Re: Adaptive JI/RI tuning and neo-Gothic valleys (Part IV)

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Adaptive rational intonation and neo-Gothic valleys
A 24-note scheme a la Vicentino
(Essay in honor of John deLaubenfels)
Part IV: Comma shifts, nuances, special sonorities (2)
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For earlier parts of this article, please see:

http://www.egroups.com/message/tuning/16640 (Part I)
http://www.egroups.com/message/tuning/17034 (Part II)
http://www.egroups.com/message/tuning/17339 (Part III)

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4.2. Alternative melodic nuances
--------------------------------

Although these "necessary" comma shifts are often desirable and indeed
delectable, we often have available alternative forms of progressions
featuring the same vertical sonorities but altered melodic cadential
steps not calling for such shifts. The result is a yet expanded
universe of choices.

To illustrate these alternative forms, let us consider the three comma
shift sequences we have just explored, beginning with our first
sequence with a chain of expansive resolutions from sixth sonorities
to trines:

1 2 | 1 2 & | 1 1 2 | 1 2 & | 1
F4 G^4 A^4 G^4 F#^4 G4 F4 G^4 A4 G4 F#^4 G4
F4 D^4 E^4 D^4 C#^4 D4 F4 D^4 E4 D4 C#^4 D4
Bb3 A^3 A3 G3 Bb3 A3 G3

(standard steps, comma shift) (alternative steps)

Both our original version with standard melodic steps and our new
alternative version share identical tunings for vertical sonorities:
trines are at 2:3:4, cadential sixth sonorities at 7:9:12, and the
mildly unstable quintal/quartal sonority in the second measure
(A^3-D^4-G^4 or A3-D4-G4) at 9:12:16.

The subtle difference, which makes a comma shift unnecessary in our
new version, occurs in the first remissive resolution involving the
sixth sonority Bb3-D^4-G^4, which progresses in the standard manner to
A^3-E^4-A^4 at the beginning of the second measure in our original
example, but to A3-E4-A4 on the lower keyboard in this variant.

With the lowest voice now poised on A3, we can move through the
connecting sonority A3-D4-G4 (a septimal comma lower than in the
original verson) to the sixth sonority A3-C#^4-F#^4, available at the
desired 7:9:12 without the need for any comma shift, and followed by a
standard intensive resolution to G3-D4-G4 as in the first version.

Looking more closely at the two versions of the remissive resolution
of Bb3-D^4-G^4, we can see the altered melodic steps in the second
version which have made it possible to maintain vertical sonorities at
the simplest ratios while avoiding any comma shifts:

G^4 A^3 G^4 A4
D^4 E^3 D^4 E4
Bb3 A^3 Bb3 A3

(standard steps) (alternative steps)

In the standard 7-flavor version, we have the descending semitone
Bb3-A^3 in the lowest voice at 28:27, and ascending whole-tones
D^4-E^4 and G^4-A^4 in the two upper voices at 9:8.

In our alternative version, however, we have a descending semitone
Bb3-A3 at the usual Pythagorean 256:243 or a rounded 90 cents, a
septimal comma larger than 28:27 at 63 cents; and whole-tone steps of
D^4-E4 and G^4-A4, a septimal comma narrower than the standard 9:8, or
about 177 cents rather than the usual 204 cents. More precisely, this
small whole-tone has a ratio equal to 9:8 divided by the 64:63 comma,
or 567:512 (~176.65 cents).

A yet closer look at fine intonational structure reveals how both
versions involve identical vertical intervals, but with differently
allocated melodic motions. Numbers in parentheses show vertical
intervals between each voice and any higher voices, while signed
numbers show ascending or descending melodic steps in each voice:

standard 7-flavor alternative 7-flavor

G^4 ------ +204 ----- A^4 G^4 ----- +177 ----- A4
(498) (498) (498) (498)
D^4 ------ +204 ----- E^4 D^4 ----- +177 ----- E4
(933,435) (1200,702) (933,435) (1200,702)
Bb3 ------ -63 ----- A^3 Bb3 ----- -90 ----- A3

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

Both versions feature two-voice resolutions expanding from a 9:7 major
third at a rounded 435 cents to a fifth, and from a 12:7 major sixth
at 933 cents to an octave, in either case involving expansion over a
total distance of 267 cents (the size of the 7:6 minor third, ~266.87
cents).

