Re: The e-based tuning and metachromatic progressions (Part 2)

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The e-based tuning and metachromatic progressions:
A feast of neo-Gothic flavors
(Part 2: Flavors and metachromaticism)
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[Please see /tuning/topicId_20573.html#20573 for
Part I of this article.]

To survey some general characteristics of the e-based tuning in the
neo-Gothic type of musical setting for which it was designed, we may
begin with a consideration of the main intonational "flavors" which
this tuning offers. For an introduction to neo-Gothic sonorities and
flavors, please see my series "A Gentle Introduction to neo-Gothic
progressions," with some material on the e-based tuning included in
Part 2C:

/tuning/topicId_15038.html#15038 (1/Pt 1)
/tuning/topicId_15630.html#15630 (1/Pt 2A)
/tuning/topicId_15685.html#15685 (1/Pt 2B)
/tuning/topicId_16134.html#16134 (1/Pt 2C)

Here we first consider regular major and minor thirds of the usual
"11-flavor," and diminished fourths and augmented seconds of the
submajor/supraminor "17-flavor," before turning to the sonorities and
standard resolutions of the "7-flavor" with their routine element of
metachromaticism.

---------------------------------
3.1. Regular 11-flavor sonorities
---------------------------------

As discussed in Part I, the e-based tuning is located in a central
region of the spectrum between Pythagorean and 17-tET where major and
minor thirds have ratios in the general area of 14:11 and 13:11,
therefore known as the "11-flavor" region. This region extends from
around 29-tET (fifths ~703.45 cents, ~1.49 cents wide) to around the
e-based tuning (fifths ~704.61 cents, ~2.65 cents wide).

In the vicinity of 29-tET, the more "Pythagorean-like" end of the
region, we have a "mild" 11-flavor with minor thirds close to 13:11
(~289.21 cents) and major thirds about midway between the Pythagorean
81:64 (~407.82 cents) and 14:11 (~417.51 cents>.

In the neighborhood of the e-based tuning, the more "17-tET-like" end,
we have a "strong" 11-flavor: the e-based major third at ~418.43 cents
is slightly larger than 14:11, while the minor third at ~286.18 cents
is closer to 33:28 (~284.45 cents), the fifth complement of 14:11,
than to 13:11.

To accommodate this "strong" 11-flavor, somewhat more delicate timbral
adjustments may be called for with the e-based tuning than with
Pythagorean or 29-tET: I have found with a synthesizer that milder
plucked-string textures (a "harpsichord lute-stop" kind of effect) and
flute-like registrations can be very pleasing, as can some "choir"
textures.

The following standard 11-flavor cadences may illustrate these points,
also demonstrating the efficient 76.97-cent diatonic semitones of our
tuning. Here C4 is middle C, numbers in parentheses show vertical
intervals above a given voice in rounded cents, and signed numbers
show the ascending (positive) or descending (negative) melodic motion
of each of the voices:

E4 ------ +77 ----- F4 F4 ----- -77 ----- E4
(209) (495) (286) (0)
D4 ------ -209 ----- C4 D4 ----- +209 ----- E4
(495,286) (495,0) (705,418) (705,705)
B3 ------ +77 ----- C4 Bb3 ----- -77 ----- A3
(914,705,418) (1200,705,705) (991,705,286) (705,705,0)
G3 ------ -209 ----- F3 G3 ----- +209 ----- A3

(M6-8 + M3-5 + m3-5 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Our first cadence, an "intensive" form with ascending melodic
semitones, shows the standard resolutions expanding from major sixth
to octave, major third to fifth, and major second to fourth; the
middle two voices contract from minor third to unison. Each resolution
involves in this tuning a total expansion or contraction of about 286
cents, the size of the minor third.

Our second cadence similarly shows the standard resolutions by
contraction from minor seventh to fifth and minor third to unison,
with the two middle voices expanding from major third to fifth. Here
we have a "remissive" form with descending semitones.

