Re: A plethora of thirds: From Vicentino 24 to Pepper 24

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A plethora of thirds: From Vicentino 24 to Pepper 24
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When Nicola Vicentino explored his archicembalo dividing the octave
into a meantone cycle of 31 notes (likely 1/4-comma temperament with
pure major thirds) he noted the diverse types of thirds available on
this "superharpsichord." In his treatise of 1555, _Ancient Music
Adapted to Modern Practice_, he provides a catalogue of these thirds
and where to find them on his expanded keyboard.

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Vicentino's name Tones Cents Vicentino's Description
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minimal third 7/5 ~269.21 Tends toward dissonant M2
minor third 8/5 ~310.26 Usual minor third
proximate m3 9/5 ~345.25* Tends toward consonant M3; ~11:9
major third 10/5 ~386.31 Usual major third
proximate M3 11/5 ~427.37 Tends toward tension of 4th; ~9:7
----------------------------------------------------------------------
* This interval might also be ~351.32 cents in 1/4-comma tuning.[1]
======================================================================

This month a year ago, I was reenacting this process of discovery on
an electronic kind of 24-note archicembalo using two 12-note keyboards
tuned in 1/4-comma meantone (Eb-G#) a diesis or fifthtone apart
(128:125, ~41.06 cents).

A year later, I find myself engrossed in the exploration of another
24-note tuning offering among its many attractions a plethora of
thirds: Keenan Pepper's neo-Gothic tuning based on Phi.

Like a Renaissance meantone, Pepper's neo-Gothic temperament reveals
much of its beauty even in a simple 12-note tuning. The near-pure
fifths and fourths, the regular major and minor thirds with ratios
close to 14:11 and 13:11, and the diminished fourths and augmented
seconds with a "submajor/superminor" flavor close to 21:17 and 17:14,
are all there to be heard and relished.

However, as Vicentino demonstrated, a Renaissance meantone such as
1/4-comma (almost equivalent to the 31-tone equal temperament or
31-tET associated with Vicentino's keyboard by Lemme Rossi in 1666)
also has wonderful secrets and surprises which only a larger tuning
can discover. With Pepper's tuning, also, an archicembalo of 24 or
more notes brings us new musical treasures and surprises.

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2. Exploring 24 notes: Navigating the diesis
--------------------------------------------

While neo-Gothic tunings have in many ways a radically different
structure than Renaissance meantones, for tunings in the range of
around 29-tET to 46-tET and slightly beyond I curiously find
Vicentino's accidental notations to be quite apt at the brink of the
21st century.

Going beyond 12 or 17 notes, in either type of tuning we (like
Vicentino) may find it convenient to have a symbol for indicating a
note raised by a _diesis_, in one definition the difference between
two accidentals such as Ab and G#. In another definition, it is the
amount by which 12 fifths differ (in either direction) from 7 pure
octaves.

If Vicentino used 1/4-comma meantone with pure major thirds, then this
diesis would have a size of 128:125 or ~41.06 cents, with Ab _higher_
than G#, and 12 fifths falling short of an octave by this amount.
Vicentino describes this diesis as equal to 1/5-tone. In Rossi's
31-tET, it would be precisely 1/31 octave, ~38.71 cents.[2]

Interestingly, the diesis in the neo-Gothic neighborhood including
Pepper's tuning has a size comparable to Vicentino's, but with Ab
_lower_ than G#, and 12 fifths exceeding an octave by this amount. In
Pepper's tuning, it is equal to ~49.15 cents.

To show a note raised by a diesis, Vicentino placed a dot above the
note, here represented as an ASCII asterisk (*).

Vicentino's 36-note archicembalo, with two manuals each having three
"ranks" of keys, has its first and fourth ranks -- the seven diatonic
notes or "white keys" of each manual -- tuned a diesis apart. Thus C
on the first manual would have the corresponding note C* on the second
manual.

Drawing inspiration from this feature and generalizing it, we can
arrange a 24-note archicembalo in either a Renaissance meantone such
as Vicentino's or a neo-Gothic temperament such as Pepper's by tuning
the first manual as usual, and tuning the second manual in identical
fashion a diesis higher or lower. Following Vicentino's convention, I
here have the second manual tuned a diesis above the first:

C#* Eb* F#* G#* Bb*
C* D* E* F* G* A* B* C*
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C# Eb F# G# Bb
C D E F G A B C

In a neo-Gothic context, or more generally in any regular tuning with
fifths larger than 700 cents, sharps are _higher_ than corresponding
flats by a diesis: thus Eb* on the upper keyboard is equivalent to D#,
and Bb* likewise to A#.[3] We are free to use either spelling. Some
people may find it more natural to speak of "the fifth G#-D#," for
example, but "the regular major third Eb*-G* on the upper manual."

