Re: Gentle introduction to neo-Gothic (I, Pt. 2A) -- corrected

-----------------------------------------------------------
A gentle introduction to neo-Gothic progressions (1):
Trines, quads, and intonational flavors
Part 2A of 2: Some flavors of most proximal quads
(Corrected version -- 17 November 2000)
-----------------------------------------------------------

[Please note that this is a corrected version of this installment,
with some material added on the derivations and 24-note keyboard
locations of some sonorities, as well as the intended complete URL for
a Web link to an earlier installment. My apologies for the extra
bandwidth involved -- and warmest thanks to Robert Walker for scores
and audio files of musical examples in these articles.]

-----------------------------------------
2. Some flavors of the most proximal quad
-----------------------------------------

As already mentioned (Section 1.2), the concept of a neo-Gothic flavor
tends to focus on the tuning of the unstable intervals found in most
proximal quads and their three-voice subsets or triples (Section 4):
especially M3, m3, and M6 for finer distinctions, and also M2 and m7
for larger ones.

Here is a table of some common flavors or intonational regions for the
tuning of the most proximal quad, with examples from specific tuning
systems. No attempt is made to delimit exact borders for these
overlapping and very pleasantly "fuzzy" regions of the continuum:

----------------------------------------------------------------------
Flavor Ratio zone Sample Expansive Contractive Steps
M3 m3 Tuning (1-M3-5-M6) (1-m3-5-m7) T S
----------------------------------------------------------------------
3-flavor 81:64 32:27 Pythag 0-408-702-906 0-294-702-996 204 90
......................................................................
11-flavor 14:11 13:11 Noble5th 0-416-704-912 0-288-704-992 208 80
......................................................................
23-flavor 23:18 27:23 17-tET 0-424-706-918 0-282-706-988 212 71
......................................................................
7-flavor 9:7 7:6 22-tET 0-436-709-927 0-273-709-981 218 55
Pythag 0-431-702-929 0-271-702-973 204 67
......................................................................
13-flavor 13:10 15:13 29-tET 0-455-703-952 0-248-703-952 207 41
----------------------------------------------------------------------

As a legend on the chart may suggest, a flavor is an impressionistic
"ratio zone" where unstable intervals of a most proximate quad
approximate certain ratios. For example, the "11-flavor" is a region
where major thirds are in the general vicinity of 14:11 (~417.51
cents), and/or minor thirds in the vicinity of 13:11 (~289.20 cents).

Our example of the 11-flavor may especially well illustrate why
neo-Gothic flavors should not be confused with the "limits" of just
intonation (JI) theory. The ratios 14:11 and 13:11 both belong to the
11-flavor because they conveniently share the common factor of 11, but
would have different prime-limits as well as odd-limits (11 and 13
respectively).

The 11-flavor also illustrates another aspect of the "flavor as
ratio-zone concept": with regular tunings, the characteristic ratios
for major and minor thirds may actually occur or be approximated most
closely at slightly different points in the spectrum, suggesting a
region with subtle shadings leaning toward one ratio or the other.

For example, in the "lower" or "milder" 11-flavor region around
29-tET, minor thirds are close to 13:11 (~289.66 cents in 29-tET) but
major thirds (~413.79 cents) somewhat narrower than 14:11.

In the "central" 11-flavor region exemplified by Keenan Pepper's
"Noble Fifth" tuning[6] shown on the chart (fifths ~704.10 cents,
~2.14 cents wide), both regular thirds (~416.38 cents, ~287.72 cents)
are quite close to these ratios -- within 1.5 cents in this tuning.

In the "upper" or "stronger" 11-flavor region, as in the "e-based
tuning"[7] (fifths ~704.61 cents, ~2.65 cents wide), major thirds are at
or slightly larger than 14:11 (here ~418.43 cents) while minor thirds
(~286.18 cents) are rather smaller than 13:11.

Like fine shadings within a given flavor, fuzzy "border regions" of
transitions between flavors are characteristic of the system. Regular
tunings with fifths at around 703 cents (roughly 1 cent wide), for
example, might share qualities of both the 3- and 11-flavors; while
such tunings with fifths at around 705 cents (roughly 3 cents wide)
might likewise be transitional between the 11- and 23-flavors.

The different flavors and shadings differ not only the sizes of
vertical intervals and tunings of most proximal quads, the latter
shown in rounded cents for some sample tuning systems, but also in the
sizes of cadential whole-tones (T) and semitones (S).

