Re: A neo-Gothic approach to 20-tET (Part 2)

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A neo-Gothic approach to 20-tET:
Essay in honor of Brian McLaren and Dan Stearns
Part 2
---------------------------------------------------

For Part 1 of this article, please see:

/tuning/topicId_25475.html#25475

For the reader's convenience, I reproduce a keyboard diagram showing
the system of 20-tET spelling used in these articles, which I hope may
assist in navigating the cadences discussed here:

300 420 780 900 1080
C#+4/ Eb+4/ F#+4/ Ab+4/ Bb+4
Eb4 E4 Ab4 A4
C+4 D+4 E+4 F+4 G+4 A+4 B+4 C+5
120 360 540 600 840 1020 1260 1380
240 180 60 240 180 240 60
-----------------------------------------------------------------
180 300 660 780 960
C#4 Eb4 F#4 Ab4 Bb4
C4 D4 E4 F4 G4 A4 B4 C5
0 240 420 480 720 900 1140 1200
240 180 60 240 180 240 60

--------------------------------
3. Standard cadences and manners
--------------------------------

In approaching the fine permutations of neo-Gothic cadences in 20-tET,
we may find it helpful to approach "standard" forms in terms of two
main categories.

"Closest approach" cadences involve resolutions of unstable intervals
by contrary motion with one voice moving by a whole-tone, and the
other by a semitone (e.g. m3-1, M3-5, M6-8, m7-5, M2-4). Such
progressions are especially favored in typical 14th-century Western
European styles, and derivative neo-Gothic styles.

In contrast, "omnitonal" cadences feature resolutions of unstable
intervals with whole-tone motion in both voices (e.g. M3-1, m3-5,
m6-8, M7-5, m2-4). These progressions are very common in a
13th-century or derivative style, and take on special nuances in a
20-tET mode such as D Dorian -- or sometimes, in terms of the keyboard
spelling scheme described in Part 1, C# Dorian.

For readers who may be unfamiliar with Gothic and derivative cadences,
or who may wish to compare the following discussion of such
progressions in 20-tET with the situation in more "conventional"
neo-Gothic tunings, the series "A Gentle Introduction to neo-Gothic
progressions" offers some general background:

/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)

------------------------------
3.1. Closest approach cadences
------------------------------

One convenient place to start our exploration of 20-tET cadences for
three or four voices is with a standard progression where an outer
major sixth expands to an octave while a major third expands to a
fifth, here G3-B3-E4 (0-420-900 cents) to F3-C4-F4 (0-720-1200 cents).
The latter sonority of resolution is known as a complete _trine_ with
outer octave, lower fifth, and upper fourth:

E4 F4
B3 C4
G3 F3

(M6-8 + M3-5)

MIDI example: <http://value.net/~mschulter/20tei001.mid>

Here the lowest voice descends by a 240-cent whole-tone G3-F3, while
the two upper voices ascend by usual 60-cent semitones (B3-C4, E4-F4).
These two motions together add up to the 300 cents of expansion needed
for the 420-cent major third G3-B3 to resolve to the 720-cent fifth
F3-C4, or for the 900-cent major sixth G3-E4 to resolve to the
1200-cent octave F3-F4.

An enticing feature of 20-tET is that for any major sixth sonority
such as G3-B3-E4, there are no fewer than _four_ resolutions to a
complete trine involving stepwise motion in all voices. Here is a
possible scheme for describing and naming these resolutions:

Intensive/Polarized Remissive/Polarized
(-240/+60) (-60/+240)

E4 F4 E4 F#4
B3 C4 B3 C#4
G3 F3 G3 F#3

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

Intensive/Pastel Remissive/Pastel
(-180/+120) (-120/+180)

E4 E+4 E4 F+4
B3 B+3 B3 C+4
G3 E+3 G3 F+3

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

Intensive/Polarized: <http://value.net/~mschulter/20tei001.mid>
Remissive/Polarized: <http://value.net/~mschulter/20ter001.mid>
Intensive/Pastel: <http://value.net/~mschulter/20tei002.mid>
Remissive/Pastel: <http://value.net/~mschulter/20ter002.mid>

In an "intensive/polarized" cadence such as our opening example of
G3-B3-E4 to F3-C4-F4, we have an intensive resolution featuring
ascending melodic semitones and descending whole-tones, with
strongly "polarized" sizes of 60 cents and 240 cents.

