Yahoo Tuning Groups Ultimate Backup tuning Re: Neo-Gothic tunings (Part 2 of 2)

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Neo-Gothic tunings and temperaments:
Meantone through a looking glass
(Part 2 of 2)
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2.2. The neo-Gothic region and R
--------------------------------

Our survey suggests one possible definition for the neo-Gothic
spectrum as the portion of the continuum of regular tunings
("meantone" in the most generic sense) where the value of T/S, or
Blackwood's R, ranges from 2.25 (53-tet) to 4 (22-tet).

We can subdivide this Neo-Gothic region into a "central" neo-Gothic
zone from 53-tet or Pythagorean through 17-tet, where R ranges from
2.25 through 3; and a "far" neo-Gothic zone beyond 17-tet through
22-tet, with R ranging from 3 through 4.

The following chart may illustrate this possible mapping; equal
temperaments are identified by the number of steps per octave,
Pythagorean tuning as "Py," and exponential meantone as "Ex":

Py
53 41 29 Ex 17 39 22
|--------|--------|--------|--------|--------|---------|--------|
R 2.25 2.5 2.75 3 3.25 3.5 3.75 4
|--------------------------|------------------------------------|
Central neo-Gothic Far neo-Gothic
(2.25 <= R <= 3) (3 < R <=4)

As is the case with many such mappings ranging from music history
periodizations to stratigraphic boundaries in paleobiology, different
schemes for "drawing lines" can bring out interesting concepts,
whatever scheme we choose to adopt at a given moment or for a given
purpose.

It might seem natural to adopt Pythagorean (or the almost identical
53-tet) as the lower boundary of the neo-Gothic region, given our
focus on "reverse meantone temperaments" with fifths wider than pure.
Pythagorean tuning, or "zero-comma meantone" with pure fifths is the
lower limit of this zone, just as it is the upper limit of meantone
temperaments with fifths narrower than pure.[14]

If we focus on Blackwood's R, then 53-tet has the attraction of
placing the lower boundary at the neat integer ratio of 9/4, or 2.25.
Since 53-tet so closely resembles Pythagorean (when its steps are used
to define intervals in a Pythagorean manner, of course), and has such
a great potential for neo-Gothic music, both tunings seem to belong in
the same category.[15]

Interestingly, Blackwood suggests a musical basis for setting a lower
boundary slightly _below_ Pythagorean, but not much lower than 53-tet.
In his view, 406 cents is about the maximize size at which major
thirds can acceptably serve as stable concords; beyond this point,
they become too acoustically complex and active to form stable 5-limit
triads.[16]

Since the use of thirds as unstable although _relatively_ blending
intervals is a cardinal feature both of historical Gothic polyphony
and of its neo-Gothic offshoots, Blackwood's observation suggests a
lower boundary at a fifth size of around 701.5 cents (~0.46 cents
narrower than pure), producing a major third of 406 cents. Such a
tuning would have a whole-tone of 203 cents and a diatonic semitone of
92.5 cents, yielding an R of ~2.195.

Taking 53-tet as our lower limit provides a bit of artistic margin: in
53-tet or Pythagorean, thirds have a distinctly active and dynamic
quality fitting Gothic and neo-Gothic styles.

Setting the high end of the neo-Gothic spectrum at around 22-tet may
reflect mainly the problem of the increasing temperament of the fifths
(~7.14 cents in 22-tet) in styles where fifths and fourths are the
primary concords. Disregarding this constraint, we might engage in
interesting dialogues regarding the point at which major thirds
growing into narrow fourths, or diatonic semitones shrinking into
comma-like intervals difficult to recognize as "half-steps," might
clearly place us in a new musical terrain.[17]

If we do take the quality of fifths as a governing constraint, and
draw an upper boundary at 22-tet, then these other factors may become
more academic. In a neo-Gothic spectrum running from 53-tet and
Pythagorean to 22-tet, diatonic semitones remain larger than 50 cents
throughout the region, and major thirds range from a Pythagorean 81:64
(or minutely smaller in 53-tet) to just larger than 9:7. Minor thirds
range from around a Pythagorean 32:27 to slightly larger than 7:6, and
major seconds from around a Pythagorean 9:8 to a kind of "mean-tone"
in 22-tet about midway between a 9:8 and an 8:7.[18]

Following this approach, tunings and temperaments along a neo-Gothic
spectrum of R=2.25-4 share a common family resemblance to medieval
Pythagorean, giving the region a somewhat unified quality, and yet
featuring an impressive range of variation.

