---------------------------------------------
Neo-Gothic tunings and temperaments:
Meantone through a looking glass
(Part 1 of 2)
---------------------------------------------
One artistic source for tunings and temperaments both old and new is
the Western European musical tradition of the Gothic era, and
especially the complex polyphony of the 13th and 14th centuries. This
music, and "neo-Gothic" styles drawing inspiration from it, invite
approaches to interval aesthetics quite distinct from those of
European music from the Renaissance to the Romantic era.
The historical Gothic tradition itself offers a consummate tuning
system: Pythagorean tuning, or 3-limit just intonation (JI), which
results in a subtle "balance of power" between the stable and unstable
intervals and sonorities of Gothic polyphony. This system deserves
attention both for its intrinsic beauty and for its historical role as
the one JI system to win widespread acceptance on standard keyboard
instruments.
Neo-Gothic tunings and temperaments strive to develop this historical
tradition further in one of at least two directions: either the
further extension of Pythagorean tuning beyond the 17 notes recognized
by medieval theorists (Gb-A#); or the use for Gothic and neo-Gothic
music of "reverse meantone" temperaments with fifths somewhat _wider_
than a pure 3:2. Both approaches maintain a Pythagorean flavor while
offering new types of intervals or artfully altered interval sizes,
blending tradition with innovation.
Starting with Pythagorean tuning, or the almost identical 53-tone
equal temperament (53-tet), the neo-Gothic spectrum moves through a
universe of "reverse meantone" temperaments with the fifths becoming
increasingly larger than pure: for example 41-tet, 29-tet, exponential
meantone (see Section 2), and 17-tet. Thirds and sixths become even
more active than in Pythagorean, and their resolutions to stable
3-limit intervals even more economic and efficient, involving diatonic
semitones increasingly smaller than the already compact Pythagorean
limma at 256:243, or about 90.22 cents.
The region from around Pythagorean or 53-tet to 17-tet (with fifths
about 3.93 cents wider than pure) might be considered the central or
"quintessential" neo-Gothic zone, where (as the latter term may
suggest) fifths and fourths are pure or reasonably close to pure.
These 3-limit concords coexist with and provide resolutions for a
kaleidoscopic variety of unstable intervals and sonorities, some
approximating higher-prime ratios such as 13:11 or 17:14. While the
progressions retain a Gothic logic, these sonorities add a color and
flavor radically distinct from that of historical 3-limit or 5-limit
European practice.
Beyond 17-tet, we enter a "far neo-Gothic" zone ranging out to around
22-tet, where fifths and fourths are less smooth but still acceptable
primary concords, especially with timbral adjustments (Darregization
or Sethareanization). At 22-tet, where fifths are about 7.14 cents
wide, the diatonic semitone is reduced to only 1/4-tone (~54.55
cents), and we encounter a Wonderland of interval spellings and
alterations -- yet standard Gothic progressions and some modern
offshoots still succeed musically, and indeed delightfully.
From one viewpoint, the neo-Gothic spectrum ranging from around
Pythagorean to 22-tet is a kind of mirror-reversed image of the
historical European meantone spectrum from Pythagorean to 1/3-comma
meantone or 19-tet (with fifths tempered in the _narrow_ direction by
about 7.17 cents and 7.22 cents respectively). Thus the subtitle of
this article, "Meantone through a looking-glass."
-------------------------------------------------------------------
2. The neo-Gothic spectrum, Blackwood's R, and exponential meantone
-------------------------------------------------------------------
Neo-Gothic temperaments with fifths larger than pure may be seen (and
heard) as artful variations and distortions of classic medieval
Pythagorean tuning, the amount of distortion increasing as we move
from 53-tet or Pythagorean to 17-tet, and from there to 22-tet.
At the outset of this exploration of the "reverse meantone" continuum,
it would be well to distinguish between the scales themselves and the
specific Gothic or neo-Gothic applications which are the focus of this
article. Musicians who use 53-tet as a system of 5-limit or 7-limit
rather than Pythagorean JI, or who use 22-tet for Paul Erlich's
tetradic 7-limit system developed by analogy to 5-limit major/minor
tonality[1], will be well aware of this distinction.