In the standard version, this distance is allocated between a
supercompact descending semitone of 28:27 or 63 cents, and a usual
Pythagorean whole-tone of 9:8 or 204 cents. In the alternative
version, we have a usual Pythagorean semitone of 256:243 or 90 cents,
and small or compressed whole-tones at 567:512 or 177 cents.

By using these alternative melodic steps for remissive cadences, we
can extend our "septimal comma pump" sequence of sixth sonorities
while avoiding comma shifts -- or, for a hypothetical ensemble of
variable-pitch performers, comma drift:

1 2 | 1 2 & | 1 2 | 1 2 | 1 2 | 1
F4 G^4 A4 G4 F#^4 G4 E^4 F4 D^4 E4 C#^4 D4
F4 D^4 E4 D4 C#^4 D4 B^3 C4 A^3 B3 G#^3 A3
Bb3 A3 G3 F3 E3 D3

Interestingly, this style of adaptive JI involves regular Pythagorean
steps (9:8, 256:243) in the lowest voice, with three varieties of
whole-tone steps in the upper voices: the usual 9:8 at 204 cents; the
large 8:7 at 231 cents, a septimal comma wider (F4-G^4 in the highest
voice, measure 1); and the small 567:512 at 177 cents, a septimal
comma narrower (e.g. D^4-E4 and G^4-A4, measures 1-2). Intensive
resolutions retain standard supercompact 28:27 semitones at 63 cents
in the upper voices (e.g. C#^4-D4, F#^4-G4, measures 2-3).

Here is the same alternative approach applied to our second sequence:

1 2 3 | 1 2 3 | 1 1 2 3 | 1 2 3 | 1
G^4 F4 E^4 F#^4 G4 G^4 F4 E4 F#^4 G4
D^4 Bb3 A^3 C#^4 D4 D^4 Bb3 A3 C#^4 D4
G^3 A^3 A3 G3 G^3 A3 G3

(standard version) (alternative version)

The first remissive cadence in the sequence involving the contractive
minor seventh sonority G^3-Bb3-F4 now resolves to the fifth A3-E4,
instead of the standard A^3-E^4 on the upper keyboard, again making
possible a move to the next cadential sonority A3-C#^4-F#^4 without a
comma shift, resolving in either version to G3-D4-G4. Here is an
intonational diagram of the remissive seventh sonority cadence in its
two versions:

standard 7-flavor alternative 7-flavor

F4 ------ -63 ----- E^4 F4 ----- -90 ----- E4
(702) (702) (702) (702)
Bb4 ------ -63 ----- A^3 Bb4 ----- -90 ----- A3
(969,267) (702,0) (969,267) (702,0)
G^3 ------ +204 ----- A^3 G^3 ----- +177 ----- A3

(m7-5 + m3-1) (m7-5 + m3-1)

In either version, the two-voice resolutions from a 7:6 minor third to
a unison, and from a 7:4 minor seventh to a fifth, involve a total
contraction of 267 cents -- allocated between motions of a 63-cent
descending semitone and a 204-cent ascending whole-tone in the
standard version, and a 90-cent semitone and a 177-cent whole-tone in
the alternative version.

The same altered steps also give us the option of avoiding comma
shifts in our third sequence:

1 2 3 | 1 2 3 | 1 1 2 3 | 1 2 3 | 1
G^4 F4 E^4 F#^4 G4 G^4 F4 E4 F#^4 G4
D^4 A^3 E3 D4 D^4 A3 D4
G^3 A^3 A3 G3 G^3 A3 G3

(standard version) (alternative version)

The contrast between these versions may facilitate meaningful musical
statements and structures. For example, one might use the alternative
version without comma shifts for the first statement of a passage, and
the standard version with its striking double comma shift for a
repetition with a new and distinctive element.

