Yahoo Tuning Groups Ultimate Backup tuning Re: The e-based tuning and metachromatic progressions (Part 3)

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The e-based tuning and metachromatic progressions:
A feast of neo-Gothic flavors
(Part 3: Metachromatic shifts and the near-Phi sixth)
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For Parts 1 and 2 of this article, please see:
/tuning/topicId_20573.html#20573 (Part 1)
/tuning/topicId_23881.html#23881 (Part 2)

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)

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4.3. Metachromatic shifts: intonational warp space
--------------------------------------------------

One "special effect" in the e-based tuning is a familiar progression
diverted by a metachromatic 7-flavor cadence leading to an unexpected
destination. While metachromatic sequences such as those discussed in
Section 4.1 seem to have their own internal logic and sense of
equilibrium, shifts of the kind we now consider can have an effect of
dramatic surprise, as if a warp space had suddenly opened in the usual
diatonic fabric.

As a basis for the examples that follow, let us consider a routine
progression such as the following:

A4 G4 F4 E4 F4
E4 D4 C4 B3 C4
C4 Bb3 A3 G3 F3

MIDI example: <http://value.net/~mschulter/eb7sh000.mid>

In this type of idiom, characteristic of some 14th-15th century
European styles and also common in neo-Gothic music, a series of sixth
sonorities leads to a usual expansive cadence (M6-8 + M3-5) arriving
at a complete 2:3:4 trine (here G3-B3-E4 to F3-C4-F4). Our version in
the e-based tuning features the usual 11-flavor intervals and
resolutions (see Section 3.1).

As Johannes Boen writes in 1357, the unstable thirds and sixths are
like "forerunners and handmaidens" of the stable concord eventually
following [12].

In the following variation, however, we discover that these
forerunners have announced an unexpected guest:

A4 G4 F4 F*4
E4 D4 C4 C*4
C4 Bb3 A3 G*3 F*3

MIDI example: <http://value.net/~mschulter/eb7sh001.mid>

As an intonational closeup will show, the familiar diatonic degrees C4
and F4 have suddenly become the upper notes of the 7-flavor cadential
sonority G*3-C4-F4 (a rounded 0-440-936 cents, ~7:9:12), with an
intensive resolution to F#*3-C*4-F*4 in which each of these upper
voices ascends by a diesis (C4-C*4, F4-F*4).[13]

F4 ------------------ +55 --- F*4
(495) (495) (495)
C4 ------------------ +55 --- C*4
(782,286) (936,440) (1200,705)
A3 --- -154 -- G*3 -- -209 --- F*3

This metachromatic "surprise" is made possible by the pivotal motion
of the lowest voice from A3 to G*3, a descending step of a diminished
third (~153.93 cents, equal to two diatonic semitones of ~76.97 cents).
The following cadence in itself is a standard 7-flavor resolution: it
is the sudden shift in getting there that produces the "warp" effect.

A somewhat milder variety of metachromatic shift may be the following,
in which our initial routine progression often leading to F is
diverted to a remissive 7-flavor cadence on G, with the upper voices
progressing by usual diatonic steps (C4-D4, F4-G4):

A4 G4 F4 G4
E4 D4 C4 D4
C4 Bb3 A3 G*3 G3

MIDI example: <http://value.net/~mschulter/eb7sh002.mid>

Here is an intonational close-up:

F4 ------------------ +209 --- G4
(495) (495) (495)
C4 ------------------ +209 --- D4
(782,286) (936,440) (1200,705)
A3 --- -154 -- G*3 -- -55 --- G3

In this more "gentle" shift, the upper voices have familiar diatonic
patterns of steps: E4-D4-C4-D4 or A4-G4-F4-G4.

