31-et tetrads

7-limit

[0, 6, 13, 21]

[1, 8/7, 4/3, 8/5]
[8/7, 1, 7/6, 7/5]
[4/3, 7/6, 1, 6/5]
[8/5, 7/5, 6/5, 1]

[0, 7, 15, 23]

[1, 7/6, 7/5, 5/3]
[7/6, 1, 6/5, 10/7]
[7/5, 6/5, 1, 6/5]
[5/3, 10/7, 6/5, 1]

9-limit

[0, 5, 13, 21]

[1, 9/8, 4/3, 8/5]
[9/8, 1, 6/5, 10/7]
[4/3, 6/5, 1, 6/5]
[8/5, 10/7, 6/5, 1]

[0, 5, 13, 20]

[1, 9/8, 4/3, 14/9]
[9/8, 1, 6/5, 7/5]
[4/3, 6/5, 1, 7/6]
[14/9, 7/5, 7/6, 1]

[0, 5, 11, 18]

[1, 9/8, 9/7, 3/2]
[9/8, 1, 8/7, 4/3]
[9/7, 8/7, 1, 7/6]
[3/2, 4/3, 7/6, 1]

[0, 5, 15, 21]

[1, 9/8, 7/5, 8/5]
[9/8, 1, 5/4, 10/7]
[7/5, 5/4, 1, 8/7]
[8/5, 10/7, 8/7, 1]

[0, 5, 10, 15]

[1, 9/8, 5/4, 7/5]
[9/8, 1, 9/8, 5/4]
[5/4, 9/8, 1, 9/8]
[7/5, 5/4, 9/8, 1]

[0, 5, 11, 16]

[1, 9/8, 9/7, 10/7]
[9/8, 1, 8/7, 9/7]
[9/7, 8/7, 1, 9/8]
[10/7, 9/7, 9/8, 1]

[0, 5, 11, 21]

[1, 9/8, 9/7, 8/5]
[9/8, 1, 8/7, 10/7]
[9/7, 8/7, 1, 5/4]
[8/5, 10/7, 5/4, 1]

[0, 5, 10, 16]

[1, 9/8, 5/4, 10/7]
[9/8, 1, 9/8, 9/7]
[5/4, 9/8, 1, 8/7]
[10/7, 9/7, 8/7, 1]

[0, 11, 16, 26]

[1, 9/7, 10/7, 9/5]
[9/7, 1, 9/8, 7/5]
[10/7, 9/8, 1, 5/4]
[9/5, 7/5, 5/4, 1]

[0, 5, 10, 20]

[1, 9/8, 5/4, 14/9]
[9/8, 1, 9/8, 7/5]
[5/4, 9/8, 1, 5/4]
[14/9, 7/5, 5/4, 1]

[0, 5, 13, 23]

[1, 9/8, 4/3, 5/3]
[9/8, 1, 6/5, 3/2]
[4/3, 6/5, 1, 5/4]
[5/3, 3/2, 5/4, 1]

[0, 5, 10, 18]

[1, 9/8, 5/4, 3/2]
[9/8, 1, 9/8, 4/3]
[5/4, 9/8, 1, 6/5]
[3/2, 4/3, 6/5, 1]

[0, 13, 18, 26]

[1, 4/3, 3/2, 9/5]
[4/3, 1, 9/8, 4/3]
[3/2, 9/8, 1, 6/5]
[9/5, 4/3, 6/5, 1]

[0, 7, 13, 20]

[1, 7/6, 4/3, 14/9]
[7/6, 1, 8/7, 4/3]
[4/3, 8/7, 1, 7/6]
[14/9, 4/3, 7/6, 1]

11-limit

[0, 7, 16, 20]

[1, 7/6, 10/7, 14/9]
[7/6, 1, 11/9, 4/3]
[10/7, 11/9, 1, 11/10]
[14/9, 4/3, 11/10, 1]

[0, 4, 10, 18]

[1, 11/10, 5/4, 3/2]
[11/10, 1, 8/7, 11/8]
[5/4, 8/7, 1, 6/5]
[3/2, 11/8, 6/5, 1]

[0, 4, 11, 18]