Not only our aesthetic appreciation of a scale, but our reckoning of
its vital statistics, can change with musical viewpoint. From our
Pythagorean perspective, 53-tet has whole-tones of 9 steps and
diatonic semitones of 4 steps; in 22-tet, these intervals are 4 steps
and 1 step. For a musician using 53-tet to approximate 5-limit JI, or
using Erlich's tetradic 22-tet system, these basic intonational
metrics can and will vary.[2]
To describe the aesthetics of neo-Gothic tunings and temperaments in a
nutshell, we might focus on three main themes: (1) Pure or near-pure
fifths and fourths, the prime concords; (2) Active and dynamic thirds
and sixths inviting efficient resolutions to stable intervals; and (3)
Large whole-tones and small diatonic semitones, facilitating
expressive melody and incisive cadential action.
In medieval Pythagorean intonation, we have a classic balance between
these elements; neo-Gothic temperaments with fifths wider than pure
compromise the first element in order to accentuate the second and
third. Unstable thirds and sixths become yet more active, and can yet
more efficiently resolve to stable intervals; the contrast between
large whole-tones and small semitones becomes yet greater.
Looking more closely at some specific tunings in the context of Gothic
or neo-Gothic parameters of musical style may help in understanding
what happens as we move along the reverse meantone continuum from
Pythagorean to 22-tet. For a more thorough discussion of medieval
Pythagorean tuning in the context of Gothic musical style, and of
medieval sonorities and cadences, see
[32]http://www.medieval.org/emfaq/harmony/pyth.html
[33]http://www.medieval.org/emfaq/harmony/13c.html
------------------------------------------------
2.1. Artistic parameters and Blackwood's R (T/S)
------------------------------------------------
The following table surveys a few tunings and temperaments at various
points along the neo-Gothic continuum, and may become more meaningful
as we relate interval sizes to traits and constraints of musical
style:
----------------------------------------------------------------------
tuning/ fifth M2 M3 m3 m2 R=T/S
temperament (+/-3:2) (+/-9:8)
======================================================================
Central or quintessential neo-Gothic
53-tet 701.89 203.77 407.55 294.33 90.57 2.25
(-0.07) (-0.14) (~81:64) (~32:27) 9/4
----------------------------------------------------------------------
Pythagorean 701.96 203.91 407.82 294.13 90.22 ~2.26
(0.00) (0.00) (81:64) (32:27)
----------------------------------------------------------------------
41-tet 702.44 204.87 409.76 292.68 87.80 2.33...
(+0.48) (+0.96) (~19:15) 7/3
----------------------------------------------------------------------
29-tet 703.45 206.90 413.79 289.66 82.76 2.50
(+1.49) (+2.99) (~13:11) 5/2
----------------------------------------------------------------------
exponential 704.61 209.21 418.43 286.18 76.97 ~2.71828
meantone (+2.65) (+5.30) (~14:11) (e)
----------------------------------------------------------------------
17-tet 705.88 211.76 423.53 282.35 70.59 3.00
(+3.93) (+7.85) 3/1
======================================================================
Far neo-Gothic
39-tet 707.69 215.38 430.77 276.92 61.54 3.50
(+5.73) (+11.47) 7/2
----------------------------------------------------------------------
22-tet 709.09 218.18 436.36 272.72 54.55 4.00
(+7.14) (+14.27) (~9:7) 4/1
======================================================================
As even this small sampling shows, neo-Gothic tuning systems represent a
variety of approaches. Pythagorean intonation is a "tuning" in the
strict sense, a JI system based on integer ratios only; 17-tet,
22-tet, 39-tet, 41-tet, and 53-tet all belong to the family of equal
temperaments or "n-tet's." Exponential meantone, like more familiar
historical meantone temperaments not based on an equal division of the
octave, would be classified in some schemes as "non-just, non-equal."
"Exponential meantone" is defined as having a ratio between its
whole-tone and diatonic semitone equal to Euler's exponential _e_,
~2.71828.[3] The result is a temperament with qualities somewhere
between those of 29-tet and 17-tet on our chart, with the versatile
46-tet providing a yet closer approximation, and 109-tet a nearly
exact one.[4]
More generally, as we shall see, the ratio between whole-tone and
diatonic semitone, termed "R" by tuning theorist and composer Easley
Blackwood[5], provides one measure of the extent of the Neo-Gothic
continuum and its place in the larger intonational universe. We may
also express this ratio as T/S, using medieval initials for "tone" and
"semitone." For neo-Gothic tunings and temperaments, as the last
column of our chart shows, R or T/S varies from around 2.25 to 4.
