Linear Temperaments cont.

< A slightly lower unweighted sum of sq'ed errors, duh, by definition.

I think the "duh" is unnecessary. The mathematics are obvious; my
point was about the appropriateness of the weightings which is
a psychoacoustic question whose answer I don't really know.

You mentioned (ironically) "sharp or flat unisons." I was pointing out
that octaves, fifths, and unisons modified by a 1/4 tone were used
by Wyschnegradsky for expressive purposes, so the statement is not as
humorously vacuous as it might appear at first sight.

I too agree that the 7/4 is not always the best tuning for the
7th of a dom7th chord. Bosanquet thought that it was disturbing
in melodic passages, though beautiful in full chords (in meantone
systems).

I'm using Bosanquet's nomenclature in which a positive tuning is
one in which the fifth is sharper than 700 cents (some contemporary
writers use the term for fifths sharper than 3/2). The degree to which
a tuning is positive or negative is the number of steps 12 of its fifths
exceeds or falls short of 7 octaves. For untempered systems, this
definition may be modified by approximating their fifths to those of
a tempered system. Thus 1/4-meantone may be treated as a -1 system
by approximating it to 31-tet (others are 7,19,43,55,67). Doubly
negative systems are 14, 26,38,50,62, 74... Primary positive systems
are 5,17,29,41,53... and doubly positive systems are 10,22,34,46....
Triply negative systems are 9, 21, 33, 45, 57,69,81 .... and triply
positive, 3, 15, 27, 39...

As the chain length for major thirds and harmonic sevenths is so
long in positive systems, more than 12 tones is mandatory for harmonic
music.

I wouldn't put too much faith in the precision of the tunings I posted.
I calculated them in double precision, but I'm not sure the accuracy
is anywhere near the number of decimal places shown. I simply didn't
have the time to round them down to 8 places as I did for the
negative ones prior to posting them.

I am aware that errors may cancel when the ABS(Sum) is taken rather
than the Sum(ABS) and I thought the resulting tunings might be
interesting for exactly this reason. Thus in the 1/3-comma tuning the
6/5 is just, in the 1/5 comma, the 15/8 is.

I stress these tunings, for the most part, were generated as a
theoretical study many years ago. I am not proposing them seriously
today as I think the corresponding equal temperaments would be
perceptually equivalent and easier to implement and use, though
Eduardo Sabat-Garibaldi has successfully embodied the 1/9th skhisma
tuning on a specially fretted guitar.

The negative tunings might have some value in realizing early music, but
I'm not sure one could really tell the difference from 1/4-comma meantone
or 31-tet within a 12-tone gamut. For a larger series of notes, perhaps..

Bosanquet himself decided that tuning his organ to the 1/7 skhisma
system was not worth the extra trouble and henceforth used 3/2's.

XH17 will appear when I get all the promised articles.

--John


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On Thu, 13 Mar 1997, Aline Surman wrote:
> So, here's a question I've always wanted to ask some physicist sort, and
> since a lot of the great tuning theorists are exactly that, now's the
> time to ask. Why is it that the Moon always keeps one face turned to us?
> I've long thought that seemed sort of odd. What sort of math is at work,
> on a very basic level, to make that happen, and why? Is that common with
> Moons in general, or is Earth an exception? It seems to me that it makes
> more sense if the two spheres were off a bit, so to speak, and that the
> Moon would gradually show all of it's surface. Any ideas?

This is a fairly basic phenomenon called "Tidal locking". Ask an
astrophysicist if you want a really good explanation, but basically:
the near side of the moon is closer to the earth than the far side, so
the earth's gravity pulls harder on it. This exerts a radial stretching
force on the moon. As the moon turns (turned, early in its history),
the tidal force stretches it in different directions. This constant
distortion creates friction which dissipates the rotational energy of
the moon. Eventually it's all gone, so the same face of the moon ends
up oriented toward Earth all the time.

Before you get too caught up exploring the mystical significance of
ultra-high harmonics of the period of the moon, though, I'd point out
Carl Sagan's refutation of astrology: the gravitational force of the
obstetrician is greater than that of a moon or planet, at the moment of
your birth. 8-)>

--pH http://library.wustl.edu/~manynote
O
/\ "'Jever take'n try to give an ironclad leave to
-\-\-- o yourself from a three-rail billiard shot?"


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