Metachromatic shifts of these kinds, ranging from the more "exuberant"
or "warplike" to the more "gentle," invite much exploration in
practice and theory. They might be seen as a further development of
the bold medieval chromaticism of Marchettus of Padua (1318).[14]

---------------------------------
4.4. Subdiesis or 17-comma shifts
---------------------------------

A more subtle kind of nuance in metachromatic progressions and
cadences involves shifts by the small interval of the subdiesis or
"17-comma" of around 21.68 cents. As discussed in Section II (Part 1),
this subdiesis is equal to the difference between a regular diatonic
semitone (e.g. E-F) at ~76.97 cents, and a diesis or small semitone
(e.g. E-E*) at ~55.28 cents.

Additionally, the subdiesis is equal to the amount by which 17 fifths
in the e-based tuning fall short of 10 pure octaves, thus the
alternate name of "17-comma."[15]

Here is a diagram showing the division of the regular semitone E-F
into diesis and subdiesis, a division with immediate relevance to the
musical examples which follow:

55.3 21.7
|------------diesis-----------|--subdiesis--|
E E* F
|-------------------------------------------|
diatonic semitone
77.0

In various progressions, notes a subdiesis apart such as E* and F may
serve as alternative versions of the "same" degree or cadential goal,
and this "near-equivalence" can facilitate motion from one 12-note
manual to the other. Here we consider some routine illustrations.

Let us begin with the same standard progression as in our last
examples, concluding with a cadence on F:

A4 G4 F4 E4 F4
E4 D4 C4 B3 C4
C4 Bb3 A3 G3 F3

MIDI example: <http://value.net/~mschulter/eb7sh000.mid>

Here is the same basic progression, but with the conclusion modified
to an equivalent (or "near-equivalent") 7-flavor cadence on E*:

A4 G4 F4 E4 E*4
E4 D4 C4 B3 B*3
C4 Bb3 A3 F#*3 E*3

MIDI example: <http://value.net/~mschulter/eb7cs001.mid>

Here the trine E*3-B*3-E*4 serves as a near-equivalent of F3-C4-F4.
As an intonational closeup of the concluding cadence will show, this
substitution makes possible a standard 7-flavor metachromatic
resolution:

F4 ---- -77 --- E4 --- +55 --- E*4
(495) (495) (495)
C4 ---- -77 --- B3 --- +55 --- B*4
(782,286) (936,440) (1200,705)
A3 --- -231 -- F#*3 -- -209 --- E*3

In this intensive cadence from F#*3-B3-E4 to E*3-B*3-E*4, the lower
voice descends by a regular whole-tone (F#*3-E*3) while the upper
voices each ascend by a diesis (B3-B*3, E4-E*4).

Although no direct melodic motion or "shift" of a subdiesis occurs,
there is a downward "drift" of a subdiesis, as may be seen in the
melodic progressions of the upper parts: C4-B3-B*3 and F4-E4-E*4.
These figures involve an approach to the 7-flavor sixth sonority by
descents of a regular 77-cent diatonic semitone (C4-B3, F4-E4),
followed by resolving ascents of a 55-cent diesis in the usual
metachromatic manner. The difference between these motions measures
the 22-cent downward drift.

An example very concisely illustrating this kind of drift is the
following:

F4 E4 E*4
C4 B3 B*3
F3 F#*3 E*3

MIDI example: <http://value.net/~mschulter/eb7cs002.mid>

Here we begin at the stable trine F3-C4-F4, and move to a 7-flavor
cadential sonority F#*3-B3-E4, again resolving to E*3-B*3-E*4, with a
downward drift of a subdiesis from our starting point. In its approach
to this cadential sonority, the lowest voice ascends (F3-F#*3) by a
kind of small whole-tone of around 187.53 cents, a subdiesis narrower
than the usual step F3-G3.

Again, each upper voice descends by a regular semitone (C4-B3, F4-E4)
and then cadentially ascends by a diesis (B3-B*3, E4-E*4). The
difference of 22 cents between these steps measures the drift.

Direct subdiesis shifts can also be a pleasant nuance of style, as in
this idiom:

D4 E4 E*4
C4 B3 B*3
G3 F#*3 E*3

MIDI example: <http://value.net/~mschulter/eb7cs003.mid>

Here a relatively concordant but unstable G3-C4-D4 sonority (~6:8:9)
builds up to the sixth sonority F#*3-B3-E4, resolving again to
E*3-B*3-E*4.

