Yahoo Tuning Groups Ultimate Backup tuning Re: Counterpoint (for Graham Breed and Joseph Pehrson)

Hello, there, Joe Pehrson and Graham Breed and everyone.

Here I'd like to present my own viewpoint on the question of whether
and how systems of Western European discant and counterpoint between
around 850 and 1640 may be related to the question of open or closed
tuning systems.

Since an adage says that "where you stand may depend on where you
sit," I should begin by saying something about my own predilections
and arguable biases:

(1) The great bulk of my playing is done in regular tunings:
medieval Pythagorean, Renaisance meantones, and neo-Gothic
temperaments with fifths rather gently tempered in the wide
direction (see my definition of "equitone" tunings below);

(2) These are generally open systems of 12-24 notes, although
many would precisely or "virtually" close in somewhere
between 29 and 53 notes[1];

(3) An especially characteristic arrangement for me seems to be
two 12-note keyboards a "12-comma" apart in the specific
tuning, e.g. the Pythagorean comma or meantone diesis.

Given this experience, where circularity is more of a curiosity than a
typical musical feature, much less a main imperative, what I'm about
say may not be much of a surprise.

First of all, I agree with you both that counterpoint is very
flexible, and that systems of counterpoint may very nicely fit tunings
which don't seem to mesh very well with either trinic or triadic
verticality, for example 11-tone equal temperament (11-tET) or 13-tET.
Composers such as Ivor Darreg and Brian McLaren have suggested that
lively contrapuntal textures may be especiall well suited to such
systems.

However, although counterpoint may fit all kinds of tuning systems
included n-tET's or "virtually" closed systems (e.g. Pythagorean 53,
1/4-comma meantone 31), I'm not sure that there's any special
connection between counterpoint and circulating systems.

What I might propose as very tentative hypotheses, subject to lots of
debate (some it likely quite impassioned!) are that basic schemes of
counterpoint may find their simplest "mappings" to regular tunings
here styled "equitones," and that there may be some tendency of
counterpoint to pull toward Ervin Wilson's Moment of Symmetry (MOS)
tunings, whether open or closed.

Note that "simplest mapping" doesn't mean _only_ mapping or even
necessarily _best_ mapping, and that the popularity of split-key
instruments with 13-16 notes of the kind discussed here in depth by
Ibo Ortgies should caution us against taking the MOS idea as some kind
of test for a "useful" tuning.

From an historical perspective, I would say that the 8-17 note
Pythagorean systems of the medieval era, and also Renaissance
meantones in 12-19 notes (apart from 19-tET or the almost identical
1/3-comma meantone), are fine examples of open regular tunings central
to the practice of counterpoint as we're discussing it in this thread.

By "regular tuning," I here mean more specifically an "equitone"
defined in terms not only of the tuning structure itself but of how
that structure is used musically. The term fits such systems as
medieval Pythagorean intonation (pure fifths), Renaissance meantones
(narrow fifths), and neo-Gothic equitones (wide fifths) as used in
these stylistic contexts.

In an equitone, for example, a major third is derived from four fifths
up less two octaves (e.g. C-E from C-G-D-A-E), and consists of two
equal whole-tones (e.g. C-E from the equal steps C-D-E).

While the term "meantone" historically implies narrow fifths and
thirds near 5:4 and 6:5, other types of equitones old and new may
feature such thirds at 81:64 and 32:27 (Pythagorean tuning), for
example, or at or near 14:11 and 33:28 (a neo-Gothic favorite).[2]

From an MOS perspective, 12-note and 17-note Pythagorean tunings (the
first common by the mid-14th century or so, the latter advocated in
the early 15th century) are examples of how MOS does not necessarily
mean "closed." For a circulating Pythagorean system, we need to go to
53 notes, as was reportedly recognized in China but not to my
knowledge proposed in Europe before the 17th century, well after the
Gothic Pythagorean era.

Likewise, what 12-note, 19-note, and 31-note meantone keyboards share
is an MOS status. While Costeley (19-tET, 1570) and Salinas (1/3-comma
meantone, 1577) were drawn to such a shade of meantone because of its
circular features, open 19-note systems such as Zarlino's 2/7-comma
meantone (1558) or 1/4-comma meantone (e.g. likely the Neapolitan
"chromatic harpsichords" of around 1600 in Gb-B#) may well have been
at least equally popular.

While Vicentino (1555, 1561) indeed advertises the circulating feature
of his 31-note archicembalo cycle as an important advantage, his very
small sampling of known enharmonic settings use no more than 24 notes
in a given piece or excerpt. I'm tempted to observe that 31 was the
smallest MOS providing other features he needed for his music such as
enharmonic dieses or fifthtones in lots of positions and new intervals
such as "proximate minor thirds" near 11:9. The fact that 31 happens
to be a "virtually closed" system in 1/4-comma meantone or something
very close would be an additional premium.

