Yahoo Tuning Groups Ultimate Backup tuning Re: Gentle Introduction to neo-Gothic (I, Pt. 2C)

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A gentle introduction to neo-Gothic progressions (1):
Trines, quads, and intonational flavors
Part 2C of 2: Proximal triples -- the split fifth
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5. Proximal triples: The "split fifth" and 17-flavor cadences
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Another subset of the most proximal quad produces a milder variety of
triple described by Jacobus of Liege around 1325 as a _quinta fissa_
or "split fifth": an outer fifth divided by the middle voice into the
unstable but relatively blending intervals of a major third and minor
third. Jacobus expresses a preference for the arrangement with the
major third below and the minor third above (e.g. G3-B3-D4), but cites
a motet (happily preserved in the Bamberg and Montpellier Codices)
with an opening sonority of A3-C4-E4 to demonstrate that the converse
arrangement is also acceptable.[13]

Either variety of _quinta fissa_ or "split fifth," as we may
conveniently term it in English, is open to three varieties of
directed cadential progressions involving resolutions of both unstable
thirds by stepwise contrary motion: intensive (ascending semitonal
motion), remissive (descending semitonal motion), and omnitonal
(whole-tone motion in all voices). Here are these three resolutions
for G3-B3-D4:

Intensive Remissive Omnitonal

D4 C4 D4 C#4 D4 E4
B3 C4 B3 C#4 B3 A3
G3 F3 G3 F#3 G3 A3

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

Similarly, the form with the minor third below and major third above
is open to these three resolutions:

Intensive Remissive Omnitonal

E4 F4 E4 F#4 E4 D4
C4 Bb3 C4 B3 C4 D4
A3 Bb3 A3 B3 A3 G3

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

In practice, the availability of all three resolutions in these
regularly spelled forms assumes a tuning set including any required
accidentals: for the above examples, as it happens, a typical 12-note
set of Eb-G# in any regular neo-Gothic tuning would suffice.

With its ideally euphonious fifth and two relatively blending thirds,
the _quinta fissa_ or split fifth is one of the mildest unstable
sonorities in Gothic or neo-Gothic music[14], often inviting free
"coloristic" treatments as well as directed cadential resolutions.

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5.1. A cornucopia of flavors: 17-flavor cadences
------------------------------------------------

The following table may give some idea of the protean flavors which
split fifth triples may assume, with an asterisk (*) identifying
flavors most commonly favored in directed cadential resolutions as
penultimate sonorities proceeding immediately to a stable fifth as
shown in the above examples:

----------------------------------------------------------------------
Flavor Ratio zone Sample M3 below m3 below Steps
M3 m3 Tuning (1-M3-5) (1-m3-5) T S
----------------------------------------------------------------------
49-flavor 49:40 49:40 17-tET 0-353-706 0-353-706 212 141
......................................................................
17-flavor* 21:17 17:14 Noble 5th 0-367-704 0-337-704 208 129
......................................................................
5-flavor 5:4 6:5 Pythag 0-384-702 0-318-702 204 114
......................................................................
3-flavor* 81:64 32:27 Pythag 0-408-702 0-294-702 204 90
......................................................................
11-flavor* 14:11 13:11 Noble 5th 0-416-704 0-288-704 208 80
......................................................................
23-flavor* 23:18 27:23 17-tET 0-424-706 0-282-706 212 71
......................................................................
7-flavor* 9:7 7:6 22-tET 0-436-709 0-273-709 218 55
Pythag 0-431-702 0-271-702 204 67
......................................................................
13-flavor* 13:10 15:13 29-tET 0-455-703 0-248-703 207 41
----------------------------------------------------------------------
* This flavor typically serves as a penultimate cadential sonority
----------------------------------------------------------------------

Our intonational spectrum ranges from a neutral "49-flavor" where the
fifth is split into two precisely or approximately equal intervals
near 49:40 or 11:9 (around 353 cents in 17-tET), to the dramatic
polarization of the 13-flavor with major and minor thirds near 13:10
and 15:13 (around 455 cents and 248 cents in 29-tET), respectively
some 100 cents larger or smaller.

