Re: EXPERIMENT: 22-tet for Neo-Gothic

Hello, there, Paul Erlich, and thanks for a very helpful correction:

>> Thus the normal 17-tet intervals have a kinship to the so-called
>> "near-7-limit" intervals of 24-note Xeno-Gothic, but one importance
>> difference for our testing purposes: they are much further from
>> 7-based ratios than the ~3.80 cents of the septimal comma:

> You mean septimal schisma?

Of course, and this shows how reliably I sometimes proofread my own
material. One of the advantages of posting here is getting the benefit
of some assistance in this department. Currently I'm planning to post
an article on notation of 53-note Pythagorean or 53-tet -- including
some corrections to my first version offered by Ken Moore on Usenet's
rec.music.theory.

* * * *

Please note that I wrote some of what follows _before_ tuning up
22-tet on a TX-802 and trying it both in a full 22-note version mapped
to two keyboards 1/11 octave apart (both in a usual medieval Eb-G#
tuning), and in a 12-out-of-22 configuration with both keyboards in
unison (Eb-G#).

Both keyboards in the 12-out-of-22 tuning have the pattern of the
lower keyboard in the full 22-note tuning (C=0/22):

1 3 10 14 16
Upper: 20 2 6 7 11 15 19 20

3 5 12 16 18
Lower: 0 4 8 9 13 17 21 22

Here's the same layout in a conventional Pythagorean note notation,
with an asterisk (*) showing a note raised by a single 22-tet step,
actually not so far from a diesis:

Db C# Gb Ab G#
A# Db* Eb* D# Gb* Ab* Bb* A#
------------------------------------------------------------------------
C# Eb F# G# Bb
C D E F G A B C

Each keyboard has a usual diatonic pattern C-C of 4 4 1 4 4 4 1, a
pattern certainly explifying the high contrast in tone/semitone sizes
when 22-tet is interpreted as a Pythagorean-like "Neo-Gothic" tuning.

In what follows, I'll put curly braces { } around material I've added
after actually playing in 22-tet, maybe an interesting sample of how
views of a tuning before trying it can compare with the actual
experience.

* * * *

[Your table of 7-limit approximations in 22-tet]

> M2 (4 steps) ~218.18 8:7 ~231.17 ~12.99
> m3 (5 steps) ~272.73 7:6 ~266.87 ~5.86
> M3 (8 steps) ~436.36 9:7 ~435.08 ~1.28
> M6 (17 steps) ~927.27 12:7 ~933.13 ~5.86
> m7 (18 steps) ~981.82 7:4 ~968.83 ~12.99

> Related to these better approximations is the fact that 17-tET is
> not even consistent in the 7-limit (or 9-limit), while 22-tET is.

While the "inconsistency" of 17-tet makes me feel all the more
affectionate toward this tuning -- "Equal Rights for Inconsistent
Tunings" -- more dispassionately I might say that each tuning offers
its own advantages for this kind of "neo-medieval" style, with 22-tet
featuring some very interesting ones.

On the question of "n-limit consistency," I looked up and found and
found a definition in your rightfully famous 22-tet article (a kind of
transitive property for best ratio approximations with a given limit).

Here I would say that while the 7-limit accuracy and consistency of
22-tet might give it some unique qualities for "neo-Gothic"
applications, the "7-limit inconsistency" of 17-tet is not a problem
from my point of view, at least until someone demonstrates to me some
musical embarrassment in playing neo-Gothic sonorities and
progressions based on a 3-limit of stability.

In other words, the only "consistency" I ask from a temperament is
that it be more or less consistently pleasing for the music I'm
playing. For neo-Gothic music, 17-tet satisfies this test very nicely
with a bit of Sethareanizing or timbral finesssing. Indeed, I would
say that the way that the unstable intervals of this tuning
approximate a various of higher prime limits (11, 13, 17, even 23 if
we want to go that high) may be one of its charms.