Of course, for Fabio Colonna's "example of circulation" (1618), all 31
notes of his _Sambuca Lincea_ were needed. However, this piece may
stand out in part precisely because such circularity may be a rather
tangential theme in most experimental 16th-17th century music,
including chromatic and enharmonic compositions.

From Trabaci, at any rate, we know that 19-tET or the like was not an
overwhelming imperative for 19-note keyboards, although musicians such
as Costeley and Salinas did favor it. When Trabaci writes D#-F##-A# for
an archicembalo or the like, but gives the player of a usual 19-note
chromatic keyboard the option of playing D#-F#-A# with a minor third
above the lowest part, he implies an open tuning, in contrast to
19-tET or the like where F##=Gb and the major third would be
available.

To say that discant and counterpoint in their "basic" form may lean
toward MOS-oriented formulations is to state a tendency, not a "law,"
much less a restraint on the growth of practice and theory alike.

For example, 13th-century theory postulates 13 or 14 generally
recognized intervals ranging from unison to octave -- the intervals
occurring in a 7-note diatonic Pythagorean system (fifth chain F-B).
The count varies depending on whether one considers the general
category "tritone" (_tritonus_) to include the diminished fifth, or
places the latter interval in its own class, based on its distinct
ratio (e.g. the _semitritonus_ of Jacobus of Liege).

However, Jacobus and Johannes Boen (c. 1357) discuss other augmented
and diminished intervals also. These may have arisen in part from
accidentals introduced because of such motivations as the desire to
have stable consonances in more places (e.g. B-F#, F#-C#, Eb-Bb), and
in part from cadential patterns prevailing by the early 14th century
calling for specifically major or minor thirds and sixths in certain
progressions (e.g. the major third E-G# or Eb-G before the fifth D-A).

Whatever the reasons for these accidentals, however, once they are
introduced, they give rise to new kinds of intervals: the apotome or
chromatic semitone Bb-B (or German B-H) is already potentially present
in the usual Guidonian gamut including Bb along with the seven
diatonic notes. In the early 14th century, Marchettus of Padua makes
the most of this "theoretical" interval in very active chromatic
practice, at the same time suggesting a yet greater intonational
contrast between semitones in cadential progressions.

From one perspective, while Marchettus could have used the Pythagorean
apotome in direct chromaticism as one feature already supported by the
emerging 12-note Pythagorean MOS of the epoch, he preferred to
introduce intensified cadences calling for more flavors of unstable
intervals as well as semitones. Given that he was addressing vocal
intonation, such refinements could be viewed as elaborations at once
based on a "groundwork" of "closest approach" progressions expressible
in a 12-note MOS ("prefer E-G# rather than E-G before D-A"), but also
transforming that groundwork, giving it a new topology.

To reach an MOS providing the "superefficient" cadences of Marchettus
for the usual diatonic cadential centers, we would need to go to 29
notes, whether in something like 29-tET (with "adaptive tuning" to
achieve pure 2:3:4 trines and 9:8 whole-tones, as vocalists might do),
or in conventional Pythagorean tuning.

My own solution is a system of _two_ 12-note MOS tunings a Pythagorean
comma or 29-tET diesis apart -- which I'm tempted to describe in the
manner of Dave Keenan's multiple "chain-of-fifths" tunings as a
"two-MOS-set" tunings, both sets here generated by a single chain of
fifths. Note that in such 24-note schemes, either Pythagorean or
29-tET is an open tuning, although they would be precisely closed (29
of 29-tET) or "virtually closed" (Pythagorean 53) with more notes.

Similarly, the meantone diesis of 128:125 (~41.06 cents) in 1/4-comma
might originally have been viewed as more of a quirk or complication
than an esteemed "feature," but Vicentino made the most of it both as
a new melodic interval and as a factor in differentiating new flavors
of familiar intervals, vertical or melodic.

Again, while Vicentino's 31-note cycle is the smallest meantone MOS
providing the enharmonic features integral to his style, I've found
that a "two-MOS-sets-of-12" system provides a very pleasing kind of
enharmonic system with its own kind of symmetries (and some
asymmetries also mirroring those of a usual 12-note meantone).

This isn't to say that Vicentino's 31 doesn't have its practical
advantages apart from the possibilities of circularity. For example,
unlike my 24-note "dual 12-note MOS" scheme, it nicely accommodates
the full "chromatic" subset of Gb-B#, as used for example in various
of his sacred and secular collections featuring compositions with
13-15 notes per octave sometimes favoring D# and A#.

In open systems or closed, MOS or otherwise, counterpoint may reflect
a human tendency or even irrestible inclination toward play: seeking
to get "useful" intervals in more positions, etc., leads to more
"unusual" intervals in the system, also, which may take on a value all
their own, providing a motivation for yet further expansion of the
system to get _them_ in more positions also, etc.

One complication of counterpoint systems is that they embrace many
different kinds of musical phenomena at different levels, so that two
notes may seem possibly "equivalent" for one purpose, but not for
another.