Certain progressions permit three alternative melodic interpretations
all sharing the same vertical intervals, as with the following
neo-Gothic resolution from a quintal/quartal sonority of 8:9:12 to a
complete trine featuring a directed resolution between the lower
voices from a vertical major second to a fifth:

(standard 3-flavor) (variant 1) (variant 2)

C4 -- +408 -- E4 C4 -- +435 -- E^4 C^4 -- +381 -- E4
(498) (498) (498) (498) (498) (498)
G3 -- +408 -- B3 G3 -- +435 -- B^3 G^3 -- +381 -- B3
(702,204) (1200,702) (702,204) (1200,702) (702,204) (1200,702)
F3 -- -90 -- E3 F3 -- -63 -- E^3 F^3 -- -117 -- E3

(M2-5) (M2-5) (M2-5)

The melodic semitones and major thirds, suggesting a certain
"pelog-like" affinity, are open to three nuances. In a standard
Pythagorean or 3-flavor version, they have sizes of 256:243 and 81:64,
or 90/408 cents. In a version with 7-flavor steps, moving from the
lower to the upper manual, they have an accentuated polarity at sizes
of 28:27 and 9:7, or 63/435 cents. In a version where they are altered
by a septimal comma in the opposite direction, moving from the upper
to the lower manual, they have sizes of 16384:15309 and 5103:4096, or
117/381 cents (more precisely 117.49 cents and 380.56 cents).[21]

Comma shifts and nuances, in addition to providing what can be
beautiful "special effect" progressions, add another dimension of
musical choice and contrast, along with such dimensions of choice as
3-flavor/7-flavor, expansive/contractive, or intensive/remissive.
"Intonational necessity," in the 21st century as at other points in
history, can often be synonymous with artistic opportunity.

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4.3. Special sonorities: the Erlichan ninth and near-11:8
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Along with cadential changes of flavor, comma shifts, and related
melodic nuances, our neo-Gothic JI system features some special
vertical sonorities. Here we consider two: a major ninth sonority of
the kind described by Paul Erlich, having in this tuning pure ratios
of 4:6:7:9 (e.g. A^3-E^4-G4-B^4); and a cadential major sixth sonority
featuring a near-11:8 superfourth, to use Dave Keenan's fitting term,
between the two upper voices (e.g. B3-Eb4-G#^4).

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4.3.1. The Erlichan ninth (4:6:7:9)
-----------------------------------

Growing out of a suggestion by Paul Erlich[22], the neo-Gothic
"Erlichan ninth" sonority consists of a fifth, minor seventh, and
major ninth above the lowest voice. In a pure 7-flavor tuning of the
kind available in our 24-note system, this sonority features just
ratios of 4:6:7:9 (a rounded 0-702-969-1404 cents), for example
A^3-E^4-G4-B^4.

From a neo-Gothic point of view, this sonority has a complex and
"different" sound, one might say "xenharmonic" in a very enchanting
sense. Its beguiling complexity is coupled with the property that all
six intervals have some degree of "compatibility" by 13th-century
standards: two ideally blending fifths; a relatively blending major
third and minor third; and a relatively tense minor seventh and major
ninth. There are no acutely tense intervals (m2, M7, A4, d5).

Although various resolutions are possible, the following directed
progression is especially favored and characteristic:

B^4 C5
G4 F4
E^4 F4
A^3 F3

(M9-12 + m3-1 + M3-5)

In this resolution, the outer 9:4 major ninth expands to a twelfth,
while the upper 7:6 minor third E^4-G3 and 9:7 major third G4-B^4
respectively contract to a unison and expand to a fifth.

