Re: Apology on ~17.095-tet confusion (Dan Stearns, Paul Erlich)

Hello, there, Dan Stearns and Paul Erlich, and my congenial apologies
to you both for my rather comical misunderstanding of ~17.095-tet.

For whatever reason, when I saw that number, I quickly confirmed that
something very close to 17.095 pure Pythagorean _fifths_ at 3:2 would
be exactly equivalent to 10 pure octaves. Of course, this interesting
statistic quite leaves aside the fact that Pythagorean is 3-limit just
intonation (JI), not an equal temperament of any variety. If I
translate "17.095-tet" to a Gary Morrison style "~70.196-cet," then
the obvious difference becomes more clear: a Pythagorean diatonic
semitone is around 90 cents, not 70 cents!

Interestingly, ten steps make a pure 3:2 fifth -- sort of like two
mean-tones make a 5:4 major third.

To both of you, I would say that I now realize that my interpretation
of "consistency" is obviously something very different than the
concept relating to equal temperaments. All I mean -- and I should use
a differen term -- is that in Pythagorean 24, for example, a narrow
minor third (~7:6) plus a fifth (3:2) makes a narrow minor seventh
(~7:4); and likewise that a wide major third (~9:7) plus a fourth
(4:3) makes a wide major sixth (~12:7), etc. These intervals "add up"
as expected.

Curiously, when I talk about what I should maybe call something like
"7-limit congruency" (if that term isn't also already in use), I
typically mean a 9-odd-limit tetrad like 12:14:18:21 or 14:18:21:24,
both with regular resolutions featuring expansion of the 9:7 to a
stable 3:2. Of course, three-note subsets like 12:18:21 or 14:21:24
would remain with the 7-odd-limit. The resolutions, a feature of
Xeno-Gothic (Pythagorean 24) are based on a 3-stability-limit.

Dan, quite apart from my confusion on 17.095-tet, I was really excited
to see how your idea for this temperament led to all kinds of lattices
and musical concepts -- it looks the kind of thing that could lead to
some very striking music. It's curious how these dialogues can unleash
creativity in various unexpected directions.

While I may not understand all of your fine points about this scale,
please let me urge you to make some beautiful music based on it.

Paul, just as you called to my attention that Partch's
otonality/utonality was meant to apply only to the 5-limit or higher,
so you've helpfully filled me in on my misunderstanding of
"consistency." I want to use these terms correctly -- or use others
which may better fit my intended concepts.

Dan, here's one of those "unusual" neo-Gothic progressions in 17-tet
which might be of interest, an augmented sixth expanding to an octave:

C#4 - D4
G3 - A3
Eb3 - D3

Here the lower major third to fifth is standard, but the augmented
sixth is "something different" (~24:13), and the upper augmented
fourth very close to 13:9. At this point, I'm only reporting a
mathematical proximity; I'm not sure about musical implications of
"13-ness," although as someone used to large Pythagorean integer
ratios, I have a taste for higher primes as well, I guess <grin>.

Something like

C#4 D4
G3 A3
D3

with the two upper voices resolving by oblique motion over a
stationary lowest voice would be quite like Perotin (e.g. opening of
_Viderunt omnes_, maybe around 1198) -- a transposition of the more
likely

E4 F4
Bb3 C4
F3

That augmented sixth in the first example has an effect similar and
quite different -- one of the attractions of 17-tet.

Most respectfully,

Margo Schulter
mschulter@value.net

Margo and others,

I haven't been following this thread too closely, but upon
reading a bit of TD 699.10, I thought that perhaps I'd point
out, if it hasn't been already, that 53-EDO or -tET is the
smallest EDO that accurately approximates both 3-limit
Pythagorean and 5-limit JI tuning.

This is because 2^(1/53), the 'step-size' of 53-EDO, is
almost exactly midway between the two relevant commas,
being ~1.1 cent smaller than the Pythagorean and ~0.8 cent
larger than the syntonic:

~21.5 cents - syntonic comma
~22.6415 cents - 2^(1/53)
~23.46 cents - Pythagorean comma

Therefore, any combinations of tones in 53-EDO will yield
pitches or intervals that are reasonably close to any of the
rational ones within a 5-limit system of 53 tones or less.

An important point to note is that 53-EDO distinguishes between
the two different sizes of semitone in both the Pythagorean
and 5-limit JI tunings; it would thus be inappropriate
(or at least, radical) for performance of works written with
a 12-note meantone tuning in mind. An extended meantone is
perhaps another matter...

If one's interest is to find EDOs which provide good approximations
to 'Xeno-Gothic' Pythagorean tunings, 53-EDO is the prime candidate.

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
---------------------------------------------------

________________________________________________________________
YOU'RE PAYING TOO MUCH FOR THE INTERNET!
Juno now offers FREE Internet Access!
Try it today - there's no risk! For your FREE software, visit:
http://dl.www.juno.com/get/tagj.