Yahoo Tuning Groups Ultimate Backup tuning Re: FAQ (?) -- _Sumer

Hello, there, everyone, and I might just offer a few comments on the
Gesualdo and meantone discussions, especially in response to some
remarks by Ibo Ortgies and Graham Breed.

This may be largely tangential to the meantone section of the FAQ and
the Gesualdo thread, both of which give me occasion for more writing,
but maybe I can at least address a couple of points which deal as much
with general Gothic/Renaissance music theory as with tuning -- but in
an intonationally relevant way, I hope.

[Inserted reflection on concluding what follows: actually the latter
part of what follows on 16th-century cadences and meantone may be
relevant, and I post it to invite comments before writing my FAQ item
on meantone from around 1450 to 1640. One of my conclusions is that
although Aaron's tuning of 1523 certainly invites a 1/4-comma reading,
the first compelling although nonrigorous description of such a
regular tuning may date to 1555.]

First of all, _Sumer is icumen in_ is a very striking English piece
likely from around the 13th century (proposed dates range from around
1240 to 1310), showing a trait described by the theorist now known to
the world, thanks to an ordering of treatises by a 19th-century editor
named Coussemaker, as Anonymous IV (c. 1275?): the preference in the
"Westcountry" of England for thirds as "the best consonances,"
sometimes used in concluding sonorities.

However, while the _Sumer_ canon uses thirds as conclusive intervals
(or at any rate intervals occurring at any point of conclusion an
ensemble chooses for what might be taken as a perpetual canon), this
does not mean that the piece must be or should be analyzed in terms of
18th-century key concepts. While I realize that this is a natural
language for many people, I'd like to offer my own "natural"
impression, not necessarily more or less accurate.

From my perspective, the _Sumer_ canon maybe relates to conventional
13th-century verticality of a kind found on the Continent and also in
some other English schools somewhat in the way that 20th-century jazz
may often relate to Classic/Romantic progressions, with African and
African-American elements also involved.

If we take the two supporting parts of the _pes_ (Latin for "foot," or
in later terms the "ground bass"), we get what might be a repeated
and quite orthodox 13th-century progression, indeed a common cadential
figure, given here in a simplified form:

C4 Bb3 C4
F3 G3 F3

5 m3 5

In Continental terms, we have a stable fifth F-C, a mildly unstable
minor third G3-Bb3, and the resolution of that third by expansion back
to the stable fifth F3-C4.

We could use either Bb or B-natural for the unstable middle sonority;
both B and Bb are regular steps of the medieval gamut, and both forms
occur frequently in conventional 13th-century writing, so that the
third expanding to a fifth might be either minor or major. Fluid
shifts between Bb and B-natural in the same piece are common, for
example in various French motets of this era.

While modal theory in the 13th century is applied mainly to the
formulas of plainsong (liturgical chant) rather than polyphony, we
could describe the mode here as _tritus_ or the F-mode in its variant
with consistent use of Bb. Tritus means the "third" variety of mode,
the four usual varieties concluding respectively on resting notes or
_finals_ of D, E, F, and G. A medieval musician might also describe
this as one color of F Lydian, and a 16th-century theorist such as
Glareanus or Zarlino as F Ionian (distinguishing the diatonic mode of
C-C, or of F-F with consistent Bb, from F-F with prominent use of
B-natural).

Anyway, we can add a third part to get a complete Continental-style
sound and also a _Sumer_-like melodic motive. Without getting into
the medieval metrical patterns or rhythmic modes, why don't I notate
this in a lively 6/8 or 6/4:

1 2 3 4 5 6 | 1

F4 E4 D4 E4 F4
C4 Bb3 C4
F3 G3 F3

(M6-8 + m3-5)

Here we start at a complete trine (2:3:4) of F3-C4-F4, move to an
unstable "split fifth" divided by the middle voice into two thirds,
G3-Bb3-D4, with the upper voice moving from the fifth to the major
sixth to form a momentary sonority of G3-Bb3-E4 which expands by
stepwise contrary motion back to our stable "home" or final of repose,
again our complete trine F3-C4-F4.

In addition to the m3-5 resolution between the two lowest parts, we
now have an M6-8 resolution between the outer parts -- and these two
expanding resolutions very nicely unite in a mutually reinforcing way,
bringing us to a full trine. Incidentally, through most of the 13th
century, in England as on the Continent sixths may have been deemed
rather more tense than thirds; at least one scholar has concluded that
while pervasive thirds are typical of English writing, sixths tend to
get used more in cadential resolutions to the octave. By around 1300,
sixths come more into vogue in England as pervasive intervals.

