Re: EXPERIMENT: Xeno-Gothic, 17-tet, and "near-7-limit"

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EXPERIMENT: Unstable Xeno-Gothic tetrads in 17-tet
"Near-7-limit" sonorities reconsidered
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In presentations of an approach to complex 3-limit music based on a
24-note Pythagorean tuning which I term "Xeno-Gothic," because of its
roots in the Gothic polyphony of Western Europe during the 13th and
14th centuries, I have often discussed the "near-7-limit" intervals
which this system makes available. While such a description is
mathematically accurate, I have come to question the _musical_
significance of 7-limit approximations in Xeno-Gothic style.

In a Xeno-Gothic tuning, two 12-note keyboards are mapped with
identical Pythagorean tunings (typically Eb-G#) a Pythagorean comma
apart (531441:524288, ~23.46 cents). From a mathematical point of
view, this arrangements makes available intervals a comma larger or
smaller than usual which closely approximate 7-based ratios.

Using the symbol @ to show a note lowered by a Pythagorean comma, and
C4 for middle C (with higher note numbers for higher octaves), we thus
have narrow minor thirds at ~7:6 (e.g. E3-G@3), narrow minor sevenths
at ~7:4 (e.g. C3-Bb@3); and wide major seconds at ~8:7 (e.g. D@-E),
wide major thirds at ~9:7 (e.g. G3@-B3), and wide major sixths at
~12:7 (e.g. A@3-F#4).

While it may be mathematically interesting that these intervals are
only a "septimal schisma" or ~3.80 cents from 7-limit ratios -- the
difference between the Pythagorean comma and the septimal comma of
64:63 (~27.26 cents), the question remains as to how _musically_
significant this approximation is in a neo-medieval style where the
3-limit, not the 7-limit, is the basis for stability.

In an experiment which may lend a bit of Aristoxenian pragmatism to
this question, I tried certain typical Xeno-Gothic cadences resolving
"near-7-limit" sonorities in 17-tone equal temperament (17-tet), where
the relevant unstable intervals likewise have an "exaggerated
Pythagorean" quality but are 11-19 cents removed from 7-based ratios.

My tentative conclusion is that these progressions have a similar and
comparably pleasing "Xeno-Gothic" quality in 17-tet, and that
"near-7-limit" terminology may run the risk of obscuring both the
musical significance and aesthetic intent of this neo-medieval style.

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1. Xeno-Gothic tetrads and "cadence-stacking"
---------------------------------------------

In a recent dialogue with such helpful contributors as Paul Erlich and
Graham, I learned a very important lesson in discussing Xeno-Gothic.
There is a need to place due emphasis on the _Gothic_ aspect of this
style, the 13th-14th century practices and conventions which serve as
a basis this approach to using a 24-note Pythagorean tuning.

For example, as Paul Erlich demonstrated in a very helpful example
relating various schismas, the term "full 7-limit tetrad" suggests a
4:5:6:7 sonority. In Xeno-Gothic, however, a sonority such as
G3-B3-D4-F4 is not especially characteristic; and when it does occur,
a regular 3-limit Pythagorean tuning (576:729:864:1024) is usual.

A typical Xeno-Gothic tetrad including a major third and minor
seventh in "near-7-limit" ratios, in contrast, would be ~12:14:18:21
-- taking the 7-limit notation with a large grain of salt, as we may
see by the end of this article. Incidentally, were we to speak in such
terms, this would actually be a "9-odd-limit tetrad."

Here is such an unstable Xeno-Gothic tetrad and its standard
resolution to a stable 3-limit concord:

D@4 C@4
B3 C@4
G@3 F@3
E3 F@3

This unstable tetrad and its directed resolution may be said to
reflect a process of "cadence-stacking" in which two or more two-voice
progressions are superimposed to built a unifying multi-voice
progression. Before considering the "Xeno" aspect, the comma
alterations, let's consider a Gothic version of this cadence in usual
Pythagorean:

D4 C4
B3 C4
G3 F3
E3 F3

(m7-5 + m3-1 + M3-5 + m3-1)

Here we have four two-voice progressions each moving from an unstable
to a stable 3-limit interval by stepwise contrary motion. The outer
minor seventh contracts to a fifth, and the lower and upper minor
thirds to unisons; the middle major third expands to a fifth.