As we move from the 3-flavor to the 13-flavor, expansive intervals of
the most proximal quad (M2, M3, M6 in M2-4, M3-5, M6-8) become larger,
while contractive intervals (m3 and m7 in m3-1, m7-5) become smaller.
At the same time, cadential whole-tones become larger while semitones
become yet narrower and more compact.

--------------------------------------------------------
2.1. Differing flavors between and within tuning systems
--------------------------------------------------------

Distinctions between flavors may arise in basically two ways: between
different neo-Gothic tuning systems, and within a single tuning
system. Let us consider these two forms of variation, both illustrated
on the chart.

Focusing on variations between tuning systems, we find that the
spectrum of flavors correlates closely for the most part with the
continuum of regular neo-Gothic tunings. Glancing at the chart, we see
that Pythagorean exemplifies the 3-flavor; a tuning such as the "Noble
Fifth" temperament, the 11-flavor; 17-tET, the 23-flavor; and 22-tET,
the 7-flavor.

These associations involve regular diatonic intervals: M3 (three
fifths up); m3 (three fourths up or fifths down); M6 (three fifths
up); M2 (two fifths up); and m7 (two fourths up or fifths down).

Additionally, however, the same tuning may offer multiple flavors of
the same unstable intervals and cadential progressions. With regular
tunings such as Pythagorean JI or 29-tET, such variations occur when
intervals such as "major thirds" of diverse sizes are derived from
differing chains of fifths.

For example, as our chart shows, Pythagorean exemplifies the 3-flavor,
but also includes a 7-flavor. In the latter option, unstable intervals
of the most proximate quads are derived from chains an extra 12 fifths
or fourths long, excellently approximating the characteristic 7-flavor
ratios:

------------------------------------------------------------
Interval Chain Cents 7-flavor ratio Cents
------------------------------------------------------------
M3 16 5ths up ~431.28 9:7 ~435.08
m3 15 4ths up ~270.67 7:6 ~266.87
M6 15 5ths up ~929.33 12:7 ~933.13
M2 14 5ths up ~227.37 8:7 ~231.17
m7 14 4ths up ~972.63 7:4 ~968.83
------------------------------------------------------------

Similarly, as we saw in our discussion of the 11-flavor, 29-tET
represents a mild variety of this flavor with regular minor thirds
very close to 13:11. On the chart, however, it is listed as an example
of the 13-flavor, whose characteristic ratios it superbly approximates
through either the same chains of fifths or fourths as in the
Pythagorean 7-flavor, or another set of comparably long chains
precisely equivalent in this tuning:

----------------------------------------------------------------------
Interval Equivalent Chains Cents 13-flavor ratio Cents
----------------------------------------------------------------------
M3 16 5ths up = 13 4ths up ~455.17 13:10 ~454.21
m3 15 4ths up = 14 5ths up ~248.27 15:13 ~247.74
M6 15 5ths up = 14 4ths up ~951.72 26:15 ~952.26
M2 14 5ths up = 15 4ths up ~248.27 15:13 ~247.74
m7 14 4ths up = 15 5ths up ~951.72 26:15 ~952.26
----------------------------------------------------------------------

As this listing shows, the 13-flavor is a region where vertical major
seconds and minor thirds have precisely or approximately the same
size, and likewise major sixths and minor sevenths! We explore this
striking state of affairs in the next section.

From a practical standpoint, the 7-flavor option in Pythagorean or the
13-flavor option in 29-tET obviously requires a large tuning to
accommodate the elongated chains of fifths. A 24-note tuning provides
these altered flavors of intervals in about as many positions as their
regular or "native" counterparts in a 12-note tuning -- precisely as
many in Pythagorean tuning or the 29-tET realization of the 13-flavor
using the same chains of fifths or fourths.

Moving from the question of _how_ diverse flavors arise in a single
tuning to the question of _why_ they are used, we come to a leading
motivation: the amplification of directed cadential progressions.

------------------------------------------------------------
2.2. A closer "closest approach": the 7-flavor and 13-flavor
------------------------------------------------------------

In Part I (Section 1.2), we surveyed the usual "3-flavor" resolutions
of most proximate quads in medieval Pythagorean tuning, a point of
departure for neo-Gothic practice, noting three intonational traits:

(1) Major and minor thirds at 81:64 (~408 cents) and 32:27 (~294
cents), and major sixths at 27:16 (~906 cents), have a rather complex
quality which enhances a sense of directed cadential tension;

(2) There is a pleasing melodic contrast between generously large
whole-tones at 9:8 (~204 cents) and compact ascending or descending
semitones at 256:243 (~90 cents); and

(3) Cadential action is quite efficient and economic, with unstable
intervals needing to expand (M2-4, M3-5, M6-8) or contract (m3-1,
m7-5) by a total distance of only about 294 cents (equal to the size
of a 32:27 minor third) in order to attain their stable goals.

