Re: [tuning-math] A 5-limit, "geometric" temperament list

  • From: Herman Miller <hmiller@xxxxxx>
  • To: tuning-math@xxxxxxxxxxxxx
  • Date: Mon, 22 Jul 2002 23:51:59 -0400

On Sun, 21 Jul 2002 21:37:27 -0700, Gene W Smith <genewardsmith@xxxxxxxx>
wrote:

>The following is a complete list of all 5-limit temperaments satisfying
>the requirements that rms error be less than 15, geometric complexity
>less than 40, and the badness calculated from these less than 3000. If
>people feel something valuable has been left off (e.g, 135/128, 25/24, =
or
>16875/16384) we could raise the error limit.

I think that 135/128 (aka "pelogic") is one of the more interesting /
useful temperaments despite its errors. Once you get beyond a certain =
level
of complexity in the unison vector, you might as well use JI. It's hard =
to
imagine a piece of music that wanders around the lattice enough to be =
able
to take advantage of the (3)^35/(2)^16/(5)^17, repeated enough times to
magnify the 1 cent difference into something that would have been easily
audible. Pelogic on the other hand has a nice 9-note subset with =
musically
interesting melodic and harmonic properties.

Any reason why 531441/524288 doesn't show up? (Is it really that bad? :-)=
 )

It's also a bit surprising that Ampersand doesn't show up on this list, =
but
it's really better as a 7- or 11-limit (Miracle) temperament anyway. =
Still,
as long as way-out temperaments like parakleismic are on the list, it's =
odd
that it doesn't show up.

My short list of "best" / most useful 5-limit temperaments would go
something like this:

meantone 81/80
augmented 128/125
diminished 648/625
Blackwood decatonic 256/243
porcupine 250/243
diaschismic 2048/2025
pelogic 135/128
MAGIC 3125/3072
schismic 32805/32768
kleismic 15625/15552
orwell 2109375/2097152

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