While the pure 7-flavor tuning of 4:6:7:9 may be of special interest
here, this sonority and its majestic resolution are very beautiful
also in regular Pythagorean intonation, where the Erlichan ninth has a
more complex ratio of 36:54:64:81 (0-702-996-1404 cents), and in a
variety of neo-Gothic temperaments.

To understand the musical logic of this characteristic resolution, we
may approach it as superimposing or uniting two three-voice
progrssions of the Gothic era, here shown in their traditional
Pythagorean tuning:

B4 C5 B4 C5
E4 F4 G4 F4
A3 F3 E4 F4

(M9-12) (m3-1 + M3-5)

The first progression involves a sonority favored by Jacobus of Liege
(c. 1325) featuring an outer major ninth "split" into two fifths. In
this resolution, the outer ninth expands to a twelfth by near-conjunct
contrary motion, the upper voice ascending by a semitone and the lower
voice descending by a major third.

The second progression is a very common resolution of the _quinta
fissa_ or "split fifth" of Jacobus (a fifth "split" into a major and
minor third), with the minor third contracting to a unison and the
major third expanding to a fifth.

Since the "Erlichan ninth" (4:6:7:9 in its simplest tuning) can be
seen as a variant on the _nona fissa_ or "split ninth" of Jacobus with
its two ideally concordant fifths (4:6:9), it has the fuller name in
neo-Gothic theory of _nona fissa Erlichana_ or "Erlichan split ninth."

The Erlichan ninth is an example of a neo-Gothic sonority I might
never have conceived on my own, but which adorns the style in a very
special way, a xenharmonic "stranger" very happily adopted.

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4.3.2. A hybrid sixth sonority: the near-11:8 superfourth
---------------------------------------------------------

In addition to the usual complement of 3-flavor and 7-flavor cadential
resolutions, our adaptive rational tuning includes a special form of
expansive sixth sonority combining a Pythagorean diminished fourth or
schisma major third very close to 5:4 with a pure 12:7 major sixth,
e.g. B3-Eb4-G#^4 (a rounded 0-384-933 cents). This arrangement results
in an interval of about 549 cents between the two upper voices, a
near-pure 11:8 "super fourth" as Dave Keenan[23] aptly describes it.

The standard resolution of this hybrid sixth sonority follows a
familiar pattern, with the major third expanding to a fifth and the
major sixth to an octave:

G#^4 A4
Eb4 E4
B3 A3

(M6-8 + M3-5)

Here the diminished fourth B3-Eb4 (8192:6561, ~384.36 cents) is a
Pythagorean comma narrower (531441:524288, ~23.46 cents) than a
regular major third at 81:64, while the 7-flavor major sixth B3-G#^4
at 12:7 (~933.13 cents) is a septimal comma or 64:63 (~27.26 cents)
wider than a regular Pythagorean major sixth at 27:16.

The fourth between the two upper voices is enlarged beyond a pure 4:3
by the sum of these two commas, 59049:57344 or ~50.72 cents, yielding
an interval of 19683:14336 or ~548.77 cents, happily close to a pure
a pure 11:8 at ~551.32 cents, falling short by about 2.55 cents.[24]

This superfourth can have a quality at once tart, lending impetus to
cadential action, and yet engagingly smooth and tranquil. Together
with the other intervals of this combination, it offers a variation on
the theme of expansive sixth sonorities at once recognizable and
quite unique in flavor.

A closer intonational look shows how the contrast between the narrow
major third and wide major sixth is reflected in the disparity of
melodic semitonal motions in this cadence:

G#^4 ------ +63 ----- A4
(549) (498)
Eb4 ------ +114 ----- E4
(933,384) (1200,702)
B3 ------ -204 ----- A3

The diminished fourth B3-Eb4 is a Pythagorean comma narrower than a
regular major third, and must expand by this extra distance to reach
the resolving fifth A3-E4 -- a total expansion of about 318 cents.[25]
Its upper voice ascends by a 114-cent chromatic semitone or apotome
Eb4-E4 (2187:2048, ~113.69 cents) rather than the regular 90-cent
diatonic semitone at an incisive 256:243, while the lowest voice
descends by a usual 204-cent whole-tone at 9:8 (B3-A3).