All three traits, and especially (2) and (3), may be seen as aspects
of a medieval and neo-Gothic cadential ethos of "closest approach," in
which unstable intervals should "approach" their stable goals as
closely as possible, resolving by maximally parsimonious and efficient
expansion or contraction. From a melodic viewpoint, as Mark Lindley
has aptly stated, such progressions involve narrow or "incisive"
diatonic semitones.

As we move along our continuum from the 3-flavor to the 13-flavor, as
shown in the table near the beginning of this article segment (opening
of Section 2), this cadential closest approach becomes yet closer.
Expansive intervals become yet wider, contractive intervals yet
narrower, resulting in yet more parsimonious resolutions with yet
keener or more incisive melodic semitones.

To illustrate these themes, we focus here on the "superefficient"
cadential progressions of the 7-flavor as realized in Pythagorean
tuning, and the 13-flavor as realized in 29-tET.

----------------------------------------------------
2.2.1. The Pythagorean 7-flavor: streamlined valleys
----------------------------------------------------

In Pythagorean tuning, the altered cadential intervals of most
proximate quads in the 7-flavor have generating chains 12 fifths or
fourths longer than their usual 3-flavor counterparts (Section 2.1).
This makes expanding intervals (M2, M3, M6) wider than usual by a
Pythagorean comma of 531441:524288 (~23.46 cents), and contracting
intervals (m3, m7) a comma narrower.

The following diagrams of standard resolutions in this 7-flavor may
invite comparison with those given in Part I (Section 1.2) for the
usual Pythagorean 3-flavor cadences:

http://www.egroups.com/message/tuning/15038

Here I have used the ASCII ^ symbol to show a note raised by a comma,
and the @ symbol to show a note lowered by a comma[8]:

Expansive/Intensive Contractive/Intensive

E^4 ----- +67 ----- F4 D4 ----- -204 ----- C4
(227) (498) (271) (0)
D4 ------ -204 ----- C4 B^3 ----- +67 ----- C4
(498,271) (498,0) (702,431) (702,702)
B^3 ----- +67 ----- C4 G3 ----- -204 ----- F3
(929,702,431) (1200,702,702) (973,702,271) (702,702,0)
G3 ------ -204 ----- F3 E^3 ----- +67 ----- F3

Expansive/Remissive Contractive/Remissive

E4 ---- +204 ----- F#4 D@4 ----- -67 ----- C#4
(227) (498) (271) (0)
D@4 ---- -67 ----- C#4 B3 ----- +204 ----- C#4
(498,271) (498,0) (702,431) (702,702)
B3 ---- +204 ----- C#4 G@3 ----- -67 ----- F#3
(929,702,431) (1200,702,702) (973,702,227) (702,702,0)
G@3 ----- -67 ----- F#3 E3 ----- +204 ----- F#3

In comparing these progressions with the usual 3-flavor ones, we might
first note what has _not_ changed. All fifths and fourths, in unstable
quads as well as stable trines, remain pure; and melodic whole-tones
remain at a pure 9:8 (~204 cents).

Cadential semitones, however, have shrunk from around 90 cents to a
"supercompact" 67 cents -- more precisely 66.76 cents, the difference
being equal to the Pythagorean comma.

From a vertical perspective, expansive unstable intervals have become
a comma larger, and contractive intervals a comma smaller, thus
reducing the total motion required for their resolution to stability
from around 294 cents to 271 cents, the size of the Pythagorean
7-flavor minor third (e.g. B3-D@3 or E3-G@3).

Thus these progressions offer a very impressive accentuation of the
"closest approach" ideal while retaining just intonation for the
stable concords, 3:2 fifths and 4:3 fourths.

At the same time, intriguingly, this intensified and "superefficient"
7-flavor cadential action may actually produce simpler and more
"consonant" ratios for the expansive and contractive quads than the
usual 3-flavor tuning. In the 7-flavor, these quads are very close
respectively to 14:18:21:24 and 12:14:18:21, while in the 3-flavor
they have more complex ratios of 64:81:96:108 and 54:64:81:96.

