In-Reply-To: <abmetp+mul3@eGroups.com>
jonszanto wrote:

> So great and so important you couldn't take the time to include a link?
> Sure, we can find it from our collection of links, or do a search, etc.
> - I thought you were the consummate cross-referenced poster.

Everybody does or doesn't include links according to their mood. This one
is <http://www.microtonal.co.uk/catalog.htm>.

I personally would like somebody to include the definitions of the
temperaments I'm supposed to be adding. It's been quite difficult to keep
up over the past year.

> > (by contrast, check out joe
> > monzo's definition of linear temperament if you want to turn red).

After following the links, I thought the "temperament" definition was
lacking, and decided to clarify it. Then scrolled down, and saw I already
had. Although I now disagree with my own definition -- temperaments can
approximate "ideal" tunings other than just intonation.

It looks like the problem with the "linear temperament" definition is that
there isn't one! You simply say it's "another term for unequal
temperaments" which isn't the case. Not all unequal temperaments are
linear.

A linear temperament is fully described by a generator and equivalence
interval. The mapping from generators to consonances is fixed. For
example, meantones are defined by the generator approximating 3:2 and four
generators approximating 5:4. The generator can have different sizes,
giving 1/3-, 1/4-, 1/6- comma meantone, but they're all the same
temperament class. 53-equal, however, isn't a meantone because the best
5:4 is 8 fourths rather than 4 fifths.

Planar temperaments are like linear temperaments, but with two generators
and an equivalence interval. The difference between a generator and an
equivalence interval is only a matter of formalism.


Graham