In a "remissive/polarized" cadence such as G3-B3-E4 to F#3-C#4-F#4,
we have descending semitones and ascending whole-tones, again with
sizes of 60 cents and 240 cents.

In an "intensive/pastel" cadence such as G3-B3-E4 to E+3-B+3-E+4, we
have ascending large semitones at 120 cents, and descending small
whole-tones at 180 cents -- a more "pastel" shading of these melodic
steps.

In a "remissive/pastel" cadence such as G3-B3-E4 to F+3-C+4-F+4, we
have descending semitones at 120 cents and ascending whole-tones at
180 cents, again a more "pastel" effect.

Since all four varieties of resolutions are freely transposible to any
scale step, 20-tET offers many shades of expression, and also an
assortment of ways to move between manuals.

The same four basic patterns are available for other standard cadences
such the following where an unstable four-voice sixth sonority or
"quad" resolves to a trine, e.g. G3-B3-D4-E4 (0-420-720-900 cents):

Intensive/Polarized Remissive/Polarized
(-240/+60) (-60/+240)

E4 F4 E4 F#4
D4 C4 D4 C#4
B3 C4 B3 C#4
G3 F3 G3 F#3

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

Intensive/Pastel Remissive/Pastel
(-180/+120) (-120/+180)

E4 E+4 E4 F+4
D4 B+4 D4 C+4
B3 B+4 B3 C+4
G3 E+3 G3 F+3

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

Intensive/Polarized: <http://value.net/~mschulter/20tei003.mid>
Remissive/Polarized: <http://value.net/~mschulter/20ter003.mid>
Intensive/Pastel: <http://value.net/~mschulter/20tei004.mid>
Remissive/Pastel: <http://value.net/~mschulter/20ter004.mid>

In addition to the M6-8 and M3-5 resolutions, these cadences include
resolutions with the 300-cent minor third B3-D4 contracting to a
unison, and the 180-cent small major second D4-E4 expanding to a
480-cent fourth. Again, the total distance of contraction or expansion
is 300 cents.

A related type of closest approach progression in 20-tET involves a
large minor seventh at 1020 cents contracting to a 720-cent fifth, and
a 300-cent minor third contracting to a unison (m7-5 + m3-1). Here are
some examples in four voices showing the resolutions available for an
unstable quad such as A3-C4-E4-G4 (0-300-720-1020 cents):

Intensive/Polarized Remissive/Polarized
(-240/+60) (-60/+240)

G4 F4 G4 F#4
E4 F4 E4 F#4
C4 Bb3 C4 B3
A3 Bb3 A3 B3

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

Intensive/Pastel Remissive/Pastel
(-180/+120) (-120/+180)

G4 E+4 G4 F+4
E4 E+4 E4 F+4
C4 A+3 C4 Bb+3
A3 A+3 A3 Bb+3

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

Intensive/Polarized: <http://value.net/~mschulter/20tei005.mid>
Remissive/Polarized: <http://value.net/~mschulter/20ter005.mid>
Intensive/Pastel: <http://value.net/~mschulter/20tei006.mid>
Remissive/Pastel: <http://value.net/~mschulter/20ter006.mid>

These "standard" cadential patterns may help to explain why I prefer a
two-keyboard layout for 20-tET with manuals of Ab-C# and Ab+-C#+, in
contrast to my usual neo-Gothic schemes based on an Eb-G# arrangement.

Of course, in any complete mapping of 20-tET, all the accidentals will
be available, and the question here is party one of spelling. However,
my choice of Ab-C# may highlight some interesting properties of the
tuning and its neo-Gothic cadences.