Taking the columns of our table from left to right, let us consider
how the intonational qualities of these tunings interact with the
artistic parameters of medieval or neo-medieval styles.
--------------------------------
2.1.1. Smooth fifths and fourths
--------------------------------
In Pythagorean tuning, fifths and fourths, the choice medieval
concords rightfully having pride of place on the first column of our
chart, have pure ratios of 3:2 and 4:3; in the almost identical
53-tet, fifths are very slightly narrow (~0.07 cents).
As we move out along the central neo-Gothic zone through 41-tet and
29-tet and exponential meantone, this ideal is rather mildly
compromised; at 17-tet, fifths are about 3.93 cents wide. As we move
into the far neo-Gothic zone, this compromise becomes more pronounced,
with fifths at 22-tet about 7.14 cents wide.
One approach might be to compare these temperaments with historical
meantones where the fifths are narrowed. With 41-tet and 29-tet, the
tempering is less than in 12-tet (~1.95 cents), whose fifths are often
considered "near-pure"; exponential meantone is comparable to
1/8-comma meantone (~2.69 cents), and 17-tet to 2/11-comma meantone
(~3.91 cents). Further out, 39-tet compares to something between
1/4-comma (~5.38 cents) and 1/7-comma (~6.14 cents)[6], and 22-tet to
1/3-comma meantone (~7.17 cents).
With neo-Gothic or reverse meantones, as with historical meantones, a
bit more than 7 cents of tempering seems to mark the limit of tenable
compromise for the fifths, and this constraint provides one motivation
for placing the far end of our spectrum around 22-tet.[7]
To avoid confusing stylistic norms with universal values, we might add
that other world musics such as as Balinese or Javanese gamelan quite
pleasingly use fifths and fourths much further from 3:2 or 4:3, while
musics based on 11-tet or 13-tet get along without any intervals
resembling these ratios.
--------------------------------------------------
2.1.2. Compatible major seconds and minor sevenths
--------------------------------------------------
In addition to pure fifths and fourths, Pythagorean features pure
ratios for major seconds or ninths and minor sevenths (9:8, 9:4, 16:9)
with ideal ratios to form relatively concordant sonorities in
combination with fifths or fourths: e.g. 4:6:9, 6:8:9, 8:9:12,
9:12:16.[8] In a medieval setting, I term these combinations "mildly
unstable quintal/quartal sonorities"; in certain neo-medieval styles,
they might be treated not merely as relatively blending but as stable.
As the second column of our table shows, the variance of major seconds
(and likewise of major ninths and minor sevenths) from their ideal
Pythagorean ratios is equal to twice the tempering of the fifths (a
relationship sometimes slightly obscured by rounding adjustments). At
17-tet, this variance is around 7.85 cents; by 22-tet, it is around
14.27 cents.
Although the just intonation of relatively concordant quintal/quartal
sonorities is a special charm of medieval Pythagorean tuning, and
these combinations are better within the quintessential neo-Gothic
zone from Pythagorean to 17-tet, I agree with Paul Erlich that they
remain acceptable in 22-tet[9] -- especially, I would add, with some
Darregian/Setharean timbre adjustments.
-------------------------------
2.1.3. Active thirds and sixths
-------------------------------
In medieval Pythagorean tuning, major thirds at ~407.82 cents (81:64)
and minor thirds at ~294.13 cents (32:27) have an active and unstable
but relatively blending or "imperfectly concordant" quality; 53-tet
(interpreted in a Pythagorean manner) offers almost identical ratios.
Major sixths at ~905.87 cents (27:16) are regarded in the 13th century
as somewhat more tense, and minor sixths at ~792.18 cents (128:81) as
yet more tense, but play a vital role in cadential progressions, often
expanding by contrary motion to octaves. In the 14th century, major
and minor sixths gain a status as "imperfect concords" on par with the
thirds.
To borrow the term of modern composer and theorist Ludmila Ulehla[10],
these intervals act as "dual-purpose" sonorities (as distinguished in
her terminology from stable "concords" or urgent "discords"), at once
inviting directed resolutions to stable intervals and serving as
moments of diverting vertical color.