One purpose for proposing such boundaries, of course, should be not to
discourage but to provoke experimentation beyond recognized metes and
bounds. If this discussion leads to more exploration of the world
beyond R=4, it will have served its purpose.

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3. Alternative thirds and 17-tet symmetry
-----------------------------------------

If we provisionally accept R=2.25-4 as a range for distinctively
neo-Gothic tunings -- as opposed to _meritorious_ tunings! -- there
remains the question of why 17-tet (R=3) should serve as a line of
demarcation between the central and far neo-Gothic zones.

One might reply that 17-tet is a well-known "exaggerated Pythagorean"
temperament roughly in the middle of our spectrum[19], that it is the
point where a diatonic semitone is equal to precisely 1/3-tone, or
roughly the point beyond which the tempering of the fifth becomes a
more substantial issue, or simply that R=3 is a nice round number.

There is, however, another basis for regarding 17-tet as a point of
symmetry in relation to the neo-Gothic spectrum, a basis providing a
connection between musical developments of the early 15th century and
a possible "reenvisioning of history" through the use of new
intonational variations on familiar 14th-century progressions.

In the decades around 1400, as Mark Lindley[20] has documented through
theoretical sources and actual music, musicians became intrigued with
a variation of Pythagorean intonation on keyboards where sharps were
tuned at the flat end of the chain of fifths. In this tuning, thirds
involving written sharps -- in such sonorities as D-F#-A, A-C#-E,
E-G#-B -- had a distinctively smooth quality differing from the active
flavor of regular Pythagorean thirds.

These alluring major and minor thirds were actually Pythagorean
diminished fourths at 8192:6561 (~384.36 cents) and augmented seconds
at 19683:16384 (~317.60 cents) -- e.g. D-Gb-A, A-Db-E, E-Ab-B --
intervals removed only by a schisma of 32805:32768 from pure ratios of
5:4 (~386.31 cents) and 6:5 (~315.64 cents). Thus they are often known
today as "schisma thirds."

By around 1450, the appetite of musicians for these "alternative"
thirds evidently led to meantone temperaments seeking them in as many
places as possible. This intonational shift may be seen as one aspect
of the transition from Gothic to Renaissance musical style, often
placed somewhere in the early to middle 15th century.

Reflecting on the 15th-century role of Pythagorean augmented seconds
and diminished fourths as schisma thirds leads us to a new world of
possibilities: the varied flavors these "alternative thirds" take on
at different points along the neo-Gothic spectrum.

In Pythagorean tuning, as we have just seen, augmented seconds are
very close to 6:5 and diminished fifths to 5:4; and this situation is
almost identical in 53-tet.[21] More generally, the augmented second
is smaller than the diminished fourth.

Let us see what happens to these alternative thirds as we move along
the neo-Gothic spectrum. In the following table, the last column shows
for equal temperaments the number of steps in these intervals, along
with the arithmetic of their derivation: an augmented second from a
whole-tone plus a chromatic semitone (equal to the difference between
T and S); and a diminished fourth from a fourth minus a chromatic
semitone.

