Re: Neo-Gothic tunings (Part 1 of 2)

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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

http://www.medieval.org/emfaq/harmony/pyth.html
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
mschulter@value.net