As we move across this spectrum from the somewhat "Impressionistic" or
"mysterious" 49-flavor[15] to the superefficient 13-flavor, cadential
semitones shrink from a "near-neutral" second at around 141 cents
(2/17 octave in 17-tET) to a diesis of around 41 cents (1/29 octave in
29-tET), likewise a range of about 100 cents.

As the table indicates, the 49-flavor and 5-flavor are rather
specialized. In early 15th-century and derivative styles, 5-flavor
versions of the "split fifth" serve as smooth and beguiling
noncadential points of diversion or pausing, and sometimes also in
cadential resolutions. Neutral or 49-flavor thirds occur as augmented
seconds or diminished fourths in regular tunings with fifths in the
range of around 705-706.5 cents, with 17-tET (fifths ~705.88 cents) in
the center of this region.

Additionally, a characteristic neo-Gothic treatment of 5-flavor split
fifths in various tunings where they occur (e.g. Pythagorean, Noble
Fifth, 22-tET) is to use them as antepenultimate cadential sonorities,
with one of the voices moving by a comma or diesis or the like to a
more active flavor of split fifth which then resolves in a usual
manner. This idiom invites discussion in coming articles.

Focusing on the remaining flavors of split fifths most typically
receiving direct cadential resolutions, we might approach these in two
groups. Split fifths in the range from the 3-flavor to the 13-flavor
invite efficient "closest approach" resolutions or more stately
omnitonal resolutions much like those for full proximate quads or
other types of proximate triples (see Parts 2A and 2B); no radically
new principles are involved.

Additionally, some of the most characteristic neo-Gothic tunings
feature 17-flavor versions of the split fifth with submajor thirds at
around 21:17 and supraminor thirds at around 17:14, with a distinctive
kind of cadential resolution involving strikingly large chromatic
semitones in the neighborhood of 14:13 (~128.30 cents). These
17-flavor cadences are a distinctive mark of neo-Gothic as opposed to
known historical medieval practice and theory.

Let us first briefly survey some resolutions along the familiar
portion of the spectrum from the 3-flavor to the 13-flavor, and then
the distinctive 17-flavor.

-----------------------------------------
5.1.1. From the 3-flavor to the 13-flavor
-----------------------------------------

As we have seen, either basic arrangement of the split fifth (with the
major third below and minor third above, or the converse) permits
three types of resolutions: intensive, remissive, or omnitonal. Here
are examples of these resolutions in the usual 3-flavor of Pythagorean
just intonation for G3-B3-D4 (major third below) and A3-C4-E4 (minor
third below):

Pythagorean 3-flavor

Intensive -- M3 below m3 Remissive -- M3 below m3

D4 ------ -204 ----- C4 D4 ----- -90 ----- C#4
(294) (0) (294) (0)
B3 ------ +90 ----- C4 B3 ----- +204 ----- C#4
(702,408) (702,702) (702,408) (702,702)
G3 ------ -204 ----- F3 G3 ----- -90 ----- F#3

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

Omnitonal -- M3 below m3

D4 ------ +204 ----- E4
(294) (702)
B3 ------ -204 ----- A4
(702,408) (702,0)
G3 ------ +204 ----- A3

(M3-1 + m3-5)

Intensive -- m3 below M3 Remissive -- m3 below M3

E4 ------ +90 ----- F4 E4 ----- +204 ----- F#4
(408) (702) (294) (702)
C4 ------ -204 ----- Bb3 C4 ----- -90 ----- B3
(702,294) (702,0) (702,294) (702,0)
A3 ------ +90 ----- Bb3 A3 ----- -90 ----- B3

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

Omnitonal -- m3 below M3

E4 ------ -204 ----- D4
(408) (0)
C4 ------ +204 ----- D4
(702,294) (702,702)
A3 ------ -204 ----- G3

(m3-5 + M3-1)

From a vertical perspective, the Pythagorean 3-flavor split fifth with
its rather complex JI ratios of 64:81:96 (major third below) or
54:64:81 (minor third below) has a quality at once relative blending
and somewhat tense, with a "forward-going" feeling adding impetus to
the invited cadential resolution.