At the same time, 17-tet gives a quite nice approximation of our
primary concords in a Gothic or neo-Gothic setting: the stable 3-limit
intervals. It also nicely approximates relatively blending sonorities
such as 6:8:9 or 4:6:9.

From this point of view, 17-tet seems to be an obvious choice for
neo-Gothic styles; 22-tet is more of an alternative that "pushes the
envelope," and thus is well worth some exploration.

{After playing in 22-tet, and comparing it with 17-tet, I would
generally still go along with the above: 17-tet is a more "mainstream"
choice within the universe of "Neo-Gothic" tunings beyond Pythagorean,
and 22-tet a more "far out" choice. Both tunings invite some
Setharianizing -- or might one also say Darregizing -- in the matter
of timbre, but here 22-tet may be a bit more sensitive than 17.

Since 17-tet is charming, and the extra "unusual" or "neutral"
intervals can have an alluring and mysterious air when I want to mix
them with the usual medieval ones, I'm inclined to consider "lack of
consistency" another term for "variety."

However, 22-tet is fascinating also, both musically and
intellectually, and with the right timbre, that temperament of the
fifth by over 7 cents isn't as much a problem as it might look on
paper. It's "pushing the envelope," but with the right timbre, you
_can_ have a scale with the whole-tone four times as large as the
semitone and still make more or less "conventional" European medieval
music. To borrow Easley Blackwood's terminology, the intervals can be
not only recognizable but acceptable.

Using Blackwood's concept of "R" as the ratio between the whole-tone
and diatonic semitone in a tuning (W/H, or in my more medieval
notation T/S -- L/s to others), the "neo-Gothic" region might go from
around R=2.25 (53-tet or Pythagorean) to R=4 (22-tet). Thus 22-tet is
very important for marking a possible "outer limit" of this continuum,
maybe a role a bit analogous to that of 1/3-comma meantone among
historical 5-limit tunings. Both the tempering of the fifth by about
the same amount in these tunings (in opposite directions, of course!),
and the point where very wide major thirds tend to shade close to
narrow fourths, etc. (a bit beyond 9:7?) may suggest R=4 as around the
"high end" of the neo-Gothic zone.

Exploring various tunings within this zone is very important, and
22-tet is definitely a "can't miss" destination.

To sum up on "consistency": If you or others were to argue that the
7-limit consistency of 22-tet gives it a special cadential flavor in
neo-Gothic styles, that sounds like a good argument, and certainly the
flavor can be very pleasing. However, I would want to emphasize that I
don't find lack of "consistency" to be a flaw of 17 _in a neo-medieval
kind of setting_.

Of course, for composing 7-limit music, 22-tet consistency would be a
vital advantage.}

A curious feature of 22-tet is that it provides both the
"near-5-limit" and "near-7-limit" variants of 24-note Pythagorean or
Xeno-Gothic, but not any very close approximations of regular 3-limit
intervals (e.g. ~5:4 and ~9:7, but not ~81:64). Since the 22-tet step
is very close to 32:31, it isn't too far from 36:35, the difference
for example between 5:4 and 9:7.

With 17-tet, in contrast, we get a kind of "meantone-like" compromise
between the usual 3-limit and "near-7-limit" values in Xeno-Gothic,
with the comma getting factored out; there's no 5-odd-limit
approximation.

{If one wants schisma thirds in the usual sense of near-5-limit
intervals, then either something close to Pythagorean or something in
the 22-tet region seems to be the choice. Curiously, in 22-tet, the
major schisma third is spelled as an augmented second (e.g. Eb-F#),
and the minor schisma third as a diminished fourth (e.g. C#-F) -- the
opposite of Pythagorean. Interestingly, the minor schisma third at
around 327 cents is part of the 11-tet set. For most of the music I
play, these are "special effect" intervals, and the usual ones the
norm, and I tend quickly to get "acclimated" to minor thirds of 5
steps and major thirds of 8 steps as the norm.}

