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Defining "microtemperament" (and various other bits and peices)

🔗David C Keenan <>

2/15/2000 1:56:42 AM

Dear Friends,

I need to leave the list again for a while. Cold turkey. But I want to tidy
up a few loose ends first. I'll lurk for a couple more days but will try to
resist posting.

Monz, brilliant work setting up all those I-IV-V7-I tunings. Sorry I won't
be listening to them myself. (I'd have to switch the modem to my Mac and

JdL, keep up the great work on the adaptive tuner.

Paul E., I know that people pay you money to point out their errors. Thanks
for doing it for us for free!

Jerry E., just because your theory is beautiful doesn't mean it necessarily
bears any resemblance to reality. This may be so in Physics, but not in
Psychoacoustics. I've had to adjust mine many times since joining this
list. Please take advantage of the generous people on this list to help you
perform experiments to test your theories with accurate measurements, or
accurate synthesis and good ears.

Graham Breed, I'm glad to be able to leave the defense and development of
the "Fokker-plus" interval names in your capable hands. When you get to
write it up including the wide/narrow extension, please email it to me, or
its URL. Could you post your schismic? letter # b / \ notation for 22-tET
alongside your Fokker-plus interval names.

Anyone else waiting for a response from me on something?

Thanks to everyone.

About the definition of "microtemperament"

Note that I intended it to be a (more generally acceptable and
self-explanatory) synonym for "wafso-just". I earlier wrote:

"The point of microtempering an otherwise strictly-just scale is to
increase the number of available consonances while keeping the errors close
to the limit of tuning accuracy for non-electronic instruments and less
than half those of meantone (which is sometimes considered quasi-just)."

Two important points in there.
1. A microtemperament is a microtemperament _of_ something, namely some
strictly-Just scale (or at least some odd-limit lattice).

2. The errors (possibly weighted in some way) are less than some maximum
value. I earlier wrote, "less than half those of meantone". This was for a
minimax errors tempering of an 11-limit scale. I might want to let this
creep up a bit.

Maybe it's too early to lay down a precise upper limit. But I think that no
ET less than 72-tET should be considered a microtemperament of anything
7-limit or higher. I would say that 72-tET is a microtemperament of 7-limit
(2.98 c max error), but not 9-limit (3.91 c max error).

I want my 11-limit tunings with distributed 224:225 and 384:385
(particularly the one that has minimax beats in the 4:5:6:7:9:11, TD
485.12) to be considered a microtemperament.

So if it's to be unweighted, the cutoff's gotta be between 3.3 c and 3.9 c.
But It really needs some weighting that doesn't allow such large errors in
the more complex intervals. Minimax beats in a complete (stack-of-"thirds")
otonality, means the errors are effectively weighted by the highest LCM of
the interval as it occurs in the extended ratio of that chord. e.g. 2:3's
occur as 4:6 and 6:9 so they would be weighted by LCM(6,9) = 18, while 9:11
would be weighted by LCM(9,11) = 99.

Graham, the definition you proposed is a definition of what might better be
called a "planar temperament", by extension from "linear temperament". In
that way of thinking, ETs are "point temperaments", but "closed
temperaments" is probably a better term.

Of course most microtemperaments are planar temperaments. This should
certainly be mentioned in any definition of either term, but I see no need
to identify the two.


-- Dave Keenan