Yahoo Tuning Groups Ultimate Backup tuning Re: Keenan Pepper's Noble Tuning (Part 2 of 2)

-------------------------------------------
Keenan Pepper's Exquisite Neo-Gothic Tuning
Noble Mediants and Walks on Plateaus
(Part 2 of 2)
-------------------------------------------

---------------------------------------------------------
2. Walking the gentle plateau: Mediants and intermediates
---------------------------------------------------------

A leading feature of neo-Gothic temperaments in their most
characteristic range between Pythagorean and 17-tET is the use of
complex major and minor thirds as unstable intervals typically
inviting resolutions to unisons or fifths, for example.

Major thirds are located on a "plateau" not too far from midway
between the simpler ratios or "valleys" of 5:4 (~386.314 cents) and
9:7 (~435.084 cents). Minor thirds are likewise located on a plateau
between 7:6 (~266.871 cents) and 6:5 (~315.641 cents).

While all intervals in these plateau regions are rather complex,
different neo-Gothic temperaments offer finely distinguished degrees
of "shading." For example, a Pythagorean major third at 81:64
(~407.820 cents) may have a somewhat different flavor or "color" than
a 29-tET major third at ~413.793 cents, or a 17-tET major third at
~423.529 cents.

In placing Keenan's beautiful neo-Gothic temperament on this
continuum, we can use a technique of exploration I term "walking the
gentle plateau," based in part on the same Classic and Noble Mediants
we encountered in defining the tuning itself (Section 1).

Let us here focus on the plateau region for major thirds poised
between 5:4 and 9:7. One reference point for this intermediate region
is the Classic Mediant, defined simply as the sum of the two
numerators over the sum of the two denominators:

(5 + 9)
ClassicMediant = ------- = 14:11 = ~417.508 cents
(4 + 7)

We may write this Classic Mediant either as 14:11, or in a mediant
notation as (5+9):(4+7). This complex integer ratio is one of the
"Four Convivial Ratios" of neo-Gothic theory closely approximated by
Keenan's tuning (see Section 3 below).[9]

In exploring the plateau, a cardinal landmark of special interest is
the rough "midpoint" of maximal complexity between 5:4 and 9:7, the
point of "gravitational balance" as it were about equally remote from
either simpler interval. Using Dave's application of the Noble Mediant
to this problem, we can estimate the point of maximal complexity
between the ratios i:j and m:n, where i:j is the simpler of the two
ratios, as follows:

(i + m Phi)
NobleMediant = -----------
(j + n Phi)

Here 5:4 is somewhat simpler than 9:7, so we treat 5:4 as i:j and 9:7
as m:n, with the following result:

(5 + 9 Phi) ~19.5623
NobleMediant = ----------- = -------- = ~1.2764 = ~422.487 cents
(4 + 7 Phi) ~15.3262

Thus our theoretical point of maximal complexity is about 422.487
cents, not far from the major third of 17-tET (~423.529 cents). Other
ratios in the plateau region represent various shades or degrees of
complexity which may "slope" slightly toward either 5:4 or 9:7.

In exploring these fine shadings, or "walking the plateau," we can use
a generalized variation of the mediant formulas to generate a variety
of other ratios or "weighted intermediates." As with the Noble
Mediant, let us define i:j as the simpler of the two "valley" ratios
(here 5:4) and m:n as the less simple ratio (here 9:7):

(xi + ym)
WeightedIntermediate (y/x) = ---------
(xj + yn)

Here "y" is the weighting factor by which we multiply the terms of the
more complex ratio (9:7), and "x" the factor by which we multiply the
terms of the simpler ratio (5:4).

In the case of y/x=1, we have the Classic Mediant between 5:4 and 9:7
of (5*1 + 9*1):(4*1 + 7*1), or simply (5+9):(4+7) or 14:11.

In the case of y/x=Phi, we have the Noble Mediant estimating the
region of maximal complexity: (5*1 + 9*Phi):(4*1 + 7*Phi), or ~422.487
cents.

The special case of the Noble Mediant where y/x=Phi, or ~1.618, serves
as a cardinal point of orientation, the point of rough "gravitational
balance." As y/x becomes smaller, our plateau may tend to slope
somewhat toward the simpler valley 5:4; as it becomes larger, we may
be sloping somewhat toward the less simpler valley, 9:7.

Choosing other values for y/x allows us to explore other shades of
complexity along the plateau, but with an important caution! If we
wish to stay within the plateau region rather than inadvertantly
wander into a valley, we need some sense of "metes and bounds" for
this region.