As the third and fourth columns of our table show, major and minor
thirds (and sixths also, their octave complements) become even more
active and dynamic as we move from Pythagorean into the realm of
neo-Gothic temperaments with fifths wider than pure. By 17-tet, major
thirds have expanded to ~423.53 cents while minor thirds have
contracted to ~282.35 cents. In 22-tet, these intervals have sizes of
~436.36 cents (~9:7) and 272.72 cents (not far from 7:6).
As we move beyond Pythagorean, Darregian/Setharian timbre adjustments
can help in keeping a _relatively_ blending quality for these
intervals while enjoying the superefficient cadential action featured
by these reverse meantone temperaments (see Section 2.1.5).
The expansion of major thirds (and sixths), and contraction of minor
thirds (and sixths), may also serve as a possible constraint placing
the far end of the neo-Gothic spectrum not too far from 22-tet. At
this point, major and minor thirds are near 9:7 and 7:6 respectively,
still quite distinct (to my ears) from narrow fourths or wide major
seconds. Likewise, major sixths are near 12:7, and minor sixths near
14:9.
Going well beyond 22-tet, at a fifth size of around 712 cents we would
find major thirds expanding into the region near 450 cents, and minor
thirds yet later contracting into the region near 250 cents, etc.,
where questions of categorical ambiguity could become more
important. This is by no means to suggest that such temperaments are
undesirable, only to suggest that they may belong to a somewhat
different realm than neo-Gothic from Pythagorean to 22-tet.[11]
--------------------------------------------------------------------
2.1.4. Large whole-tones and small diatonic semitones: Blackwood's R
--------------------------------------------------------------------
In Pythagorean tuning, whole-tones are a generous 9:8 (~203.91 cents),
and diatonic semitones a compact 256:243 (~90.22 cents). This contrast
facilitates expressive melody and efficient cadential action.
A useful measure of this contrast is the ratio between the sizes of
these two intervals, T/S or Blackwood's R, about 2.26 for Pythagorean
and precisely 2.25 or 9/4 for the almost identical 53-tet, where a
whole-tone is equal to 9 steps and a diatonic semitone to 4 steps.
As we move into the spectrum of neo-Gothic temperaments where fifths
are tempered increasingly wide of pure, the second column of our table
(already met in Section 2.1.2) shows how major seconds increase in
size; column 5 shows how diatonic semitones shrink even more
rapidly. Column 6 follows the consequent accentuation of the contrast
between these intervals as measured by T/S or R.
Note that for equal temperaments, R is given both as a decimal and as
a fraction showing the number of steps for each interval, e.g. 9/4 for
53-tet and 5/2 for 29-tet.
Moving from Pythagorean to 29-tet, we find that whole-tones have
expanded rather moderately to ~206.90 cents while diatonic semitones
have contracted to ~82.96 cents, with R increasing from ~2.26 to 2.5.
In exponential meantone, these intervals are ~209.21 cents and ~76.97
cents, with the defining ratio R of Euler's e, ~2.71828. At 17-tet,
whole-tones and diatonic semitones are at ~211.76 cents and ~70.59
cents -- 3 steps and 1 step respectively -- so that R is 3.
Travelling into the far neo-Gothic zone, we find that at 39-tet, these
intervals are ~215.38 cents and ~61.54 cents, with R at 3.5; the
diatonic semitone has become slightly smaller than the _chromatic_
semitone of 19-tet (~63.16 cents, very close to 28:27). At 22-tet, the
whole-tone has grown to ~218.18 cents and the diatonic semitone has
contracted to ~54.54 cents, with R at 4. While this semitone --
literally a "diatonic quartertone" -- may look very small on paper, I
find that my ears can routinely accept it as a regular semitone.
Thus the contrast between large whole-tones and concise diatonic
semitones, already a notable attraction of Pythagorean, becomes yet
more accentuated as we progress along the reverse meantone spectrum:
the "minor semitone" of 53-tet or Pythagorean (at or around 4/9-tone)
shrinks to the thirdtone of 17-tet and the literal quartertone of
22-tet.
This contrast, in its continuum of Neo-Gothic shades, can lend an
expressive air to melodic lines and vertical progressions alike, and
leads to our fifth artistic theme of efficient cadences.