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3. Four families of thirds in 24 notes
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Emulating Vicentino's exploration of thirds on his archicembalo, let's
see how many flavors of thirds we can find on our 24-note neo-Gothic
archicembalo in Pepper's tuning. Here it's helpful to keep in mind
that raising a note by a diesis changes the size of an interval by
about 49.15 cents.

In previous articles, I have discussed the two kindred families of
thirds available within a 12-note tuning[4], and now also available in
some new positions involving notes on both manuals. These families
consist of regular major and minor thirds near 14:11 and 13:11; and
diminished fourths and augmented seconds serving as alternative
"submajor/superminor" thirds near 21:17 and 17:14. Both families have
a complex and active quality fitting their unstable and often
cadentialy directed role in neo-Gothic music.

To find the two intervals of our third family, "maximal/minimal"
thirds, we play a fourth reduced by a diesis (e.g. C*-F), and a major
second enlarged by a diesis (e.g. F-G*). The first interval is about
(496 - 49) or 447 cents, while the second is about (208 + 49) or 257
cents.

These maximal and minimal thirds, more precisely at about 446.76 and
257.34 cents, have a striking quality, strongly inviting cadential
resolutions in which the minimal third contracts to a unison and the
maximal third expands to a fifth. They have some degree of "category
warping" effect, approaching the domain of large major seconds and
small fourths respectively (as their spellings suggest).

At the same time, they seem to me maybe a bit "kinder and gentler"
than their counterparts near 13:10 (~454.21 cents) and 15:13 (~247.74
cents) in 29-tET at 11/29 octave and 6/29 octave (~455.17 cents,
~248.275 cents); or in 24-tET at 9/24 octave and 5/24 octave, an even
450 cents and 250 cents.

To find our fourth family of "schismalike" thirds, we play an
augmented second enlarged by a diesis (e.g. F-G#*), and its
counterpart, a diminished fourth reduced by a diesis (e.g. G#*-C).
The first interval has a size of about (337 + 49) or 386 cents, and
the second about (367 - 49) or 318 cents.

Like Pythagorean diminished fourths and augmented seconds, or "schisma
thirds" at about 384.36 and 317.60 cents, these intervals at around
386.01 and 318.09 cents very closely approach pure ratios of 5:4
(~386.31 cents) and 6:5 (~315.64 cents).

In a neo-Gothic style, these schismalike thirds might serve as moments
of coloristic diversion; or they might move by diesis shifts and the
like to other flavors of thirds resolving in a usual cadential
manner.

Here is a quick summary of our four families of thirds in Keenan
Pepper's tuning, with some of the examples of regular and alternative
thirds illustrating how these intervals can be found in some new
positions. As noted at the end of Section 2, the conventional spelling
B-D# for the second example of a regular major third is equivalent to
B-Eb* in the 24-note keyboard diagram there given:

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Family Type Examples Cents Description
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Regular M3 F-A, B-D# 416.38 Usual major third, ~14:11
m3 D-F, A*-C* 287.71 Usual minor third, ~13:11
......................................................................
Alternative dim4 C#-F, F*-A 367.24 Submajor, ~21:17
Aug2 F-G#, D-F* 336.86 Superminor, ~17:14
......................................................................
Maximal/ Mx3 D*-G, F*-Bb 446.76 4th-less-diesis, ~22:17
Minimal mn3 G-A*, Bb-C* 257.34 M2-plus-diesis, ~22:19
......................................................................
Schismalike S3 F-G#*, Bb-C#* 386.01 Aug2-plus-diesis, ~5:4
s3 G#*-C, C#*-F 318.09 dim4-less-diesis, ~6:5
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4. Our four families in perspective: The 704-cent neighborhood
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Having identified four families of thirds on our 24-note archicembalo,
we may find it interesting to compare the subtly shifting qualities of
these intervals in the general neighborhood of Pepper's tuning, the
neighborhood with fifths of around 704 cents or slightly larger.

From Pepper's tuning at ~704.10 cents (~2.14 cents wide), we move to
46-tET (~704.34 cents, ~2.39 cents wide) and an almost identical
tuning with pure 14:11 major thirds (~704.38 cents, ~2.42 cents wide),
and from there to an "e-based" tuning (~704.61 cents, ~2.65 cents
wide) where the ratio of the whole-tone to the diatonic semitone is
equal to Euler's famous _e_, 2.71828182845904523536029.