In this version, we have a tritone between the two upper voices,
Bb3-E4, which acts rather like a "counterfeit fourth," proceeding by
parallel motion to the upper fourth of the complete trine F3-C4-F4.
While this tritone certainly adds some color and extra tension, it's
the m3-5 and M6-8 resolutions by contrary motion which mainly guide
the progression. Here's another very common 13th-century version of
this kind of figure, this time with M3-5 and M6-8:

1 2 3 4 5 6 | 1

F4 E4 D4 E4 F4
C4 B4 C4
F3 G3 F3

(M6-8 + M3-5)

What happens in the English style of _Sumer_, however, is that other
voices are constantly adding a third A3 to our trine on the final F,
so that, in orthodox 13th-century Continental terms, the expected
resolution is avoided or "obscured." As Anonymous IV notes, this is a
distinctive trait of music in Western England. Then, as now, it would
seem that one musician's "thwarted resolution" can be another's "full
concord."

Some remarks around 1300 by the English theorist Walter Odington that
major and minor thirds at 81:64 and 32:27 are close to 5:4 and 6:5,
and that singers can make them fully concordant, suggest in connection
with such stylistic traits an intonation leaning toward these simple
ratios for thirds.

How this may have affected English keyboard tunings is an open
question, and one interesting piece of evidence is a treatise of 1373
recommending a basic tuning of the seven diatonic notes in usual
Pythagorean, with accidental semitones added as the arithmetic mean of
a whole-tone, i.e. with a pipe having a length equal to the average of
the two notes of the whole-tone it is dividing, e.g. 18:17:16.

FAQ aside: maybe this is the first practical instance of a "Rational
Intonation" (RI) tuning, with "tempering by integer ratio"; one result
is that major thirds involving accidentals are somewhere between
Pythagorean and 5:4, possibly one of the attractions of the tuning in
an English setting, apart from the simplified mathematics.

Anyway, what _Sumer_ illustrates for me in part is how standard
progressions of one style can take on a very different quality if
musical parameters such as the definition of stable saturation are
changed. Just as a Classic European music enthusiast might recognize
jazz progressions -- but with "strange" added sevenths or ninths -- so
in _Sumer_ I can recognize familiar 13th-century Continental patterns
and progressions, but with curious added thirds.

This brings us to a point you rightly raise in a Renaissance setting,
Ibo Ortgies: the concept of a "discant clause" (Latin _clausula_, with
an Elizabethan English equivalent "close").

While _clausula_ in this Renaissance context means a "close" or
"cadence," the unwary bystander may find it helpful to know that
"discant" in Gothic terms means the art of part-music for two or more
voices, and in Renaissance terms it may mean the art of two-voice
counterpoint, often improvised.

Lending meaning to both words in "discant clausula," Renaissance
music features various two-voice cadential formulas often superimposed
or combined to build multi-voice cadences. For example, one favorite
formula is a resolution from major sixth to octave (carried over from
Gothic practice), introduced by the characteristic Renaissance element
of a 7-6 suspension:

1 2 & | 1
F4 E4 F4
A3 G3 F3

7-M6 - 8

Here I warmly agree that a meantone or meantone-like intonation makes
the minor seventh G3-F4 a somewhat more complex or decisive
dissonance, and the resolving major sixth G3-E4 close to a pure 5:3.
The theorists of this era tell us how the tension of the dissonance
makes the following concord (here the major sixth) all the more sweet,
and an intonation at or near 5:3 optimizes this ideal in a Renaissance
setting.

I am inclined to agree that singers, like keyboardists, of the 16th
century may have leaned toward large cadential semitones -- 16:15
(~112 cents) in classic just intonation under the syntonic diatonic,
and about 117.1 cents in 1/4-comma meantone (likely with some adaptive
variations for singers tending toward pure vertical concords and
quasi-regular melodic steps).

The same considerations would apply to multi-voice cadences, for
example this four-part version:

1 2 & | 1
F4 E4 F4
C4 C4 C4
A3 G3 F3
F3 C3 F2

Here our two-voice 7-6-8 close or cadence between the next-to-lowest
and highest voices (tenor and superius or soprano) gets combined with
other intervals and resolutions. In addition to the 7-6 suspension,
we have a 4-3 suspension between the outer voices.

My reading of 16th-century writers such as Zarlino (1558) and Morley
(1597) is that the resolving major tenth C3-E4 should be _sweet_,
which in Renaissance terms means 5:4 or something very close.

