Re: Correction on "Gentle Introduction to neo-Gothic," pt. 2A

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"RIGHT INTERVAL SIZES, WRONG SPELLINGS"
CORRECTION TO:
A gentle introduction to neo-Gothic progressions (1):
Trines, quads, and intonational flavors
Part 2A of 2: Some flavors of most proximal quads
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Hello, there, and part of the excitement of participating on the
Tuning List is to discover a mistake after one has posted, at once
providing an occasion to document one's own fallibility and possibly
to explore some interesting intonational facets in the process.

Writing what seems a routine if slightly embarrassing correction can
sometimes itself be a serendipitous experience of discovery. In this
case, while "proofreading" revised cadential note spellings at the
musical keyboard, I discovered an unsuspected nuance involving the
size of certain melodic steps in these progressions.

In Part 2A of the "Gentle introduction to neo-Gothic progressions"
series, Section 2.3, I gave the following correct table for interval
sizes of an intermediate cadential flavor somewhere between the
7-flavor and 13-flavor in Keenan Pepper's "Noble Fifth" regular tuning
with the fifth at around 704.10 cents:

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Interval Chain Cents 7-ratio/cents 13-ratio/cents
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M3 13 4ths up ~446.76 9:7 (~435.08) 13:10 (~454.21)
m3 14 5ths up ~257.34 7:6 (~266.87) 15:13 (~247.74)
M6 14 4ths up ~942.66 12:7 (~933.13) 26:15 (~952.26)
M2 15 4ths up ~238.57 8:7 (~231.17) 15:13 (~247.74)
m7 15 5ths up ~961.43 7:4 (~968.83) 26:15 (~952.26)
---------------------------------------------------------------------

Unfortunately, my musical examples gave the wrong spellings for these
intervals -- and also failed to note some ramifications for a keyboard
scheme mentioned in the article.

Here are the incorrect examples: as explained in the article, the
ASCII asterisk or * shows a note raised by a diesis, ~49.15 cents in
this tuning, while a "d" symbol shows a note lowered by a diesis:

Expansive/Intensive Contractive/Intensive

E*4 ----- +49 ----- F4 D4 ----- -208 ----- C4
(239) (496) (257) (0)
D4 ------ -208 ----- C4 B*3 ----- +49 ----- C4
(496,257) (496,0) (704,447) (704,704)
B*3 ----- +49 ----- C4 G3 ----- -208 ----- F3
(943,704,447) (1200,704,704) (961,704,257) (704,704,0)
G3 ------ -208 ----- F3 E*3 ----- +49 ----- F3

Expansive/Remissive Contractive/Remissive

E4 ---- +208 ----- F#4 Dd4 ----- -49 ----- C#4
(239) (496) (257) (0)
Dd4 ---- -49 ----- C#4 B3 ----- +208 ----- C#4
(496,257) (496,0) (704,447) (704,704)
B3 ---- +208 ----- C#4 Gd3 ----- -49 ----- F#3
(943,704,447) (1200,704,704) (961,704,257) (704,704,0)
Gd3 ----- -49 ----- F#3 E3 ----- +208 ----- F#3

The obvious glitch here is that while these spellings would be
intuitive and practical in 29-tET, from which I borrowed them, they
actually are based on different chains of fifths and fourths: for
example, 16 fifths up for the large major third G3-B*3 or Gd3-B3.

Either spelling succeeds in 29-tET, where 16 fifths up and 13 fourths
up happen to be equivalent -- i.e. a regular major third plus a diesis
(10 + 1 = 11 steps) or a fourth minus a diesis (12 - 1 = 11 steps).

In the Noble Fifth tuning, however, these spellings and interval sizes
are definitely not equivalent. Our desired 447-cent major third is
equal to a fourth minus a diesis (13 fourths up), as the table shows,
for example G*-C or G-Cd, or a rounded (496 - 49) cents. However, the
given spelling G-B* or Gd-B would give us a regular major third plus a
diesis (16 fifths up), or a rounded (416 + 49) cents, about 465 cents.

(As it happens, this "wrong" spelling yields a near-21:16, known
informally in neo-Gothic theory as the _quarta Pepperiana_ because of
Keenan Pepper's praise of this ratio for its "crunchiness." For the
above examples, however, it would be the right interval in the wrong
place.)

