Re: For Dan Stearns -- neo-Gothic

Hello, there, Dan Stearns, and thank you immensely for what I might
call, borrowing your own word, a "heartwarming" post making me feel a
responsibility to make some actual music in neo-Gothic tunings and
temperaments available.

While improvising in one of these tunings, of which I hope to post
more in the next few days, I was thinking of you; I wish I had some
kind of recording to share. Given the distances involved, inviting you
and others over to hear a bit of improvisation might not be so
practical, although this setting might best reflect where I am at the
moment.

It's curious: over the last 30 years or so, I've developed a fairly
standard stock of improvisational figures or "riffs" in a more or less
14th-century European style, maybe better heard than described in
words. Trying them out, along with some medieval compositions, is one
way I "get acquainted" with a new tuning.

Sometimes I may find it easy to write about theory or mathematics,
harder to communicate _enthusiasm_ for the music itself which provides
an occasion for the diversions of theory and mathematics. You remarks
lend me encouragement in focusing on the musical experience itself.

Before replying to an interesting question you raise about 17-tone
equal temperament (17-tet) and Pythagorean, please let also express my
warm regards to Paul Erlich and Graham, who have shared their own
musical experiences and viewpoints here. Thus Paul's enthusiastic
advocacy of 22-tet, and Graham's of schismic temperaments, have set an
example for me, and their sometimes divergent views serve as a very
creative irritant to spur me on.

> One thing that might be interesting to note here is that if 17 is
> set so that the fifth is just -- i.e., ~17.095-tET -- the max error
> at the octave is only ~6.7�, and if you were to call this margin of
> error something close to naturally occurring tuning errors, it
> should follow that you could also conceivably see 17 as consistent
> all the way through the 15-odd limit (though according to the
> integer consistency measure as I understand it, an actual
> ~17.095-tET would be consistent through seven).

Here your "17.095-tet" is a neat description of Pythagorean tuning
with pure fifths, and 17-note tunings (Gb-A#) are described and
advocated by the early 15th-century theorists Prosdocimus of
Beldemandis (1413) and Ugolino of Orvieto (c. 1430-1440). I use this
tuning, or typically a 15 or 16 note subset, for some keyboard and
other compositions of the era, and such tunings occur in various other
world musics, including the Arabic and Persian traditions.

However, it is Pythagorean 53 rather than 17 which permits a
"virtually closed" system, with a differenece of only ~3.6 cents
between 53 pure fifths and 18 pure octaves.

Indeed 17 (16 pure fifths) is what Ervin Wilson calls a "Moment of
Symmetry" (MOS), but adding a 17th fifth breaks the symmetry by giving
rise to a new interval of around 66.76 cents, a Pythagorean comma
smaller than the usual diatonic semitone or limma generated from five
fifths down or fourths up, which I affectionately call the "sublimma."

This interval of about 1/3-tone occurs in a Xeno-Gothic tuning
(Pythagorean 24) at a number of places, along with "near-7-limit"
intervals whose resolutions to stable 3-limit concords feature this
"sublimma-nal" melodic semitone.

In Pythagorean 17, we have a few near-7-limit intervals involving
notes near the ends of the tuning chain: 14 fifths or fourths produce
a ~8:7 or ~7:4; 15 fifths or fourths, a ~12:7 or ~7:6; and 16 fifths
or fourths, as between Gb-A#, a ~9:7 or ~14:9. The reason in
Xeno-Gothic for going to 24 notes (the maximum number on two 12-note
keyboards with usual octaves) is to make these intervals available in
more places, and also to make available some sublimmas for their
optimally efficient resolution.

Having made these somewhat mathematical points, I would add that
hearing some musical examples would explain what I am talking about
much more effectively than words and numbers alone. If only you were
here, I would play some of these resolutions first and _then_ try to
"explain" them.

While I'm not sure how one goes about systematically testing or
confirming the "consistency" of a scale, my first blush guess might be
to agree with you that Pythagorean 24 could be consistent to the
7-limit at least. With only 17 notes, the main limitation might be
that only a few near-7-limit intervals are available, and no
sublimma-nal semitones for the ideally efficient resolutions of these
intervals.

Your mention of Pythagorean 17 or "17.095-tet" and 17-tet leads nicely
into the question of how these tunings might be placed in a larger
context. In another article I hope to post very soon, I would like to
address this issue, proposing a "neo-Gothic" or "reverse meantone"
spectrum in some ways a mirror image of the more familiar historical
European spectrum from Pythagorean to around 1/3-comma meantone. The
neo-Gothic spectrum may be said to range from Pythagorean to about
"-1/3-comma meantone" (22-tet) with fifths increasingly tempered in
the _wide_ direction.

For now, however, I want mainly to offer my thanks for your post,
which moves me to new efforts in practice as well as theory.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

> While I'm not sure how one goes about systematically testing or
> confirming the "consistency" of a scale, my first blush guess might
be
> to agree with you that Pythagorean 24 could be consistent to the
> 7-limit at least.

Consistency is only defined (so far) for equal temperaments, and I'm
sure Dan knows this, so in "agreeing" with him you must be
misunderstanding something).

> Your mention of Pythagorean 17 or "17.095-tet"

By 17.095-tET, Dan Stearns did not mean Pythagorean tuning. In what
sense would you equate the two?