In addition to the concluding 6-8 progression, we also have a
characteristic 16th-century progression between the bass and one of
the upper voices which Vicentino (1555) illustrates in his examples of
common practice and Zarlino describes as typical of multi-voice
writing: a progression from the major third or tenth to the unison or
octave, with the upper voice ascending by a semitone and the bass
falling a fifth (as here) or rising a fourth. Vicentino's language in
related discussions suggest that this bass motion might help in
defining a mode, classically analyzed as the division of an octave
into a species of fifth and fourth.

In this kind of setting, a cadential semitone at around 16:15 seems
fine to me, although Mark Lindley has persuasively suggested that the
late 15th and early 16th centuries may have been an era of some mixed
feelings about this use of less concise diatonic semitones, a
consequence of the quest for more smooth and restful thirds.

Thus while the transition from Pythagorean to meantone tuning for
keyboards might reasonably be dated around 1450-1480, with Ramos
(1482) as interpreted by Lindley providing hints of its prevalence,
and Gaffurius (1496) expressly declaring that fifths are tempered,
singers may have continued to lean toward Pythagorean intonation in
performing Ockeghem around 1480, or even Josquin around 1500.

As I read Aaron, his remarks on keyboard temperament in the
_Toscanello_ (1523) describe 1/4-comma meantone or something very like
it (at least for the first five notes) in nonmathematical but very
poetic terms. Some of his examples of modal cadences concluding on
sonorities with thirds -- with accidental alterations in some cases
where called for to obtain closing _major_ thirds, which he advocates
in the _Toscanello_ as more pleasing at certain points -- seem very
much in keeping with this kind of tuning system for keyboards.

Speaking as a very enthusiastic advocate for Pythagorean and also
super-Pythagorean tunings in a Gothic or neo-Gothic context, I would
be inclined much to agree with Lindley that the continued reference to
Pythagorean theory in the early meantone era, as late as Aaron, may
reflect in part a kind of conceptual conservatism. The reversed role
of major and minor semitones (respectively now diatonic and chromatic,
in contrast to Pythagorean) may have taken some time to "codify."

Recognizing Lindley's thoughtful and enriching treatment of Aaron, I
would add that this interpreter also has his own theoretical agenda
(as do we all): he evidently wishes to take Aaron as a cautionary
example on two themes. First, Lindley wants to challenge the
conception that 1/4-comma was _the_ near-universal 16th-century
standard; secondly, he seeks to counter the widespread modern
presupposition that 16th-century "meantone" tunings were necessarily
regular or conceived by tuners in a rigorous mathematical way.

Demonstrating that Aaron's instructions do not define 1/4-comma -- or
any other flavor of meantone! -- in mathematically rigorous terms, nor
do they require that all fifths be tempered by the same amount, nicely
fits in with both items of this agenda. It is an ingenious argument,
ably presented.

However, I also find it natural to read Aaron as at least permissive
evidence for something like 1/4-comma meantone in 1523. I agree that
the first _mathematical_ definitions do not come until Zarlino (1571)
and Salinas (1577), and also with Lindley's witty observation that we
need not assume that either of these theorists was the first to place
a keyboard in this temperament.

If we regard Aaron's description as defining some form of meantone,
but not _necessarily_ 1/4-comma, is there any evidence before 1571
that 1/4-comma was in use?

Here I would propose the date of 1555, and the treatise of Nicola
Vicentino, as a source pointing decisively to the use of 1/4-comma or
something very close, albeit without any rigorous mathematical
definition in terms of fractions of the syntonic comma or the like
(Zarlino does define his 2/7-comma system rigorously in 1558, but
other meantones such as 1/4-comma only in 1571).

The compelling evidence, for me, is the nature of both of Vicentino's
two tuning systems for his archicembalo. I agree with Lindley that
Vicentino's statement that the first 12 notes are tuned as in common
keyboard instruments does not in itself address the question of the
precise temperament, but his other musical parameters do.

His first system divides the octave into 31 conceptually equal parts,
with all intervals available from all steps -- thus it must closely
approximate 31-tone equal temperament, or 31-tET, which in turn is
almost identical to 1/4-comma.

His second system combines "perfect fifths and perfect thirds,"
implying that the major thirds on each manual are "perfect," or pure,
and thus also describing in nonrigorous terms what Zarlino in 1571
will more formally define as a "1/4-comma" temperament.

If we regard Aaron's instructions as inviting but not compelling
evidence for a 1/4-comma temperament, then one might draw the
interesting conclusion that this tuning or something almost identical
was first compellingly defined -- albeit nonrigorously -- in the
context of a full 31-note cycle.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: FAQ (?) -- _Sumer_ canon, discant cadences, meantone

🔗M. Schulter <MSCHULTER@VALUE.NET>

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