Re: Comments on Xeno-Gothic

Hello, there.

Recently there have been some interesting replies to my post about
Xeno-Gothic tuning and music. Here I'd just like to address a couple
of points.

First of all, Daniel Wolf correctly noted that Xeno-Gothic is based on
the tradition of European composition -- more specifically, I might
add, medieval composition, with an emphasis on the 13th-early 15th
centuries. He also made the vital point that in various musical
traditions -- gamelan, for example -- it is musical conventions, not
only scales and tunings themselves, which determine musical usage.

This last point bears emphasis. In a medieval context, I tend to hear
a sonority with an outer major sixth divided into lower major third
and upper fourth -- a "6\3 sonority" in one common parlance[1], and
M6|M3-4 in my medieval-inspired notation -- as suggesting an expansion
of the sixth to the octave and the third to the fifth. Thus:

f#'-g'
c#'-d'
a -g

This is a property of the musical language, not of any specific
tuning, and in fact this "syntax" can hold with various possibilities:
the M3 and M6 might be the usual Pythagorean 81:64 and 27:16 (~408
cents and ~906 cents); or schisma intervals at a more "softened"
8192:6561 and 32768:19683 (~384 cents and ~882 cents, very close to
5:4 and 5:3); or even "accentuated" septimal schisma intervals (~431
cents and ~930 cents -- a Pythagorean comma _wider_ than the usual).

At least for me, this can be demonstrated by some experiments. One of
the ideas of a Xeno-Gothic tuning (one form of Pythagorean 24) is to
have these three kinds of options available -- and I find that they
are all "allophones" of the same basic progression, to borrow a
linguistic concept.

Similarly, using a 17-note Pythagorean tuning of the kind proposed by
Ugolino of Orvieto in the early 15th century, or typically a 15-note
or 16-note subset, I can nicely hear in practice that a-c#'-f#' and
a-db'-gb' (the latter with schisma intervals close to 5:4 and 5:3)
both expand nicely to g-d'-g' -- but with a difference in color.

Maybe even more dramatically, I can sit down at a keyboard tuned in
1/4-comma meantone -- a tuning which compromises fifths by 5.38 cents
while making major thirds a pure 5:4 -- and find that when I play
in medieval style, a major third still wants to expand to a stable
fifth, etc.

This is not to say that tunings are irrelevant to musical syntax: the
close association between a shift in the Western European musical
language during the first half of the 15th century toward a pervasive
tertian texture, and the shift in tuning toward Pythagorean schemes
with prominent schisma thirds, and then meantone starting by around
1450, is a persuasive example.

However, it's easy to come up with examples of how styles can shift in
ways that seem somewhat less obviously linked to tuning. For example,
why the sonority of major third, fifth, and major sixth (the "added
sixth chord" or "6\5 sonority") should be unstable in 17th-18th
century music based on meantone or unequal well-temperaments, but
stable in 20th-century music often oriented toward 12-tone equal
temperament (12-tet), may involve stylistic considerations not so
obviously related to the tunings themselves.

One pattern, at least in European composed music, is that sonorities
deemed as unstable in one era may be taken as richly stable in the
next, and then as "incomplete" or "too simple fully to satisfy the
ear." Of course, this tendency may in turn have an effect on tunings,
as composers and performers seek to "optimize" emerging or even
long-recognized current "concords."

Also, Eduardo Sabat-Garibaldi asked about the similarity between the
24-note Pythagorean scale of Xeno-Gothic and the Indian system of 22
srutis. In fact, although the Indian system now uses some 5-based
ratios and derivatives (e.g. 135:128), it has been appealingly argued
that the system may have originally been purely Pythagorean, with
schisma intervals later redefined in simpler 5-limit terms.

Arabic and Persian systems of Pythagorean tuning using, for example,
17 notes, also invite citation in comparisons of this kind.

As it happens, my inspiration was from medieval European music
practice and theory. On the one hand, the 17-note tunings of
Prosdocimus and Ugolino in the early 15th century provided a model
which I had occasion to research in the course of writing a
Pythagorean FAQ now available at

http://www.medieval.org/emfaq/harmony/pyth.html

Around the same time, two threads of tuning theory came together. The
advocacy by Marchettus of Padua (1318) of major thirds and sixths
considerably _wider_ than Pythagorean before fifths and octaves had
caught my imagination, and John Chalmers, Jr., pointed out to me that
the Pythagorean series of fifth can approximate not only 5-limit but
7-limit intervals. I soon put these trains of thought together, and
realized that another way of viewing the near-7-limit intervals -- for
example, M3 and M6 a comma wider than the usual Pythagorean forms --
was as "Marchettan" cadential intervals with extra harmonic tension
and supernarrow leading tones.

Curiously, after I had formulated this basic idea for a 24-note
tuning, a scholar called to my attention that the scheme was in fact
very similar to the Indian srutis (especially if one posits an
original Pythagorean version with all intervals 3-based), the Persian
17-note scheme, and so on.

Of course, the content of a music depends on more than the general
scheme of tuning -- Arabic, Persian, Indian, and medieval European
music all have their own styles and ways of using Pythagorean or
quasi-Pythagorean intervals.

Having attempted a reply to some of the helpful comments and
questions, I warmly thank people such as Daniel Wolf and Eduardo
Sabat-Garibaldi for their interest and hope that this post may explain
some of the motivations for my "Xeno-Gothic" scheme, as well as
illustrate the way in which tuning concepts can have striking
cross-cultural analogues.

----
Note
----

1. One complication of communication in ASCII, or even in a typeset
medium, is that the symbol "6/3" can mean either "a continuo symbol
for a six-three chord, with a third and sixth above the lowest note"
or "a tuning ratio of 6:3." One possible solution in ASCII is to use a
backslash for the first meaning, as I have done here.

Most respectfully,

Margo Schulter
mschulter@value.net