[1, 11/10, 9/7, 3/2]
[11/10, 1, 7/6, 11/8]
[9/7, 7/6, 1, 7/6]
[3/2, 11/8, 7/6, 1]

[0, 4, 9, 15]

[1, 11/10, 11/9, 7/5]
[11/10, 1, 9/8, 9/7]
[11/9, 9/8, 1, 8/7]
[7/5, 9/7, 8/7, 1]

[0, 4, 9, 18]

[1, 11/10, 11/9, 3/2]
[11/10, 1, 9/8, 11/8]
[11/9, 9/8, 1, 11/9]
[3/2, 11/8, 11/9, 1]

[0, 7, 18, 27]

[1, 7/6, 3/2, 11/6]
[7/6, 1, 9/7, 14/9]
[3/2, 9/7, 1, 11/9]
[11/6, 14/9, 11/9, 1]

[0, 7, 14, 22]

[1, 7/6, 11/8, 18/11]
[7/6, 1, 7/6, 7/5]
[11/8, 7/6, 1, 6/5]
[18/11, 7/5, 6/5, 1]

[0, 4, 8, 14]

[1, 11/10, 6/5, 11/8]
[11/10, 1, 11/10, 5/4]
[6/5, 11/10, 1, 8/7]
[11/8, 5/4, 8/7, 1]

[0, 4, 9, 14]

[1, 11/10, 11/9, 11/8]
[11/10, 1, 9/8, 5/4]
[11/9, 9/8, 1, 9/8]
[11/8, 5/4, 9/8, 1]

[0, 4, 8, 15]

[1, 11/10, 6/5, 7/5]
[11/10, 1, 11/10, 9/7]
[6/5, 11/10, 1, 7/6]
[7/5, 9/7, 7/6, 1]

[0, 5, 13, 22]

[1, 9/8, 4/3, 18/11]
[9/8, 1, 6/5, 16/11]
[4/3, 6/5, 1, 11/9]
[18/11, 16/11, 11/9, 1]

[0, 6, 13, 22]

[1, 8/7, 4/3, 18/11]
[8/7, 1, 7/6, 10/7]
[4/3, 7/6, 1, 11/9]
[18/11, 10/7, 11/9, 1]

[0, 6, 14, 22]

[1, 8/7, 11/8, 18/11]
[8/7, 1, 6/5, 10/7]
[11/8, 6/5, 1, 6/5]
[18/11, 10/7, 6/5, 1]

[0, 7, 14, 21]

[1, 7/6, 11/8, 8/5]
[7/6, 1, 7/6, 11/8]
[11/8, 7/6, 1, 7/6]
[8/5, 11/8, 7/6, 1]

[0, 4, 13, 22]

[1, 11/10, 4/3, 18/11]
[11/10, 1, 11/9, 3/2]
[4/3, 11/9, 1, 11/9]
[18/11, 3/2, 11/9, 1]

[0, 14, 21, 25]

[1, 11/8, 8/5, 7/4]
[11/8, 1, 7/6, 9/7]
[8/5, 7/6, 1, 11/10]
[7/4, 9/7, 11/10, 1]

[0, 5, 16, 20]

[1, 9/8, 10/7, 14/9]
[9/8, 1, 9/7, 7/5]
[10/7, 9/7, 1, 11/10]
[14/9, 7/5, 11/10, 1]

[0, 11, 17, 25]

[1, 9/7, 16/11, 7/4]
[9/7, 1, 8/7, 11/8]
[16/11, 8/7, 1, 6/5]
[7/4, 11/8, 6/5, 1]

[0, 9, 17, 23]

[1, 11/9, 16/11, 5/3]
[11/9, 1, 6/5, 11/8]
[16/11, 6/5, 1, 8/7]
[5/3, 11/8, 8/7, 1]

[0, 9, 16, 25]

[1, 11/9, 10/7, 7/4]
[11/9, 1, 7/6, 10/7]
[10/7, 7/6, 1, 11/9]
[7/4, 10/7, 11/9, 1]

[0, 7, 13, 27]

[1, 7/6, 4/3, 11/6]
[7/6, 1, 8/7, 14/9]
[4/3, 8/7, 1, 11/8]
[11/6, 14/9, 11/8, 1]

[0, 4, 9, 20]