---------------------------------
2.1.5. Efficient cadential action
---------------------------------
In Gothic music, cadential progressions are typically guided by
directed resolutions from unstable intervals to stable ones by
stepwise contrary motion (e.g. 2-4, 3-1, 3-5, 6-8, 7-5). In the 14th
century, such resolutions where one voice moves by a whole-tone and
the other by a semitone are especially favored (e.g. m3-1, M3-5, M6-8,
m7-5, M2-4). Late medieval theorists tell us that the unstable
interval should "approach" its stable goal as closely as possible,
resolving with an ideally efficient motion.
This cadential aesthetic nicely fits both the melodic and vertical
parameters of Pythagorean tuning. A major third at 81:64 (~407.82
cents), for example, has a large size which at once lends it a degree
of dynamic tension because of its acoustical complexity, and permits
it to expand more economically to a stable fifth, as one voice moves
by a whole-tone and the other by an incisive diatonic semitone.[12]
In neo-Gothic temperaments with fifths wider than pure, both aspects
of this musical equation are further accentuated. As major thirds and
sixths grow larger and larger (Section 2.1.3), they take on an even
more active and dynamic quality, with this tension released by yet
more efficient expansion to fifths and octaves (M3-5, M6-8) involving
yet more narrow and incisive diatonic semitones (Section 2.1.4).
One measure of cadential efficiency or incisiveness is the total
distance an unstable interval must expand (M3-5, M6-8, M2-4) or
contract (m3-1, m7-5) in order to reach its stable goal. Since in
these "closest approach" progressions one voice moves by a whole-tone
and the other by a diatonic semitone -- whose sum is a minor third --
this distance will be equal to a minor third.
Thus column 4 of our table, showing the size of a minor third, can
also serve as an index of cadential efficiency; as this size gets
smaller, cadences become more efficient, involving smaller and more
incisive semitonal motions (column 5).
In Pythagorean tuning, our "closest approach" progressions are already
admirably efficient, involving only ~294.13 cents of expansion or
contraction (the size of a 32:27 minor third). By 17-tet, it has
decreased to ~282.35 cents, and by 22-tet to ~272.72 cents. This trend
correlates intimately with the shrinking of the diatonic semitone from
~90.22 cents in Pythagorean to ~54.55 cents in 22-tet.[13]
The "closest approach" aesthetic, as realized by medieval Pythagorean
tuning, may combine the satisfying contrast between a tense interval
and its stable resolution; the release of this tension through
economically directed motion; and the melodic as well as vertical
appeal of concise cadential semitones. Neo-Gothic temperaments with
fifths larger than pure offer accentuated variations on these themes
in assorted shades of intonational Mannerism.
---------------
Notes to Part 1
---------------
1. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament,"
_Xenharmonikon_ 17 (Spring 1988), pp. 12-40.
2. From a 5-limit JI perspective, 53-tet would mix large whole-tones
of 9 steps (~203.77 cents, ~9:8) and small-whole tones of 8 steps
(~181.13 cents, ~10:9), with usual diatonic semitones of 5 steps
(~113.21 cents), quite close to 16:15 -- an interval which in
Pythagorean terms closely approximates the _chromatic_ semitone or
apotome at 2187:2048 (~113.69 cents). In Erlich's 22-tet system, see
n. 1 above, pp. 22-25, a "large" interval (L) is equal to 3 steps
(~163.64 cents), and a "small" interval (s) to 2 steps (~109.09
cents); from a Pythagorean viewpoint, 3 steps is a chromatic semitone,
and 2 steps a curious "intermediate semitone" between this and the
diatonic semitone of 1 step.
3. This temperament, with fifths about 2.65 cents wider than pure,
might be taken as a kind of neo-Gothic counterpart to the "Golden
Meantone" developed by Thorvald Kornerup and advocated by Jacques
Dudon, where this same ratio is equal to the golden mean, ~1.61834
(and fifths are about 5.74 cents narrow). The latter temperament is
about midway between 1/4-comma meantone (~5.38 cents, major thirds
pure) and Zarlino's 2/7-comma meantone (~6.14 cents, major and minor
thirds equally impure).