Like closest approach resolutions of other proximal quads or triples,
intensive or remissive cadences of the split fifth in this usual
Pythagorean flavor involve melodic motions by generously large 9:8
whole-tones and incisive 256:243 semitones. The more stately omnitonal
progressions involve 9:8 whole-tone motions in all voices.

In simple and direct cadences of this kind, split fifth sonorities
with the major third below and the minor third above, and omnitonal
resolutions generally, tend to occur in the usual or native flavor for
a given tuning system: the 3-flavor of Pythagorean, 11-flavor of
29-tET, 23-flavor of 17-tET, or 7-flavor of 22-tET, etc.

With closest approach (intensive or remissive) resolutions of the
split fifth arrangement with the minor third below, there is often a
tendency to accentuate cadential action by choosing a superefficient
flavor of the kind favored for other types of most proximal quads or
triples: the Pythagorean 7-flavor or 29-tET 13-flavor, for example.
Here are examples in the Pythagorean 7-flavor, with split fifth
sonorities approximating 6:7:9 rather closely; the ^ and @ signs show
a note raised or lowered by a Pythagorean comma (~23.46 cents)[16]:

Intensive Remissive

E^4 ------ +67 ----- F4 E4 ----- +204 ----- F#4
(431) (702) (431) (702)
C4 ------ -204 ----- Bb3 C@4 ----- -67 ----- B3
(702,271) (702,0) (702,271) (702,0)
A^3 ------ +67 ----- Bb3 A3 ----- +204 ----- B3

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

Here are the same progressions in the usual 11-flavor and accentuated
13-flavor of 29-tET, with the signs "*" and "d" for the latter flavor
showing notes raised or lowered by a diesis of 1/29 octave or ~41.38
cents[17]:

29-tET 11-flavor

E4 ------ +83 ----- F4 E4 ----- +207 ----- F#4
(414) (703) (414) (703)
C4 ------ -207 ----- Bb3 C4 ----- -83 ----- B3
(702,290) (703,0) (703,290) (703,0)
A3 ------ +83 ----- Bb3 A3 ----- -83 ----- B3

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

29-tET 13-flavor

Intensive Remissive

E*4 ------ +41 ----- F4 E4 ----- +207 ----- F#4
(455) (703) (455) (703)
C4 ------ -207 ----- Bb3 Cd4 ----- -41 ----- B3
(703,248) (703,0) (703,248) (703,0)
A*3 ------ +41 ----- Bb3 A3 ----- +207 ----- B3

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

These closest approach cadences present familiar themes: high melodic
contrast between whole-tones and narrow diatonic semitones, and
efficient expansion or contraction of unstable intervals to stable
ones (here m3-1 and M3-5).

With the 17-flavor, however, we move to a different kind of vertical
color and cadential action also based on the directed resolution of
relatively blending yet somewhat tense sonorities, but featuring a
strikingly distinct kind of melodic motion: the direct chromatic
semitone.

---------------------------------------
5.1.2. Split fifths: 17-flavor cadences
---------------------------------------

In regular neo-Gothic tunings featuring an 11-flavor for regularly
spelled major and minor thirds (~14:11, ~13:11), ranging roughly from
29-tET (fifths ~703.45 cents) to the e-based tuning where whole-tones
and diatonic semitones have a ratio between their sizes equal to
Euler's e, ~2.71828 (fifths ~704.61 cents), diminished fourths and
augmented seconds have ratios in the general neighborhood of 21:17
(~365.83 cents) and 17:14 (~336.13 cents) respectively.

These somewhat complex thirds might be described as "submajor" and
"supraminor" (or "superminor") -- or, to use a more medieval
terminology, as "subditonal" and "suprasemiditonal."

While these thirds, and also 17-flavor major and minor sixths, lend
themselves to various combinations and progressions, we focus here on
the most common cadential pattern involving the intensive or remissive
resolution of a split fifth with the supraminor third or augmented
second below and the submajor third or diminished fourth above.

