Re: Pythagorean intonation 1200-1450 -- Response to Ed Foote

Hello, there, and in Tuning Digest 436, Ed Foote wrote:

> I would have considered everything up to maybe 1200 easily defined
> by the Pythagorean era, but after that things get murky.

For the composed music of Continental Europe as opposed to England, I
would say that the era of around 1200-1400 remains mostly regular
Pythagorean (e.g. Perotin-Machaut), although voices and other
non-fixed-pitched instruments obviously take any intonational model such
as Pythagorean as an ideal rather than a precise measurement.

The era of around 1400-1450 on the Continent could be described as the
period of "modified Pythagorean tunings," where sharps are often tuned
as Pythagorean flats, meaning that major thirds notated with sharps
(typically A-C#, D-F#, and E-G# in the 12-note Gb-B tuning) are
realized as Pythagorean diminished fourths or schisma thirds.

Sometime around 1450, meantone temperament evidently started getting
applied, and by 1482, Ramos (as read by Mark Lindley) seems to offer
comments about "good" and "bad" intervals on fixed-pitch instruments
which point to its general acceptance. By 1496, Gafurius tells us that
on organs, the fifths are narrowed by a certain "small and hidden
quantity" -- one way of describing meantone at a time when the theory
of intervals smaller than a comma was still a novel topic.

(Incidentally, the popularity in the early 15th century of tunings
which made written major thirds with sharps into diminished fourths
may be one indication pointing to the usual conclusion that meantone
tuning was _not_ yet the norm. One would hardly want to go out of the
way to make certain prominent thirds into _meantone_ diminished
fourths, at least in a typical 15th-century musical context! Lindley
similarly reasons that the keyboards described by Ramos in 1482 likely
_were_ meantone, because in 15th-century Pythagorean, Ramos would be
less likely to describe something like A-Gb (a schisma sixth) as "bad"
or unusable.)

> Walter of Odington wrote in the 14th century that English
> choirs were singing harmony that used pure thirds. I have seen this
> referred to in other sources as being one of the first instances on
> record of the third being regarded a concord rather than a discord.
> If choirs were singing it, I am going to assume that peasant
> folksong had been there first. Perhaps the blues artists of the
> first millineum?

Indeed Walter Odington suggests that Pythagorean major and minor
thirds at 81:64 and 32:27 are close to 5:4 and 6:5, and this comment
may reflect the predilection for thirds in some medieval English
dialects of polyphony, a trait also noted by the theorist known as
Anonymous IV (c. 1275), who reports that in the "Westcountry" of
England, these intervals are esteemed as "the best consonances," and
that singers will conclude pieces on them.

However, while the Pythagorean thirds of Continental theory and
practice are definitely regarded as unstable -- unlike the English
practice as illustrated by a piece such as _Sumer is icumen in_
(c. 1240?) -- theorists of the 13th and 14th centuries typically
classify these intervals as "imperfect concords" (e.g. Johannes de
Garlandia, Franco of Cologne), or "medial concords" (Jacobus of
Liege).

For more both on the role of these thirds in 13th-century Continental
music as _relatively_ blending but unstable, and on how the nuances of
Pythagorean tuning may affect 13th-14th century vertical style, please
see

http://www.medieval.org/emfaq/harmony/13c.html (13th-century style)
http://www.medieval.org/emfaq/harmony/pyth.html (Pythagorean tuning)

In other words, Pythagorean thirds fit a style where these intervals
are treated as what I might term "partial concords" -- but tend to
yield to tunings at or closer to 5:4 and 6:5 when thirds approach the
status of _full_ and restful concords, as happens in some English
styles as early as the 12th or 13th century, and on the Continent in
the course of the early 15th century.

In the latter case, a modified Pythagorean system provides _some_
prominent schisma thirds, but only meantone can satisfy the appetite
for smooth thirds at all possible points in the gamut.

> It must be assumed that this new resource in vocal music was so
> attractive that the added complexity of its use was no deterrent to
> motet composers of the time. Four voice harmony harmony was know by
> then (Perotin etal.) and I be we can Get Margo To Tell Us More
> (GMTTUM) about that?