[1, 11/10, 11/9, 14/9]
[11/10, 1, 9/8, 10/7]
[11/9, 9/8, 1, 9/7]
[14/9, 10/7, 9/7, 1]

[0, 13, 17, 27]

[1, 4/3, 16/11, 11/6]
[4/3, 1, 11/10, 11/8]
[16/11, 11/10, 1, 5/4]
[11/6, 11/8, 5/4, 1]

[0, 11, 17, 21]

[1, 9/7, 16/11, 8/5]
[9/7, 1, 8/7, 5/4]
[16/11, 8/7, 1, 11/10]
[8/5, 5/4, 11/10, 1]

[0, 11, 15, 25]

[1, 9/7, 7/5, 7/4]
[9/7, 1, 11/10, 11/8]
[7/5, 11/10, 1, 5/4]
[7/4, 11/8, 5/4, 1]

[0, 6, 10, 27]

[1, 8/7, 5/4, 11/6]
[8/7, 1, 11/10, 8/5]
[5/4, 11/10, 1, 16/11]
[11/6, 8/5, 16/11, 1]

[0, 10, 15, 24]

[1, 5/4, 7/5, 12/7]
[5/4, 1, 9/8, 11/8]
[7/5, 9/8, 1, 11/9]
[12/7, 11/8, 11/9, 1]

[0, 10, 16, 24]

[1, 5/4, 10/7, 12/7]
[5/4, 1, 8/7, 11/8]
[10/7, 8/7, 1, 6/5]
[12/7, 11/8, 6/5, 1]

[0, 5, 22, 27]

[1, 9/8, 18/11, 11/6]
[9/8, 1, 16/11, 18/11]
[18/11, 16/11, 1, 9/8]
[11/6, 18/11, 9/8, 1]

[0, 15, 20, 24]

[1, 7/5, 14/9, 12/7]
[7/5, 1, 9/8, 11/9]
[14/9, 9/8, 1, 11/10]
[12/7, 11/9, 11/10, 1]

[0, 5, 11, 20]

[1, 9/8, 9/7, 14/9]
[9/8, 1, 8/7, 7/5]
[9/7, 8/7, 1, 11/9]
[14/9, 7/5, 11/9, 1]

[0, 9, 16, 24]

[1, 11/9, 10/7, 12/7]
[11/9, 1, 7/6, 7/5]
[10/7, 7/6, 1, 6/5]
[12/7, 7/5, 6/5, 1]

[0, 4, 10, 20]

[1, 11/10, 5/4, 14/9]
[11/10, 1, 8/7, 10/7]
[5/4, 8/7, 1, 5/4]
[14/9, 10/7, 5/4, 1]

[0, 18, 22, 27]

[1, 3/2, 18/11, 11/6]
[3/2, 1, 11/10, 11/9]
[18/11, 11/10, 1, 9/8]
[11/6, 11/9, 9/8, 1]

[0, 16, 21, 25]

[1, 10/7, 8/5, 7/4]
[10/7, 1, 9/8, 11/9]
[8/5, 9/8, 1, 11/10]
[7/4, 11/9, 11/10, 1]

[0, 4, 8, 13]

[1, 11/10, 6/5, 4/3]
[11/10, 1, 11/10, 11/9]
[6/5, 11/10, 1, 9/8]
[4/3, 11/9, 9/8, 1]

[0, 16, 20, 26]

[1, 10/7, 14/9, 9/5]
[10/7, 1, 11/10, 5/4]
[14/9, 11/10, 1, 8/7]
[9/5, 5/4, 8/7, 1]

[0, 13, 20, 24]

[1, 4/3, 14/9, 12/7]
[4/3, 1, 7/6, 9/7]
[14/9, 7/6, 1, 11/10]
[12/7, 9/7, 11/10, 1]

[0, 6, 13, 20]

[1, 8/7, 4/3, 14/9]
[8/7, 1, 7/6, 11/8]
[4/3, 7/6, 1, 7/6]
[14/9, 11/8, 7/6, 1]

[0, 9, 13, 27]

[1, 11/9, 4/3, 11/6]
[11/9, 1, 11/10, 3/2]
[4/3, 11/10, 1, 11/8]
[11/6, 3/2, 11/8, 1]