In any of these tunings, a "closest approch" progression like M3-5 or
M6-8, etc., involves total expansion or contraction of the unstable
interval by a minor third, here in 22-tet equal to 5 steps (~272.73
cents). We have a motion of 4 steps in one voice (the whole-tone) and
1 step (the Pythagorean diatonic semitone, as it were) in the other:

F#3 ------- +1 = + 54.55 ------- G3
(8 = 436.36) (13 = 709.09)
D3 ------- -4 = -218.18 ------- C3

A cadential "semitone" of 54.55 cents is actually rather close to the
size which the cadential "diesis" of Marchettus seems to imply, about
half of a usual semitone (~45.11 cents in Pythagorean).

At the same time that we get something close to a melodic "diesis," an
unstable interval like M3 at 436 cents or ~9:7 still stays in the
range of what might be considered clearer "recognizablility" than in
24-tet, say (450 cents).

{Indeed I find 5/22 octave and 8/22 octave easily "recognizable" as
minor and major thirds in a medieval or neo-medieval context and the
right timbre. Maybe I should experiment with 24-tet or the like to see
how I would react to 450 cents as a major third.

Also, I should add that in practice, having a diatonic semitone equal
to the difference between a regular interval like a major sixth and
its schisma version can have weird and wonderful consequences, a topic
for more discussion.}

Out of curiosity, I used GNU Emacs Calc to see what would happen if we
tried to use a 54.55-cent diesis in this way while sticking to a
Pythagorean whole-tone of 9:8 (as Marchettus does in theory). In that
case, since one voice will move by a 9:8 and the other by this diesis,
we have a total expansion in M3-5 of ~258.45 cents. Subtract this from
a 3:2 fifth, and we get a major third of ~443.50 cents.

Paul, you have a good argument that if neo-Gothic "Mannerism" is the
intonational ideal, then 22-tet with its 4:1 ratio between whole-tone
and diatonic semitone, as well as thirds and sixths right around the
outer region of my "septimal comma region," may be an even more
dramatic expression of this ideal than 17-tet with a 3:1 ratio and
thirds and sixths around the middle of the region.

{After hearing both 17 and 22, I would 17 sounds somehow more
"familiar," 22 more "xenharmonic." In the Neo-Gothic zone, they are
maybe comparable to 1/5-comma meantone and 1/3-comma meantone in a way
for 5-limit music.}

Another issue, of course, is the fifths -- ~7.14 cents wide in 22-tet,
as opposed to ~3.93 cents in 17-tet. A first crude reaction might be,
"Yes, 19-tet is off by more in the other direction, but that isn't
necessarily an ideal temperament for medieval music." However, a fifth
at 709.09 cents might have a new sound of its own.

{In practice, I would say that the kind of timbre I use for this
tuning tends to smooth out the beats on the fifths, except maybe in
the higher range of my keyboard, say the usual top octave C5-C6
(c'-c''). It is more noticeable than in 17-tet, of course.}

This tuning might bring out the "close approximations to 5-limit and
7-limit, but not to 3-limit" feature that I mentioned: thus we can
play either a major third either as D-F# (~9:7) or D-Gb (~5:4, in
effect a 22-tet "schisma third"), but without anything closer to the
regular Pythagorean 81:64.

{Thus for medieval styles where schisma thirds are not an issue,
17-tet seems a more obvious choice for a "Neo-Gothic" tuning of the
super-Pythagorean variety with fifths tempered in the wide direction;
but 22-tet has its own character, and serves as an alternative tuning
among alternative tunings, so to speak. While the tuning might not
have been intended mainly as Pythagorean-like, it amazingly succeeds,
fifths and all. When it comes to large whole-tones, small semitones,
and wide major thirds and sixths, etc., 22-tet may be indeed be a tour
de force in theory and practice.}

Most respectfully

Margo Schulter
mschulter@value.net