----------------------------------------------------------------------
tuning/ fifth aug2 dim4 T/S=R aug2/dim4
temperament (+/-3:2) steps
======================================================================
53-tet 701.89 316.98 384.91 2.25 14 17
(-0.07) (~6:5) (~5:4) 9/4 9+5 22-5
----------------------------------------------------------------------
Pythagorean 701.96 317.60 384.86 ~2.26
(0.00) (~6:5) (~5:4)
----------------------------------------------------------------------
41-tet 702.44 321.95 380.49 2.33... 11 13
(+0.48) 7/3 7+4 17-4
----------------------------------------------------------------------
29-tet 703.45 331.03 372.41 2.50 8 9
(+1.49) (~17:14) 5/2 5+3 12-3
----------------------------------------------------------------------
exponential 704.61 341.46 363.14 ~2.71828
meantone (+2.65) (~17:14) (~21:17) (e)
----------------------------------------------------------------------
17-tet 705.88 352.94 352.94 3.00 5 5
(+3.93) (~11:9) (~11:9) 3/1 3+2 7-2
======================================================================
39-tet 707.69 369.23 338.46 3.50 12 11
(+5.73) (~21:17) (~17:14) 7/2 7+5 16-5
----------------------------------------------------------------------
22-tet 709.09 381.82 327.27 4.00 7 6
(+7.14) (~5:4) 4/1 4+3 9-3
======================================================================

As we move outward from 53-tet or Pythagorean, our near-6:5 augmented
seconds grow larger while our near-5:4 diminished fifths grow smaller.
At 41-tet, this process is not too pronounced, so that we can still
speak of "schisma thirds" in a quasi-Pythagorean sense at ~6.31 cents
wider than 6:5 and ~5.83 cents narrower than 5:4.

By 29-tet or exponential meantone, however, we have moved into a
different region where augmented seconds and diminished fourths take
on a flavor of "alternative thirds" quite different from 5-limit
intervals. There remains a certain polarity between these thirds,
~341.46 cents and ~363.17 cents in exponential meantone: we might term
them "superminor/submajor," or in a more medieval Pythagorean fashion
"suprasemiditonal/subditonal." Such thirds may suggest various integer
ratios such as 17:14 and 21:17, for example.

At 17-tet, our expanding augmented second and shrinking diminished
fourth converge into a single "neutral" interval at ~352.94 cents,
rather close to 11:9 and even closer to the 4-step interval of Gary
Morrison's 88-cent equal temperament (88-cet). Thus 17-tet (R=3)
represents a special "moment of convergence" on our spectrum.[22]

Beyond 17-tet, the augmented second represents the _larger_
alternative third, and the diminished fourth the _smaller_ one. Thus
at 39-tet, we have a situation somewhat comparable to exponential
meantone, but with the roles of these two intervals reversed.

At 22-tet (R=4), our augmented second has expanded to an interval
almost equivalent to the Pythagorean diminished fourth, a major third
at ~381.82 cents (~4.49 cents narrower than 5:4); our diminished
fourth has shrunk to a minor third of ~327.27 cents (~11.73 cents
wider than 6:5), somewhat akin to a Pythagorean augmented second.

From a composer's perspective, the diverse alternative thirds of the
neo-Gothic spectrum present a resource permitting us to "reenvision
history" by experimenting with familiar Gothic progressions in new
shades of intonation. The "suprasemiditonal/subditonal" thirds found
in the portion of the central neo-Gothic zone around exponential
meantone, and again in the far zone around 39-tet (with the roles of
diminished fourth and augmented second reversed), offer special
possibilities in this direction.

For example, it is quite possible to take standard 14th-century
progressions where minor thirds contract to unisons and major thirds
expand to fifths, and substitute these alternative thirds. Like
Pythagorean thirds, they seem to have an active and passionate quality
lending intensity to cadential action; yet their color is very
different. Also, such progressions involve melodic motions by
strikingly large _chromatic_ semitones, giving then a definite
contrast in flavor to usual Gothic or neo-Gothic progressions with
compact diatonic semitones.[23]

From the perspective of theory, these alternative thirds also provide
a basis for distinguishing between a central neo-Gothic zone where
augmented seconds are smaller than diminished fourths (R=2.25-3), and
a far neo-Gothic zone where augmented seconds are the larger intervals
(R=3-4). At 17-tet, R=3, these two thirds converge into one, a moment
of symmetry (in a general as opposed to Wilsonian sense) serving as a
kind of "continental divide" for the neo-Gothic spectrum.