Here is an example in the "Noble Fifth" tuning of Ervin Wilson and
Keenan Pepper with fifths at ~704.10 cents, where these intervals are
within 1.5 cents of 21:17 and 17:14; note the spellings of the
augmented seconds and diminished fourths, and of the chromatic
semitonal motions:

Noble Fifth -- 17-flavor

Intensive Remissive

Bb3 ------ +129 ----- B3 Bb3 ----- +208 ----- C4
(367) (704) (367) (704)
F#3 ------ -208 ----- E3 F#3 ----- -129 ----- F3
(704,337) (704,0) (704,337) (704,0)
Eb3 ------ +129 ----- E3 Eb3 ----- +208 ----- F3

(Aug2-1 + dim4-5) (Aug2-1 + dim4-5)

Like usual neo-Gothic thirds in closest approach progressions,
17-flavor thirds have a rather complex quality which lends a dynamic
and passionate quality to these resolutions. Strikingly distinct,
however, are the ascending or descending motions by large chromatic
semitones: Eb3-E3 and Bb-B3 (intensive version) or F#-F (remissive
version).

The two-voice progressions of augmented second to unison (Aug2-1) and
diminished fourth to fifth (dim4-5) -- intonational variations on the
usual m3-1 and M3-5 -- involve around 337 cents of total contraction
or expansion, or more precisely ~336.86 cents, the size of the
augmented second or supraminor third in this tuning. In these
progressions one voice moves by a regular 208-cent whole-tone, the
other by a 129-cent chromatic semitone.

To appreciate the special qualities of these "not-so-close approach"
resolutions, let us compare them with corresponding resolutions in the
usual 11-flavor of the same tuning, featuring incisive diatonic semitones
of around 80 cents and total expansion or contraction (m3-1, M3-5) of
around 288 cents, the size of the regular minor third:

Noble Fifth -- 11-flavor

Intensive Remissive

B3 ------ +80 ----- C4 B3 ----- +208 ----- C#4
(416) (704) (416) (704)
G3 ------ -208 ----- F3 G3 ----- -80 ----- F#3
(704,288) (704,0) (704,288) (704,0)
E3 ------ +80 ----- F3 E3 ----- +208 ----- F#3

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

In this tuning the diesis -- the difference in size between the
diatonic and chromatic semitones (known in more formal medieval terms
as the limma and apotome) -- is around 49.15 cents, giving chromatic
17-flavor resolutions a quality that is indeed special.

With various fine nuances of intonational shading, this creative
contrast between regular 11-flavor and chromatic 17-flavor resolutions
of the split fifth -- the latter flavor sometimes known as the _quinta
fissa alternativa_ or "alternative split fifth" -- is a main feature
and attraction of regular neo-Gothic temperaments from 29-tET to the
e-based tuning.[18]

-----------------------------------------------
5.2. The split fifth and intonational diversity
-----------------------------------------------

As the table of flavors at the opening of Section 5.1 and the above
examples may suggest, the split fifth in neo-Gothic music has an
amazing range of intonational variations. To illustrate this theme, we
consider three diverse flavors of cadences from a single tuning
system, the e-based tuning devised in honor of Leonhard Euler.

We focus on the form of split fifth cadence especially inviting such
diversity: the version with the minor third of the split fifth placed
below the major third, with an intensive or remissive resolution
involving some variety of semitonal melodic motion.