[0, 5, 11, 25]

[1, 9/8, 9/7, 7/4]
[9/8, 1, 8/7, 14/9]
[9/7, 8/7, 1, 11/8]
[7/4, 14/9, 11/8, 1]

[0, 9, 17, 27]

[1, 11/9, 16/11, 11/6]
[11/9, 1, 6/5, 3/2]
[16/11, 6/5, 1, 5/4]
[11/6, 3/2, 5/4, 1]

[0, 10, 16, 25]

[1, 5/4, 10/7, 7/4]
[5/4, 1, 8/7, 7/5]
[10/7, 8/7, 1, 11/9]
[7/4, 7/5, 11/9, 1]

[0, 10, 17, 27]

[1, 5/4, 16/11, 11/6]
[5/4, 1, 7/6, 16/11]
[16/11, 7/6, 1, 5/4]
[11/6, 16/11, 5/4, 1]

[0, 11, 17, 22]

[1, 9/7, 16/11, 18/11]
[9/7, 1, 8/7, 9/7]
[16/11, 8/7, 1, 9/8]
[18/11, 9/7, 9/8, 1]

[0, 4, 8, 18]

[1, 11/10, 6/5, 3/2]
[11/10, 1, 11/10, 11/8]
[6/5, 11/10, 1, 5/4]
[3/2, 11/8, 5/4, 1]

[0, 10, 17, 23]

[1, 5/4, 16/11, 5/3]
[5/4, 1, 7/6, 4/3]
[16/11, 7/6, 1, 8/7]
[5/3, 4/3, 8/7, 1]

[0, 11, 16, 25]

[1, 9/7, 10/7, 7/4]
[9/7, 1, 9/8, 11/8]
[10/7, 9/8, 1, 11/9]
[7/4, 11/8, 11/9, 1]

[0, 6, 10, 16]

[1, 8/7, 5/4, 10/7]
[8/7, 1, 11/10, 5/4]
[5/4, 11/10, 1, 8/7]
[10/7, 5/4, 8/7, 1]

[0, 4, 11, 21]

[1, 11/10, 9/7, 8/5]
[11/10, 1, 7/6, 16/11]
[9/7, 7/6, 1, 5/4]
[8/5, 16/11, 5/4, 1]

[0, 8, 17, 22]

[1, 6/5, 16/11, 18/11]
[6/5, 1, 11/9, 11/8]
[16/11, 11/9, 1, 9/8]
[18/11, 11/8, 9/8, 1]

[0, 9, 13, 26]

[1, 11/9, 4/3, 9/5]
[11/9, 1, 11/10, 16/11]
[4/3, 11/10, 1, 4/3]
[9/5, 16/11, 4/3, 1]

[0, 4, 11, 20]

[1, 11/10, 9/7, 14/9]
[11/10, 1, 7/6, 10/7]
[9/7, 7/6, 1, 11/9]
[14/9, 10/7, 11/9, 1]

[0, 9, 13, 18]

[1, 11/9, 4/3, 3/2]
[11/9, 1, 11/10, 11/9]
[4/3, 11/10, 1, 9/8]
[3/2, 11/9, 9/8, 1]

[0, 4, 10, 17]

[1, 11/10, 5/4, 16/11]
[11/10, 1, 8/7, 4/3]
[5/4, 8/7, 1, 7/6]
[16/11, 4/3, 7/6, 1]

[0, 4, 11, 15]

[1, 11/10, 9/7, 7/5]
[11/10, 1, 7/6, 9/7]
[9/7, 7/6, 1, 11/10]
[7/5, 9/7, 11/10, 1]

[0, 4, 9, 17]

[1, 11/10, 11/9, 16/11]
[11/10, 1, 9/8, 4/3]
[11/9, 9/8, 1, 6/5]
[16/11, 4/3, 6/5, 1]

[0, 4, 8, 17]

[1, 11/10, 6/5, 16/11]
[11/10, 1, 11/10, 4/3]
[6/5, 11/10, 1, 11/9]
[16/11, 4/3, 11/9, 1]

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> 7-limit
>
> [0, 6, 13, 21]
>
> [1, 8/7, 4/3, 8/5]
> [8/7, 1, 7/6, 7/5]
> [4/3, 7/6, 1, 6/5]
> [8/5, 7/5, 6/5, 1]

I don't understand your notation. What do these ratios mean?