Typically these regular 11-flavor sonorities and cadences, with their
intriguingly "accentuated" variation on a usual medieval Pythagorean
tuning, make up the bulk of a musical texture, also providing an
intonational backdrop for other flavors and special effects.

--------------------------------------------------
3.2. 17-flavor or "submajor/supraminor" sonorities
--------------------------------------------------

Tunings in the 11-flavor region from around 29-tET to the e-based
temperament also feature diminished fourths and augmented seconds with
ratios in the general vicinity of 21:17 and 17:14, thus known as
alternative "17-flavor" thirds. These thirds, like their regular
counterparts, vary in hue or shading as we move through different
parts of the region.

Around 29-tET, we have a "mild" 17-flavor where these thirds differ
from the simple or "valley" ratios of 5:4 and 6:5 about as much as
12-tET thirds do in the opposite direction: supraminor thirds are
narrow of 17:14 (~336.13 cents), and submajor thirds wide of 21:17
(~365.83 cents), by about 5 or 6 cents. With fifths of around 704
cents, these intervals are at or close to 17-based ratios.

In the e-based tuning, we have a "strong" 17-flavor leaning rather
toward the neutral thirds found in the environs of 17-tET, with the
diminished fourth or submajor third at ~363.14 cents, and the
augmented second or supraminor third at ~341.46 cents.

These "neutralish" 17-flavor thirds can be quite beautiful. Especially
characteristic is the resolution of a sonority with an outer fifth,
supraminor third below, and submajor third above (~14:17:21):

Bb3 ------ +132 ----- B3
(363) (705)
F#3 ------ -209 ----- E3
(705,341) (705,0)
Eb3 ------ +132 ----- E3

(m3-1 + M3-5)

Here the supraminor and submajor third behave much like their regular
counterparts, resolving respectively to the unison and fifth in a
typical 14th-century fashion. In addition to these altered thirds, a
striking feature of this cadence is its of melodic use of the large
chromatic semitone or apotome at ~132.25 cents (Eb3-E3, Bb3-B3) in
place of the usual 76.97-cent diatonic semitone or limma.

A related type of progression features the resolution of submajor
third to fifth and submajor sixth to octave, with the 132-cent
chromatic semitone again lending a distinctive melodic as well as
vertical quality to the cadence. Here are alternative resolutions for
the sonority F#3-Bb3-Eb4 (~17:21:28) available within a usual 12-note
tuning of Eb-G#:

Eb4 ---- +132 ---- E4 Eb4 ---- +209 ---- F4
(495) (495) (495) (495)
Bb3 ---- +132 ---- B3 Bb3 ---- +209 ---- C4
(859,363) (1200,705) (859,363) (1200,705)
F#3 ---- -209 ---- E3 F#3 ---- -132 ---- F3

(M6-8 + M3-5) (M6-8 + M3-5)

This sonority may resolve either intensively to E3-B3-E4, or
remissively to F3-C4-F4. In the course of a piece, the first
resolution can contrast very surprisingly and effectively with the
routine 11-flavor remissive cadence to the same goal, F3-A3-D4 to
E3-B3-E4, the second likewise contrasting with the usual intensive
cadence G3-B3-E4 to F3-C4-F4.

A very special 17-flavor idiom featuring the supraminor sixth or
"Phi-sixth," so known because of its proximity to the Golden Ratio or
Phi (~833.09 cents), deserves a section of its own (see Section 5,
to appear in Part 3).

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3.3. The 7-flavor and standard metachromatic cadences
-----------------------------------------------------

Extending the e-based tuning to 24 notes reveals, in addition to the
regular 11-flavor and alternative 17-flavor intervals, a set of
7-flavor intervals all within about 5 cents of pure ratios (9:7, 14:9,
7:6, 12:7, 7:4, 8:7). These intervals give the e-based tuning and its
immediate neighborhood a distinctive place on the neo-Gothic spectrum;
built from chains of 13, 14, or 15 fifths or fourths, they resolve
in a characteristic fashion here termed "metachromatic."