For example, as you comment, Graham, notes separated by a chromatic
semitone such as F/F# (the minor semitone in meantone) might have been
considered as forms of the "same step" for _some_ purposes --
specifically, in relation to the rules concerning parallels. It may be
a moot question whether the successive fifths F-C, F#-C# would be
considered "fifths by parallel _motion_" contrary to the usual
16th-century rules of counterpoint (as with F#-C#, G-D), or
repetitions of basically the same fifth with a change of "coloring."

However, F and F# are very different notes in the sixths A-F and A-F#,
with F# often being favored in theory and practice (whether written or
inferred by the performers) in order to obtain the major sixth before
the octave G-G. Some theorists such as Prosdocimus (1409, 1413) and
Zarlino (1558) seem to say that a sixth expanding to an octave should
_always_ be major, while others such as Gaffurius (1496) regard it as
a general norm for significant cadences rather than for all 6-8
progressions.

Vicentino (1555) says that the minor sixth expanding to the octave is
"gloomy," causing me to infer that he might reserve it for use with a
fitting text -- possibly a hint that a famous passage in a motet of
Gombert should _not_ automatically have a cadential sixth made major
despite the interesting melodic figure Bb-A-G# which would result.
This motet on the theme of Rachel's lament would fit Vicentino's
adjective about as well as any, and the "understatement" of a
cadential minor rather than major sixth has for me a special
eloquence. Tablatures from the epoch suggest that "major sixth before
expansion to an octave" is a usual pattern, not an absolute law.

Here I touch on some questions of accidentals mainly to illustrate how
degrees a chromatic semitone apart (or enharmonic diesis, in
Vicentino) might arguably be equivalent for _some_ purposes, but
definitely distinguished for others.

A Spanish writer around 1540 or so discusses the use of the diminished
fourth as a suspension, and Johannes Boen in 1357 treats it as an
interval "consonant by situation" when placed above a major third.
Modern counterpoint texts note the use of the augmented fifth in
16th-century style between two upper parts in a combination such as
D-Bb-F#, and Bernhard (around 1655?) offers a similar observation.

From the viewpoint of structural simplicity, one approach in the
intonational implementation of couterpoint is to extend an equitonal
tuning as far as desired in order to keep its "familiar" intervals and
regularity of the chain of fifths while adding new intervals as well
as more positions for the accustomed ones.

Both Vicentino's meantone archicembalo, and neo-Gothic Pythagorean and
tempered equitones, take this basic approach.

However, even if contrapuntal theory may "map" most neatly to an
equitonal structure where all similarly spelled intervals have
identical sizes, intricate just intonation systems such as Zarlino's
syntonic diatonic (from Ptolemy) are by no means excluded -- nor
adaptive tuning, a subtle refinement in the very fine structure of the
typology.

Similarly, considerations of economy or facility might favor using
some number of notes other than an MOS: 13-16 note split-key
instruments, for example, or 24-note systems which might be explained
as "two-12-note-MOS systems" but nevertheless are distinct from usual
MOS tunings such as Pythagorean 12 or 17 -- or meantone 12, 19, or 31.

To conclude, counterpoint may be seen and often heard as a kind of
dialogue between symmetry and innovation, with equitonal or MOS models
as a possible tendency, but by no means a rigorous constraint.

-----
Notes
-----

1. The main qualifications to this generalization might be 17-tET or
22-tET when I tune their complete sets rather than some subset
(typically 12, Eb-G#), and 20-tET and 24-tET. Also, one of my favorite
Pythagorean schemes is not _strictly_ "regular," since the two
keyboards are a septimal rather than Pythagorean comma apart in order
to obtain supplementary intervals with pure 7-based ratios (e.g. 9:7,
7:6, 12:7, 7:4, 8:7), but each keyboard is regular in itself and the
scheme has much the same "look and feel" as a conventional 24-note
Pythagorean tuning. I also sometimes use a 17-note neo-Gothic
well-temperament, and an unequal 24-note temperament I hope to
describe soon (owing lots of inspiration to 20-tET and Dan Stearns).

2. To illustrate the element of _musical usage_ in defining an
equitone, I might quote from a hypothetical guidebook to neo-Gothic
temperaments: "A special advantage and premier feature of 46-tET is
its virtually pure 14:11 major thirds, this tuning being almost
identical to a 14:11 equitone; these excellent thirds are combined
with rather moderately tempered fifths, about 2.39 cents wide." Such
a statement reflects the simple fact that four fifths up in 46-tET are
almost exactly equal to a 14:11, and the stylistic assessment that
such an interval makes an excellent regular major third (a kind of
variation on the usual Pythagorean 81:64). In such a setting, any
convenient subset of 46-tET (or an exact 14:11 equitone) is fine: 12
notes, or 13-16, or 17, or 24, etc. For styles using the same tuning
to obtain some other size of major third as the "normal" form,
however, 46-tET is not an equitone (major third from four fifths up or
two equal whole-tones).

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Counterpoint (for Graham Breed and Joseph Pehrson)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗David J. Finnamore <daeron@bellsouth.net>