Re: Distinction between thirds -- 3-limit and 5-limit

Hello, there, and the question has been raised at to whether "major
and minor thirds" are distinguished in the 3-limit music of medieval
Europe.

If we are talking about the vertical as opposed to melodic intervals
of the semiditone (32:27) and ditone (81:64), then I would say that
the distinction goes back to at least Guido d'Arezzo's treatment of
"diaphony" or polyphony in the _Micrologus_ (c. 1030).

The popular Latin term _tertia_ or third for both intervals --
sometimes called "major" and "minor," or "perfect" and "imperfect"
(more below on the 3-limit dynamics of the latter terms) -- goes back
maybe to around the 12th century.

Jacobus of Liege (1325) uses both the Greek terms and _tertia_, for
example.

Now for a critical distinction, which Paul Erlich suggests, between
major and minor thirds (under whatever names) as distinct intervals,
in 3-limit or 5-limit systems; and the specific 5-limit distinction
between the quality of these intervals as stable concords.

The latter distinction, of course, does presume a 5-limit system, and
Vicentino (1555) and Zarlino (1558) may be among the first to state
it. They are also, interestingly, the first authors I am aware of who
specifically give what might be termed the rule of 5-limit "stable
saturation" -- wherever possible, use a complete sonority with the
third plus the fifth or sixth above the bass. This complete sonority
is Zarlino's _harmonia perfetta_, and latter (1610, 1612) the _trias
harmonica_ or triad of Lippius.

However, the distinction between the two usual 3-limit or Pythagorean
thirds is basic in medieval polyphony, and these intervals are treated
in ways at once distinct but related.

In Guido, the unstable semiditone (m3) and ditone (M3) and also the
tone (M2, 9:8) are permissible intervals in his free diaphony or
organum along with the stable unison and fourth. He makes the
distinction that in _occursus_ or the "coming together" of the voices
in a cadence to the unison, a third contracting to a unison should be
a ditone (major), not a semiditone (minor). Interestingly, by around
1300, the opposite preference is becoming a standard rule and a
motivation for accidental inflections (written or left to the
performers).

As often happens, the subtle 13th-14th century continuum of
concord/discord in Continental Europe can easily be distorted if we
try to describe the 3-limit thirds simply as either "concords" or
"discords." They are unstable "semi-concords," or in the terminology
such theorists as Johannes de Garlandia and Franco of Cologne,
"imperfect concords," at once relatively blending but somewhat tense.
Their complex ratios very much fit this role.

Major and minor thirds -- whether called semiditone and ditone, or
"thirds" -- are classed together as "imperfect concords" of this
kind. They invite resolutions to a unison or fifth, and are often used
more freely also. For much more on this, see

http://www.medieval.org/emfaq/harmony/13c.html
http://www.medieval.org/emfaq/harmony/pyth.html

In discussing multi-voice sonorities, Jacobus describes the common
_quinta fissa_ or "split fifth" where an outer 3:2 fifth is "split"
by a third middle voice into a major third below and minor third
above (e.g. F3-A3-C3 or f-a-c') or the converse (e.g. A3-C3-E3 or
a-c'-e'). He prefers the first arrangement, but cites a motet
(transmitted to us in the Montpellier and Bamberg Codices) where the
second variation with the minor below appears as the opening
sonority.

Jacobus speaks of _tertia in ditono_ and _tertia in semiditono_ to
distinguish these two forms of thirds, classes them like Garlandia and
Franco as the mildest unstable intervals (adding a few others beyond
the octave), and also notes a distinction absolutely vital to late
medieval practice and theory.

By around 1300, there is a tendency to favor specifically the minor
third before a unison, and the major third before a fifth. These
"closest approach" progressions involve contrary motion with one voice
moving by a 9:8 whole-tone, and the other by a concise 256:243
semitone (~90.22 cents).

This is basic 14th-century theory and practice, and accidentals are
regularly used to make thirds minor before unisons and major before
fifths -- and likewise sixths major before octaves.

The polarity of a contractive 32:27 and an expansive 81:64 (and
27:16), vital but distinct from later 5-limit distinctions regarding
5:4 and 6:5 as stable concords with different affective qualities,
leads to an interesting terminology for these thirds used by some
theorists.

The 32:27 is sometimes called an "imperfect third" (_tertia
imperfecta_) which tends to contract to a unison. The 81:64 is
likewise a "perfect third," with an expansive quality seeking its
"perfection" in the fifth.

Thus as Italian theorists say, and other theorists agree, a naturally
minor third expanding to a fifth should be "perfected" by an
accidental inflection making it "perfect" or major. Likewise a
naturally major third contracting to a unison should be "colored" by
an inflection making it minor. Ugolino of Orvieto (c. 1435) describes
a 17-note octave (Gb-A#) which could be applied to organs so that the
organist could "perfect" or "color" these intervals on the various
degrees of the gamut with ideal intonation.

As the Monz has noted, these inflections involving accidentals other
than the traditional Bb are termed _musica ficta_ (literally "feigned"
or "invented" music -- as opposed to the regular gamut).

Marchettus of Padua (1318), in my view, exemplifies this basic 3-limit
perspective on cadential dynamics and "closest approach," and in my
reading and that of some others seeks indeed to accentuate it by
making cadential semitones even narrower than in Pythagorean (and thus
major thirds and sixths wider).

Interestingly, he refers to thirds (and sixths) as "tolerable
dissonances," while earlier and contemporary French theorists speak of
"imperfect concords" -- likely a mainly semantic distinction, like
describing a glass of water as half-empty or half-full.

He suggests the use of a sign like the modern sharp, rather than the
traditional "square-B" or "mi" sign (like the modern natural), to show
an accidental inflection to obtain a major third before a fifth or a
major sixth before an octave. This special "chromatic semitone" marked
by the sharp leaves an extra-small cadential "diesis" which he
suggests is considerably smaller than the usual mi-fa semitone (E-F,
B-C), which is in term smaller than the usual apotome (e.g. Bb-B, or
as Germans would say, B-H).

However we interpret what might called this "microtonal notation," his
aesthetics seem to me clearly 3-limit, realized by more conventional
theorists within the usual Pythagorean system.

During the 15th century, there is a gradual shift from a complex
3-limit to a new 5-limit perspective -- something happening in certain
English styles earlier, at least in terms of the use of thirds as
stable concords.

Interestingly, the 3-limit "closest approach" progressions involving
the two thirds (and sixths) continue to play an important role in
early 5-limit practice and theory, just as modern jazz may combine
some traditional 18th-19th century harmonic progressions with a new
approach to sonorities.

It's fascinating that two theorists interested in fine distinctions of
intonation -- Marchettus in a 3-limit setting, and Vicentino in a
5-limit setting -- should have both discussed the contrast between the
two thirds, whether viewed as "tolerable dissonances" or as primary
concords.

To sum up, a distinction between the two 3-limit thirds as vertical
intervals goes back to Guido (c. 1030), and the vital distinction of
m3-1 and M3-5 ("closest approach") to around 1300 (with some
tendencies toward this pattern in the late 13th century).

However, a fully developed distinction between the qualities of
5-limit major and minor thirds, and between the stable sonorities with
the major third below the minor or vice versa, goes back to around
Vicentino and Zarlino -- although as early as 1523 and 1529, for
example, Pietro Aaron notes a preference for the major third or tenth
at certain points, and in 1525 or so shows cadences in modes such as D
Dorian or E Phrygian with the final third raised.

Note that this latter distinction applies to the fluid modality of the
middle 16th to early 17th centuries, as well as later major/minor
tonality.

Most respectfully,

Margo Schulter
mschulter@value.net