An accidental mapping with an Ab-C# arrangement makes cadences of a
"remissive/polarized" variety such as the following very conveniently
accessible:

F4 G4
F4 G4 Eb4 D4
C4 D4 C4 D4
Ab3 G4 Ab3 G3

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

First cadence: <http://value.net/~mschulter/20ter007.mid>
Second cadence: <http://value.net/~mschulter/20ter008.mid>

While Ab would also be useful in more typical neo-Gothic tuning
systems for remissive cadences on G, in these systems G# has the
advantage of facilitating very frequently used intensive cadences
on D such as the following, here given in Pythagorean intonation:

C#4 D4 G#4 A4
G#3 A3 E4 D4
E3 D3 C#4 D4

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

First cadence: <http://value.net/~mschulter/py3ei003.mid >
Second cadence: <http://value.net/~mschulter/py3i001.mid>

In 20-tET, at least using the spelling described in these articles,
however, an interval such as G#3-C#4 or C#4-G#4 is not a usual fourth
or fifth; the first interval is actually 540 cents (A3-C#4 forming a
480-cent fourth), and the second 660 cents (C#4-A4 forming a 720-cent
fifth).

Thus, given the premises of the spelling system, a keyboard layout
with Ab-Eb (a 720-cent fifth) rather than C#-G# might be said to
promote the principle of "least astonishment."

However, it bears emphasis that "standard" closest approach cadences
are only one subset of the rich assortment of progressions which
20-tET offers, and that an interval such as 540 cents can be not only
"astonishing" but beautiful in such progressions (see Section 5).

---------------------------------------------------
3.1.1. Standard closest approach cadences in Dorian
---------------------------------------------------

With vertical cadences, as with melody, the Dorian mode or D-mode
presents some special nuances and complications in our 20-tET spelling
system.

As we just saw in the last examples, a typical intensive cadence on D
such as E3-G#3-C#4 to D3-A3-D4 runs into the complication that G#3-C#4
is not a usual fourth, but 1/20 octave wider at 540 cents.

Additionally, both this formula and a typical remissive formula such
as Eb3-G3-C4 to D3-A3-D4 run into the yet more basic complication that
D3-A3-D4 is not a usual trine at 0-720-1200 cents, but an "altered"
sonority of 0-660-1200 cents!

However, if we made our cadential goal a sonority such as either
C#3-A3-C#4 from "Hard Dorian" or D3-Bb3-D4 from "Soft Dorian" (see
Section 1.1), either forming a usual trine, then all four cadential
"manners" described above are available.

To illustrate this point, here are cadences of the (M6-8 + M3-5)
variety arriving at these centers:

C#3-A3-C#4 ("Hard Dorian")

Intensive/Polarized Remissive/Polarized
(-240/+60) (-60/+240)

C+4 C#4 B3 C#4
G+3 A3 F#3 A3
E3 C#3 D3 C#3

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

Intensive/Pastel Remissive/Pastel
(-180/+120) (-120/+180)

B+3 C#4 C4 C#4
Ab3 A3 G3 A3
D+3 C#3 Eb3 C#3

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

Intensive/Polarized: <http://value.net/~mschulter/20tei007.mid>
Remissive/Polarized: <http://value.net/~mschulter/20ter009.mid>
Intensive/Pastel: <http://value.net/~mschulter/20tei008.mid>
Remissive/Pastel: <http://value.net/~mschulter/20ter010.mid>

D3-Bb3-D4 ("Soft Dorian")

Intensive/Polarized Remissive/Polarized
(-240/+60) (-60/+240)

C#4 D4 C4 D4
A3 Bb3 G3 Bb3
F3 D3 Eb3 D3

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

Intensive/Pastel Remissive/Pastel
(-180/+120) (-120/+180)

C+4 D4 B+4 D4
G+3 Bb3 F#+3 Bb3
E3 D3 D+3 D3

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

Intensive/Polarized: <http://value.net/~mschulter/20tei009.mid>
Remissive/Polarized: <http://value.net/~mschulter/20ter011.mid>
Intensive/Pastel: <http://value.net/~mschulter/20tei010.mid>
Remissive/Pastel: <http://value.net/~mschulter/20ter012.mid>

As these examples may remind us, 20-tET is _not_ a regular diatonic
tuning. The descending 240-cent step of the lowest voice in the
intensive/polarized cadence in Soft Dorian, for example, is spelled
F3-D3, actually a melodic "quasi-third" in the context of this mode.

