9 & 41 & 58 & 46

I was cooking up something for pele temperament, which tempers out 196/195, 352/351 and 364/363, but it ended up with the 352/351 comma not playing any role. This left me with the rank four temperament which x31eq calls 9 & 41 & 58 & 46. I'm not sure I like the name, but I'm also not sure I like the idea of starting in on names for rank four temperaments. Should I just decide this is a special case, or what?

"genewardsmith" <genewardsmith@...> wrote:
> I was cooking up something for pele temperament, which
> tempers out 196/195, 352/351 and 364/363, but it ended up
> with the 352/351 comma not playing any role. This left me
> with the rank four temperament which x31eq calls 9 & 41 &
> 58 & 46. I'm not sure I like the name, but I'm also not
> sure I like the idea of starting in on names for rank
> four temperaments. Should I just decide this is a special
> case, or what?

A time will surely come when those special cases add up and
we start using rank 4 names. In this case I'm not sure
because it doesn't have any names rank 3 followers other
than Pele. The highest scoring rank 2 fans are
Diaschismic, Cassandra, Hemififths, and Rodan whether or
not the 352:351 is there.

Graham

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

> The highest scoring rank 2 fans are
> Diaschismic, Cassandra, Hemififths, and Rodan whether or
> not the 352:351 is there.

Finding commas which TE tuning shrinks to less than quarter-size leads to 352/351 and 896/891, either of which gives pele, but more accurate commas which shrink even more in percentage terms are 1001/1000 and 4000/3993, either of which gives an unnamed 29&58&8d temperament. The top rated rank two temperament on the page is the unnamed 58&8d, which throws 126/125 into the mix. That has a low enough badness figure it should have been noted and named. Anyone have any thoughts?

Gene wrote:

> The top rated rank two temperament on the page is the unnamed 58&8d,
> which throws 126/125 into the mix. That has a low enough badness
> figure it should have been noted and named. Anyone have any thoughts?

Nonbohpier. :-D

Petr

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > The top rated rank two temperament on the page is the unnamed 58&8d,
> > which throws 126/125 into the mix. That has a low enough badness
> > figure it should have been noted and named. Anyone have any thoughts?
>
> Nonbohpier. :-D

Where did that come from?

Gene wrote:

> > > it should have been noted and named. Anyone have any thoughts?
> >
> > Nonbohpier. :-D
>
> Where did that come from?

The first thing which almost struck me was the 7/5 mapped to 4 generators. Then, if 9 generators plus half an octave make a 3/1, then it clearly tells me it must be something similar to the BP neutral second as 9 steps in the BP chromatic are apped to 15/7 (which is ~17.5 cents wider). And if 11 of them are mapped to 5/2 here and 63/25 in BP, then I just can't help comparing the similar step sizes.
Also, if 8 generators minus half an octave make 11/8, this clearly shows that these 8 generators are approximately a quartertone narrower than an octave; again, similar to BP.
Finally, if you only need to add two more generators to that to map 13/8, then it's reminiscent of BP once again since two steps of the chromatic BP are only a few cents away from 13/11.

Petr

"genewardsmith" <genewardsmith@...> wrote:
> I was cooking up something for pele temperament, which
> tempers out 196/195, 352/351 and 364/363, but it ended up
> with the 352/351 comma not playing any role. This left me
> with the rank four temperament which x31eq calls 9 & 41 &
> 58 & 46. I'm not sure I like the name, but I'm also not
> sure I like the idea of starting in on names for rank
> four temperaments. Should I just decide this is a special
> case, or what?

I'm wondering why it should be the 352/351 that's
disposable. It's a superparticular, and between the other
two in complexity, so the theory doesn't point to it as
being the worst of the three.

I'll also come forward with the shocking revelation that I
already have rank 4 names in the database. Like this:

http://x31eq.com/cgi-bin/rt.cgi?ets=9+31+41+58&limit=7

Graham

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

> I'm wondering why it should be the 352/351 that's
> disposable.

It wasn't required for any of the essentially tempered chords, and doesn't seem to lend itself to producing them. So for what I was doing with my Tablets article on the Xenwiki, it seemed to be sort of a fifth wheel. On the other hand, adding it in to produce the pele I started with doesn't produce much tuning damage, and drops the rank by one, giving the more pleasant 2.3.5 for a transversal subgroup, rather than 2.3.5.13/11, so I think I'll use it anyway.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> It wasn't required for any of the essentially tempered chords, and doesn't seem to lend itself to producing them.

Actually, it does give essentially tempered chords if I remove a restriction on how I generate candidate chords. Perhaps I should convert the whole project into one for mystery.