A useful lower bound for the neo-Gothic plateau region between 5:4 and
9:7 is the classic Pythagorean major third at 81:64 or ~407.820 cents.
As it turns out, this represents a y/x ratio of 4/9:

(5*9 + 9*4) (45 + 36) 81
WeightedIntermediate (4/9) = ----------- = --------- = --
(4*9 + 7*4) (36 + 28) 64

One possible upper bound occurs at y/x=2, or 23:18, ~424.364 cents or
slightly beyond 17-tET:

(5*1 + 9*2) (5 + 18) 23
WeightedIntermediate (2/1) = ----------- = -------- = --
(4*1 + 7*2) (4 + 14) 18

Walking the central plateau between these two ratios, we find a range
of intermediate ratios and shades of complexity, of which some of the
familiar and not so familiar ones are shown in the following "map."

Here the main number line shows the sizes of major thirds in cents,
with the notations above this line indicating Keenan's tuning (KP) and
some equal temperaments. Fractions and interval ratios below the
number line show weighted values for y/x and resulting integer ratios,
with the Noble Mediant (NM) occurring at y/x=Phi. At the top of the
diagram, two number lines show approximate distances from the "valley"
ratios of 5:4 and 9:7.

Distance from 5:4 (~386.314)
+21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
Distance from 9:7 (~435.084)
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
-28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11

407.55 409.76 413.79 416.38 417.39 423.53
53 41 29 KP 46 17
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
407 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
4/9 2/3 3/4 1/1 4/3 Phi 2/1
81:64 19:15 33:26 14:11 51:40 NM 23:18
407.82 409.24 412.75 417.51 420.60 422.49 424.36

Our neo-Gothic plateau presents a thriving community of both complex
integer ratios (e.g. 81:64, 14:11) and irrational ratios such as those
of the 29-tET and 17-tET major thirds, the major third of Keenan's
tuning at ~416.382 cents belonging to the latter group.

We are free to approximate an irrational ratio by integer ratios as
closely as desired. For example, the 29-tET major third has an
excellent approximation at y/x=8/11, or 127:100 (~413.794 cents), as
compared to 10/29 octave at ~413.793 cents.

Now that our "walk" has situated Keenan's tuning in a fertile
neo-Gothic plateau region, and we have a general map of the
area, let us explore this tuning and its immediate neighborhood in
more detail.

--------------------------------------------------
3. Keenan's tuning and the "four convivial ratios"
--------------------------------------------------

A fascinating feature of Keenan's Phi-based tuning is its close
approximation of interval sizes for regular major and minor thirds,
and also "alternative thirds" (diminished fourths and augmented
seconds) which I affectionately term the "Four Convivial Ratios."

For regular major and minor thirds, these ratios are 14:11 and 13:11.
For alternative major and minor thirds, they are 21:17 and 17:14 --
these intervals (or their approximations) being represented in the
central neo-Gothic plateau region by diminished fourths and augmented
seconds respectively.

The description of these four ratios as "convivial" alludes to the
fact that all four occur or are closely approximated in a common
region of the neo-Gothic spectrum of regular tunings.

This region ranges from slightly beyond 29-tET (fifth ~703.448 cents,
~1.493 cents wide) to slightly beyond 46-tET (fifth ~704.348 cents,
~2.393 cents wide), centered at a fifth size of around 704 cents, or 2
cents wider than pure. A table may make this "conviviality" clearer by
showing the size of fifth involved in a pure tuning of each of these
four complex intervals:

--------------------------------------------------
Interval Ratio Cents Fifth Temperament
--------------------------------------------------
m3 13:11 ~289.210 ~703.597 ~+1.641
aug2 17:14 ~336.130 ~704.014 ~+2.059
dim4 21:17 ~365.825 ~704.272 ~+2.317
M3 14:11 ~417.508 ~704.377 ~+2.422
--------------------------------------------------

With its fifth at around 704.096 cents, or ~2.141 cents wider than
pure, Keenan's tuning seems ideally situated to approximate all four
ratios. Before confirming this point, let us consider the nature of
the four ratios themselves.

Most obviously, all four ratios define _complex_ thirds, intervals
fitting an active role in neo-Gothic cadential progressions, and also
serving as "pregnant pauses" at the end of a musical phrase or section
giving a feeling of momentarily suspended motion.