4. Emphasizing that the table represents an arbitrary sample of
tunings and temperaments at a few points on the neo-Gothic spectrum,
I might out of psychological curiosity note my selection of 17-tet,
29-tet, 41-tet, and 53-tet -- possibly because Pythagorean tunings of
17, 29, 41, and 53 notes represent "Moments of Symmetry" as described
by theorist Ervin Wilson. As for 39-tet, it is the one equal
temperament with 53 or fewer notes having a ratio of whole-tone to
diatonic semitone (T/S, or Blackwood's R, see text below and n. 5)
greater than 3 but less than 4. The purpose for including exponential
meantone is, of course, unabashed promotion. Equally meritorious
temperaments such as 46-tet might just as well have been included.
5. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_
(Princeton: Princeton University Press, 1985).
6. As it happens, 39-tet involves almost exactly the same amount of
tempering in the wide direction as Golden Meantone (see n. 3) in the
narrow direction.
7. While "mirror-image" comparisons can be engaging, there is an
important artistic asymmetry. In historical meantones, fifths are
compromised (in the narrow direction) in order to optimize thirds and
sixths, the primary Renaissance-Romantic concords. In Gothic or
neo-Gothic music, where fifths and fourths are the primary concords
and pure Pythagorean intonation provides a superb solution (except for
special "neo-medieval" styles where circularity might be sought in
less the 53 notes), temperament is more of an artistic liberty, and
indeed an artful Manneristic distortion. To tune 19-tet for 5-limit
music may be motivated in good part by a desire to optimize overall
consonance; to tune 22-tet for Neo-Gothic music is likely more an
expression of calculated xenharmonic zest.
8. Examples of these medieval sonorities, using a MIDI-style notation
where C4 indicates middle C and higher note numbers show higher
octaves, would be C3-G3-D4 (M9 + 5 + 5); C3-F3-G3 (5 + 4 + M2);
C3-D3-G3 (5 + 4 + M2); and C3-F3-Bb3 (m7 + 4 + 4).
9. Erlich, see n. 1 above, p. 26 and n. 31. Erlich's 22-tet sonorities
forming "the decatonic equivalent of 'quartal' or 'quintal' harmony,"
ibid. n. 31, involve four notes, e.g., in a conventional Pythagorean
spelling, C3-F3-G3-Bb3 or C3-D3-G3-A3. However, I find typical Gothic
quintal/quartal sonorities of the kind we have been discussing with
three notes and intervals (see n. 8 for examples) to be also
satisfactory, at least in apt timbres.
10. Ludmila Ulehla, _Contemporary Harmony: Romanticism through the
Twelve-Tone Row_ (New York, 1966), p. 428.
11. In practice, the limit of acceptable temperament for fifths and
fourths (especially in styles where they are the main concords) may
take priority as a constraining factor, and this limit is arguably
reached around 22-tet. With gamelan-like timbres, however, some
experiments exploring and possibiy circumventing such constraints
might be very interesting. See also n. 17 below on the sizes of major
thirds and diatonic semitones in tunings with fifths ranging from 710
to 715 cents.
12. For classic statements of this Pythagorean cadential aesthetic
with its "incisive" melodic semitones, see Mark Lindley, "Pythagorean
Intonation and the Rise of the Triad," _Royal Musical Association
Research Chronicle_ 16:4-61 (1980), ISSN 0080-4460; and "Pythagorean
Intonation," _New Grove Dictionary of Music and Musicians_ 15:485-487,
ed. Stanley Sadie, Washington, DC: Grove's Dictionaries of Music
(1980), ISBN 0333231112.
13. Since "closest approach" progressions involve motion of a
whole-tone in one voice and a semitone in the other, the expansion of
the major second from Pythagorean to 22-tet by ~14.27 cents (Section
2.1.2, column 2 of table) partially offsets the shrinking of the
diatonic semitone by ~35.68 cents, resulting in a net gain in
efficiency of ~21.41 cents. Increasing the size of the fifth produces
a twofold expansion of the major second (formed from two fifths up
minus an octave) but a fivefold reduction of the diatonic semitone
(formed from five fifths down).
Most respectfully,
Margo Schulter
---------------------------------------------
Neo-Gothic tunings and temperaments:
Meantone through a looking glass
(Part 2 of 2)
---------------------------------------------
--------------------------------
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.
-----------------------------------------
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
[