To use the terms of harmonic entropy or consonance/dissonance
theorists such as Paul Erlich and David Keenan, unstable 7-flavor
intervals are in the "valley" regions of 8:7 (M2), 7:6 (m3), 9:7 (M3),
12:7 (M6), and 7:4 (m7), differing from these ratios by only about
3.80 cents. Their usual Pythagorean 3-flavor counterparts other than
M2 at 9:8, in contrast, are in "plateau" regions of somewhat greater
complexity between such simple valley ratios.

Thus if one set out to optimize the "closest approach" ideal (narrow
cadential semitones, minimal distance of expansion or contraction),
and at the same time to maximize the simplicity or "consonance" of
unstable quads as well as stable trines, the Pythagorean 7-flavor
would seem a near-ideal solution.

Both the efficiency of these resolutions and the simplicity of the
unstable quads lend to the 7-flavor a certain "streamlined" quality:
the contractive quad, especially, can be quite "mellow," while the
expansive quad is, not so surprisingly, rather more outgoing.

In short, the Pythagorean 7-flavor seems to have these features:

(1) Unstable most proximate quads actually have simpler ratios,
~14:18:21:24 (expansive) and ~12:14:18:21 (contractive), than in the
usual 3-flavor, making them somewhat "smoother," but still decidedly
more complex than a stable 2:3:4 trine, so that the dynamic tension is
still sufficient to impel superb cadential action;

(2) While melodic whole-tones remain at 9:8 (~204 cents), cadential
semitones shrink from around 90 cents to a superkeen 67 cents,
heightening the contrast between these steps; and

(3) The total amount of expansion or contraction required in two-voice
resolutions also shrinks from an already efficient 294 cents to a
superefficient 271 cents, contributing along with the simpler "valley"
ratios for the unstable quads to a "streamlined" impression for these
7-flavor cadences.

----------------------------------------------------
2.2.2. The 13-flavor in 29-tET: an impressive summit
----------------------------------------------------

Carrying the themes of superwide expansive intervals, supercompact
contractive ones, and extra-narrow cadential semitones a step further,
we arrive at a "13-flavor" where major and minor thirds have ratios at
or near 13:10 (~454 cents) and 15:13 (~248 cents).

In the regular tuning of 29-tET, these strikingly impressive cadences
are combined with near-pure fifths and fourths. The following
examples, based on the same chains of fifths or fourths as the
Pythagorean 7-flavor (e.g. wide major third as 16 fifths up), use the
ASCII asterisk (*) to show a note raised by a diesis of 1/29 octave or
1/5-tone (~41.38 cents), and a "d" symbol to show a note lowered by a
diesis[9]:

Expansive/Intensive Contractive/Intensive

E*4 ----- +41 ----- F4 D4 ----- -207 ----- C4
(248) (497) (248) (0)
D4 ------ -207 ----- C4 B*3 ----- +41 ----- C4
(497,248) (497,0) (703,455) (703,703)
B*3 ----- +41 ----- C4 G3 ----- -207 ----- F3
(952,703,455) (1200,703,703) (952,703,248) (703,703,0)
G3 ------ -207 ----- F3 E*3 ----- +41 ----- F3

Expansive/Remissive Contractive/Remissive

E4 ---- +207 ----- F#4 Dd4 ----- -41 ----- C#4
(248) (497) (248) (0)
Dd4 ---- -41 ----- C#4 B3 ----- +207 ----- C#4
(497,248) (497,0) (703,455) (703,703)
B3 ---- +207 ----- C#4 Gd3 ----- -41 ----- F#3
(952,703,455) (1200,703,703) (952,703,248) (703,703,0)
Gd3 ----- -41 ----- F#3 E3 ----- +207 ----- F#3

A distinguishing mark of the 13-flavor, noted in Section 2.1, is that
expansive major sixths and contractive minor sevenths are about the
same size (here both precisely equal to 23/29 octave or ~952 cents);
and likewise expansive vertical major seconds and contractive minor
thirds (here both 6/29 octave, or ~248 cents).

This convergence between extra-wide and extra-narrow cadential
intervals dramatizes their categorical ambiguity and complexity, also
shared by the major third at 11/29 octave or around 455 cents, in the
zone where wide major thirds approach the size of narrow fourths.

This heightened vertical tension resolves through "hyperefficient"
cadential action, with a melodic semitone or diesis equal to only
1/5-tone, or about 41 cents, and a contrasting whole-tone motion equal
to the usual 29-tET interval of 5/29 octave or ~206.90 cents, quite
close to the standard Pythagorean 9:8.

The total expansion or contraction involved in two-voice resolutions
is reduced to about 248 cents (~15:13), the size of the small vertical
minor third in these progressions.

