17496/16807 comma pump and terminology question

I tried to post a Hanson comma pump earlier, but accidentally screwed
it up and stumbled on something else - a 17496/16807 comma pump. The
musical example is here:

http://soundcloud.com/mikebattagliamusic/17496-16807-comma-pump-subminor-34-equal

The chord progression takes advantage of the fact that 5 7/6's is
equated with 9/4, which means that 17496/16807 vanishes. Looks like
the rank-3 7-limit temperament is here:

http://x31eq.com/cgi-bin/rt.cgi?ets=26_9_8d&limit=7

The best rank-2 option is here...

http://x31eq.com/cgi-bin/rt.cgi?ets=9_8d&limit=7

And if we bump this up to the 13-limit, we suddenly get the name
"Progression" on it:

http://x31eq.com/cgi-bin/rt.cgi?ets=9_8d&limit=13

OK, so we're in "Progression" temperament now, cool. Now, the question
is, what's the best way to label this comma pump? Should I really call
a Progression comma pump? The problem in this case is that when we're
dealing with something like a 7-limit rank-2 temperament, two commas
vanish, so the phrase "Progression comma pump" is indeterminate and
could be any one of an infinite number of commas. Would a good way to
describe this be that the above chord progression is a 17496/16807
comma pump, which is supported by Progression temperament? I need to
name this something on my Soundcloud, and "Progression Comma Pump" is
a lot catchier than "17496/16807 Comma Pump," but it might be pretty
misleading.

-Mike

Mike wrote:

> The chord progression takes advantage of the fact that 5 7/6's is
> equated with 9/4, which means that 17496/16807 vanishes. Looks like
> the rank-3 7-limit temperament is here:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=26_9_8d&limit=7

Two years ago, I discovered a "fiveless" 2D temperament which tempered out this very same interval. But back then, I wasn't thinking a lot about how much musical significance such a temperament might have.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> Two years ago, I discovered a "fiveless" 2D temperament which tempered out
> this very same interval.

So does this interval deserve a name?

On Mon, May 9, 2011 at 2:43 PM, Petr Parízek <petrparizek2000@...> wrote:
>
> Mike wrote:
>
> > The chord progression takes advantage of the fact that 5 7/6's is
> > equated with 9/4, which means that 17496/16807 vanishes. Looks like
> > the rank-3 7-limit temperament is here:
> >
> > http://x31eq.com/cgi-bin/rt.cgi?ets=26_9_8d&limit=7
>
> Two years ago, I discovered a "fiveless" 2D temperament which tempered out
> this very same interval. But back then, I wasn't thinking a lot about how
> much musical significance such a temperament might have.
>
> Petr

It's a good'un. It also opens up the question of how we're supposed to
describe comma pumps for codimension > 1 temperaments. We can talk
about something like a "hanson comma pump" or a "negri comma pump"
because there's only one comma that we could possibly be talking
about. Talking about a "progression comma pump" is indeterminate,
however, as would be talking about a "miracle comma pump" (which
miracle comma is getting pumped)?

-Mike

Mike wrote:

> Talking about a "progression comma pump" is indeterminate,
> however, as would be talking about a "miracle comma pump" (which
> miracle comma is getting pumped)?

In the 5-limit field, I think it's called the "Ampersand's comma". The strange thing here is that if you really wanted to do a 5-limit comma pump with major and minor triads, the shortest possible pump would take, if I'm not mistaken, as much as 13 chords.
Obviously, if we want to go 7-limit, which is what miracle was originally "designed for", this number would be lower since we would probably use chords of more than 3 tones as well as 7-limit intervals.

BTW: I'm planning to make an example recording of a "semi-comma pump" in 5-limit orwell -- another one that seems to be understood mostly as a 7-limit temperament.

Petr

On Mon, May 9, 2011 at 4:10 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> > Two years ago, I discovered a "fiveless" 2D temperament which tempered out
> > this very same interval.
>
> So does this interval deserve a name?

Yes, I think so. It's the 7-limit evil twin to porcupine, in a few
ways. Firstly, if we ignore the existence of 5 completely, we end up
with this temperament:

http://x31eq.com/cgi-bin/rt.cgi?ets=8_9&limit=2.3.7

Just like how porcupine has a generator that's half a 6/5, this one
has a generator that's half a 7/6. Secondly, while in porcupine, three
6/5's plus a 9/8 gets you to the octave, in this temperament four
7/6's minus a 9/8 gets you to an octave. Lastly, porcupine is the 7&8
temperament, and this is the 8&9 temperament, meaning they also
generate similarly sized MOS's.

