Searching for JI scales

I wanted to look for scales that had no "wolf" intervals like 27/20 that are too large to be a melodic step but too dissonant to use in harmony, so I did a semi-systematic search for all the scales in which every interval wider than 10/9 was inside the 9 odd limit. I didn't get any scales with more than 6 notes (although it's possible I missed some), but I did get this gem:

1/1 6/5 9/7 7/5 3/2 9/5 2/1

It turns out it's a hexany on {7, 9, 15, 21}. Played in that mode it sounds almost like a blues scale on some kind of drug, and the mode starting on the 6/5 is also quite interesting. The only weird interval is the 49/45 neutral second (147 cents), which gives it a little exotic touch. It's rapidly becoming one of my all-time favorite JI scales.

The other scales I got with 6 notes are:

1/1 28/27 10/9 7/6 4/3 14/9 2/1
1/1 28/27 10/9 4/3 14/9 16/9 2/1
1/1 21/20 9/8 6/5 3/2 9/5 2/1
1/1 21/20 7/6 7/5 3/2 7/4 2/1
1/1 16/15 8/7 4/3 8/5 16/9 2/1

and their inversions. I haven't really explored them all yet, but so far they sound pretty good. It's satisfying to know that any two pitches you play will make something consonant as long as they're far enough apart.

I'm also trying to find the best way to fit all these into 12 notes so I can tune a real physical piano to it.

Keenan

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...> wrote:
>
> I wanted to look for scales that had no "wolf" intervals like 27/20
that are too
> large to be a melodic step but too dissonant to use in harmony, so I
did a
> semi-systematic search for all the scales in which every interval
wider than
> 10/9 was inside the 9 odd limit.

How did you do this semi-systematic search?

I didn't get any scales with more than 6 notes
> (although it's possible I missed some), but I did get this gem:
>
> 1/1 6/5 9/7 7/5 3/2 9/5 2/1
>
> It turns out it's a hexany on {7, 9, 15, 21}.

For some reason Scala does not find this, but it does find that the
scale is epimorphic.

> I'm also trying to find the best way to fit all these into 12 notes
so I can
> tune a real physical piano to it.

I wonder how well you could fit two of these hexatonic scales together
to make one 12 note scale?

They sound very good, especially the narrower minor thirds and the wider
major thirds.

----- Original Message -----
From: "Keenan Pepper" <keenanpepper@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 26 Aral�k 2005 Pazartesi 9:28
Subject: [tuning] Searching for JI scales

> I wanted to look for scales that had no "wolf" intervals like 27/20 that
are too
> large to be a melodic step but too dissonant to use in harmony, so I did a
> semi-systematic search for all the scales in which every interval wider
than
> 10/9 was inside the 9 odd limit. I didn't get any scales with more than 6
notes
> (although it's possible I missed some), but I did get this gem:
>
> 1/1 6/5 9/7 7/5 3/2 9/5 2/1
>
> It turns out it's a hexany on {7, 9, 15, 21}. Played in that mode it
sounds
> almost like a blues scale on some kind of drug, and the mode starting on
the 6/5
> is also quite interesting. The only weird interval is the 49/45 neutral
second
> (147 cents), which gives it a little exotic touch. It's rapidly becoming
one of
> my all-time favorite JI scales.
>
> The other scales I got with 6 notes are:
>
> 1/1 28/27 10/9 7/6 4/3 14/9 2/1
> 1/1 28/27 10/9 4/3 14/9 16/9 2/1
> 1/1 21/20 9/8 6/5 3/2 9/5 2/1
> 1/1 21/20 7/6 7/5 3/2 7/4 2/1
> 1/1 16/15 8/7 4/3 8/5 16/9 2/1
>
> and their inversions. I haven't really explored them all yet, but so far
they
> sound pretty good. It's satisfying to know that any two pitches you play
will
> make something consonant as long as they're far enough apart.
>
> I'm also trying to find the best way to fit all these into 12 notes so I
can
> tune a real physical piano to it.
>
> Keenan
>
>

Gene Ward Smith wrote:
> How did you do this semi-systematic search?

I was going to write a program to do it, but I decided that would probably take longer and be more error-prone than doing it by hand. I basically made a big table of which pitches in a 9-limit tonality diamond "clash" and used backtracking to exhaust all the scales without clashes.

Maybe now I will write a program, so I can confirm the results and extend them to higher limits.

>>It turns out it's a hexany on {7, 9, 15, 21}.
> > > For some reason Scala does not find this, but it does find that the
> scale is epimorphic.

It seems like it should be pretty easy to identify a hexany. Isn't a scale a hexany iff it's symmetric and has 6 pitches?

>>I'm also trying to find the best way to fit all these into 12 notes
> > so I can > >>tune a real physical piano to it.
> > > I wonder how well you could fit two of these hexatonic scales together
> to make one 12 note scale?

I think it's possible to fit all 11 of them (including inversions) in a set of 12 notes. Doesn't this do it?:

1/1 15/14 10/9 7/6 5/4 4/3 7/5 10/7 3/2 14/9 5/3 25/14

I just want to make sure and fine the best possible way before I get out my tuning hammer. =P

BTW, is there a name for this general CS problem? You have a group acting on a set (in this case the 7-limit numbers) and you have a bunch of subsets (the scales) and you want to find the smallest subset that contains an image of each of the subsets under some element of the group (transposed versions of the scales). Is it NP-complete?

Keenan

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...> wrote:

Along the same lines, here are some septimal hexatonic scales. They
all have the following characteristics:

(1) They are the union of two triads, both of which are of one of the
forms 1-5/4-3/2, 1-6/5-3/2, 1-6/5-7/5, or 1-7/6-7/5.

(2) The two triads are separated by an interval of 4/3, 7/5, 10/7 or 3/2.

(3) Each of the scales is epimorphic.

<6 10 14 17|-epimorphic
[21/20, 5/4, 7/5, 3/2, 7/4, 2]
[15/14, 5/4, 10/7, 3/2, 25/14, 2]
[21/20, 5/4, 7/5, 3/2, 42/25, 2]
[15/14, 5/4, 10/7, 3/2, 12/7, 2]
[21/20, 6/5, 7/5, 3/2, 42/25, 2]
[15/14, 6/5, 10/7, 3/2, 12/7, 2]
[21/20, 6/5, 7/5, 3/2, 7/4, 2]

<6 10 15 18|-epimorphic
[6/5, 4/3, 7/5, 14/9, 28/15, 2]