If 7-flavor cadences have a streamlined quality, 13-flavor cadences
may take us on a kind of musical journey through hyperspace, with
unstable intervals "warping" or blurring the usual categories, an
effect at once accentuating cadential tension and captivating the
listener's imagination.

In sum:

(1) The complexity and categorical ambiguity of unstable 13-flavor
intervals serves to maximize cadential tension and drama;

(2) The polarity between melodic whole-tone and semitone motions,
respectively around 207 cents and 41 cents in 29-tET, is also
maximized; and

(3) Two-voice resolutions are even more parsimonious or efficient than
in the 7-flavor, with a total expansion or contraction in 29-tET of
only around 248 cents.

If 7-flavor resolutions place us in an intriguing "valley," then
13-flavor resolution place us on a musical "summit" of maximal
cadential tension resolved with maximal efficiency.

------------------------------------------------------
2.3. Transition zones between flavors: an illustration
------------------------------------------------------

This introduction to 7-flavor and 13-flavor cadences provides an
opportunity to illustrate the "fuzzy" transitions between such
flavors. A fine example is Keenan Pepper's "Noble Fifth" tuning.

In this tuning with fifths at ~704.10 cents, usual 11-flavor cadences
may be supplemented with "superefficient" progressions where unstable
intervals of most proximate quads are derived from chains of 13, 14,
or 15 fifths or fourths:

---------------------------------------------------------------------
Interval Chain Cents 7-ratio/cents 13-ratio/cents
---------------------------------------------------------------------
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)
---------------------------------------------------------------------

These intervals of the Noble Fifth tuning have an intermediate
character, with M3 slightly closer to 13:10 than to 9:7, and the other
intervals leaning slightly toward 7-flavor ratios. Here M6 and m7, or
M2 and m3, differ in size by around 18.77 cents, in contrast to their
equivalence or near-equivalence in the 13-flavor and their difference
of 49:48 (~35.70 cents) when given simplest 7-flavor ratios.

Considering our standard cadential resolutions confirms this
impression of an intermediate quality, and also illustrates the
curious interval spellings resulting from the chains of fifths or
fourths used in generating this flavor. As in 29-tET, the signs "*"
and "d" show notes raised or lowered by a diesis, here equal to ~49.15
cents, the difference between Ab and G#:

Expansive/Intensive Contractive/Intensive

Fd4 ----- +49 ----- F4 D4 ----- -208 ----- C4
(239) (496) (257) (0)
D4 ------ -208 ----- C4 Cd4 ----- +49 ----- C4
(496,257) (496,0) (704,447) (704,704)
Cd4 ----- +49 ----- C4 G3 ----- -208 ----- F3
(943,704,447) (1200,704,704) (961,704,257) (704,704,0)
G3 ------ -208 ----- F3 Fd3 ----- +49 ----- F3

Expansive/Remissive Contractive/Remissive

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

My first impression of these progressions was that the expansive quads
have a stretched or somewhat "warped" feeling suggesting a kinder
and gentler 13-flavor, while the contractive quads are "cooler" or
mellower.

Note that in this scheme, the wide major third (13 fourths up) is
derived as a fourth narrowed by a diesis, e.g. G3-Cd4 or F#*3-B4, or a
rounded (496 - 49) = 447 cents. Likewise the wide major sixth,
e.g. G-Fd4 or F#*3-E4, is equal to a minor seventh less a diesis,
about (992 - 49) = 943 cents; and the wide major second, e.g. D4-Fd4 or
C#*4-E4, to a minor third less a diesis or about (288 - 49) = 239
cents.

Conversely, the narrow minor third, e.g. Cd4-D4 or B3-C#*4, is equal
to a major second plus a diesis, or about (208 + 49) = 257 cents; and
a narrow minor seventh, e.g. Fd3-D4 or E3-C#*4, to a major sixth plus
diesis, or about (912 + 49) = 961 cents.[10]

Such curious spellings may have more of a "real-world" meaning if we
consider how these progressions might be realized on a 24-note
keyboard instrument with two 12-note manuals tuned a diesis apart,
with the lower-pitched manual taken as the "standard" one:

Expansive/Intensive Contractive/Intensive

F4 ----- +49 ----- F*4 D*4 ----- -208 ----- C*4
(239) (496) (257) (0)
D*4 ----- -208 ----- C*4 C4 ----- +49 ----- C*4
(496,257) (496,0) (704,447) (704,704)
C4 ----- +49 ----- C*4 G*3 ----- -208 ----- F*3
(943,704,447) (1200,704,704) (961,704,257) (704,704,0)
G*3 ----- -208 ----- F*3 F3 ----- +49 ----- F*3

