What temperament would this be?

Hi all,

I started messing around with some random temperaments and came across
this one, which subdivides 3/2 into 4 parts for 11-limit harmony. I've
been using 7/48 to tune it, and it has two beautiful MOS's at 7 and 13
notes. It sounds pseudo-diatonic, yet much different. The thirds of
the 7-note MOS alternate between 350 cents and 325 cents, which gives
the scale some kind of ancient neutral-plus-minor feel.

Is there a name for this temperament, and how do people map this?
Would the 175-cent interval map to 10/9, and the 525 cent interval to
11/8? If so, the error ends up pretty high, but you still get a really
interesting sound of it. I get the feeling that whatever the mapping,
it would work best with wider fifths.

-Mike

Just wanted to add that I notice 5/34 is a much better generator for
this. With this, the 5/4's are 388 cents, the 6/5's are 318 cents, and
the 3/2's are 706 cents. And also that with 5/34, there is a much
better approximation for 11/8 at 10 stacked generators.

-Mike

On Sun, Aug 1, 2010 at 4:30 AM, Mike Battaglia <battaglia01@...> wrote:
> Hi all,
>
> I started messing around with some random temperaments and came across
> this one, which subdivides 3/2 into 4 parts for 11-limit harmony. I've
> been using 7/48 to tune it, and it has two beautiful MOS's at 7 and 13
> notes. It sounds pseudo-diatonic, yet much different. The thirds of
> the 7-note MOS alternate between 350 cents and 325 cents, which gives
> the scale some kind of ancient neutral-plus-minor feel.
>
> Is there a name for this temperament, and how do people map this?
> Would the 175-cent interval map to 10/9, and the 525 cent interval to
> 11/8? If so, the error ends up pretty high, but you still get a really
> interesting sound of it. I get the feeling that whatever the mapping,
> it would work best with wider fifths.
>
> -Mike
>

On 1 August 2010 09:36, Mike Battaglia <battaglia01@...> wrote:
> Just wanted to add that I notice 5/34 is a much better generator for
> this. With this, the 5/4's are 388 cents, the 6/5's are 318 cents, and
> the 3/2's are 706 cents. And also that with 5/34, there is a much
> better approximation for 11/8 at 10 stacked generators.

Looks like Tetracot but something's wrong with those cents.

http://x31eq.com/cgi-bin/rt.cgi?ets=34+41&limit=11

Graham

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi all,
>
> I started messing around with some random temperaments and came across
> this one, which subdivides 3/2 into 4 parts for 11-limit harmony. I've
> been using 7/48 to tune it, and it has two beautiful MOS's at 7 and 13
> notes.

I don't know why you want 48edo as a tuning, but this sounds like tetracot stuff, either monkey or bunya:

http://xenharmonic.wikispaces.com/Tetracot+family

On Sun, Aug 1, 2010 at 5:46 AM, Graham Breed <gbreed@...> wrote:
>
> On 1 August 2010 09:36, Mike Battaglia <battaglia01@...> wrote:
> > Just wanted to add that I notice 5/34 is a much better generator for
> > this. With this, the 5/4's are 388 cents, the 6/5's are 318 cents, and
> > the 3/2's are 706 cents. And also that with 5/34, there is a much
> > better approximation for 11/8 at 10 stacked generators.
>
> Looks like Tetracot but something's wrong with those cents.
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=34+41&limit=11

So it is. What's wrong with the cents though? That's what it comes out
to if you use 5/34 as a generator...

Also, so I don't have to keep spamming the tuning list with these
requests - what's the easiest way to figure this out? Just calculate
the unison vectors from whatever seems like a logical way to map
things and and plug them into your uv finder?

-Mike

On 1 August 2010 18:50, Mike Battaglia <battaglia01@...> wrote:

> So it is. What's wrong with the cents though? That's what it comes out
> to if you use 5/34 as a generator...

There's a big enough discrepancy that I couldn't be sure that I had it
right. If you've checked it, I'm sure you're correct.