Actually, the use of Pythagorean thirds as integral intervals in
polyphony was recognized by Guido d'Arezzo around 1030 in his
_Micrologus_, where he recommends a progression from a major third to
a unison as a good cadence or "coming together" (_occursus_) of the
voices, and by the time of Perotin, they are being treated as
the "partial concords" I describe above.

Around 1325, Jacobus of Liege makes a number of mentions of two kinds
of multi-voice sonorities we know to be featured in Perotin's music.
Of special interest here is the _quinta fissa_ or "split fifth" in
which the outer voices form a fifth and the middle voice "splits" it
into a major and minor third: e.g. F3-A3-C4 (major third below, minor
third below), or A3-C3-E4 (the converse arrangement, which he notes is
used to open a motet which has come down to use in the Montpellier and
Bamberg Codices). Note that here I use C4 to indicate middle C, and
higher numbers to show higher octaves, etc.

Jacobus gives string-ratios for these intervals as 81:64:54 for the
version with major third below; and 96:81:64 for the version with
minor third below. The frequency-ratios are the converse, of course.

Jacobus endorses another favored sonority of Perotin: an outer ninth
"split" into two fifths, e.g. G3-D4-A4, which the 14th-century
composer Machaut (c. 1300-1377) also appears to favor, and even uses
as the opening sonority for one of his ballades. Here a Pythagorean
tuning permits an ideally euphonious tuning of 9:6:4.

It may be only natural that these sonorities should be prominent in
Perotin (c. 1200), because as soon as we have composers writing
regularly for three and four voices, such mildly unstable sonorities
fit the scale of concord/discord for basic two-voice intervals.

Major and minor thirds are regarded as the mildest unstable intervals,
while major seconds or ninths and minor sevenths are regarded as
somewhat more tense, but still somewhat "compatible" -- and when
presented in sonorities with a preponderance of ideal fifths and
fourths, may indeed contribute to an overall impression of energetic
blend.

Sonorities of this kind in Gothic style on the Continent are points of
intriguing euphony mixed with definite tension; "partial concord" or
"relative blend" may be terms as good as any, and Ludmila Ulehla's
"dual-purpose" category (as opposed to stable concord or urgent
discord) may be apposite.

Of course, those of us who favor a Pythagorean model for the
Continental European music of around 1200-1400 can and should
recognize that especially with non-fixed-pitch instruments (including
voices), variations must have taken place. Major thirds, for example,
may have leaned sometimes in the direction of 5:4, and at other times
(e.g. cadences) in the direction of 9:7.

Marchettus of Padua (1318) suggests the use of very narrow semitones
at cadences which would seem to call for larger-than-Pythagorean major
thirds and sixths before fifths and octaves. Around 1375, one treatise
in the Berkeley Manuscript suggests the use of regular diatonic
semitones equal to 2/3-tone, leading modern editor Oliver Ellsworth to
suggest a reading of 19-tet.

Anyway, while preferring the typical recent chronology which regards
the era of 1200-1400 on the Continent as basically Pythagorean, I
would certainly recognize that variations may have taken place.

However, it would seem to be the early 15th century when changes in
musical style set the stage on the Continent for meantone; this is the
era of composers such as Dufay and Binchois, who interestingly were
described by one poet around 1440 as following the "English
countenance" epitomized by the composer John Dunstable.

In other words, the Pythagorean-meantone transition might be seen as
one feature of the shift from Gothic to Renaissance style on the
Continent, with the music of Dufay's youth (c. 1420-1430), and the
modified Pythagorean tunings with their alluring schisma thirds,
providing a kind of stylistic and intonational bridge.

However, the intonational practices of 12th-14th century England --
especially in genres which do use thirds as virtually full and stable
concords rather than the partial and unstable ones of Continental
practice from Perotin to Machaut -- remains a moot question. A reading
of Odington to suggest that these thirds were often tuned at or near
5:4 and 6:5 seems quite likely to me.

Most respectfully,

Margo Schulter
mschulter@value.net