105/104 temperaments?

Hi all,

I've been playing around in Graham's temperament finder. Searching the
2.3.7.13/5 (actually 2.3.7.78/5*) subgroup for rank 3 temperaments
yields the following two at the top of the badness list:

http://x31eq.com/cgi-bin/rt.cgi?ets=31_5_19&limit=2_3_7_78%2F5
http://x31eq.com/cgi-bin/rt.cgi?ets=22_5_19&limit=2_3_7_78%2F5

The second is biome, which equates 9/7 with 13/10. The first is an
unnamed temperament that instead goes the other direction and equates
13/10 with 21/16 by eliminating 105/104 - certainly a very interesting
approach! I'm not sure exactly how it might be used, although it makes
possible chords like 7/4-8/7-13/10-7/6 that span a 3/1 on the outside.
You can play around with subminor 11 chords and then "resolve" them
into 10:12:13:15:18 chords. Not as conceptually simple as 1/(6:7:9) ->
10:13:15, but interesting nonetheless.

Does anyone know of any good linear temperaments that eliminate
105/104? Mixing this with 91/90 gives you a negri-ish temperament,
which isn't as accurate as either since the two commas drag the size
of the 7 in opposite directions. I guess 36-equal and 60-equal are
about optimal for this?

-Mike

*This subgroup is equivalent to 2.3.7.13/5, but I'm using it to weight
Graham's temperament finder properly.

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

> Does anyone know of any good linear temperaments that eliminate
> 105/104?

One is the 13-limit meantone extension which Graham does, and Herman does not, think should be the main extension. Why not help us all out by suggesting a better name than "Meaningless"?

On 4/30/2011 11:10 AM, genewardsmith wrote:
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia<battaglia01@...>
> wrote:
>
>> Does anyone know of any good linear temperaments that eliminate
>> 105/104?
>
> One is the 13-limit meantone extension which Graham does, and Herman
> does not, think should be the main extension. Why not help us all out
> by suggesting a better name than "Meaningless"?

Meaningless is 12&31 [<1 2 4 7 11 -3|, <0 -1 -4 -10 -18 16|], which I was temporarily calling "meantone B" for lack of a better name.

http://x31eq.com/cgi-bin/rt.cgi?limit=13&ets=12p+31

The 105/104 meantone is the one I'd labeled as "meantone A", which Graham suggested as "meantone".

12f&31 [<1 2 4 7 11 10|, <0 -1 -4 -10 -18 -15|]

http://x31eq.com/cgi-bin/rt.cgi?limit=13&ets=12f+31

There's also the mohajira that came up in the same thread:

7&24 [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|]

A few other 105/104 temperaments:

9&31 winston (orwell) [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]
19&41 magic [<1 0 2 -1 6 -2|, <0 5 1 12 -8 18|]
7&34 monkey [<1 1 1 5 2 4|, <0 4 9 -15 10 -2|]
15&26 superkleismic [<1 4 5 2 4 8|, <0 -9 -10 3 -2 -16|]
10&31 miraculous (miracle) [<1 1 3 3 2 4|, <0 6 -7 -2 15 -3|]