Illustrating this fashion, the following two 7-flavor progressions
correspond to the regular 11-flavor versions of Section 3.1, with a
near-14:18:21:24 major sixth sonority in the first cadence and a
near-12:14:18:21 minor seventh sonority in the second:

F4 ------ +55 ----- F*4 E*4 ---- -55 ----- E4
(264) (495) (264) (0)
D*4 ----- -209 ----- C*4 D4 ---- +209 ----- E4
(495,231) (495,0) (705,440) (705,705)
C4 ------ +55 ----- C*4 A*3 ---- -55 ----- A3
(936,705,440) (1200,705,705) (969,705,264) (705,705,0)
G*3 ----- -209 ----- F*3 G3 ---- +209 ----- A3

(M6-8 + M3-5 + m3-5 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

In the first progression, the near-9:7 major third and near-12:7 major
sixth are actually spelled as a fourth (G*3-C4) and minor seventh
(G*3-F4) each reduced by the 55.28-cent diesis separating the two
manuals. The expansion of these large major thirds and sixths to
stable fifths and octaves involves the use of this diesis as a
cadential semitone, here ascending (C4-C*4, F4-F*4), while the other
voices descend by regular whole-tones (G*3-F*3, D*4-C*4).

Similarly, in the second progression, the near-7:6 minor third G3-A*3
and near-7:4 minor seventh G3-E*4 are spelled as a major second or
sixth enlarged by a 55-cent diesis. These small minor thirds and
sevenths contract to unisons and fifths by way of cadential diesis
motions, here descending (A*3-A3, E*3-E3), while the other voices
again move by usual 209-cent whole-tone steps (G3-A3, D3-E3).

The defining feature of a "metachromatic" progression is that the
interval separating the two keyboards of a 24-note tuning, here the
usual e-based diesis, serves as a small cadential semitone.[9] This
arrangement leads to engaging cadential geometries sometimes enhancing
usual patterns of musical structure or modality, and sometimes
inviting sequences with a logic of their own, quite distinct from that
of accustomed Gothic patterns.

Surveying first some standard cadential patterns, let us consider
alternative resolutions for the 7-flavor sonorities of our previous
example, with G*3-C4-D*4-F4 here resolving in a remissive manner
(descending diesis motions), and G3-A*3-D4-E*4 in an intensive manner
(ascending diesis motions):

F4 ------ +209 ----- G4 E*4 ---- -209 ----- D*4
(264) (495) (264) (0)
D*4 ----- -55 ----- D4 D4 ---- +55 ----- D*4
(495,231) (495,0) (705,440) (705,705)
C4 ------ +209 ----- D4 A*3 ---- -209 ----- G*3
(936,705,440) (1200,705,705) (969,705,264) (705,705,0)
G*3 ----- -55 ----- G3 G3 ---- +55 ----- G*3

(M6-8 + M3-5 + m3-5 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Comparing these progressions with the previous ones will demonstrate a
general trait of metachromatic progressions: intensive progressions
resolve to sonorities on the upper manual, and remissive progressions
to sonorities on the lower manual.

In full four-voice resolutions of the kind we are now considering with
all unstable intervals resolving by stepwise contrary motion, one pair
of voices move together in fifths or fourths while remaining on the
same manual, ascending or descending by a regular whole-tone: in our
current example, C4-F4 to D4-G4 (first cadence) or A*3-E*3 to G*3-D*3
(second cadence).

The other pair of voices, also in parallel fifths or fourths, move
together by a diesis from one keyboard to the other: here G*3-D*4 to
G3-D4 (first cadence) or G3-D4 to G*3-D*4 (second cadence). This
motion thus takes us to a stable sonority on the lower keyboard in a
remissive cadence, and on the upper keyboard in an intensive cadence.