Further, these progressions fit the 14th-century ideal of "closest
approach," or of "moving to the nearest concord": one voice of the
unstable interval moves by a whole-tone (9:8, ~203.91 cents), the
other by a compact diatonic semitone (256:243, ~90.22 cents). Such
resolutions have an especially efficient quality.

Resolutions fitting this "closest approach" paradigm include M2-4,
M3-5, and M6-8 (by expansion); and m3-1 and m7-5 (by contraction).
By "mixing and matching" these elementary resolutions, we can build
powerful multi-voice cadences to stable 3-limit sonorities.

In Xeno-Gothic, we have the option of making these standard Gothic
progressions "superefficient" by adding a comma to expanding major
intervals (M2-4, M3-5, M6-8), bringing them yet closer to the stable
goal of their expansion; and likewise subtracting a comma from
contracting intervals (m3-1, m7-5).

To describe these altered unstable intervals, I refer to "maximal"
seconds, thirds, and sixths (Mx2-4, Mx3-5, Mx6-8) and "minimal" thirds
and sevenths (mn3-1, mn7-5).

Let us now return to our Xeno-Gothic tetrad and its resolution:

D@4 C@4
B3 C@4
G@3 F@3
E3 F@3

(mn7-5 + mn3-1 + Mx3-5 + mn3-1)

In each of our four "superefficient" two-voice resolutions, one voice
still moves by a usual 9:8 whole-tone, but the other has a narrowed
semitone a Pythagorean comma smaller than the already compact 256:243
-- about 66.76 cents, or roughly 1/3-tone. Here the ascending
semitones E3-F@3 and B3-C@4 illustrate this "supernarrow" semitone.

Thus whether the unstable thirds and sevenths of such "Xeno-Gothic
tetrads" are tuned in a usual Pythagorean fashion, or in a
superefficient "near-7-limit" manner, the progression is based on
3-limit logic.

While speaking of "near-7-limit tetrads" may rather accurately report
the size of the relevant unstable intervals, it may also obscure their
musical significance in a complex 3-limit setting.

----------------------------
2. A comparison test: 17-tet
----------------------------

While Xeno-Gothic tuning uses two 12-note Pythagorean keyboards a
comma apart to provide altered versions of the standard intervals,
17-tet features regular major seconds (3/17 octave, ~211.77 cents),
thirds (6/17 octave, ~423.53 cents), and sixths (13/17 octave, ~917.65
cents) somewhat larger than Pythagorean. Minor thirds (4/17 octave,
~282.35 cents) and sevenths (14/17 octave, ~988.24 cents) are likewise
somewhat smaller than Pythagorean.

The normal diatonic semitone of 17-tet (1/17 octave, ~70.59 cents) is
considerably smaller than in usual Pythagorean, and not far from the
Xeno-Gothic "superefficient" semitone a comma smaller (~66.76 cents),
both intervals having a "thirdtone" quality.

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:

17-tet interval cents 7-based ratio cents difference
--------------- ------- ------------- ------- ----------
M2 (3 steps) ~211.76 8:7 ~231.17 ~19.41
m3 (4 steps) ~282.35 7:6 ~266.87 ~15.48
M3 (6 steps) ~423.53 9:7 ~435.08 ~11.55
M6 (13 steps) ~917.65 12:7 ~933.13 ~15.48
m7 (14 steps) ~988.24 7:4 ~968.83 ~19.41

In a rather casual and quite unscientific survey of one, I find that the
17-tet intervals seem just as satisfying for Xeno-Gothic progressions
as the "near-7-limit" intervals of 24-note Pythagorean.

Thus in describing either tuning as used for a Gothic or neo-Gothic
3-limit style, it might be more apt to speak simply of minimal thirds
and sevenths smaller than regular Pythagorean, and maximal seconds,
thirds, and sixths larger than Pythagorean, rather than to bring in
"near-7-limit" concepts.

----------------------------------------
3. Super-Pythagorean intervals and zones
----------------------------------------

While 7-limit intervals and septimal commas and schismas may not be
especially significant in explaining the musical effect of
"superefficient" 3-limit progressions in a neo-Gothic style, the
septimal comma may nevertheless provide one possible landmark in
mapping a rough "Super-Pythagorean zone" for unstable intervals in
such progressions.

Thus we might describe maximal seconds, thirds, and sixths as having
sizes appreciably larger than regular Pythagorean, but not larger by
too much more than a septimal comma (~27.26 cents). Similarly, minimal
thirds and sevenths might be described as smaller than Pythagorean,
but not smaller by too much more than a septimal comma.