Thus while all cadential permutations are symmetrically available from
all steps, the musical effect in context may be quite asymmetrical
depending on the intervallic expectations established before the
cadence itself.

This kind of contextual intricacy further enriches the expressive
possibilities of 20-tET.

-----------------------
3.2. Omnitonal cadences
-----------------------

In the standard closest approach cadences we have just considered, two
basic conditions hold:

(1) All vertical fifths and fourths have usual sizes
of 720 cents and 480 cents; and

(2) All directed resolutions of unstable intervals
by stepwise contrary motion involve a total
expansion or contraction of 300 cents, e.g.
m3-1 (300-0 cents)
M3-5 (420-720 cents)
M6-8 (900-1200 cents)
large m7-5 (1020-720 cents)
small M2-4 (180-480 cents).

We may define "standard" omnitonal cadences as those in which the
first condition also applies that all fifths and fourths have usual
sizes, and where all voices move by either a large whole-tone of 240
cents or a small whole-tone of 180 cents.

Such cadences are very typical of a 13th-century kind of style,
illustrated by some omnitonal progressions arriving at a complete
trine or fifth on G:

F4 G4 C4 D4 F4 G4 E4 D4
C4 D4 A3 G3 E3 D4 C4 D4
A3 G3 F3 G3 A3 G3 F3 G3

(m6-8 + m3-5) (M3-1 + m3-5) (m6-8 + m2-4) (M7-5 + M3-1)

First cadence: <http://value.net/~mschulter/20to001.mid>
Second cadence: <http://value.net/~mschulter/20to002.mid>
Third cadence: <http://value.net/~mschulter/20to003.mid>
Fourth cadence: <http://value.net/~mschulter/20to004.mid>

In the first cadence, for example, the 300-cent minor third A3-C4
expands to a 720-cent fifth G3-D4, while the 780-cent minor sixth
A3-F4 expands to a 1200-cent octave G3-G4. Here the lower voice
descends by a small whole-tone A3-G4 (180 cents), while the upper
voices ascend by large whole-tones of 240 cents (C4-D4, F4-G4).

More generally, two-voice resolutions in these omnitonal progressions
involve a total expansion or contraction of 420 cents, the size of the
usual major third in 20-tET. The above examples illustrate the
following resolutions:

M3-1 (420-0 cents)
m3-5 (300-720 cents)
m6-8 (780-1200 cents)
m2-4 (60-480 cents)
M7-5 (1140-720 cents)

While omnitonal resolutions with a total expansion or contraction
equal to the size of a usual major third are typical of neo-Gothic
tunings generally, 20-tET differs from a regular neo-Gothic tuning in
that these resolutions involve motions by _unequal_ whole-tones of 240
cents and 180 cents.

In contrast, Pythagorean intonation and most neo-Gothic temperaments
are "eventone" tunings where all regularly spelled whole-tones have
the same size, and the usual major third is a "ditone" equal precisely
to two such equal whole-tones.[1]

Curiously, while 20-tET might not be the first association one would
have with the term "just intonation," the kind of "diatonic" scale
here explored with its unequal whole-tones could be taken as having
some resemblance to either Ptolemy's syntonic diatonic with its
alternating steps of 9:8 and 10:9, or the diatonic of Archytas and
al-Farabi with its steps of 9:8 and 8:7.