While the regular thirds at or near 14:11 and 13:11 may be seen as
somewhat exaggerated versions of the usual Pythagorean 81:64 and
32:27, the "alternative" thirds at or near 21:17 and 17:14 have
a quality at once comparably complex and yet different. They might be
described respectively as "submajor" and "superminor" thirds
(following the interval names of Manuel op de Coul's Scala
program).[10]

It is from this general musical perspective that we might consider
some fine points of possible mathematical interpretations.

As we have seen, 14:11 is the Classic Mediant between 5:4 and 9:7, or
(5+9):(4+7). Interestingly, 13:11 is likewise the Classic Mediant
between 7:6 and 6:5, written (6+7):(5+6) if we wish to follow the
convention of placing the terms of the simpler "valley" ratio (here
6:5) first.

With 21:17 and 17:14, mediant-oriented interpretations may be somewhat
more tentative and ambiguous.

One possible interpretation might be that as a "superminor" third,
17:14 represents an intermediate form between the minor third "valley"
at 6:5 and a possible neutral third "valley" at 11:9 (~347.408 cents).
If so, it is the Classic Mediant (6+11):(5+9).

Such an explanation raises the question of whether and to what extend
11:9 is a distinct "valley" in itself.[11] A different interpretation
might consider 17:14 to be a weighted intermediate between the clearer
valleys of 6:5 and 5:4, leaning more toward 6:5. If so, then y/x=2/1
will produce this interval:

(5 + 6*2) (5 + 12) 17
WeightedIntermediate (2/1) = --------- = -------- = --
(4 + 5*2) (4 + 10) 14

Interestingly, 17:14 (~336.130 cents) is quite close to the Noble
Mediant between 6:5 and 5:4, in theory a region of maximal complexity,
where y/x=Phi:

(5 + 6*Phi) ~14.7082
NobleMediant = ----------- = -------- = ~1.2165 = ~339.344 cents
(4 + 5*Phi) ~12.0902

For the submajor third at 21:17, at least two interpretations are
likewise possible. If we take this interval to be a weighted
intermediate between 11:9 (regarded as a "valley") and 5:4, then a
value of y/x=1/2 would result in this ratio:

(5*2 + 11) (10 + 11) 21
WeightedIntermediate (1/2) = ---------- = --------- = --
(4*2 + 9) (8 + 9) 17

As this weighting somewhat toward the simpler 5:4 ratio indicates, we
have a "submajor" rather than more neutral third, a conclusion also
congenial to an alternative interpretation of 21:17 as a weighted
intermediate between 6:5 and 5:4 with y/x=1/3:

(5*3 + 6) (15 + 6) 21
WeightedIntermediate (1/3) = ---------- = -------- = --
(4*3 + 5) (12 + 5) 17

At an intuitive level, viewing 17:14 and 21:17 as weighted
intermediates between 5:4 and 6:5 may give at least a basic idea of
their character. At y/x=2/1, 17:14 leans in a "superminor" direction,
while at y/x=1/3, 21:17 has a "submajor" quality. The Classic Mediant
of 5:4 and 6:5 where y/x=1, (5+6):(4+5) or 11:9, is the familiar
"neutral" third.

Whatever the mathematical fine points of these four "convivial" and
complex ratios for neo-Gothic thirds, Keenan's tuning felicitously
approximates all four ratios within 1.5 cents:

-------------------------------------------------------------------
Interval Ratio Cents Example Actual Size Variance
-------------------------------------------------------------------
m3 13:11 ~289.210 D-F ~287.713 ~-1.497
M3 14:11 ~417.508 F-A ~416.382 ~-1.126
aug2 17:14 ~336.130 Bb-C# ~336.860 ~+0.731
dim4 21:17 ~365.825 C#-F ~367.235 ~+1.410
-------------------------------------------------------------------

As might not be unfitting in a tuning itself based on Phi, we find
that the augmented fifth, e.g. F-C#, at ~832.765 cents, gives an
outstanding approximation of Phi itself (~833.090 cents), a variance
of only ~0.325 cents in the narrow direction.

In a standard 12-note tuning of Eb-G#, we might use this interval in a
variant of the familiar 13th-century cadence where minor third expands
to fifth and minor sixth to octave, with whole-tone motion in all
voices. Here some of the usual intervals are altered, with vertical
intervals above the lowest voice shown in parentheses and melodic
intervals as signed values showing ascending (positive) or descending
(negative) motion. In this MIDI-style notation, C4 is middle C:

C#4 -- +159.0 -- Eb4
(832.8) (1200)
G#3 -- +159.0 -- Bb3
(336.9) (704.1)
F3 -- -208.2 -- Eb3

The vertical intervals of the augmented second and augmented fifth are
resolved by a usual descending whole-step in the lowest voice coupled
with ascending melodic steps of a diminished third in the upper voices
(G#-Bb, C#-Eb), an interval of ~159.044 cents or around 3/4-tone.