The major sixth B3-G#^4 at a pure 12:7 expands in standard 7-flavor
fashion to the outer octave of the resolving trine A3-E4-A4, its upper
voice ascending by a supercompact 63-cent semitone at 28:27 (G#^4-A4).
Here a total expansion of only about 267 cents is required.

An idiom highlighting the unique color of this progression is to
combine it with a variant on the "commatic genus" of melody described
by Johannes Boen in 1357 and used by Guillaume le Grant in the early
15th century (Section 4.1.2) featuring two consecutive semitones:

Bb4 A4 G#^4 A4
F4 E4 Eb4 E4
D4 C4 B3 A3

The disparate chains of melodic semitones in the two upper voices
(Bb4-A4-G#^4, 90-63 cents; F4-E4-Eb4, 90-114 cents) add to the drama
of the cadential sonority B3-Eb4-G#^4, with its superfourth mirroring
this disparity.

Idioms of this kind grow out of an intonational serendipity common to
various musical styles and tuning systems. Although our neo-Gothic JI
system is based on ratios of 2, 3, and 7, approximations of other
ratios such as 5:4 or 11:8 do arise in the "cracks" of the system.
These "unusual" intervals, like the augmented and diminished intervals
of meantone during the Manneristic Era around 1600, can serve as the
figures of a musical rhetoric opening new vistas of expression.

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4.4. Oblique resolutions: 27:14 and 14:9
----------------------------------------

In this survey, we have focused mainly on cadences involving directed
progressions by contrary motion, but 7-flavor intervals such as the
major seventh at 27:14 (~1137.04 cents) and minor sixth at 14:9
(~764.92 cents) can have an especially evocative effect in certain
resolutions by oblique motion. Here we consider two such resolutions
favored by composers such as Perotin and some of his 13th-century
successors, and available either in traditional 3-flavor versions or
new 7-flavor versions.

A most ear-catching opening in Perotin commences on a sonority with
fifth and major seventh above the lowest voice (128:192:243, a rounded
0-702-1110 cents), the major seventh obliquely resolving to the octave
of a complete trine:

E4 F4
C4
F3

This compelling flourish, with the acute Pythagorean major seventh and
the bright major third between the upper voices at 81:64 incisively
resolved to outer octave and upper fourth by the 90-cent semitone
motion of the highest voice, sets the stage for a composition in
Perotin's architectonic style.

Jacobus of Liege may be alluding to this kind of figure when he
follows 13th-century tradition in placing the major seventh among the
unequivocal or "incompatible" discords (along with m2, A4 or d5, and
also m6), but adds the qualification, "unless it proceeds to the
octave."[26)

The 7-flavor variation on this classic figure seems yet more intense
and with its "superacute" 27:14 major seventh and 9:7 major
third (14:21:27, 0-702-1137 cents):

E^4 F4
C4
F3

These unstable intervals get a "superincisive" resolution by way of
the 28:27 semitone E^4-F4 at 63 cents, a septimal comma narrower than
in the classic Pythagorean version.

A different kind of oblique resolution, this time with a descending
rather than ascending melodic semitone, proceeds from the minor sixth
to the fifth. Often ranked together with the major seventh in
13th-century theory as acutely tense, the minor sixth at its usual
Pythagorean size of 128:81 (~792.18 cents) can have a quality at once
edgy and languid[27], as in this final cadence:

F4 E4
A3

The 7-flavor variation with a 14:9 minor sixth seems to me more gentle
and mellow in its languidity, this interval resolving by the 63-cent
semitone step F4-E^4:

F4 E^4
A^3

The tension of this resolution in either flavor can be enhanced by
adding the fifth above the lowest voice, with a 90-cent (3-flavor) or
63-cent (7-flavor) vertical semitone between the upper parts resolved
to a unison as the outer minor sixth resolves to a fifth:

(3-flavor) (7-flavor)

F4 E4 F4 E^4
E4 E^4
A3 A^3

These idioms are meant only as a sampler: 13th-century sonorities with
intervals such as minor seconds, minor sixths, and major sevenths
participate in a variety of resolutions by directed contrary motion as
well as oblique motion[28], and invite enthusiastic interpretations in
a range of flavors.