Here are omnitonal cadences for four voices in 20-tET featuring the
basic two-voice progressions listed above:

F4 G4 E4 D4
E4 D4 C4 D4
C4 D4 A3 G3
A3 G3 F3 G3

(m6-8 + m3-5 + M3-1 + m2-4) (M7-5 + M3-1 + m3-5 + M3-1)

First cadence: <http://value.net/~mschulter/20to005.mid>
Second cadence: <http://value.net/~mschulter/20to006.mid>

In these progressions, we again have the characteristic mixture of
melodic motions by large whole-tones of 240 cents (F-G, C-D) and small
whole-tones of 180 cents (A-G, E-D).

Omnitonal cadences are especially a feature of neo-Gothic styles
following a 13th-century tradition, with some progressions of this
kind in the Hard Dorian (C#-C#) providing an example:

G4 A4 C5 C#5 C5 C#5 B4 A4
E4 C#4 G4 A4 B4 A4 G4 A4
C4 C#4 E4 C#4 E4 C#4 C4 C#4

(M3-1 + m3-5) (m6-8 + m3-5) (m6-8 + m2-4) (M7-5 + M3-1)

First cadence: <http://value.net/~mschulter/20to007.mid>
Second cadence: <http://value.net/~mschulter/20to008.mid>
Third cadence: <http://value.net/~mschulter/20to009.mid>
Fourth cadence: <http://value.net/~mschulter/20to010.mid>

Starting with the same unstable sonorities, each of these progressions
can also resolve to a stable trine or fifth on D, the final of the
Soft Dorian (D-D, with Bb), sometimes at least to my ears suggesting a
more "slendro-like" effect reflecting the 5-tET affinities of 20-tET:

G4 Bb4 C5 D5 C5 D5 B4 Bb4
E4 D4 G4 Bb4 B4 Bb4 G4 Bb4
C4 D4 E4 D4 E4 D4 C4 D4

(M3-1 + m3-5) (m6-8 + m3-5) (m6-8 + m2-4) (M7-5 + M3-1)

First cadence: <http://value.net/~mschulter/20to011.mid>
Second cadence: <http://value.net/~mschulter/20to012.mid>
Third cadence: <http://value.net/~mschulter/20to013.mid>
Fourth cadence: <http://value.net/~mschulter/20to014.mid>

More generally, any unstable sonority fitting an omnitonal pattern
of resolution has available two such resolutions differing in the
arrangement of 240-cent and 180-cent whole-tone steps, for example:

C4 -- +240 -- D4 C4 -- +180 -- C#4
A3 -- -180 -- G3 A3 -- -240 -- F#3
F3 -- +240 -- G3 F3 -- +180 -- F#3

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

First cadence: <http://value.net/~mschulter/20to015.mid>
Second cadence: <http://value.net/~mschulter/20to016.mid>

To this point, we have considered only the "regular" cadences of
20-tET; but other progressions involving such intervals as
quasi-thirds (240 cents) or quasi-sixths (960 cents), or the
diminished fifth (540 cents), also play a routine role in giving this
tuning its distinctive charm for neo-Gothic music. We next turn to
these progressions.

----
Note
----

1. The term "eventone," by analogy to the familiar term _meantone_,
includes tunings based on a chain of identically sized fifths where
four such fifths up less two octaves yield a regular major third
(e.g. F-A from F-C-G-D-A) formed from two equal whole-tones or
"even-tones" (e.g. F-A from F-G-A). In meantones, a subset of
eventones, the major third more specifically is at or reasonably near
to a ratio of 5:4 (~386.81 cents). Other types of eventones include
Pythagorean or 3-based just intonation based on a chain of pure fifths
at a ratio of 3:2 (major third 81:64, ~407.82 cents), and neo-Gothic
tunings with fifths wider than pure (e.g. 46-tET, major third ~417.39
cents, almost identical to 14:11 at ~417.51 cents). The term "septimal
eventone" might apply to a tuning such as 22-tET when used in a
neo-Gothic style, with a regular major third at ~436.36 cents (very
close to 9:7, ~435.08 cents). The yet more general term "evenstep"
might include any tuning where a repeated generator of any size
produces a regular interval of interest.

Most respectfully,

Margo Schulter
mschulter@value.net