> Also, so I don't have to keep spamming the tuning list with these
> requests - what's the easiest way to figure this out? Just calculate
> the unison vectors from whatever seems like a logical way to map
> things and and plug them into your uv finder?

I don't know what's easiest. What I did was to guess pairs of equal
temperaments and see what looked right. I did that in a Python
interpreter, but there is a script online that does the same job.

And, because the window's still open, I can tell you exactly what I
did. First I looked at 34&48, because these were the best two equal
temperaments you mentioned in your message. That almost worked but
there's a half-octave period that isn't consistent with 7 or 13. Then
I tried 11&34, which was a mistake. So I went for 13&34 and got a
3-mapping of 12 instead of 4. Which left 7&34, and that looked right.
And the readout told my that 7+34=41, so I thought 34&41 would be a
good way of defining it. Last, to check, I made sure the generator
rounded off to 5 steps of 34.

Graham

Graham Breed wrote:
> On 1 August 2010 18:50, Mike Battaglia <battaglia01@...> wrote:
> >> So it is. What's wrong with the cents though? That's what it comes out
>> to if you use 5/34 as a generator...
> > There's a big enough discrepancy that I couldn't be sure that I had it
> right. If you've checked it, I'm sure you're correct.
> >> Also, so I don't have to keep spamming the tuning list with these
>> requests - what's the easiest way to figure this out? Just calculate
>> the unison vectors from whatever seems like a logical way to map
>> things and and plug them into your uv finder?
> > I don't know what's easiest. What I did was to guess pairs of equal
> temperaments and see what looked right. I did that in a Python
> interpreter, but there is a script online that does the same job.
> > And, because the window's still open, I can tell you exactly what I
> did. First I looked at 34&48, because these were the best two equal
> temperaments you mentioned in your message. That almost worked but
> there's a half-octave period that isn't consistent with 7 or 13. Then
> I tried 11&34, which was a mistake. So I went for 13&34 and got a
> 3-mapping of 12 instead of 4. Which left 7&34, and that looked right.
> And the readout told my that 7+34=41, so I thought 34&41 would be a
> good way of defining it. Last, to check, I made sure the generator
> rounded off to 5 steps of 34.

You can take something like 5/34 and find a generator mapping for it. First find the closest approximation to the primes in 34-ET (34, 54, 79, 95, 118). It makes it easier if you reduce to an octave (0, 20, 11, 27, 16). Then keep adding the 5/34 generator until you hit each of these (you reach 20 after 4 iterations of the generator, 45 after 9, which reduces to 11, and so on). Keep in mind that you can also go negative, so that in this case, 19 generators up is equivalent to 15 generators down. So one possible generator mapping for 5/34 is <0, 4, 9, -15, 10], which you also get if you look at 7/48.

It's easy enough to put the octaves back in for the full rank 2 generator mapping, [<1, 1, 1, 5, 2], <0, 4, 9, -15, 10]>. You can get all the information you need from this, but I don't know if there's a convenient script on Graham's page for that. You can find the TOP-RMS generators with a spreadsheet and some matrix math, if I can find the paper that describes how to do it.

On 1 August 2010 23:20, Herman Miller <hmiller@...> wrote:

> You can take something like 5/34 and find a generator mapping for it.
<snip>

Yes, I have code to do that. It's the next thing I would have tried.
It isn't in the CGI but you get it with the source code bundle.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > Also, so I don't have to keep spamming the tuning list with these
> > requests - what's the easiest way to figure this out? Just calculate
> > the unison vectors from whatever seems like a logical way to map
> > things and and plug them into your uv finder?
>
> I don't know what's easiest. What I did was to guess pairs of equal
> temperaments and see what looked right.

What I did was take 7/48 and see what the generator mapping was, by looking at the number of steps in 2, 3, 5, 7 and 11, and then dividing by 7 mod 48. That gave monkey temperament, the 34&41 temperament, but since 48 is kind of screwed up I didn't discount the possibility of bunya, the 41&75 temperament.