In three-voice versions of these cadences -- three voices being most
typical in the 13th-14th century era, and also in many neo-Gothic
textures -- two voices likewise move together in fifths or fourths by
a whole-tone while the third voice moves in contrary motion by a
diesis, or vice versa. To get an idea of some of the permutations, let
us consider intensive and remissive resolutions for the major sixth
sonority A*3-D4-G4 (near-7:9:12) and the minor seventh sonority
C4-D*4-A*4 (near-12:14:21):

intensive resolutions

G4 ----- +55 ---- G*4 A*4 ---- -209 ---- G*4
(495) (495) (705) (705)
D4 ----- +55 ---- D*4 D*4 ---- -209 ---- C*4
(936,440) (1200,705) (969,264) (705,0)
A*3 ---- -209 ---- G*3 C4 ----- +55 ---- C*4

(M6-8 + M3-5) (m7-5 + m3-1)

remissive resolutions

G4 ----- +209 ---- A4 A*4 ---- -55 ---- A4
(495) (495) (705) (705)
D4 ----- +209 ---- E4 D*4 ---- -55 ---- D4
(936,440) (1200,705) (969,264) (705,0)
A*3 ---- -55 ---- A3 C4 ----- +209 ---- D4

(M6-8 + M3-5) (m7-5 + m3-1)

All of these 7-flavor cadences in three and four voices share a
"superefficient" quality: directed two-voice resolutions involve a
total expansion (M2-4, M3-5, M6-8) or contraction (m3-1, m7-5) of only
about 264.5 cents, the sum of the regular 209.2-cent whole-tone and
superincisive 55.3-cent diesis steps by which the two parts progress
in contrary motion. This total distance of expansion or contraction is
also equal to the size of the 7-flavor minor third, an interval built
from whole-tone-plus-diesis (e.g. G3-A*3 or C4-D*3).

Often 7-flavor progressions are introduced here and there to provide a
touch of variety and accentuated cadential action in a texture
otherwise pervaded by 11-flavor or 17-flavor sonorities and
resolutions. Additionally, however, they can be used for sequences and
other special effects whose geometries unite the two keyboards in new
ways.

---------------------------------------------
4. Metachromatic sequences and related idioms
---------------------------------------------

One definitely _neo_-Gothic approach to metachromatic cadences in the
e-based and other tunings is to string a chain or sequence of such
cadences together into a kind of progression quite different from
13th-14th century practice.

Other idioms involve the "superfourth" or small diminished fifth
formed from fourth-plus-diesis, or the use of unexpected metachromatic
"shifts" to divert a familiar progression in a most unaccustomed
direction, or melodic shifts of a 21.67-cent "subdiesis" or "17-comma"
defining the distance between such "near-equivalent" notes on the two
manuals as F3 and E*3. Also, the 7-flavor or near-14:9 minor sixth
formed from a fifth-plus-diesis (e.g. G3-D*4) lends itself to some
notable progressions.

----------------------------
4.1. Metachromatic sequences
----------------------------

Chaining together a series of metachromatic cadences produces a
sequence like this, here using remissive resolutions of minor seventh
sonorities, with each note sustained until the next note or the
conclusion of the passage (||) is indicated, with a link to a MIDI
version produced with Manuel Op de Coul's Scala program (the lynx
browser for UNIX had problems with this direct URL, but successfully
downloaded the file through the "Files" menu of the group Tuning):

||
C4 D*4 D4 E*4 E4 F#*4 F#4 G#*4 G#4
C4 D4 E4 F#4 G#4
F3 G*3 G3 A*3 A3 B*3 B3 C#*4 C#4
F3 G3 A3 B3 C#4

/tuning/files/Schulter%20sound%20files/eb7cr01.mid

This kind of sequence has a logic of its own, with a "floating"
quality not necessarily tied to any particular vertical or modal
center. The lowest and third-lowest voices characteristically progress
in ascending whole-tones (e.g. F3-G3-A3-B3-C#4 in the lowest voice), a
procedure which might recall the whole-tone scales of Debussy and
other composers around 1900.[10]

The second-lowest and highest voices have a complementary pattern of
alternating motions, first ascending by a near-7:6 minor third to move
from a stable vertical fifth to an unstable ~12:14:18:21 sonority, and
then descending by a diesis in the remissive resolution of this
sonority to a new stable fifth.

