Yahoo Tuning Groups Ultimate Backup tuning Re: Keenan Pepper's Noble Tuning (Part 1 of 2)

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Keenan Pepper's Exquisite Neo-Gothic Tuning
Noble Mediants and Walks on Plateaus
(Part 1 of 2)
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In a recent article on "The Other Noble Fifth,"[1] Keenan Pepper
presented a beautiful super-Pythagorean or neo-Gothic counterpart to
Thorwald Kornerup's "Golden Meantone," both tunings derivable from a
Phi-based function which Dave Keenan has termed the "Golden Mediant"
or "Noble Mediant."

While in Kornerup's tuning the ratio of diatonic/chromatic semitones
is equal to Phi, the "Golden Ratio" of ~1.61803398874989484820459, in
Pepper's new tuning this relationship is reversed, with the chromatic
semitone larger than the diatonic by this same factor of Phi.

Upon learning of Keenan Pepper's tuning, Dave Keenan realized that the
same Golden Mediant function used to find the size of its fifth could
be used to approximate the region of maximum complexity or ambiguity
between two simpler intervals, e.g. 5:4 and 9:7, or 6:5 and 7:6. In a
recent paper posted to the Tuning List, Dave Keenan and I discussed
this application and some possible connections with the interval
aesthetics of medieval European and neo-Gothic styles.[2]

Drawing these two threads together, I would like to show how Keenan
Pepper's "noble" counterpart to Kornerup's Golden Meantone itself
exemplifies the theme of beautiful complexity, providing a tuning
exquisitely suited to neo-Gothic music.

With fifths and fourths gently tempered by only about 2.14 cents, this
tuning offers approximations to within 1.5 cents of four favorite
neo-Gothic ratios: 14:11 and 13:11 for regular major and minor thirds;
and 21:17 and 17:14 for diminished fourths and augmented seconds, used
as alternative "submajor" and "superminor" thirds.

A pleasing musical feature as well as defining property of this tuning
is the contrast between the compact diatonic semitone at ~79.52 cents
and the large chromatic semitone at ~128.67 cents, the larger semitone
having to the smaller a ratio of Phi. This contrast invites various
experiments in medieval or neo-medieval chromaticism after the example
of Marchettus of Padua (1318).

Additionally, the tuning includes an augmented fifth within 1/3-cent
of Phi itself.

In exploring how Keenan Pepper's Phi-based neo-Gothic temperament so
nicely embodies Dave Keenan's theme of complexity, one diverting
complication is the similarity of these two names. Following the
rather informal style common on the Tuning List, I have decided to
refer to Keenan Pepper as "Keenan," and to Dave Keenan as "Dave."

Section 1 presents Keenan's Golden Mediant or Noble Mediant[3] as used to
calculate both Kornerup's Golden Meantone and its super-Pythagorean or
neo-Gothic counterpart.

Section 2 introduces the practical and contemplative art of "walking
the gentle plateau" in order to explore the region of complexity
between two simpler intervals or "valleys" such as 5:4 and 9:7, with
the classic mediant and Dave's application of the Golden Mediant
serving as helpful cardinal points of reference.[4]

Section 3 applies the method of "walking the plateau" to Keenan's
tuning, with a special focus on this tuning's close approximations of
four favorite neo-Gothic ratios already mentioned: 14:11, 13:11,
21:17, and 17:14.

Section 4 briefly considers some issues regarding the use of
mathematical quantities such as Phi, pi, or Euler's _e_ to define
musical tuning schemes, suggesting that such tunings can serve as
beautiful but not _exclusively_ meritorious choices on the open
xenharmonic continuum.

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1. The Kornerup and Pepper tunings: Two "Noble Mediants"
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We might describe the "noble" tunings of both Thorwald Kornerup and
Keenan Pepper as "Noble Mediants" between 7-tone equal temperament
(7-tET) and 5-tET. These two tunings mark boundary conditions for the
spectrum of regular diatonic tunings generating all intervals, or all
intervals other than octaves, from chains of fifths having identical
sizes.

Of course, both 7-tET and 5-tET are noble tunings in their own right
featured or approximated (more or less closely) by various world
musics, e.g. 7-tET in Thai and Cambodian music[5], and 5-tET as a very
rough model for the customized _slendro_ tunings of Balinese and
Javanese gamelan ensembles.