As we expand this tuning beyond 12 notes to 17 or 24, there are other
surprises, including some "crunchy" Pepperian narrow fourths at
~465.530 cents, quite close to 21:16. These features call for an
article in themselves.

---------------------------------------------
4. Mathematical tunings and musical qualities
---------------------------------------------

In approaching "mathematically defined" tunings such as Kornerup's
Golden Meantone or Kennan Pepper's neo-Gothic counterpart, we
encounter two schools of opinion which may invite a third intermediate
approach -- a kind of philosophical mediant, as it were.

One kind of opinion presents such systems as "tunings to end all
tunings" -- or, at least, especially favored and privileged choices
when compared to more familiar tunings of the same general variety,
historical or otherwise.

Another kind of opinion regards such tunings based on mathematical
quantities such as Phi, pi, Euler's e, etc., as inherently suspect
exercises in "numerology" or the like. Here the argument seems to be
that a tuning should be defined around mathematical parameters with
some evident psychoacoustical significance, as opposed to arbitrary
constants which happen to produce interesting musical results.

A "mediant" approach might be to regard tunings based on arbitrary
mathematical constants -- a scale-building constant such as Phi or pi,
an interval ratio such as 14:11 or (5+9):(4+7), and so forth -- as
points on the xenharmonic continuum with "equal citizenship." Such
tunings supplement, but do not supplant, other tunings.

For example, Kornerup's Golden Meantone serves as a musically
interesting compromise between 1/4-comma meantone with pure major
thirds and Zarlino's 2/7-comma with equally tempered major and minor
thirds. Like these tunings, it is a shade of temperament.

John Harrison's meantone tuning based on a ratio of pi or ~3.14159
between the octave and the major third, adopted by Charles Lucy in his
system of LucyTuning, has fifths at ~695.493 cents (~6.462 cents
narrower than pure). This temperament, moving about a third of the way
from 2/7-comma to 1/3-comma or 19-tET, has as one of its properties
diminished fourths at ~436.056 cents, very close to a pure 9:7.

If one grants the validity of all points on the continuum, then
choosing an attractive mathematical constant and letting a tuning take
shape around it is a perfectly valid strategy. It produces a tuning
with qualities which can be deduced at least in part from an analysis
of interval sizes and nearby tunings, etc.

At the same time, such an approach has a charming "emblematic" or
"heraldic" quality which appeals to the intellect, and can be
appreciated while recognizing the validity of other tunings also.

Dan Stearns, in his recent articles on exploring or "hiking" various
"trails" of tunings based on Phi, suggests the attractiveness of this
kind of approach which can take us to musically interesting places
while leaving us free to roam throughout the continuum.[12]

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

9. While in theory the Classic Mediant may define a "valley" in its own
right between two simpler or "deeper" valleys -- e.g. 9:7 in relation
to 5:4 and 4:3, i.e. (5+4):(4+3) -- our ratio of 14:11 seems too
complex to serve as such a distinctly audible valley. Dave Keenan has
noted his own experimental finding that 14:11 is not tuneable by ear,
while Paul Erlich has proposed the test that if a ratio a:b has a
product of the two terms a*b>105, then the ratio is complex rather
than simple. With 14:11, a*b = 154, defining this ratio as complex
according to Paul's test as well as Dave's experiment.

10. See http://www.xs4all.nl/~huygensf/doc/intervals.html. In
medievalist terms, where regular major and minor thirds are known
respectively as ditones (equal to two whole-tones) and semiditones
(equal to a whole-tone plus a diatonic semitone), intervals at or near
21:17 or 17:14 might be called "subditones" and "suprasemiditones."

11. Applying Paul Erlich's test, 11:9 has a*b=99, so is near the upper
limit of a*b=105 for a "simple" interval ratio.

12. See, for example, Dan Stearns, "Re: The Golden Mediant: Complex
Ratios and Metastable Intervals," TD 827:13 (22 September 2000);
"takin' in the 3/5 'Phi trail'," TD 829:6 (23 September 2000); and
also "Wilsonian institute," TD 823:1 (21 September 2000).

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Keenan Pepper's Noble Tuning (Part 2 of 2)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Kraig Grady <kraiggrady@anaphoria.com>