While this notation shows the notes and intervals, another approach
can better show the "logic of the hands" involved in realizing this
sequence on a two-manual keyboard. Here notes below the dashed line
are played by the left hand on the lower manual, while notes above it
are played by the right hand on the upper manual, with the symbol "r"
indicating a rest. The indicated triple rhythm might be taken as 3/4
or 3/2 at a moderate tempo, with an "iambic" of pattern of short-long:

1 2 3 | 1 2 3 | 1 2 3 | 1 2 3 | 1 2 3 ||

r D*4 r E*4 r F#*4 r G#*4 r
r G*3 r A*3 r B*3 r C#*4 r
----------------------------------------------------------------
C4 D4 E4 F#4 G#4
F3 G3 A3 B3 C#4

In this type of vertical rhythm, common in some Gothic pieces, stable
sonorities on stressed beats alternate with unstable sonorities on the
unstressed but longer portion of an iambic foot.

In our 24-note tuning, C#4-G#4 is as far as we can carry this example
with unbroken symmetry; with a larger tuning set, the sequence would
take us next to C#4-D#*4-G#4-A#*4, resolving to D#4-A#4, and so on.

These 7-flavor sequences may involve either minor seventh sonorities
or major sixth sonorities (~14:18:21:24), with a chain of resolutions
in either a remissive or an intensive manner. For example, here is a
sequence of major sixth sonorities resolving intensively (with
ascending semitones):

||
C#*5 B4 B*4 A4 A*4 G4 G*4 F4 F*4
G#*4 F#*4 E*4 D*4 C*4
G#*4 F#4 F#*4 E4 E*4 D4 D*4 C4 C*4
C#*4 B*3 A*3 G*3 F*3

/tuning/files/Schulter%20sound%20files/eb7ei01.mid

In this sequence the stable sonorities, here complete 2:3:4 trines,
occur on the upper keyboard, alternating with ~14:18:21:24 sonorities
which expand to another complete trine. The lowest and third-lowest
voices _descend_ through four whole-tones (e.g. C#*4-B*3-A*3-G*3-F*3
in the lowest voice), while the other pair of voices alternate motion
by descending near-7:6 minor thirds and ascending cadential dieses.

These sequences feature strong cadential progressions, but an overall
structure outside the usual diatonic order. At least to my ears, they
have a certain "21st-century" sound, and can contrast with passages
more along typical 13th-14th century lines.

------------------------------------------------
4.2. The "superfourth" or small diminished fifth
------------------------------------------------

Another metachromatic progression involves a sonority combining a
usual 11-flavor minor third (~286.18 cents) plus a 7-flavor minor
third (~264.50 cents) to form an outer inteval of a fourth-plus-diesis
or "superfourth," to borrow Dave Keenan's term, at ~550.68 cents, very
close to a pure 11:8 (~551.32 cents).

Characteristic resolutions involving diesis steps between the two
keyboards are the following, with one of the minor thirds contracting
to a unison and the other expanding to a fifth:

E*4 -- +209 -- F#*4 E*4 -- -55 -- E4
(264) (705) (264) (0)
D4 -- -231 -- B*3 D4 -- +209 -- E4
(551,286) (705,0) (551,286) (705,705)
B3 -- +55 -- B*3 B3 -- -209 -- A3

(m3-1 + m3-5) (m3-1 + m3-5)

In the first progression, the regular minor third between the lower
pair of voices (B3-D4) contracts to a unison at B*3, with the lowest
voice ascending by a diesis and the middle voice descending by a large
whole-tone at ~230.90 cents (almost exactly 8:7, ~231.17 cents). The
7-flavor minor third between the upper voices (D4-E*4) expands to a
fifth (B*3-F#*4) by way of this descending motion in the middle voice
coupled with the ascent of the highest voice by a regular whole-tone
of ~209.21 cents.