In 7-tET, the fifth is equal to 4/7 octave (~685.714 cents), while in
5-tET it is equal to 3/5 octave (720 cents).

One approach for arriving at an intermediate tuning would be to find
the traditional mediant or Classic Mediant between these two ratios of
the fifth to the octave. For the ratios i:j and m:n, this Classic
Mediant is defined by taking the sum of the two numerators and
dividing by the sum of the two denominators:

i + m
ClassicMediant = -----
j + n

Thus for the 7-tET and 5-tET fifths, we have:

4 + 3 7
-------- = -- = 700 cents
7 + 5 12

Interestingly, the Classic Mediant of 7-tET and 5-tET is 12-tET,
another equal temperament having a size equal to the sum of their
sizes (5 + 7 = 12).

Additionally, following Keenan, we can use a different variety of
mediant to find both Kornerup's Golden Meantone and its ideal
neo-Gothic counterpart: the Phi-based Golden Mediant or Noble Mediant.

For two ratios i:j and m:n, this Noble Mediant is defined as follows:

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

As Keenan demonstrated in presenting his new scale, we can treat
either the 7-tET fifth (4/7 octave) or the 5-tET fifth (3/5 octave) as
the Phi-weighted m:n ratio. The first choice produces Kornerup's
Golden Meantone, with a fifth somewhat closer to 7-tET and smaller
than the Classic Mediant fifth at 700 cents. The second choice
produces Keenan's new neo-Gothic tuning, with a fifth somewhat closer
to 5-tET and larger than in 12-tET.

To find Kornerup's Golden Meantone, we have:

3 + 4 Phi
NobleMediant = --------- = ~0.5801787 = ~696.214 cents
5 + 7 Phi

This tuning has the striking mathematical property that the ratio
between the sizes of the diatonic semitone at ~118.928 cents and the
chromatic semitone at ~73.501 cents is precisely equal to Phi itself,
a relationship which carries over to other ratios in the tuning
structure such as whole-tone to diatonic semitone, minor third to
whole-tone, and fourth to minor third.

While these intricate symmetries are charming to the intellect
regardless of their possible aural effect, the tuning is very
attractive for Renaissance music in any case simply because it is
located in a choice portion of the historical meantone spectrum.

With the fifth tempered at ~696.214 cents, or ~5.741 cents narrower
than pure, Golden Meantone is located about midway between 1/4-comma
meantone with pure 5:4 major thirds (fifths ~5.377 cents narrow), and
Gioseffo Zarlino's 2/7-comma meantone of 1558 (fifths ~6.145 cents
narrow) with major and minor thirds tempered equally by 1/7 syntonic
comma (~3.072 cents).

The result in Kornerup's Golden Meantone is a fine shading slightly
toward the minor third, although more subtly than in Zarlino's
2/7-comma temperament. This mathematically intricate tuning adds its
own beauty to the meantone spectrum, and at the same time takes on a
richer musical meaning if we are familiar with its charming situation
on this spectrum.[6]

As Keenan demonstrates in his article, to find "The Other Noble Fifth"
defining the neo-Gothic counterpart to Kornerup's tuning, we need only
apply the same Noble Mediant function, this time weighting our Phi
factor in the direction of 5-tET:

4 + 3 Phi
NobleMediant = --------- = ~0.5867746 = ~704.096 cents
7 + 5 Phi

This tuning has a defining property exactly the converse of Kornerup's
tuning: the ratio of the chromatic semitone at ~128.669 cents to the
diatonic semitone at ~79.522 cents is precisely equal to Phi. This
relationship likewise holds for others ratios such as whole-tone to
chromatic semitone, augmented second to whole-tone, and doubly
augmented whole-tone or major second to minor third.[7]

Just as Kornerup's Golden Meantone is situated in a choice portion of
the meantone spectrum between 1/4-comma and 2/7-comma (and I would say
that _all_ portions of the spectrum are "choice" if appreciated for
their own unique qualities), Keenan's tuning is poised within a very
pleasant portion of the neo-Gothic spectrum where major thirds are at
or very close to the intriguing ratio of 14:11, of which more in the
next sections.