----------------------------------------
4. Through a looking glass enigmatically
----------------------------------------

If we view the neo-Gothic meantone spectrum from Pythagorean or 53-tet
to 22-tet (R=2.25-4) together with the historical European meantone
spectrum from 19-tet to Pythagorean (R=1.5-~2.26), we may see both
regions in better perspective. Here the main number line shows the
tempering of the fifth in cents in the narrow (negative) or wide
(positive) direction. As with the chart in Section 3, numbers
immediately above this line identify equal temperaments, "Py" shows
Pythagorean, and "Ex" exponential meantone; fractions represent
various historical meantone tunings:

1/3 2/7 1/4 1/5 1/6 1/8 Py Ex
19 31 12 53 41 29 17 39 22
-|----|----|----|---|---|---|---|----|---|---|---|---|---|---|-
5th -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
R 1.5 1.75 2 2.25 2.5 2.75 3 3.5 4
|--------------------------------|------------------------------|
Historical meantone Neo-Gothic meantone
(1.5 <= R <=~2.26) (2.25 <= R <=4)
|------------------| |---------------|--------------|
Characteristic Central Far
(1.5 <= R <= ~1.82) (2.25 <= R <=3) (3 < R <= 4)

Proceeding from left to right, we travel first through the zone of
"characteristic" meantone or meantone in the usual sense of
temperaments from around 1/3-comma to 1/6-comma where fifths are
narrowed in order to obtain pure or near-pure thirds, including 19-tet
and 31-tet. Here R is appreciably less than 2, with diatonic semitones
larger than chromatic semitones -- an especially colorful contrast in
the area of 1/3-1/4 comma.

We next move through the intermediate region between 1/6-comma
meantone and 53-tet or Pythagorean, including 12-tet (R=2). At about
R=2.2, we reach Blackwood's limit of acceptability for 5-limit music,
from another perspective our beckoning portal to the Gothic or
neo-Gothic world of active and dynamic thirds and sixths efficiently
resolving to stable 3-limit intervals.

At R=2.25 (53-tet) or R=~2.26 (Pythagorean) we are at the point of
symmetry on our chart where fifths are virtually or precisely pure. As
we continue through the central neo-Gothic zone, thirds and sixths
become even more active while diatonic semitones shrink from around
4/9-tone to 1/3-tone in 17-tet (R=3).

Much beyond Pythagorean, at least when seeking to maintain a 13th-14th
century balance of concord/discord, we may "Darregize/Sethareanize"
our timbres to keep our thirds _relatively_ blending while enjoying
the "superefficient" resolutions they invite and the other special
qualities of these tunings. With the right timbre for 17-tet, a
half-cadence in Machaut on a major third can have the expected quality
of a charming but pregnant pause rather than an acute clash!

Beyond 17-tet, we move through the far neo-Gothic zone where timbre
adjustments can also mitigate the increasingly pronounced temperament
of our fifths. We arrive at 22-tet (R=4), a xenharmonic outpost or
resort where many features of the classic Pythagorean world are
dramatically and intriguingly distorted, and yet the musical terrain
remains recognizable.

Just as Ivor Darreg courageously asserted that _every_ equal
temperament has its own potential for beautiful music[24], so each
point and region on our meantone continuum has its own musical virtues
and attractions. Some of these beautiful possibilities may more fully
reveal themselves if we are familiar with the whole spectrum,
including the region of Pythagorean and beyond, a region too often
represented by the cartographic legend: "And here there be Wolves."

For example, a quite familiar and conventional temperament such as
12-tet can take on new qualities when we appreciate its intriguing
ambivalence: is it a compromised 5-limit meantone, or a somewhat
subdued "semi-Gothic" 3-limit tuning? One might creatively play on
this ambiguity, with the 20th-century use of this scale for both
tertian and quartal/quintal harmony (e.g. Bartok, Hindemith) a
possible precedent. Once 12-tet is viewed as one member of a vast
society of scales, not a substitute for all the others, this
temperament and its surrounding "middle country" between Renaissance
meantone and Pythagorean may flower in ways not yet imagined.