The following revised examples show correct spellings for a 24-note
scheme with two 12-note keyboards tuned a diesis apart, taking the
lower-pitched keyboard as the "standard" one. Note that large major
thirds are spelled as small fourths (e.g. G*3-C4), large major sixths
as small minor sevenths (e.g. G*3-F4), and large major seconds as
small minor thirds (e.g. D*3-F4); conversely, small minor thirds are
spelled are spelled as large major seconds (e.g. E3-F#*3), and small
minor sevenths as large major sixths (e.g. E3-C#*4).

Expansive/Intensive Contractive/Intensive

F4 ----- +49 ----- F*4 C#*4 ----- -208 ----- B*3
(239) (496) (257) (0)
D*4 ---- -208 ----- C*4 B3 ----- +49 ----- B*3
(496,257) (496,0) (704,447) (704,704)
C4 ----- +49 ----- C*4 F#*3 ----- -208 ----- E*3
(943,704,447) (1200,704,704) (961,704,257) (704,704,0)
G*3 ---- -208 ----- F*3 E3 ----- +49 ----- E*3

Expansive/Remissive Contractive/Remissive

F4 ---- +208 ----- G4 C#*4 ----- -49 ----- C#4
(239) (496) (257) (0)
D*4 ---- -49 ----- D4 B3 ----- +208 ----- C#4
(496,257) (496,0) (704,447) (704,704)
C4 ---- +208 ----- D4 F#*3 ----- -49 ----- F#3
(943,704,447) (1200,704,704) (961,704,257) (704,704,0)
G*3 ----- -49 ----- G3 E3 ----- +208 ----- F#3

As these revised spellings show, there is an interesting ramification
on our 24-note keyboard, if we wish to use regular 208-cent melodic
whole-tones for all four progressions.

While the expansive quad G*3-C4-D*4-F4 has an intensive resolution
(ascending semitonal motion) to F*3-C*4-F*4 as expected, the
contractive quad E3-F#*3-B3-C#*3 resolves not to F3-C4 but to
E*3-B*3. The distance between these last two sonorities is about 30.37
cents, equal to the difference between 17 fifths and 10 pure octaves,
or between the diesis (~49.15 cents) and the usual diatonic semitone
(~79.52 cents).

Similarly, while the contractive quad E3-F#*3-B3-C#*4 has a remissive
resolution (descending semitonal motion) to the expected F#*3-C#*3,
the expansive quad G*3-C4-D*4-F4 resolves not to F#*3-C#*4 but to
G3-D4-G4, again a difference of around 30 cents.

In these progressions, one voice of a two-voice resolution moves by a
208-cent whole-tone and the other by a 49-cent diesis, easily heard
(at least for me) as a kind of semitone (compare the regular
neo-Gothic diatonic semitone of 22-tET at 1/22 octave or ~54.55
cents).

Trying these progressions at the musical keyboard to "proofread" my
correction, however, I realized that I had actually been playing the
following variations on the two progressions with less "obvious"
spellings, the contractive/intensive and expansive/remissive (shown in
the above diagrams at the upper right and lower left):

Contractive/Intensive Expansive/Remissive

C#*4 ----- -178 ----- C4 F4 ---- +178 ----- F#*4
(257) (0) (239) (496)
B3 ----- +80 ----- C4 D*4 ---- -80 ----- C#*4
(704,447) (704,704) (496,257) (496,0)
F#*3 ----- -178 ----- F3 C4 ---- +178 ----- C#*4
(961,704,257) (704,704,0) (943,704,447) (1200,704,704)
E3 ----- +80 ----- F3 G*3 ----- -80 ----- F#*3

These progressions, with regular 80-cent diatonic semitones, have
small whole-tones equal to a chromatic semitone or apotome (~128.67
cents) plus a 49.15-cent diesis, or ~177.82 cents. This is close to
the Pythagorean _tonus minor_ at 59049:32768 or ~180.45 cents, or to
10:9 (~182.40 cents).

While the version with a usual 208-cent whole-tone plus an
extra-compact 49-cent diesis might better fit the ideal of a
"superefficient" cadence as described for example by Marchettus of
Padua (1318), the alternative version with a usual semitone and narrow
whole-tone is also available as a subtle variant.

Most respectfully,

Margo Schulter
mschulter@value.net