An intonational closeup may show the logic of the subdiesis shift
G3-F#*3 in the lowest voice:

D4 ---- +209 --- E4 --- +55 --- E*4
(209) (495) (495)
C4 ---- -77 --- B3 --- +55 --- B*4
(705,495) (936,440) (1200,705)
G3 ---- -22 -- F#*3 -- -209 --- E*3

In the first sonority G3-C4-D4, all notes are on the same keyboard,
as they must be in order to form the regular fifth (~3:2), fourth
(~4:3), and major second (~9:8) of this ~6:8:9 sonority.

To form the following 7-flavor cadential sonority F#*3-B3-E4, near
7:9:12, however, the lowest voice must move from G3 to F#*3 on the
upper keyboard, descending by a subdiesis. From this point, a typical
resolution to E*3-B*4-E*4 follows, as in our previous examples.

In the outgoing and ebullient intonational setting of the e-based
tuning with its various flavors and colorful contrasts between
stability and instability, such subdiesis or "17-comma" shifts or
drifts can be pleasantly stimulating.

----------------------------------------------
4.5. The 7-flavor or metachromatic minor sixth
----------------------------------------------

On our 24-note keyboard, a note on the lower manual and the note
visually a fifth higher on the upper manual (e.g. F3-C*4) form the
interval of a small or 7-flavor minor sixth at ~759.89 cents, rather
close to 14:9 (~764.92 cents).

A very characteristic resolution by oblique motion (m6-5) involves the
descent of the upper voice downward by a 55.28-cent diesis, here
C*4-C4, to arrive at a stable fifth (~704.61 cents). Here is an
example in two voices:

C*4 C4
F3

<http://value.net/~mschulter/eb7ms001.mid>

An intensified three-voice version of this resolution adds a middle
voice at a fifth above the lowest voice, forming an unstable sonority
featuring the vertical interval of the diesis between the two upper
voices (~0-705-760 cents):

C*4 C4
C4
F3

<http://value.net/~mschulter/eb7ms002.mid>

As in the previous example, the highest voice descends by a diesis
(C*4-C4), resolving the small outer sixth to a fifth, and here also
the vertical diesis between the upper voices to a unison:

C*4 --- -55 --- E4
(55) (0)
C4 ------------------
(760,705) (705,705)
F3 ------------------

These examples are metachromatic variations on 13th-century
progressions in which a minor sixth resolves to a fifth, a very
expressive figure in Pythagorean tuning also.[16]

--------------------------------------------------------
5. The "near-Phi" or supraminor sixth: "Fibonacci drift"
--------------------------------------------------------

A very special idiom in the e-based tuning features the supraminor
sixth at around 836.86 cents, quite close to the ratio of the Golden
Section, or Phi, at ~1.61803398874989484820459, or ~833.09 cents.

More generally, this type of "Phi-sixth" is available in a variety of
neo-Gothic tunings including a "17-flavor" set of supraminor/submajor
thirds and sixths[17], so that most of what follows could apply to any
of these tunings, allowing for fine shadings of interval sizes and
colors.

In a neo-Gothic setting, Phi-sixths when presented in certain
"floating" contexts may have a quality at once of complexity and of
relative blend, with the overall impression one of a certain "nebulous
concord." They are very different than the primary concords, stable
fifths and fourths, and yet do not necessarily have any strong sense
of directed tension or motion. One might describe them as having a
pleasantly foggy or almost Debussyan quality, inviting a kind of
"neo-Gothic Impressionism."[18]

Accordingly, an idiom known as "Fibonacci drift"[19] features two
upper voices moving in fourths above a sustained lowest note,
"drifting" around a sonority with a supraminor or Phi-sixth plus a
supraminor third, in the e-based tuning a rounded 0-341-837 cents.
Here is an example of Fibonacci drift, illustrating one kind of
transitional progression for returning to a "usual" neo-Gothic style,
with the rhythm possibly best expressed as a mixture of 3/4 and 2/4:

(3/4) (2/4)
1 2 + 3 + | 1 + 2 + | 1 2 | 1 2 ||
F#4 G4 F#4 F4 F#4 G4 F#4 F4 F#4 F4 F#4
C#4 D4 C#4 C4 C#4 D4 C#4 C4 C#4 B3
Bb3 B3

<http://value.net/~mschulter/ebphi001.mid>

Most typically, as in the e-based tuning, the Phi-sixth is spelled as
an augmented fifth and the supraminor third as an augmented second,
here Bb3-C#4-F#4, although some tuning systems use other spellings.