I was trying to figure out what the generator for this temperament
would be, but it turned into a huge chore. After doing a lot of
reading on something called "Diophantine Equations," it turns out that
all possible mappings for the generator satisfy p_3 = 3+7t and p_7 =
-2-5t. I plugged in t=0 to get [0 3 -2> for the 2.3.7 smonzo for the
generator, which turns out to be 27/49. Convinced that I've screwed
something up, I now hereby end this stream of consciousness post in
hopes that you can tell me what exactly I did wrong here.

On second look, it seems pretty obvious that to find the generator,
you're going to want to not only solve <V|M> = 1 for the generator,
but <V|M> = 0 for the period. Since the generator mapping for 2 is
zero, we can transpose the above up to get 54/49. Some kind of
one-shot kill formalization of all of this would be great. Is it that
I end up with two systems of equations and three variables then?

-Mike

On Mon, May 9, 2011 at 6:57 PM, Petr Parízek <petrparizek2000@...> wrote:
>
> In the 5-limit field, I think it's called the "Ampersand's comma". The
> strange thing here is that if you really wanted to do a 5-limit comma pump
> with major and minor triads, the shortest possible pump would take, if I'm
> not mistaken, as much as 13 chords.

Sounds good to me!

> BTW: I'm planning to make an example recording of a "semi-comma pump" in
> 5-limit orwell -- another one that seems to be understood mostly as a
> 7-limit temperament.

What's the generator in 5-limit orwell, 75/64?

-Mike

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> BTW: I'm planning to make an example recording of a "semi-comma pump" in
> 5-limit orwell -- another one that seems to be understood mostly as a
> 7-limit temperament.

Good. I was set to do both the semicomma and ampersand on general principles if no one else gave one.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Yes, I think so. It's the 7-limit evil twin to porcupine, in a few
> ways. Firstly, if we ignore the existence of 5 completely, we end up
> with this temperament:

You don't actually need to ignore 5; you can toss 81/80 into the mix and get the 26&43 temperament, with generator a fifth of a fifth, which is what you get anyway.

> Convinced that I've screwed
> something up, I now hereby end this stream of consciousness post in
> hopes that you can tell me what exactly I did wrong here.

Can't help you. 54/49 is what I get also. Toss in 81/80, and it's still hanging about.

> Some kind of
> one-shot kill formalization of all of this would be great.

My code nearly always works for me. It hasn't bothered me enough to contrive it to always work.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What's the generator in 5-limit orwell, 75/64?

'Fraid so.

On Mon, May 9, 2011 at 8:38 PM, genewardsmith
<genewardsmith@...> wrote:
>
> > Yes, I think so. It's the 7-limit evil twin to porcupine, in a few
> > ways. Firstly, if we ignore the existence of 5 completely, we end up
> > with this temperament:
>
> You don't actually need to ignore 5; you can toss 81/80 into the mix and get the 26&43 temperament, with generator a fifth of a fifth, which is what you get anyway.

Nice, could this be the mythical pentacot?

-Mike

On 5/9/2011 8:51 PM, Mike Battaglia wrote:
> On Mon, May 9, 2011 at 8:38 PM, genewardsmith
> <genewardsmith@...> wrote:
>>
>>> Yes, I think so. It's the 7-limit evil twin to porcupine, in a
>>> few ways. Firstly, if we ignore the existence of 5 completely, we
>>> end up with this temperament:
>>
>> You don't actually need to ignore 5; you can toss 81/80 into the
>> mix and get the 26&43 temperament, with generator a fifth of a
>> fifth, which is what you get anyway.
>
> Nice, could this be the mythical pentacot?

It looks like many of the best 7-limit 17496/16807 temperaments share
that general mapping of [<1 1 ... 2|, <0 5 ... 7|]. Besides the 8d&9
progression, and the 26&43, there's a 9&17 [<1 1 3 2|, <0 5 -6 7|] and a
17&43 [<1 1 5 2|, <0 5 -23 7|]. That 9&17 is an interesting option,
since it looks about twice as good as progression in the 7-limit while
only being slightly more complex.

The other best 17496/16807 temperaments are unrelated:

17&34d [<17 27 39 48|, <0 0 1 0|] 70.404, 29.853
9&43 [<1 6 1 9|, <0 -10 3 -14|] 1198.862, 529.477