The fifths of this neo-Gothic tuning at ~704.096 cents are tempered in
the wide direction by about 2.141 cents, placing it between 29-tET
(~703.448 cents, ~1.493 cents wider than pure) and 46-tET (~704.348
cents, ~2.393 cents wider than pure), in the close vicinity of the
latter temperament.

Here is a keyboard diagram for a 17-note tuning, with the octave C-C
taken as an arbitrary reference for intervals:

128.669 287.713 624.574 832.765 991.809
c#' eb' f#' g#' bb'
_128.2|79.5_79.5|128.2_ _128.2|79.5_128.2|79.5_79.5|128.2_
c' d' e' f' g' a' b' c''
0 208.191 416.382 495.904 704.096 912.287 1120.478 1200
208.191 208.191 79.522 208.191 208.191 208.191 79.522
db' d#' gb' ab' a#'
_79.5|128.2_128.2|79.5_ _79.5|128.2_79.5|128.2_128.2|79.5_
79.522 336.861 575.426 783.618 1040.956

As we saw with Kornerup's Golden Meantone, intricate mathematical
tunings can take on a richer musical meaning as we become more
familiar with the surrounding territory along the intonational
spectrum.

While the region of Kornerup's tuning near 1/4-comma may be familiar,
at least to lovers of Renaissance music, the wonderful musical
properties of the region near 46-tET may be less familiar.[8] Thus a
friendly "walk" around this general region may provide a context for
exploring Keenan's neo-Gothic tuning in depth.

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Notes to Part 1
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1. Tuning Digest [TD] 794:8, 10 September 2000.

2. "The Golden Mediant: Complex ratios and metastable musical
intervals," TD 810:3, 18 September 2000.

3. Dave Keenan, TD 823:2, 21 September 2000, proposes that the term
"Noble Mediant" might best fit this function, and in what follows I
use this term extensively.

4. Here I am utterly indebted to Dave both for the phrase "gentle
plateau" to describe a region of maximal "harmonic entropy" or
complexity, and for an introduction to mediants which suggested to me
the approach for "walking" such a plateau presented in Section 2.
However, this technique may reflect my own roots in medieval theory
with its large Pythagorean integer ratios and tradition of _musica
speculativa_, rather than Dave's somewhat "different emphases," to
quote his own most gracious words in TD 813:25. Dan Stearns has set
his own seal on a related kind of "hiking" to explore various scales;
see, e.g., "Re: The Golden Mediant: Complex Ratios and Metastable
Intervals," TD 827:13 (22 September 2000); and "takin' in the 3/5 'Phi
trail'," TD 829:6 (23 September 2000).

5. See, e.g., Paul Erlich, "Tuning, Tonality, and Twenty-Two Tone
Temperament," _Xenharmonikon_ 17:12-40 (Spring 1998), at 21 n. 27,
quoting Daniel Wolf on the use of 7-tET in Thai and Cambodian music.

6. While Kornerup's Golden Meantone should have a pleasing effect for
just about any 16th-century or early 17th-century European composition
fitting within the size of a given tuning set (a 12-note set of Eb-G#
sufficing for most compositions of the era), works written in modes
emphasizing the minor third (e.g. Aeolian, Phrygian) might be
especially interesting to compare in this tuning and Zarlino's
2/7-comma. Mark Lindley has found the latter tuning especially
felicitous for some organ compositions of Andrea Gabrieli in these
modes

7. The doubly augmented major second, equal to a whole-tone of
~208.191 cents plus two chromatic semitones of ~128.669 cents each, or
~465.530 cents (16 fifths up, e.g. Gb-A#), is appropriately quite
close to 21:16 (~470.781 cents), an interval which Keenan has praised
for its "crunchiness."

8. Neo-Gothic temperaments may be unfamiliar in part because as this
name suggests they may indeed be "new," at least in relation to
medieval European music traditionally based on Pythagorean just
intonation. Additionally, the style of historical Gothic music may
itself be unfamiliar, so that tunings such as 29-tET or 46-tET may not
often have been approached even in the xenharmonic literature from the
perspective of this music.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Keenan Pepper's Noble Tuning (Part 1 of 2)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗graham@microtonal.co.uk