Let us first focus on the basic Fibonacci drift figure, with its
melodic pattern of diatonic and chromatic semitone steps oscillating
around the Phi-sixth sonority. Here it may simplify matters to
indicate the intervals formed by the upper voices with the sustained
lowest note as they move together in fourths (~495.39 cents):

F#4 -- +77 -- G4 -- -77 -- F#4 -- -132 -- F4 -- +132 -- F#4 ...
(837) (913) (837) (705) (837)
C#4 -- +77 -- D4 -- -77 -- C#4 -- -132 -- C4 -- +132 -- C#4 ...
(341) (418) (341) (209) (341)
Bb3

From a vertical perspective, the two upper voices start at the
Phi-sixth sonority Bb3-C#4-F#4, and ascend by 77-cent diatonic
semitones to a usual major sixth sonority Bb3-D4-G4 (0-418-913 cents),
then returning to the Phi sonority. Next they descend by 132-cent
chromatic semitones to the relatively concordant quintal/quartal
sonority Bb3-C4-F4, or ~8:9:12, combining an outer fifth and upper
fourth with a lower major second, and ascend back to the Phi sonority.

The total effect is a kind of oscillation or "drift" which can be
repeated with various forms of elaboration or ornamentation.
Melodically, the two upper voices undulate together through a cycle of
diatonic and chromatic semitone steps adjacent to their notes in the
Phi sonority: here C#4-D4-C#4-C4-C#4... and F#4-G4-F#4-F4-F#4....

In this example, the Fibonacci drift passage, involving as it does a
kind of 17-flavor sonority, not inappositely concludes with a standard
17-flavor cadence, typically marking a return in a composition or
improvisation back to a more usual neo-Gothic style:

F#4 -- -132 -- F4 --- +132 -- F#4
(495) (363) (705)
C#4 ----------------- -209 -- B3
(837,341) (705,341) (705,0)
Bb3 ----------------- +132 -- B3

To make this transition, the highest voice descends from the
supraminor sixth to the fifth above the lowest note (F#4-F4), thus
forming a usual 17-flavor cadential sonority Bb3-C#4-F4 (see Part 2,
Section 3.2), or 0-341-705 cents, which then resolves in standard
fashion, the lower supraminor third contracting to a unison and the
upper submajor third expanding to a fifth. We arrive at the stable
fifth B3-F#4.

In addition to its vertical affinity with the Fibonacci drift passage,
this 17-flavor cadence has a melodic affinity with its characteristic
chromatic semitone steps, here Bb3-B3 and F4-F#4 in the two outer
voices. There may be a pleasing contrast between the coloristic
drifting or "floating" quality of the step F4-F#4 (or F#4-F4) in the
Fibonacci figure, and its use here in directed cadential action.

Our second example concludes the same Fibonacci drift passage with a
different variety of cadential maneuver:

(3/4) (2/4)
1 2 + 3 + | 1 + 2 + | 1 2 | 1 2 ||
F#4 G4 F#4 F4 F#4 G4 F#4 F4 F#4 G4
C#4 D4 C#4 C4 C#4 D4 C#4 C4 C#4 D4
Bb3 Bb3 A3 G3

<http://value.net/~mschulter/ebphi002.mid>

Here the lower voice of the Phi sonority Bb3-C#4-F#4 descends by a
diatonic semitone (Bb3-A3) to form the regular major sixth sonority
A3-C#4-F#4 (0-418-913 cents), leading to a routine 11-flavor cadence
(Part 2, Section 3.1). Here is an intonational closeup[20]:

F#4 ---------------- +77 -- G4
(495) (495) (495)
C#4 ---------------- +77 -- D4
(837,341) (913,418) (1200,705)
Bb3 -- -77 -- A3 --- -209 -- G3

In a variation on this last example, the Fibonacci drift passage is
followed by a similar cadence, but in the 7-flavor:

(3/4) (2/4)
1 2 + 3 + | 1 + 2 + | 1 2 | 1 2 ||
F#4 G4 F#4 F4 F#4 G4 F#4 F4 F#4 F#*4
C#4 D4 C#4 C4 C#4 D4 C#4 C4 C#4 C#*4
Bb3 Bb3 G#*3 F*3

<http://value.net/~mschulter/ebphi003.mid>

An intonational closeup reveals that the cadential approach involves
a special melodic interval in the lowest voice:

F#4 ----------------- +55 -- F#*4
(495) (495) (495)
C#4 ----------------- +55 -- C#*4
(837,341) (936,440) (1200,705)
Bb3 -- -99 -- G#*3 -- -209 -- F#*3

In shifting the texture from the Phi sonority Bb3-C#4-F#4 to the
7-flavor cadential sonority G#*3-C#4-F#4 (0-440-936 cents, ~7:9:12),
this voice descends Bb3-G#*3, an interval equal to a 76.97-cent
diatonic semitone (Bb3-A3) plus a 21.68-cent subdiesis (A3-G#*3).
This interval of ~98.65 cents is curiously very close to 18:17 (~98.95
cents). The resulting 7-flavor sonority then resolves in a standard
metachromatic progression, the lowest voice descending by a regular
whole-tone (G#*3-F#*3) and the upper voices ascending by 55-cent
dieses (C#4-C#*4, F#4-F#*4).

Our three examples have shown how the Fibonacci drift idiom can be
concluded with a usual cadence in the 17-flavor, 11-flavor, or
7-flavor, sampling as it were the three principal flavors of the
e-based tuning. As it happens, all three cadences have involved an
intensive manner, with descending whole-tone and ascending semitone
motions; remissive resolutions further multiply the variety of
permutations and choices.[21]

With its richness of intonational flavors, progressions, and
metachromatic idioms, the e-based tuning may exemplify the plethora of
possibilities offered by neo-Gothic tunings and temperaments.

-----
Notes
-----

12. Wolf Frobenius, _Johannes Boens Musica und Seine
Konsonanzenlehre_, Freiburger Schriften zur Musikwissenschaft
(Musikwissenschaftliche Verlags-Gesselschaft mbH Stuttgart, 1971),
Latin text at p. 70.

13. To make this kind of rather intricate diagram more visually
appealing, I use explicit dashed lines to show that a given voice
sustains a note while other voices move, although this would follow
from my general convention that each note in a given voice is
sustained until the next note in that voice, or the end of the
example, unless a rest is indicated.

14. Marchettus describes an idiom he calls a "feigned color" -- freely
translated, a deceptive cadential inflection -- in which a major sixth
expected to expand to an octave (e.g. E3-C#4 to D3-D4) instead
contracts to a fifth with the upper voice descending by a direct
chromatic semitone (e.g. E3-C#4 to F3-C4, upper voice C#4-C4). While
this idiom involves an unexpected conclusion to an anticipated
cadence, our "metachromatic shifts" involve a cadence at an unexpected
place which itself follows standard progressions (here M6-8 and M3-5).
For more on Marchettus, see for example my paper written to celebrate
Microfest 2001, "Xenharmonic Excursion to Padua, 1318: Marchettus, the
cadential diesis, and neo-Gothic tunings" (2001),
http://value.net/~mschulter/marchetmf.txt (ASCII text version)
http://value.net/~mschulter/marchetmf.zip (ASCII text and PostScript);
Joseph L. Monzo, _Speculations on Marchetto of Padua's "Fifth-Tones"_
(1998), http://www.ixpres.com/interval/monzo/marchet/marchet.htm; and
Jay Rahn, "Practical Aspects of Marchetto's Tuning," _Music Theory
Online_ 4.6 (1998),
http://boethius.music.ucsb.edu/mto/issues/mto.98.4.6/mto.98.4.6.rahn.html.