The idea here is to propose a rough range where intervals are somewhat
larger or smaller than Pythagorean, but still are recognizable for
example as very large major sixths rather than small minor sevenths in
a resolution such as M6-8, or conversely as very small minor sevenths
rather than large major sixths in m7-5. etc.

If the septimal comma roughly approximates the size of this zone -- in
the wide direction for maximal intervals and the narrow direction for
minimal ones -- then we might suggest ranges like these:

--------------------------------------------------------------------
interval typical resolution rough range
--------------------------------------------------------------------
Mx2 Mx2-4 9:8 (~203.91) - 8:7 (~231.17)
Mx3 Mx3-5 81:64 (~407.82) - 9:7 (~435.08)
Mx6 Mx6-8 27:16 (~905.87) - 12:7 (~933.13)
--------------------------------------------------------------------
mn3 mn3-1 7:6 (~266.87) - 32:27 (~294.13)
mn7 mn7-5 7:4 (~968.83) - 16:9 (~996.09)
--------------------------------------------------------------------

This table is only a first suggestion, and it might be interesting to
see how various listeners familiar (or familiarized) with these
neo-Gothic resolutions might perceive intervals such as the supermajor
third of 440 cents in Gary Morrison's 88-cent equal temperament
(88-cet).

Another way of describing such resolutions is to say that they
typically involve narrow melodic semitones smaller than the usual
Pythagorean diatonic semitone of 256:243 (~90.22 cents) but not much
smaller than a 28:27 (~62.96 cents). In both 24-note Pythagorean
or Xeno-Gothic and in 17-tet, these semitones are in the "1/3-tone"
range -- ~66.76 cents (a Pythagorean comma smaller than 256:243) and
~70.59 cents (1/17 octave) respectively.

Here I would emphasize that the intend is descriptive, not
prescriptive: there is no reason why one might not experiment with
having a 450-cent interval expand to a 700-cent fifth in 24-tet, for
example, with motion by a usual 200-cent whole-tone in one part and a
50-cent quartertone in the other. Such a progression, with * showing a
note raised by such a quartertone, might closely approximate one
literal reading of the cadential "diesis" of Marchettus of Padua
(1318), for example:

B*3 -- +50 -- C4
(450) (700)
G3 -- -200 -- F3

However, such a cadential interval in the zone of "categorical
ambiguity" resolving by quartertone motion might have a different
effect than Xeno-Gothic or 17-tet with its characteristic thirdtone
melodic motion. More familiarity with Gothic and neo-Gothic
progressions, as well as with these tunings, may prompt many pleasant
variations.

---------------
4. More to come
---------------

If speaking of "near-7-limit" intervals in Xeno-Gothic may distract
from rather than clarify the medieval and neo-medieval 3-limit
aesthetics of this style, aesthetics aptly realized by 17-tet also,
then the question arises of how these aesthetics might better be
described and explained.

In a "prequel" to this article, I hope to develop such an explanation
based on the late Gothic concepts of "perfection" and "coloration" as
applied to unstable intervals expanding or contracting to stable ones,
with neo-medieval progressions in Xeno-Gothic or 17-tet as a kind of
manneristic amplification of these concepts -- "superperfection" and
"supercoloration."

Also, the treatment in Gothic and neo-Gothic styles of tritonic
seventh sonorities such as G3-B3-D4-F4 in a 3-limit-oriented manner
may illustrate some of the qualities of these styles; thanks to Paul
Erlich and his 4:5:6:7 for setting the stage for a coming article on
the 576:729:864:1024 and its direct or indirect resolution.

Last but not least, this experiment has intensified my passion for
17-tet, a tuning which some attention to the Setharian factor of
timbre may make delightful for conventional 13th-14th century music as
well as neo-Gothic genres. Lending a novel color to familiar
progressions, and offering some strikingly new and beautiful ones,
this tuning is a topic which warmly beckons.

Most respectfully,

Margo Schulter
mschulter@value.net

In this post, I'll use prime limits because that makes it simpler. No
chords are in the respective odd limits, although the 7-limit version is
9-limit consonant. I'll also ignore weasel words like "near" and
"approximate" so don't assume intervals are just unless I say so.