With a fifth of around 704.61 cents (~2.65 cents wide), this tuning
features regular or native 11-flavor cadences with major thirds
(~418.42 cents) slightly wider than 14:11, and minor thirds (~286.18
cents) somewhat narrower than 13:11, and closer to 33:28 (~284.44
cents). The regular whole-tone is around 209.21 cents, and the
diatonic semitone around 76.97 cents, these two intervals having the
defining ratio of Euler's _e_.

e-based -- 11-flavor

Intensive Remissive

B3 ------ +77 ----- C4 B3 ----- +209 ----- C#4
(418) (705) (418) (705)
G3 ------ -209 ----- F3 G3 ----- -77 ----- F#3
(705,286) (705,0) (704,286) (705,0)
E3 ------ +77 ----- F3 E3 ----- +209 ----- F#3

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

In contrast to these usual closest approach progressions, the
chromatic 17-flavor resolutions involve chromatic semitones at around
132.25 cents, and submajor and supraminor thirds at around 363.14 and
341.46 cents, respectively about 2.68 cents smaller than 21:17 and
5.33 cents larger than 17:14.

e-based -- 17-flavor

Intensive Remissive

Bb3 ------ +132 ----- B3 Bb3 ----- +209 ----- C4
(363) (705) (363) (705)
F#3 ------ -208 ----- E3 F#3 ----- -132 ----- F3
(705,341) (705,0) (704,341) (704,0)
Eb3 ------ +132 ----- E3 Eb3 ----- +209 ----- F3

(Aug2-1 + dim4-5) (Aug2-1 + dim4-5)

Here the difference between the diatonic and chromatic semitones is
around 55.28 cents, a measure of the contrast between these 17-flavor
progressions and the usual 11-flavor ones. The submajor and supraminor
thirds differ in size by around 21.68 cents. If we moved much further
along the spectrum of regular tunings toward larger fifths, these
thirds would approach the equal and neutral 49-flavor condition of the
17-tET neighborhood.

This variety of 17-flavor leaning somewhat toward neutrality is known
as "strong," in contrast to the "mild" 17-flavor of 29-tET and the
"central" or medium flavor found with fifths close to 704 cents
(e.g. the Noble Fifth tuning, Section 5.1.2) where thirds are at or
quite near the characteristic ratios of 21:17 and 17:14.[19]

In addition, the e-based tuning when carried to 24 notes per octave
offers superefficient 7-flavor resolutions with wide major thirds and
narrow minor thirds formed from 13 fourths up (~440.11 cents) and 14
fifths up (~264.50 cents) -- spelled respectively as a fourth minus a
55-cent diesis and a major second plus this diesis.

In a 24-note tuning on two 12-note keyboards a diesis apart, taking
the lower keyboard as the "standard" one, and using the * symbol to
show notes on the upper keyboard (raised by a diesis), we might play
and notate our sample resolutions as follows:

e-based -- 7-flavor

Intensive Remissive

B3 ------ +55 ----- B*3 B3 ----- +209 ----- C#4
(440) (705) (440) (705)
F#*3 ------ -209 ----- E*3 F#*3 ----- -55 ----- F#3
(705,264) (705,0) (705,264) (705,0)
E3 ------ +55 ----- E*3 E3 ----- +209 ----- F#3

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

The 7-flavor split fifth is quite close to 6:7:9, with superefficient
resolutions involving melodic motions by a 55-cent diesis, here E3-E*3
and B3-B*3 (intensive), or F#*3-F#3 (remissive). The other voice in a
two-voice resolution of m3-1 or M3-5 moves by a usual whole-tone of
around 209 cents, for a total expansion or contraction of 264 cents,
equal to the size of the 7-flavor minor third in this tuning.[20]

In this near-6:7:9 sonority, the outer fifth is ~2.65 cents wide; the
lower minor third only ~2.37 cents smaller than 7:6; and the upper
major third wider than 9:7 by a greater variance of ~5.03 cents.

It seems especially fitting that a tuning based on Euler's _e_ should
include a 7-flavor with a near-pure approximation of 7:6, since
Leonhard Euler (1764) describes the 7-based ratios 7:4 and 7:6, and
advocates their use in practical music, as documented in a brilliant
translation by Joe Monzo of a of a fascinating paper in French by
Patrice Bailhache on Euler's music theory, complete with Monzo's
alternative translations and commentary on Euler's original Latin:

http://www.ixpres.com/interval/monzo/euler/euler-en.htm

In neo-Gothic theory, a 7-flavor split fifth at or near 6:7:9 is
called the _quinta fissa suavis_, the "agreeable" or "attractive"
split fifth -- or, more freely, the "smooth" or "sweet" split fifth.
This name is in honor of Euler and his _gradus suavitatis_ or "degree
of attraction" (Monzo's artful translation) of a sonority.[21]

In surveying the e-based tuning, we have considered only intensive or
remissive resolutions of split fifth sonorities with the minor third
below and the major third above, the situation especially lending
itself to intonational variations. The following table may offer an
overview of the diversity we have encountered even within this limited
scope: the regular 11-flavor, chromatic 17-flavor, and streamlined
7-flavor.

