Three and four tone scales

Three tones:

1--5/4--3/2 [5/4, 6/5, 4/3] connectivity = 2

1--3/2--5/3 [3/2, 10/9, 6/5] connectivity = 1

1--3/2--8/5 [3/2, 16/15, 5/4] connectivity = 1

Four tones:

1--5/4--3/2--5/3 [5/4, 6/5, 10/9, 6/5] connectivity = 2

1--6/5--3/2--5/3 [6/5, 5/4, 10/9, 6/5] connectivity = 1

1--6/5--3/2--8/5 [6/5, 5/4, 16/15, 5/4] connectivity = 2

1--6/5--3/2--15/8 [6/5, 5/4, 5/4, 16/15] connectivity = 1

1--6/5--5/4--3/2 [6/5, 25/24, 6/5, 4/3] connectivity = 2

1--25/24--5/4--3/2 [25/24, 6/5, 6/5, 4/3] connectivity = 1

I assume these are all the 3- and 4-tone permutations with
connectivity greater than 1? This is great... sorry to reply
so much, I can't wait for the 6, 8, 9, and 10 tone results! -C.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Three tones:
> 1--5/4--3/2 [5/4, 6/5, 4/3] connectivity = 2
> 1--3/2--5/3 [3/2, 10/9, 6/5] connectivity = 1
> 1--3/2--8/5 [3/2, 16/15, 5/4] connectivity = 1
>
> Four tones:
> 1--5/4--3/2--5/3 [5/4, 6/5, 10/9, 6/5] connectivity = 2
> 1--6/5--3/2--5/3 [6/5, 5/4, 10/9, 6/5] connectivity = 1
> 1--6/5--3/2--8/5 [6/5, 5/4, 16/15, 5/4] connectivity = 2
> 1--6/5--3/2--15/8 [6/5, 5/4, 5/4, 16/15] connectivity = 1
> 1--6/5--5/4--3/2 [6/5, 25/24, 6/5, 4/3] connectivity = 2
> 1--25/24--5/4--3/2 [25/24, 6/5, 6/5, 4/3] connectivity = 1

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> I assume these are all the 3- and 4-tone permutations with
> connectivity greater than 1?

These are the 5-limit 3 and 4 tone scales, up to isomorphim by mode and inversion, with only superparticular scale steps and which are connected. The connectivity number I give is the edge-connectivity, which means the number of edges (consonant intervals) which would need to be removed in order to make it disconnected. It is therefore a measure of how connected by consonant intervals the scale in question is.

>These are the 5-limit 3 and 4 tone scales, up to isomorphim by mode
>and inversion, with only superparticular scale steps and which are
>connected. The connectivity number I give is the edge-connectivity,
>which means the number of edges (consonant intervals) which would
>need to be removed in order to make it disconnected. It is therefore
>a measure of how connected by consonant intervals the scale in
>question is.

Right. Naturally, we're all looking forward to the 7-limit.
Hopefully, there won't be too many superparticulars.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Right. Naturally, we're all looking forward to the 7-limit.
> Hopefully, there won't be too many superparticulars.

I imagine it's too much for me; however I'll try to do some of the smaller scales.

>> Right. Naturally, we're all looking forward to the 7-limit.
>> Hopefully, there won't be too many superparticulars.
>
>I imagine it's too much for me; however I'll try to do some of
>the smaller scales.

You mean too much for your computer, or too much to post, or
is there busy work involved? As I'm sure you know, scales
of 5-10 notes are most interesting from a compositional point
of view.

Perhaps you could restrict yourself to superparticulars which
have factors of 7 and scales with connectivity >= 2.

-C.