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Type Generator Examples Pepper 46-tET 14:11 e-based
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(Family 1: Regular major/minor thirds, M3/m3)
M3 4 5ths up F-A, E-G# 416.382 417.391 (16) 417.508 418.428
m3 3 4ths up E-G, C-Eb 287.713 286.957 (11) 286.869 286.179
......................................................................
(Family 2: Alternative thirds, dim4/Aug2 or "submajor/supraminor")
dim4 8 4ths up C#-F, B-Eb 367.235 365.217 (14) 364.984 363.145
Aug2 9 5ths up F-G#, F*-A 336.860 339.130 (13) 339.393 341.462
......................................................................
(Family 3: Maximal/minimal thirds, Mx3/mn3)
Mx3 14 5ths up D*-G, A*-D 446.757 443.478 (17) 443.099 440.110
mn3 13 4ths up D-E*, A-B* 257.338 260.870 (10) 261.278 264.496
......................................................................
(Family 4: Schismalike thirds, S3/s3)
S3 21 5ths up F-G#*, Bb-C#* 386.008 391.304 (15) 391.917 396.745
s3 20 4ths up G#*-C, C#*-F 318.088 313.043 (12) 312.460 307.862
----------------------------------------------------------------------

Having compared the regular and alternative thirds (Families 1 and 2)
for these tunings in an earlier post[5], I focus here on some nuances
of shading involving Families 3 and 4: the maximal/minimal and
schismalike thirds.

Since maximal/minimal thirds are formed from 13 fourths up or 14
fifths up, they are very sensitive to small changes of temperament.
Interestingly, as we move from Pepper's tuning through 46-tET and the
14:11 tuning to our e-based tuning, these intervals grow somewhat
"mellower." Maximal thirds contract from around 447 cents to 440
cents, about 5 cents wide of the simpler "valley" of 9:7; minimal
thirds expand from around 257 cents to 264 cents, only a bit more than
2 cents narrow of the "valley" of 7:6

Our schismalike thirds formed from 21 fifths up or 20 fourths up, yet
more sensitive to small distinctions of temperament, are very close to
pure in Pepper's tuning.

By 46-tET or the 14:11 temperament, the major thirds of this family
range between 391 and 392 cents, comparable to 1/5-comma or 2/11-comma
meantone, although the minor thirds are much closer to pure than in a
meantone because the large fifth allows them more "room." In our
e-based tuning, the major thirds approach 397 cents, comparable to
1/8-comma meantone.

In appreciating the qualities of these 24-note temperaments, we might
take an archicembalist's-eye view like that of Vicentino, embracing
both the more familiar intervals and the more "exotic" or remote
ones.

Keenan Pepper's tuning and its neighbors, like Vicentino's likely
archicembalo temperament of 1/4-comma and its neighbors including
Kornerup's Golden Meantone, suggest some of the possibilities for a
new millennium of music.

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Notes
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1. In 1/4-comma meantone, the size of the "proximate minor third" or
neutral third varies slightly depending on whether one adds a diesis
of ~41.06 cents to a regular minor third (310.26 + 41.06 = ~351.32
cents); or subtracts a diesis from a regular major third at 5:4
(386.31 - 41.06 = ~345.25 cents). Which of these sizes appears in
which positions on an archicembalo depends on the order of the
tuning. In 31-tET, this interval has a single size of 9/31 octave,
~348.39 cents.

2. In an extended tuning in 1/4-comma meantone, we actually have two
slightly unequal flavors of fifthtones: the familiar diesis of 128:125
(e.g. G#-Ab), plus an interval equal to the chromatic semitone of
~76.05 cents less this diesis, or ~34.99 cents (~50:49).

3. With Vicentino's meantone and other regular tunings with fifths
smaller than 700 cents, however, flats are a diesis _higher_ than
sharps, so that we have the equivalences C#*=Db, F#*=Gb, and G#*=Ab.
In a 16th-century or derivative context, these identities can be very
helpful, permitting these three keys in a 24-note tuning to serve at
once as supplementary flats (providing the fifths Db-Ab, Gb-Db, and
Ab-Eb) and as usual sharps in the "parallel universe" of the upper
manual.

4. See, e.g., "Keenan Pepper's Exquisite Neo-Gothic Tuning: Noble
Mediants and Walks on Plateaus," Tuning Digest [TD] 860:3 (Part 1) and
861:15 (Part 2), 3 October 2000;

http://www.egroups.com/message/tuning/13947
http://www.egroups.com/message/tuning/13971

and "Optimizing the 'Four Convivial Ratios': A Footnote to Keenan
Pepper's Noble Tuning," TD 871:15, 9 October 2000.

http://www.egroups.com/message/tuning/14180

5. See ibid., "Optimizing the 'Four Convivial Ratios.'"

Most respectfully,

Margo Schulter
mschulter@value.net