variants to the basic MOS scales . . .

. . . and then, when they're all done, one could to the hyper-MOS
scales of the second order: fokker periodicity blocks where two
unison vectors are not tempered out. and the omnitetrachordal
variants of those . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> could i put in a request -- when you look at these, try to find
> omnitetrachordal variants to the basic MOS scales . . .

Yipe!

> . . . and then, when they're all done, one could to the hyper-MOS
> scales of the second order: fokker periodicity blocks where two
> unison vectors are not tempered out. and the omnitetrachordal
> variants of those . . .

Double yipe. I'll have to think about this.

Beatles

[2, -9, -4, -19, -12, 16]
[1, 16/15, 15/14, 8/7, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
45/28, 12/7, 7/4, 28/15, 15/8] [14/9, 15/8, 7/6, 10/7, 7/4, 15/14,
4/3, 45/28, 1, 49/40, 3/2, 28/15, 8/7, 7/5, 12/7, 16/15, 9/7]

Tetrads in generator circle order (49/40 generator)

[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

Squares

[4, 16, 9, 16, 3, -24]
[1, 21/20, 49/45, 9/8, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
49/30, 12/7, 9/5, 49/27, 27/14] [9/8, 10/7, 49/27, 7/6, 3/2, 27/14,
49/40, 14/9, 1, 9/7, 49/30, 21/20, 4/3, 12/7, 49/45, 7/5, 9/5]

Tetrads in generator circle order (9/7 generator)

[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]

what are the generators of these scales? what can we say about the
conditions leading to this kind of uniformity of the tetradic
materials as a function of "root" scale degree number?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > could i put in a request -- when you look at these, try to find
> > omnitetrachordal variants to the basic MOS scales . . .
>
> Yipe!
>
> > . . . and then, when they're all done, one could to the hyper-MOS
> > scales of the second order: fokker periodicity blocks where two
> > unison vectors are not tempered out. and the omnitetrachordal
> > variants of those . . .
>
> Double yipe. I'll have to think about this.
>
> Beatles
>
> [2, -9, -4, -19, -12, 16]
> [1, 16/15, 15/14, 8/7, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
> 45/28, 12/7, 7/4, 28/15, 15/8] [14/9, 15/8, 7/6, 10/7, 7/4, 15/14,
> 4/3, 45/28, 1, 49/40, 3/2, 28/15, 8/7, 7/5, 12/7, 16/15, 9/7]
>
> Tetrads in generator circle order (49/40 generator)
>
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
>
>
> Squares
>
> [4, 16, 9, 16, 3, -24]
> [1, 21/20, 49/45, 9/8, 7/6, 49/40, 9/7, 4/3, 7/5, 10/7, 3/2, 14/9,
> 49/30, 12/7, 9/5, 49/27, 27/14] [9/8, 10/7, 49/27, 7/6, 3/2, 27/14,
> 49/40, 14/9, 1, 9/7, 49/30, 21/20, 4/3, 12/7, 49/45, 7/5, 9/5]
>
> Tetrads in generator circle order (9/7 generator)
>
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]

> could i put in a request -- when you look at these, try to find
> omnitetrachordal variants to the basic MOS scales . . .

Paul,

1. Omnitetrachordal just means the 4/3 doesn't have to be
cut in *four*, right? The symmetry still has to be 3:2,
right?

2. How might one find such variants?

-Carl

> what can we say about the conditions leading to this kind of
> uniformity of the tetradic materials as a function of "root"
> scale degree number?

Paul,

Could you explain what you're asking here? I'm not tracking
you.

-C.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> what are the generators of these scales? what can we say about the
> conditions leading to this kind of uniformity of the tetradic
> materials as a function of "root" scale degree number?

This uniformity is connected to the fact that 36/35 is a chroma. As
for generators:

> > Beatles

> > Tetrads in generator circle order (49/40 generator)

> > Squares

> > Tetrads in generator circle order (9/7 generator)

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > could i put in a request -- when you look at these, try to find
> > omnitetrachordal variants to the basic MOS scales . . .
>
> Paul,
>
> 1. Omnitetrachordal just means the 4/3 doesn't have to be
> cut in *four*, right?

are you sure you're phrasing that question correctly?

> The symmetry still has to be 3:2,
> right?

as opposed to 4:3? i think it's more 3:2 than 4:3.

>
> 2. How might one find such variants?

i don't know a general rule, but interestingly even scales with 3
steps sizes can be omnitetrachordal (we found a 22-tone example in
the shrutar discussion).

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > what are the generators of these scales? what can we say about
the
> > conditions leading to this kind of uniformity of the tetradic
> > materials as a function of "root" scale degree number?
>
> This uniformity is connected to the fact that 36/35 is a chroma. As
> for generators:
>
> > > Beatles
>
> > > Tetrads in generator circle order (49/40 generator)
>
> > > Squares
>
> > > Tetrads in generator circle order (9/7 generator)

oh, the uniformity is because you put the tetrads in generator circle
order, isn't it? what does 36/35 have to do with it? i thought you
were giving the tetrads in scale order. perhaps in fifths order would
be best of all.

>> 1. Omnitetrachordal just means the 4/3 doesn't have to be
>> cut in *four*, right?
>
>are you sure you're phrasing that question correctly?

Mmm...

>> The symmetry still has to be 3:2,
>> right?
>
>as opposed to 4:3? i think it's more 3:2 than 4:3.

As opposed to 5:4. I always forget the meaning of omnitetrachordal.
I think, because the tetra's still in there.

-C.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> oh, the uniformity is because you put the tetrads in generator circle
> order, isn't it?

I was for a while, but it turns out it's better to stick with
generator circles.

>what does 36/35 have to do with it?

36/35 exchanges 7/6<-->6/5, 5/4<-->9/7, 5/3<-->12/7, 7/4<-->9/5 as we
move from one kind of tetrad to another. Unfortunately, it also
exchanges 35/24<-->3/2, leading to wolves.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > oh, the uniformity is because you put the tetrads in generator
circle
> > order, isn't it?
>
> I was for a while, but it turns out it's better to stick with
> generator circles.
>
> >what does 36/35 have to do with it?
>
> 36/35 exchanges 7/6<-->6/5, 5/4<-->9/7, 5/3<-->12/7, 7/4<-->9/5 as
we
> move from one kind of tetrad to another.

right, but the uniformity would happen no matter what you used.
another appealing possibility, important in the whole version of this
discussion that we were having before with kalle, is 49:48, which
exchanges 7/4 <-> 12/7, 7/6 <-> 8/7, and in particular those systems
where 50:49 is one of the commatic unison vectors, such that the
chromatic unison vector can also be expressed as 25:24, leading to
6/5 <-> 5/4 and 8/5 <-> 5/3.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> right, but the uniformity would happen no matter what you used.
> another appealing possibility, important in the whole version of this
> discussion that we were having before with kalle, is 49:48, which
> exchanges 7/4 <-> 12/7, 7/6 <-> 8/7, and in particular those systems
> where 50:49 is one of the commatic unison vectors, such that the
> chromatic unison vector can also be expressed as 25:24, leading to
> 6/5 <-> 5/4 and 8/5 <-> 5/3.

What strikes me as more interesting than that is 21/20, leading to
10/7<-->3/2, 8/7<-->6/5, 5/3<-->7/4, 12/7<-->9/5, 10/9<-->7/6 and
which can covert a tetrad into an ass. I think I'll tackle it next.