In the second progression, the lower pair of voices expand from a
regular minor third B3-D4 to a fifth (A3-E4), each moving by a regular
whole-tone. The small or 7-flavor minor third D4-E*4 between the upper
pair of voices contracts to a unison on E4, with the highest voice
descending by a diesis (E*4-E4).

The superfourth between the outer voices B3-E*4 itself resolves in
either progression to a stable fifth (B*3-F#*4 or A3-E4) by parallel
motion rather than directed contrary motion; it is the resolutions of
the minor thirds (m3-1, m3-5) which guide the cadential action.

This is also a very typical cadential role for usual diminished fifths
or augmented fourths in Gothic or neo-Gothic music, lending a striking
color to resolutions where other unstable intervals resolving by
contrary motion play a guiding role.

To illustrate this point, let us consider progressions with regular
diminished fifths corresponding to the above examples:

F4 -- +209 -- G4 F4 -- -77 -- E4
(286) (705) (286) (0)
D4 -- -209 -- C4 D4 -- +209 -- E4
(572,286) (705,0) (572,286) (705,705)
B3 -- +77 -- C4 B3 -- -209 -- A3

(m3-1 + m3-5) (m3-1 + m3-5)

This kind of progression seems rather common in the 13th century,
where Pythagorean tuning provides the intonational norm: the outer
diminished fifth B3-F4 resolves by parallel motion to a fifth, while
one of the minor thirds contracts to a unison and the other expands to
a fifth.[11]

Substituting the superfourth for the usual diminished fifth does not
seem to change the basic logic of the progression, but does introduce
both the special color of this interval so close to a pure 11:8, and
the metachromatic element of the diesis as an ascending or descending
melodic step.

-----
Notes
-----

9. While the 7-flavor metachromatic progressions of the e-based tuning
use the "natural" 55-cent diesis between the keyboards arising in a
regular 24-note tuning, other types of metachromatic schemes may use
noncontiguous or even "artificially engineered" intervals between the
manuals. In the Pythagorean tricomma tuning, for example, the manuals
are placed at the interval of three Pythagorean commas or a "tricomma"
(~70.38 cents), or 36 pure fifths up, this interval serving in a
metachromatic fashion as a small semitone. In the 24-note variation on
17-tET discussed in Part I, n. 7, two keyboards in Eb-G# tunings are
placed at the arbitrary distance of ~55.11 cents, almost identical to
the e-based diesis, so as to attain pure 7-flavor minor thirds
(e.g. G3-A*3) and major sixths (e.g. G*3-F4) at 7:6 and 12:7.

10. In this tuning, the chain of four whole-tones F3-G3-A3-B3-C#4 or
C4-D4-E4-F#4-G#4 outlines an augmented fifth F3-C#4 or C4-G#4 with a
size of ~836.86 cents, quite close to Phi at ~833.09 cents. Around
1325, Jacobus of Liege describes a corresponding interval in
Pythagorean tuning, the _tetratonus_ equal to four 9:8 whole-tones
or 6561:4096 (~815.64 cents).

11. In Pythagorean intonation, these last examples would be:

F4 -- +204 -- G4 F4 -- -90 -- E4
(294) (702) (294) (0)
D4 -- -204 -- C4 D4 -- +204 -- E4
(588,294) (702,0) (588,294) (702,702)
B3 -- +90 -- C4 B3 -- -204 -- A3

(m3-1 + m3-5) (m3-1 + m3-5)

Most respectfully,

Margo Schulter
mschulter@value.net