Margo Schulter wrote:

> Here is such an unstable Xeno-Gothic tetrad and its standard
> resolution to a stable 3-limit concord:
>
> D@4 C@4
> B3 C@4
> G@3 F@3
> E3 F@3

I'll re-write this as:

D4 C/4
B/3 C/4
G\3 F\3
E/3 F\3

I'm using my schismic keyboard mapping as described at
<http://x31eq.com/schv21.htm> where the high D is on the key
that would normally play Bb. The / and \ mean move up and down
respectively by a comma. This isn't quite the way I normally write
schismic scales, but it's the keyboard that's important.

Note: the absolute pitches are not correct, but who cares?

I haven't worked out the minimum number of notes from the spiral of fifths
required for this to work, but I suspect it will be more than 17. Margo,
how many do you have? Is it 17 tuned to 24 keys, or do you get all 24?

> This unstable tetrad and its directed resolution may be said to
> reflect a process of "cadence-stacking" in which two or more two-voice
> progressions are superimposed to built a unifying multi-voice
> progression. Before considering the "Xeno" aspect, the comma
> alterations, let's consider a Gothic version of this cadence in usual
> Pythagorean:
>
> D4 C4
> B3 C4
> G3 F3
> E3 F3

Which for me is:

D/4 C4
B/3 C4
G3 F3
E/3 F3

and I've also tried this 5-limit version:

D/4 C4
B3 C4
G3 F3
E3 F3

Where the minor thirds are 5/6 and the major third is 5/4. So we have 3-,
5- and 7-limit tunings of the progression.

I tried these chords in three different tunings, using the Harpsichord
patch on my SB Live!. The first is Pythagorean, the second is my optimal
5-limit tuning, where 6/5 is just, and the third is my optimal 7-limit
tuning, where 7/4 and 7/6 are almost just. Fifths are always almost just,
that being the way of schismic tuning.

My exact optimum 7-limit tuning is based on the following scientifically
chosen principles:

1) Be close to the minimax tuning. I think this is where 7/4 is just, but
it could also be where 7/6 is just. Whatever, the condition states the
tuning must be somewhere in-between.

2) Slightly favour 7/4. Don't know why, it was a while back I worked this
out.

3) Make the comma, as expressed in octaves, a nice number.

All conditions are satisfied by making the comma equal to 0.022222...
octaves. This happens to be identical to 135-equal. The fourth is 56/135
octaves and the fifth is 79/135 octaves or 702.222222... cents.

In the 5-limit tuning, the 5-limit progression is best, the 3-limit okay
and the 7-limit nothing special.

In the Pythagorean tuning, the 3-limit progression is probably best, the
others okay.

In the 7-limit tuning, the 7-limit progression is really stunning. The
others are still okay, but all 7-limit definitely has my vote.

> 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:
>
> 17-tet interval cents 7-based ratio cents difference
> --------------- ------- ------------- ------- ----------
> M2 (3 steps) ~211.76 8:7 ~231.17 ~19.41
> m3 (4 steps) ~282.35 7:6 ~266.87 ~15.48
> M3 (6 steps) ~423.53 9:7 ~435.08 ~11.55
> M6 (13 steps) ~917.65 12:7 ~933.13 ~15.48
> m7 (14 steps) ~988.24 7:4 ~968.83 ~19.41

Right, normal 17-tet intervals, so this is the 3-limit progression played
in 17-equal. The 7-limit progression actually leaves no movement between
E/ and F\, B\ and C/.

As the distinction between the 3- and 7-limit progressions is lost, it
seems that this breaks the spirit of Xeno-Gothic: that you can choose the
commas. Although the 5-7 distinction is still there.

> In a rather casual and quite unscientific survey of one, I find that the
> 17-tet intervals seem just as satisfying for Xeno-Gothic progressions
> as the "near-7-limit" intervals of 24-note Pythagorean.

Yes, I find that the 5-limit progression is still good and recognisable,
and the 3-limit is good and recognizable ... as the 7-limit progression.
However, all 7-limit still has my vote. A certain amount is lost in all
cases, moving from the optimal tunings.

I also tried the first 7-limit chord, and resolving to the second chord a
step high, but that didn't really work at all.

> Thus in describing either tuning as used for a Gothic or neo-Gothic
> 3-limit style, it might be more apt to speak simply of minimal thirds
> and sevenths smaller than regular Pythagorean, and maximal seconds,
> thirds, and sixths larger than Pythagorean, rather than to bring in
> "near-7-limit" concepts.

I disagree, because using a 7-limit optimised tuning does improve the
progression. So the 7-limit interpretation must be relevant.

Errors are to be expected, it's getting late,

Graham