Tempering the 5-limit JI major scale

The 5-limit JI major scale would be great, except it has that 40/27
wolf fifth in it. One option is to temper out 81/80, turning the wolf
into 3/2, so that you end up with the meantone major scale; the
Pythagorean minor third turns into 6/5. Another option is to temper
out 135/128, turning it into 25/16; 32/27 turns into 5/4. A third
option is to temper out 25/24, turning the wolf into a more complex
but perhaps more tolerable 64/45; 32/27 stays 32/27.

A different option is to temper out 64/63, which turns the 32/27 minor
third into 7/6. This is the Archytas planar temperament. If you want,
you can then temper the 27/20 to be equal to 11/8, which eliminates
55/54 - this has the side effect of tempering 6/5 equal to 11/9, which
can profoundly alter the gestalt of the scale (if you're used to
11-limit harmony). This temperament seems to have no name and is here:

http://x31eq.com/cgi-bin/rt.cgi?ets=7p_15_5p&limit=11

A slightly more accurate route would be to temper it to be equal to
15/11, which eliminates 100/99. This one does seem to have a name, and
that name is "Ares" temperament:

http://x31eq.com/cgi-bin/rt.cgi?ets=22_12_15&limit=11

These are just some quick ideas, you can likely get far more creative
with this. Does anyone else have any clever ideas for how to temper
the 5-limit JI scale while maintaining its planar structure? e.g.
tempering out 81/80 or what not is cheating.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> The 5-limit JI major scale would be great, except it has that 40/27
> wolf fifth in it. One option is to temper out 81/80, turning the wolf
> into 3/2, so that you end up with the meantone major scale; the
> Pythagorean minor third turns into 6/5. Another option is to temper
> out 135/128, turning it into 25/16; 32/27 turns into 5/4. A third
> option is to temper out 25/24, turning the wolf into a more complex
> but perhaps more tolerable 64/45; 32/27 stays 32/27.
>
> A different option is to temper out 64/63, which turns the 32/27 minor
> third into 7/6. This is the Archytas planar temperament.

This is also a "consonant class scale" ( http://xenharmonic.wikispaces.com/Consonant+class+scale ) where the thirds are all 7-limit consonances (7/6,6/5,5/4). It's quite nice.

> If you want,
> you can then temper the 27/20 to be equal to 11/8, which eliminates
> 55/54 - this has the side effect of tempering 6/5 equal to 11/9, which
> can profoundly alter the gestalt of the scale (if you're used to
> 11-limit harmony). This temperament seems to have no name and is here:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=7p_15_5p&limit=11

Note that the error shown for this temperament is 8.822, while the error for porcupine is 8.823.

This is one of those cases Gene's often talking about where a temperament shrinks some other comma down so small that it's crazy not to temper it out too. In optimal 7p&15&5p, the porcupine comma shrinks to less than 1 cent, so might as well make it porcupine.

It might be useful as a way to think about or organize porcupine scales, but that's about it.

Keenan

On Sun, Oct 30, 2011 at 6:46 PM, Keenan Pepper <keenanpepper@...> wrote:
>
> Note that the error shown for this temperament is 8.822, while the error for porcupine is 8.823.
>
> This is one of those cases Gene's often talking about where a temperament shrinks some other comma down so small that it's crazy not to temper it out too. In optimal 7p&15&5p, the porcupine comma shrinks to less than 1 cent, so might as well make it porcupine.
//snip
> It might be useful as a way to think about or organize porcupine scales, but that's about it.

Very neat observation. I suggest we use this facet to name it. I think
that it's very useful in organizing porcupine scales, but also a bit
more than that, because Blacksmith, Pajarous, and Suprapyth both also
support this temperament, and each one has its own way of forming
MODMOS's to bridge between the temperament's MOS's and the "MOS's" of
this temperament (I assume this means that it forms some killer
decatonic scales, in addition to the 7-note "JI" major scale). These
temperaments contain this one in the same way that 22-equal contains
suprapyth and pajarous.

For example, the 5-limit JI major scale, tempered in the above
temperament, is a MODMOS of porcupine[7], forming the * . * * . * * .
* . . * set on the lattice - it's Lssssss b4 #7. This is similar to
how the diatonic scale is a MODMOS of 12EDO[7], which forms the
Halberstadt-shaped * . * . * * . * . * . * set on that lattice.

I've been exploiting the 64/63-tempering of this scale for several
months now, since the V7 chord of this scale is 4:5:6:7. I haven't
played with 11/8 over the 5/3 yet but I'll be sure to do that soon.

-Mike