22 etc.

re PAULE's comments on 22 and my original:

> >whether you mean just (as in 81:80) or not. The context usually
> >determines; in any ET, it's obvious.

> NO, IT'S NOT OBVIOUS AT ALL!

All I said and meant was that there is no just 81:80 in any ET. Terribly
obvious. But your elaboration is more interesting:

In any ET, we would agree that the perfect
> fifth is the closest approximation to the 3:2.

We might not, but let's for the moment.

We would also agree that the
> syntonic comma is NOT necessarily the closest approximation to the 81:80 --
> for example, in 22-tET the closest step size to an 81:80 is 0 steps, and yet
> we would both call 1 step the syntonic comma. Why? Because the syntonic
> comma is defined as interval obtained by tuning several successive CONSONANT
> intervals.

I agree that we may define crucial consonances as closest to just in an ET
(usually representing the pure harmonics, with a power of 2 in the
denominator, in case anyone else isn't sure what we are talking about).
After that, I would start (not necessarily conclude) by defining such
things as commas in structural terms. They are left over as differences
between multiples of harmonics. The syntonic comma has a standard
definition; often it comes out positive or zero, sometimes it is negative.
To say that it may not be the closest to 21.5 cents is certainly true but
is not affected by the structural definition. If you don't keep the same
definition across tunings, you can't compare what is being defined, as
your subsequent demonstration shows.

[two usually equivalent definitions omitted, one for m3s, one for M3s]

> So the syntonic comma is 1 step in 22-tET, even though 0 steps would be
> closer in size to the JI syntonic comma. This situation would never occur
> for a basic consonace like the perfect fifth.

Perhaps not relevant, but it would. There are plenty of tunings where
there may be a choice of P5, and we may prefer the one which is farther
from just for one of several reasons.

> Thus the "pseudo" terminology
> is warranted.

If you don't try to make dissonances define themselves like consonances,
it's not necessary.

> Now there are tunings where (1) and (2) give different answers! For example,
> in 21-tET, definition (1) gives a syntonic comma of 0 steps, while
> definition (2) gives an syntonic comma of -1 (that's minus 1) steps. Another
> example is 20-tET, where definition (1) gives a syntonic comma of 1 step,
> while definition (2) gives a syntonic comma of 2 steps. So I would argue
> that if an ET is not consistent within the 5-limit, the syntonic comma is
> undefined in that tuning.

This I find most interesting, and I must get back to your notion of
consistency when I have time to return to the theory. Of course, if a m3
is defined in terms of a M3 and P5, this discrepancy will never come up.
Still, the m3 is valuable enough that the problem you mention is worth
consideration. I tried to do so, in fact, in my 25-tET piece, because it
has two viable m3s, at 288 cents and 336 cents.

Obviously [!], if a syntonic comma is defined only one way (the difference
between four P5s and a M3), there will be only one result.

Paul R




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Denis.Atadan@mvs.udel.edu writes:

> This post doesn't have as much to do with tuning as it does with
> sound. I'm hoping to build a pair of matched leslie-type speakers
> but currently I don't know much about the acoustics of cabinet
> building. Does anyone know of a good book that would act as a
> practical reference source for this sort of thing.

It won't help with the rotating component aspects of a leslie cab,
and, in fact, I don't even know where to look for something like that
off hand, but probably the best book around for the other aspects of
speaker cabinet design is "The Loudspeaker Design Cookbook" by Vance
Dickason. I believe it is published by the folks who do "Audio
Amateur" and "Speaker Builder" magazines, but you can get it from
nearly anyplace that caters to amateur speaker builders, like
Madisound or Parts Express.

I've only designed one cabinet (yet to be assembled, so I don't know
how well I did), so there isn't much I could tell you that Dickason
wouldn't do a much better job at.

Hope this helps,

Steve


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