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

  • From: Carl Lumma <carl@xxxxxxxxx>
  • To: tuning-math@xxxxxxxxxxxxx
  • Date: Tue, 23 Jul 2002 00:51:39 -0700

>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.

I agree.  I tried to make this point some time ago -- that while flat
badness may show interesting patterns in series of temperaments, a more
musically useful badness would produces a finite list of temperaments,
favoring simple ones.

One might even say that from a musical standpoint complexity and error
should be separate.  Gene, your last 5-limit list seemed to have to hard
limit complexity and error in addition to the badness limit of 3000, yes?

>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

Is that in order?  It's a lot like my list...

augmented
meantone
diminished
pelogic
porcupine
magic
kleismic
diaschismic
quadrafourths
schismic
orwell
miracle

...which IIRC I made by sorting the temperaments in Graham's catalog
by 5-limit complexity and keeping everything with complexity under
10 (or thereabouts).

-Carl

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