Returning to our main focus on the vibrant world of Gothic and
neo-Gothic music and tunings, temperaments beyond Pythagorean could
lead in many directions. One of the most intriguing is what I might
term direct chromaticism.

While Gothic music of the 13th and 14th centuries has many routine and
unconventional uses of accidentalism, chromaticism in the proper sense
of melodic motion by a chromatic semitone is rather less common -- in
contrast to the many examples of such chromaticism in the 16th-century
ambience of tunings such as 1/4-comma meantone.

In these tunings the difference in size between the large diatonic and
small chromatic semitones is dramatic, equal to a diesis of ~41.06
cents in 1/4-comma meantone and ~63.16 cents (1/19 octave) in 19-tet,
lending a special air to 16th-century chromatic progressions
alternating between the two semitones.

While the contrast between the small _diatonic_ and large chromatic
semitones is less dramatic in Pythagorean (~23.46 cents), as fifths
get wider than pure the disparity rapidly increases to ~41.38 cents in
29-tet (about the same as in 1/4-comma meantone) and ~55.28 cents in
exponential meantone (not too far from 19-tet). A millennial era of
neo-Gothic chromaticism may be at hand.[25]

Microtonalist and tuning theorist Graham Breed[26] has offered some
interesting observations raising the question of what I might term
"direct commaticism": the use of the Pythagorean comma (e.g. Ab-G#),
or its equivalents in various neo-Gothic temperaments as direct
melodic intervals. This comma is equal to the difference between the
diatonic and chromatic semitones discussed just above.

While I have used the direct melodic Pythagorean comma, e.g. Ab-G#, in
"neo-medieval" interpretations of certain early 15th-century cadential
progressions, such idioms may take on new flavors as we move through
the neo-Gothic spectrum. By 29-tet or exponential meantone, as we have
seen, this "comma" has grown to the size of a Renaissance meantone
diesis, used as a direct melodic interval in the "enharmonic" style of
Nicola Vicentino (1511-1576). Might one emulate Vicentino by using the
diesis-like commas of these neo-Gothic tunings in amazing shifts and
variations on standard medieval progressions?[27]

Then again, in 22-tet, the normal diatonic semitone of ~54.55 cents is
around the size of a largish diesis, making the above categories
rather problematic.[28] This temperament, like a good science fiction
novel or physics book, is a feast for the intellect and imagination,
as well as actually working musically -- and beautifully -- for Gothic
or neo-Gothic music. Whether I compare the experience to travelling
near the speed of light, or observing the behavior of photons near an
extreme gravity well, 22-tet as a neo-medieval tuning deserves its own
theory of relativity.

There is also the question of the world beyond 22-tet, possibly a
region where neo-Gothic and gamelan may meet.

To consider such musical possibilities of the realm beginning with
rather than ending at Pythagorean, a realm where the high art of the
Gothic era may hold up "a distant mirror" not only to the present but
to the future, is to gaze through a looking glass enigmatically, but
the enigma is pleasant, enticing one to new music.

---------------
Notes to Part 2
---------------

14. This boundary would accord with the classification of equal
temperaments as "positive or negative (that is, fifths that fall short
of 701.955-cent third harmonic or fifths that exceed the third
harmonic)." See Brian McLaren, "A Brief History of Microtonality in
the Twentieth Century," _Xenharmonikon_ 17:57-110 (Spring 1998), at
p. 78, describing the work of M. Joel Mandelbaum.

15. Thus the Neo-Gothic spectrum begins with a tuning which has fifths
very slightly narrower than pure; in technical terms, 53-tet might be
described as ~1/315-comma meantone.

16. Blackwood, n. 5 above, pp. 202-203. Here Blackwood is suggesting a
fifth size of 701.5 cents or major third size of 406 cents (R=~2.2) as
an upper limit of acceptability for tertian music of the European
Renaissance-Romantic repertory. Note that tunings somewhat below this
limit, for example 12-tet (R=2), may be quite acceptable for Gothic or
neo-Gothic as well as tertian music, but do not have the _distinctive_
"Gothic/neo-Gothic" quality of tunings in the region of R=2.25-4 (from
53-tet and Pythagorean to 22-tet).