----------------------------------------------------------------------
Flavor Ratio zone e-based tuning Size of Steps
M3 m3 (1-m3-5) upper M3 T S
----------------------------------------------------------------------
11-flavor 14:11 13:11 0-286.18-704.61 418.42 209.21 76.97
17-flavor 21:17 17:14 0-341.46-704.61 363.14 209.21 132.25
7-flavor 9:7 7:6 0-264.50-704-61 440.11 209.21 55.28
----------------------------------------------------------------------
Total motion in m3-1 or M3-5 equals size of cadential m3, or (T + S)
----------------------------------------------------------------------

In these cadences, minor thirds vary in size from around 264 cents to
341 cents, and major thirds likewise from 363 to 440 cents -- a range
equal to the size of a full diatonic semitone in this tuning, or
around 77 cents. Yet each version is recognizable as a variation on
the basic theme of a split fifth resolving (m3-1 + M3-5).

As it happens, I originally hit upon the idea of the e-based tuning by
a kind of serendipity: if other people had based tunings on such
arbitrary mathematical quantities as pi or the Golden Section, why not
Euler's _e_?

However, the tuning illustrates the delicate kind of intonational
balance which can give such a system a special quality. Here we have
the distinctive combination of an 11-flavor with major thirds within
one cent of 14:11; a 17-flavor where submajor and supraminor thirds
retain enough difference in size to give a sense of contrast or
"polarity" rather than even neutrality[22]; and a near-pure 7-flavor (with
a close approximation of 7:6 and a virtually just 7:4).

If the fifth (~704.61 cents) were slightly larger, then the 17-flavor
would approach a neutral 49-flavor as in 17-tET; if it were slightly
smaller, then the 7-flavor would be considerably less accurate in
approximating ratios such as 7:6 and 7:4.

Of course, _any_ point on the neo-Gothic spectrum will offer its own
unique balances and opportunities; these articles are meant to suggest
a few possibilities and to encourage further exploration.

----------------
Notes to Part 2C
----------------

13. We can derive the first variety of _quinta fissa_ or split fifth
from the lowest three notes of an expansive most proximal quad,
e.g. G3-B3-D4-E4; and the second variety from the lowest three notes
of a contractive most proximal quad, e.g. A3-C4-E4-G4. Various
derivations from other types of proximal quads are also possible,
e.g. G3-B3-D4 from the contractive most proximal quad E3-G3-B3-D4; or
A3-C4-E4 from the omnitonal quad A3-C4-E4-F4.

14. The other category of mildly unstable or relatively blending
three-voice sonorities features combinations with two ideally
euphonious fifths and/or fourths plus a relatively tense major second
or ninth or a minor seventh. Examples of these "mildly unstable
quintal/quartal sonorities" include G3-D4-A4 (M9 + 5 + 5); G3-C4-F4
(m7 + 4 + 4); G3-C4-D4 (5 + 4 + M2); and G3-A3-D4 (5 + M2 + 4). While
these sonorities are very important in Gothic and neo-Gothic music,
they are not _proximal_ triples capable of resolving to stability by
way of _stepwise_ motion in all voices, and we therefore reserve them
for later discussion.

