New JI scale using a 3/1 period

   As always, I discovered this scale myself and used my own math to arrive at it, but bear no claims to being the only person who has ever come across it.

  The idea with this scale is to go beyond the 2/1 octave into use of a 3/1 octave and, in the process, concentrate on adding extra notes beyond 2/1 that do not have matching notes one octave below them when possible for compositional flexibility.

 
1/1
9/8
29/24 (near 6/5)
4/3
3/2
5/3
7/4
15/8
2/1
25/12  (not octave equivalent)
9/4
19/8  (not octave equivalent)
8/3
23/8  (not octave equivalent)
3/1

  Thoughts or suggestions?
  And, dare I ask, Gene, could you do me a favor and run a least squares optimization on it for all limits up to 9?

>    As always, I discovered this
> scale myself and used my own math to arrive at it, but bear
> no claims to being the only person who has ever come across
> it.
>
>   The idea with this scale is to go beyond the 2/1 octave
> into use of a 3/1 octave and, in the process, concentrate on
> adding extra notes beyond 2/1 that do not have matching
> notes one octave below them when possible for compositional
> flexibility.

Turns out I missed a tone of 5/2 in the scale...so the scale should be

1/1
9/8
29/24 (near 6/5)
4/3
3/2
5/3
7/4
15/8
2/1
25/12  (not octave equivalent)
9/4
19/8  (not octave equivalent)
>>5/2<<<<
8/3
23/8  (not octave equivalent)
3/1

   I'm working on a new composition with this scale as well...if you keep checking the MMM list: I will soon post it there.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
>    As always, I discovered this scale myself and used my own math to arrive at it, but bear no claims to being the only person who has ever come across it.
>
>   The idea with this scale is to go beyond the 2/1 octave into use of a 3/1 octave and, in the process, concentrate on adding extra notes beyond 2/1 that do not have matching notes one octave below them when possible for compositional flexibility.
>
>  
> 1/1
> 9/8
> 29/24 (near 6/5)
> 4/3
> 3/2
> 5/3
> 7/4
> 15/8
> 2/1
> 25/12  (not octave equivalent)
> 9/4
> 19/8  (not octave equivalent)
> 8/3
> 23/8  (not octave equivalent)
> 3/1
>
>
>   Thoughts or suggestions?
>   And, dare I ask, Gene, could you do me a favor and run a least squares optimization on it for all limits up to 9?

How would you use this in practice, if you wanted more notes below 1/1 or above 3/1?

Is the idea that the period is really supposed to be 3/1 (so like Bohlen-Pierce except it contains 2/1 and other even-number intervals...)? I'm not sure I could ever get the hang of such a scale. It's hard enough to try to perceive 3/1 as the interval of equivalence even in BP, where there are no 2/1s to grab your attention as stronger consonances.

Or perhaps the scale is intended to be extended in some other way than simple repetition at the 3/1?

Or I guess it could be intended as a finite set of 15 pitches, not to be extended at all...

Keenan

Keenan>"Is the idea that the period is really supposed to be 3/1 (so like
Bohlen-Pierce except it contains 2/1 and other even-number
intervals...)? "

Exactly...3/1 is the period or interval of equivalence.
   So you would repeat it at 3/1, starting with the first tone of the scale times 3/1, then the second tone times 3/1...and so on.  Same goes with finding lower notes...you'd take the whole scale and divide it by 3 to get the fractions for the lower tones.

>"I'm not sure I could ever get the hang of such a scale. It's hard enough
to try to perceive 3/1 as the interval of equivalence even in BP, where
there are no 2/1s to grab your attention as stronger consonances."

    Right, and the scale is intended for free form playing without the idea of any interval having to be the "strongest consonance".  It turns out this scale has a good 2/1, 3/1, 4/1, 5/1, and 6/1 harmonic if you go up that far...so you can really just take your pick.

  And it is weird and tricky...but I have spent far too long being scared of using periods that are not 2/1 or under...and I thought I was long overdue to challenge myself with this concept. :-)

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>   Thoughts or suggestions?
>   And, dare I ask, Gene, could you do me a favor and run a least squares optimization on it for all limits up to 9?

If 3 is the interval of equivalence, there is no such thing as the 9-limit. I think it would make more sense to make the interval of equivlence 2, take the part up to 2/1, and work with that.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
>
> >   Thoughts or suggestions?
> >   And, dare I ask, Gene, could you do me a favor and run a least squares optimization on it for all limits up to 9?
>
> If 3 is the interval of equivalence, there is no such thing as the 9-limit. I think it would make more sense to make the interval of equivlence 2, take the part up to 2/1, and work with that.

Alternatively, I could simply optimize the scale as a finite thing up to 3/1 and ignore intervals of equivalence.

Gene,

     Is it impossible to use 3/1 as the interval of equivalence in your least squares optimization code?  It would/should be the same thing as optimizing your least squares program to optimize the BP scale.

> If 3 is the interval of equivalence, there is no such thing as the 9-limit.
  Ah ok, since 9 has a factor of 3.  Then it would be optimized up to 7-limit, right? :-D

On Fri, Aug 12, 2011 at 5:28 PM, Michael <djtrancendance@...> wrote:
>
> Gene,
>
>      Is it impossible to use 3/1 as the interval of equivalence in your least squares optimization code?  It would/should be the same thing as optimizing your least squares program to optimize the BP scale.
>
> > If 3 is the interval of equivalence, there is no such thing as the 9-limit.
>   Ah ok, since 9 has a factor of 3.  Then it would be optimized up to 7-limit, right? :-D

8-limit, right?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > > If 3 is the interval of equivalence, there is no such thing as the 9-limit.
> >   Ah ok, since 9 has a factor of 3.  Then it would be optimized up to 7-limit, right? :-D
>
> 8-limit, right?

Why not 10-limit? =D

Keenan