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > 7-limit

> > [0, 6, 13, 21]

> > [1, 8/7, 4/3, 8/5]
> > [8/7, 1, 7/6, 7/5]
> > [4/3, 7/6, 1, 6/5]
> > [8/5, 7/5, 6/5, 1]
>
> I don't understand your notation. What do these ratios mean?

The ratios are the approximate intervals between the scale degrees, in terms of the approximation defined by the 31-et val. In this case we see that we have a standard minor tetrad. If I read the third row,
for instance, and invert the values below the 1 I would get

3/4--6/7--1--6/5

which I could normalize to

1--6/5--3/2--12/7

I can also read downwards; for instance from the last row I get

1--6/5--7/5--8/5

which is an inversion of the major tetrad. The first row

1--8/7--4/3--8/5

is of course an inversion of the minor tetrad. Inverting the first row
gives

1--7/8--3/4--5/8

which I can normalize to

1--5/4--3/2--7/4

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > > 7-limit
>
> > > [0, 6, 13, 21]
>
> > > [1, 8/7, 4/3, 8/5]
> > > [8/7, 1, 7/6, 7/5]
> > > [4/3, 7/6, 1, 6/5]
> > > [8/5, 7/5, 6/5, 1]
> >
> > I don't understand your notation. What do these ratios mean?
>
> The ratios are the approximate intervals between the scale degrees,
in terms of the approximation defined by the 31-et val. In this case
we see that we have a standard minor tetrad. If I read the third row,
> for instance, and invert the values below the 1 I would get
>
> 3/4--6/7--1--6/5

Well that would have made a lot more sense -- why didn't you do it
that way to begin with? In other words, wouldn't the 'interval
matrix' of the chord/scale be what we want to see here?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_32757.html#32757

> 7-limit
>
> [0, 6, 13, 21]
>
> [1, 8/7, 4/3, 8/5]
> [8/7, 1, 7/6, 7/5]
> [4/3, 7/6, 1, 6/5]
> [8/5, 7/5, 6/5, 1]
>

etc...

This is actually pretty cool. You do great work Gene! Keep up these
informative posts! Just out of curiousity, though, how do you do
this stuff? Is there a computer program you made?

I take it you don't use an abacus... :)

Actually, when I was in Russia last year in the outlying districts,
they were *still* using an abacus with the cash register... !

JP

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_32757.html#32757

> 7-limit
>
> [0, 6, 13, 21]
>
> [1, 8/7, 4/3, 8/5]
> [8/7, 1, 7/6, 7/5]
> [4/3, 7/6, 1, 6/5]
> [8/5, 7/5, 6/5, 1]
>

Wait a minute.... now I *may* be a dunce, but why do you have 4 of
these on the scale degree "0" ?? You must be starting on different
places or such like...?? Help!

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> This is actually pretty cool. You do great work Gene! Keep up these
> informative posts! Just out of curiousity, though, how do you do
> this stuff? Is there a computer program you made?

I did it with the Maple computer algebra package.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Wait a minute.... now I *may* be a dunce, but why do you have 4 of
> these on the scale degree "0" ?? You must be starting on different
> places or such like...?? Help!

It shows the approximate interval between each of the scale steps and the others, starting from each of them.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_32757.html#32824

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Wait a minute.... now I *may* be a dunce, but why do you have 4
of
> > these on the scale degree "0" ?? You must be starting on
different
> > places or such like...?? Help!
>
> It shows the approximate interval between each of the scale steps
and the others, starting from each of them.

Thanks, Gene!

Actually, that "came to me" shortly after I made that post.

Great work!

JP

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_32757.html#32824

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Wait a minute.... now I *may* be a dunce, but why do you have 4
of
> > these on the scale degree "0" ?? You must be starting on
different
> > places or such like...?? Help!
>
> It shows the approximate interval between each of the scale steps
and the others, starting from each of them.

Hi Gene!

So then, are they within a couple of cents off of these ratios??

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> So then, are they within a couple of cents off of these ratios??

It's 31-et, and 31-et isn't quite that good.