15. Seventeen fifths are precisely equal to 10 octaves in 17-tET,
where the fifth is defined as 10/17 octave.

16. In a standard medieval Pythagorean tuning, the regular minor sixth
is 128:81 (~792.18 cents), and the diatonic semitone 256:243 (~90.22
cents) -- with the fifth, of course, a pure 3:2 (~701.96 cents). Here
is a Pythagorean version of the striking three-voice resolution shown
above, with an unstable sonority of 162:243:256 (~0-702-792 cents),
the upper voice descending by a usual diatonic semitone or minor
second (m6-5 resolution with lowest voice, m2-1 resolution with middle
voice):

C4 --- -90 ---- B3
(90) (0)
B3 ------------------
(792,702) (702,702)
E3 ------------------

17. The term "17-flavor" suggests ratios for these intervals in the
general vicinity of 17:14 (~336.13 cents), 21:17 (~365.83 cents);
34:21 (~834.17 cents, very close to Phi), and 28:17 (~863.87 cents).

18. I am reminded of Carl Sandburg's poem about the fog coming on
little cat feet, or of the Great Nebula in Orion.

19. This idiom is named for the mathematician Leonardo of Pisa, known
by his nickname of Fibonacci, who in his _Liber abaci_ (c. 1202) made
what may be the first significant introduction of Arabic decimal
numbers to the non-Islamic portion of medieval Europe. Fibonacci's
famous series of numbers, devised to solve a problem concerning the
reproduction of rabbits, begins 1,1,2,3,5,8,13,21,34,55..., each new
term starting with the third being equal to the sum of the two
previous terms. The ratio between successive terms converges on the
Golden Section. On Phi and its possible relationship to harmonic
complexity or "entropy," see Margo Schulter and David Keenan, "The
Golden Mediant: Complex ratios and metastable musical intervals," 18
September 2000; /tuning/topicId_12915.html#12915.

20. A kind of precedent for this cadential approach might be found in
some 14th-century pieces where, in one interpretation of the
accidentals, an augmented fifth may move by descent of the lower voice
to a major sixth, expanding in usual fashion to an octave:

C#4 D4
F3 E3 D4

In 14th-16th century music, convention calls for various kinds of
accidental inflections sometimes specified in the notation, but often
left to the discretion of the performers. On the 14th-century question
of the augmented fifth in approaching an M6-8 progression, see
<http://www.medieval.org/emfaq/harmony/hex.html>, Section 2.4. In
Pythagorean tuning, this approach would look as follows:

C#4 --------------- +90 -- D4
(816) (906) (1200)
F3 -- -90 -- E3 -- -204 -- D3

Here the augmented fifth or _tetratonus_ F3-C#4, equal to four 9:8
whole-tones at 6561:4096 (~815.64 cents), proceeds by the descent of a
90-cent diatonic semitone in the lowest voice (F3-E3) to a regular
major sixth E3-C#4 at 27:16 (~905.87 cents), followed by the cadence
to the octave.

21. Standard remissive resolutions for the cadential sonorities of our
three Fibonacci drift examples: first example, Bb3-C#4-F4 to C4-G4
(17-flavor, lower voices contracting to unison on C4); second example,
A3-C#4-F#4 to G#3-D#4-G#4 (11-flavor); third example, G#*3-C#4-F#4 to
G#3-D#4-G#4 (7-flavor). The last two progressions involve the
equivalence D#4=Eb*4. "Standard" remissive or intensive resolutions
combine regular 209-cent whole-tone steps with steps of a 77-cent
diatonic semitone (11-flavor), a 132-cent chromatic semitone
(17-flavor), or a 55-cent diesis (7-flavor). It happens that our
24-note gamut makes available such resolutions in both the intensive
and remissive manners for these three examples.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: The e-based tuning and metachromatic progressions (Part 3)

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