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Notes to Part IV
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21. In each instance the sum of these two melodic steps is a pure 4:3
fourth at a rounded 498 cents, the distance by which the unstable
major second of the first sonority expands to reach a pure fifth, and
the outer fifth of the first sonority expands to reach the octave of
the resolving trine.

22. Paul Erlich's original query concerned a possible neo-Gothic use
for the more complex 11-limit sonority 4:6:7:9:11 found in 22-tone
equal temperament (22-tET). Experimentation suggested to me that this
sonority seemed a bit too complex or "dissonant" for a usual
neo-Gothic setting, but that the subset 4:6:7:9 had just the right
blend of complexity and "compatibility" to fit this musical language
in a way at once strange and beautiful.

23. See http://www.uq.net.au/~zzdkeena/Music/IntervalNaming.htm.

24. From another viewpoint, this superfourth Eb4-G#^4 is a septimal
comma wider than the Pythagorean augmented third or "Wolf fourth"
Eb4-G#4 at 177147:131072 (~521.51 cents), itself a Pythagorean comma
larger than a pure 4:3 fourth. Although the Wolf fourth is much less
smooth than the superfourth as a bare interval, proverbially being
named for the resemblance of its beating to the howling of wolves, it
can also have a pleasant effect in a sixth sonority such as B3-Eb4-G#4
(0-384-906 cents) resolving to A3-E4-A4. In the early 15th century,
when the 12-note Pythagorean tuning Gb-B was popular, this sonority
may often have occurred at written cadences of D3-F#3-B3 (realized as
D3-Gb3-B3) to C3-G3-C4. Mark Lindley suggests that the cadential
context and the tuning of D-Gb3 very close to a pure 5:4 serve to
"alleviate" the musical effect of the Wolf fourth, see "Pythagorean
Intonation and the Rise of the Triad," _Royal Musical Association
Research Chronicle_ 16:4-61 at 43 (1980), ISSN 0080-4460.

25. This total distance of expansion is equal to the size of the
Pythagorean augmented second or wide minor third, more precisely
19683:16384 (~317.60 cents).

26. Johannes de Garlandia (c. 1240?) states that any discord becomes
"equivalent to concord" when duly resolved to a stable interval. The
view of the major seventh as a strong dissonance may be common to many
eras of European composed music, and Bartok's famous piece "Major
Sevenths, Minor Seconds" presents an alternative to this tradition,
just as new or unfamiliar tunings may challenge conventions as to what
is "out of tune."

27. While any interval in the general "minor sixth" region may have a
rather high level of acoustical complexity, tension, or "entropy," the
Pythagorean 792.18 cents interestingly coincides almost exactly with
the area of maximal complexity as predicted by the "Noble Mediant" of
the simplest neighboring ratios 3:2 and 8:5, a mediant weighted by the
Golden Ratio or Phi at ~1.61803398874989484820459, i.e.

3 + 8 Phi 15.9442719
--------- = ---------- = 1.5801787 = ~792.105 cents
2 + 5 Phi 10.0901699

See, e.g., Margo Schulter and David Keenan, "The Golden Mediant:
Complex ratios and metastable musical intervals," TD 810:3, 18
September 2000; http://www.egroups.com/message/tuning/12915; the term
"Noble Mediant" is now preferred, as suggested by Dave Keenan.

28. For an overview of 13th-century sonorities and progressions, see
http://www.medieval.org/emfaq/harmony/13c.html.

Most respectfully,

Margo Schulter
mschulter@value.net