15. In choosing 49:40 (~351.34 cents) rather than the simpler 11:9
(~347.41 cents) as a ratio representing the zone of neutral thirds, I
am motivated in part by the closer approximation of the former
interval to an equal division of the fifth in regular neo-Gothic
tunings where neutral thirds are most characteristic (fifths in the
neighborhood of 706 cents). In 17-tET, an ideal example, we have a
precisely equal division of the fifth at 10/17 octave (~705.88 cents)
into two neutral thirds of 5/17 octave (~352.94 cents). In contrast,
for a meantone temperament such as 31-tET, 11:9 closely approximates
an equal division of the fifth, e.g. 9/31 octave (~348.39 cents) as
precisely half of a fifth at 18/31 octave (~696.77 cents).

16. With a 24-note Pythagorean tuning mapped to two 12-note keyboards
a Pythagorean comma apart (the Xeno-Gothic scheme), taking the lower
keyboard as the standard one, these progressions would be spelled
A^3-C4-E^3 to Bb3-Bb3-F4, and A^3-C4-E^4 to B^3-B^3-F#^4.

17. With a 24-note subset of 29-tET mapped to two 12-note keyboards a
diesis apart, taking the lower keyboard as standard, these
progressions would be spelled A*3-C4-E*4 to Bb3-Bb3-F4, and A*3-C4-E*4
to B*3-B*3-F*4.

18. In 29-tET, 17-flavor submajor and supraminor thirds have sizes of
9/29 octave (~372.41 cents) and 8/29 octave (~331.03 cents); in the
e-based tuning, they are ~363.14 cents and ~341.46 cents. These
relatively complex and somewhat polarized thirds represent a region
between the smooth 5-flavor augmented seconds and diminished fourths
("schisma thirds") of Pythagorean tuning and its immediate neighbors,
and the equal or near-equal neutral thirds of the 49-flavor around
17-tET.

19. One rough guide for placing a transitional region between
17-flavor and 49-flavor zones might propose that submajor and
supraminor thirds should differ by a ratio of at least something like
85:84 (~20.49 cents). This would place the area of transition just
beyond the e-based tuning. While 85:84 is merely one intuitive
"guesstimate" for such a guide, this ratio may have the appeal of
defining the difference between a 21:17 or 17:14 third and the
simplest ratios for major and minor thirds of 5:4 and 6:5
respectively.

20. The ~55.28-cent diesis of these progressions is curiously very
close to the regular diatonic semitone of 22-tET (1/22 octave, ~54.55
cents), where the 7-flavor is the usual or native flavor.

21. On the question of translating the Latin _suavis_ (and the noun
_suavitas_, "agreeableness" or "pleasantness," or "attraction"), see
http://www.ixpres.com/interval/monzo/euler/euler-en.htm. Most
remarkably, one might add, the 7-flavor of our e-based tuning features
a virtually pure 7:4 of 15 fifths up -- a regular major sixth plus
diesis -- of (~913.82 + ~55.28) or 969.10 cents, only ~0.28 cents wide
of a just 7:4 at ~968.83 cents. While Euler proposes an intonation of
the 18th-century dominant seventh sonority at 36:45:54:63 or 4:5:6:7,
a Neo-Gothic most proximate triple in the e-based 7-flavor such as
E3-B3-C#*4 (roughly 0-264.50-969.10 cents) resolving intensively to
E*3-B*3-B*3 or remissively to F#3-C#4-C#4 (m7-5 + m3-1), nicely
illustrates a near-pure 4:6:7 in a neo-medieval setting.

22. At around 341.46 cents, the e-based augmented second is very
slightly closer to 17:14 (~5.33 cents wide) than to 11:9 (~5.94 cents
narrow), and interestingly close to 28:23 (~340.55 cents), possibly
marking roughly the zone of transition from a "supraminor third" to a
more neutral quality. In the context of the 17-flavor split fifth,
this interval may take on more of a supraminor quality by contrast
with the submajor third (diminished fourth) at ~363.14 cents, only
~2.68 cents narrow of 21:17. Please let me thank Jacky Ligon, who in a
different context mentioned using 28:23 (personal communication), thus
bringing to my attention what proved to be a nice approximation of the
augmented second or 17-flavor supraminor third in the e-based tuning.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Gentle Introduction to neo-Gothic (I, Pt. 2C)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Monz <MONZ@JUNO.COM>