Expansive/Remissive Contractive/Remissive

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

To obtain a full expansive quad, as I have learned from experience,
one plays the lowest note and fifth on the upper keyboard, plus the
apparent fourth and minor seventh (actually the wide major third and
sixth!) on the lower keyboard, thus G*3-C4-D*4-F4 or F#*3-B3-C#*4-E4.
For the full contractive quad, one plays the lowest note and fifth on
the lower keyboard, plus the apparent major second and sixth (actually
the narrow minor third and seventh!) on the upper keyboard, here
F3-G*3-C4-D*4 or E3-F#*3-B3-C#*4.

In intensive progressions, the resolution is to a trine or fifth on
the upper keyboard, with two upper parts ascending in parallel fourths
or fifths by very compact 49-cent dieses, here from C4-F4 to C*4-F*4
(expansive) or F3-C4 to F*3-C*4 (contractive).

In remissive progressions, the resolution is to the lower keyboard,
with descending motions by dieses: here F#*3-C#*4 to F#3-C#4 in either
the expansive or contractive resolution.

Such intermediate shadings enrich the intonational possibilities of
neo-Gothic music, and can also contribute to the unique qualities of a
given tuning system, while sometimes making life considerably more
interesting and complicated for a keyboard player.

----------------
Notes to Part 2A
----------------

6. The "Noble Fifth" tuning is defined as having a fifth equal to the
Phi-based "Noble Mediant" between 4/7-octave (7-tET) and 3/5-octave
(5-tET), weighted toward the latter. That is, the "Noble Fifth" of
this tuning is equal to the octave fraction (4 + 3 Phi)/(7 + 5 Phi),
or about 704.096 cents. Phi is the "Golden Ratio" of ~1.61803. As
Keenan Pepper noted in describing this tuning, it is a counterpart to
Thorwald Kornerup's "Golden Meantone" with a fifth equal to the octave
fraction (3 + 4 Phi)/(5 + 7 Phi).

7. The "e-based" tuning has a ratio between the whole-tone and
diatonic semitone equal to Euler's _e_, ~2.71828, producing a fifth of
about 704.609 cents.

8. In a 24-note Pythagorean arrangement with two 12-note keyboards
tuned with identical intervals, but a Pythagorean comma apart -- the
Xeno-Gothic system -- we could assign a fixed note name to each key,
producing somewhat different spellings for some of these resolutions.
For example, suppose we take the lower-pitch keyboard as the "usual"
one. Then the spelling for intensive resolutions would remain as
shown; remissive resolutions would be G3-B^3-D4-E^4 to F#^3-C#^4-F#^4
(expansive) and E^3-G3-B^3-D4 to F#^3-C#^4 (contractive). More
generally, in such 24-note schemes with two manuals, intensive
progressions resolve to stable sonorities on the lower manual, and
remissive progressions to sonorities on the upper manual.

9. In a tuning of 24-out-of-29-tET mapped to two 12-note keyboards a
diesis apart, with the lower-pitched keyboard taken as the "usual" one
(compare n. 8), the remissive progressions would have the spellings
G3-B*3-D4-E*4 to F#*3-C#*4-F#*4 (expansive) and E*3-G3-B*3-D4 to
F#*3-C#*4 (contractive).

10. In Pythagorean tuning, where a 7-flavor major third is derived
from a usual major third (4 fifths up) plus a comma (12 fifths up), or
16 fifths up, this interval of about (408 + 23) = 431 cents is smaller
than a fourth (1 fourth up) minus a comma (12 fourths up), or 13
fourths up, about (498 - 23) or 471 cents. In 29-tET, these two
intervals have the property of being precisely equal 13-flavor
intervals: roughly, (413.8 + 41.4) = (496.6 - 41.4) = 455.2 cents. As
we move further along the continuum of regular tunings, the
fourth-minus-diesis becomes increasingly _smaller_ than the
major-third-plus-diesis, so that in our e-based tuning (fifth ~704.61
cents, ~2.65 cents wide), with a diesis of ~55.28 cents, the situation
is almost the converse of Pythagorean. Here 13 fourths up yield a
7-flavor major third of around (495 - 55) or 440 cents, and 16 fifths
up an intervals of around (418 + 55) or 473 cents.

Most respectfully,

Margo Schulter
mschulter@value.net