17. At a fifth size of 710 cents (~8.04 cents wider than pure), we
would have a major third of 440 cents and a minor second of 50 cents;
at 712 cents (~10.04 cents wider than pure), 448 cents and 40 cents;
at 714 cents (~12.04 cents wider than pure), 456 cents and 30 cents;
at 715 cents (~13.04 cents wider than pure), 460 cents and 25 cents.
At 710 cents, the value of R is 4.4; at 712 cents, 5.6; at 714 cents,
7.6; at 715 cents, 9.2.

18. Thus two such 22-tet whole-tones form a near-9:7 major third, just
as two whole-tones in 1/4-comma meantone (midway between 9:8 and 10:9)
form a pure 5:4.

19. My warm thanks to John Chalmers for introducing me to this tuning;
the "exaggerated Pythagorean" description may come from Ivor Darreg.

20. For Lindley's impressive thesis, see the articles cited in n. 12
above.

21. From a Pythagorean point of view, people who treat 53-tet as a
5-limit JI system are actually redefining major and minor schisma
thirds (17 and 14 steps respectively) as regular thirds.

22. Here I am inspired by Ervin Wilson's "Moment of Symmetry." Whether
there is any connection between Pythagorean 17 being a moment of
symmetry and 17-tet being a moment of convergence for augmented
seconds and diminished fourths I leave as an intriguing question.

23. In neo-Gothic temperaments with fifths wider than pure, the
greater-than-Pythagorean difference between these semitones can
heighten such a contrast. In exponential meantone, for example, the
diatonic semitone is ~76.97 cents and the chromatic semitone ~132.25
cents, a situation somewhat resembling that of Renaissance meantones
(where the diatonic semitone is the larger interval).

24. See McLaren, n. 14 above, pp. 80-81.

25. Interestingly, one medieval theorist who does demonstrate and
advocate use of the direct chromatic semitone is Marchettus of Padua
in his _Lucidarium_ (1318), a treatise also advocating that singers
use cadential semitones or "dieses" equal to only "one of the five
parts of a tone." Whether or not Marchettus should be read as
describing cadences with semitones considerably narrower than in
Pythagorean and extra-wide major thirds and sixths -- and he is
addressing vocal intonation rather than keyboard tunings -- his
treatise has played a central role in my musical odyssey leading to
this paper, which I therefore warmly dedicate to him.

26. Graham Breed, Tuning List, Tuning Digest 700 (6 July 2000),
Message 25; and Tuning Digest 703 (7 July 2000), Message 8. Breed
finds that the comma in 53-tet (1 step, ~22.64 cents) "is too small to
be clearly comprehended," and therefore has a "troubling quality," but
that the 41-tet comma (1 step, ~29.27 cents) is satisfactorily "heard
as a melodic elaboration." He finds a comma at a round 24 cents (in a
synthesizer approximation of Pythagorean with fifths tuned to an even
702 cents, ~0.04 cents wide) to be "adequate," although "it takes a
bit of getting used to." In addition to directing my attention to the
theme of "commatic" progressions, these observations invite further
investigation of listener thresholds for the "comprehension" of very
small melodic inflections.

27. This line of development may already be a reality, and in
historically informed performances of the medieval repertory as well
as in new compositions or improvisations in allied styles, since I
have seen reports that at least one noted group (Mala Punica) is using
"micro-intervals" for 14th-century Italian music.

28. Technically speaking, a diatonic semitone (or quartertone, R=4) in
22-tet is equal to _half_ of the comma, the 2-step difference between
the chromatic semitone of 3 steps and the diatonic semitone of 1 step.
However, commalike distinctions in this temperament between regular
and "schisma" thirds, for example (augmented second vs. major third,
minor third vs. diminished fourth), involve a difference of 1 step,
the same interval as the usual minor second.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Neo-Gothic tunings (Part 2 of 2)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>