Re: Gothic/neo-Gothic and "Pythagorean" jazz progressions

Hello, there, Monz and Paul and everyone.

In response to some suggestions that harmonic progressions by chains
of fifths or fourths in jazz are "Pythagorean," and therefore may have
an affinity with the progressions of Gothic Europe, I would suggest
that the most common Gothic progressions involve motion of all voices,
including the lowest, by seconds or thirds.

Certainly relations between vertical centers a fourth or fifth part
can play an important part in the structure of a piece -- for example,
in a 13th-century motet opening on the fifth A3-E4 (with C4 as middle
C), and concluding on the fifth D3-A3 or the complete 2:3:4 trine
D3-A3-D4; or one starting on C3-G3 and ending on F3-C4-F4, etc.

However, the actual progressions from one sonority to the next,
including directed cadences, are likely to involve stepwise or
near-conjunct contrary motion -- the latter meaning that one voice
moves by step and the other by a third.

For some examples of 13th-century cadences, which I wrote back in 1997
in a style using symbols such as "8/5" to show _not_ a tuning ratio of
8:5, but a "continuo-like" notation giving the intervals the upper
parts form with the lowest part (here an octave and fifth above,
i.e. a 2:3:4 trine), please see

http://www.yahoo.com/emfaq/harmony/13c.html

In both the 13th and 14th centuries, forms often focus on vertical
centers a second or third apart for what might be called
"longer-range" organization, also. For example, one 14th-century
scheme has a final cadence on F with ascending semitones, and a
sectional cadence on A, a third higher, with a descending semitone:

Final cadence (_clos_) Internal cadence (_ouvert_)

E4 F4 G4 A4
B3 C4 D4 E4
G3 F3 Bb3 A3

Here _ouvert_ is French for an "open" or inconclusive ending, and
_clos_ for a "close" or conclusive ending. Note the fluidity of the
degree B/Bb (or German H/B), fitting the "closest approach" rule that
thirds expanding to fifths or sixths to octaves should be major.

Looking at some 13th-century motets, or some 14th-century pieces by
Machaut or Landini, might illustrate these points -- it's easy to give
examples of cadences, but reading through a score or hearing a
complete piece can give a much better feeling for form and style, to
say the least. One excellent source of information on recordings and
the like is

http://www.medieval.org/emfaq/

In a paper on 14th-century vertical organization and intonation which
I'm now planning, I'll explain more about one possible interpretation
of these progressions in terms of diatonic scales -- definitely a
"_neo_-Gothic" theoretical approach, as far as I know, rather than
anything discussed in medieval sources, although hexachord theory
provides one starting point.

Note that progressions where the lowest voice moves by a fifth or
fourth are certainly not excluded from medieval polyphony -- it's just
that motion by a second or third is more common, and typical of the
most popular cadences.

Most respectfully,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_23590.html#23590

> Hello, there, Monz and Paul and everyone.
>
> In response to some suggestions that harmonic progressions by
> chains of fifths or fourths in jazz are "Pythagorean," and
> therefore may have an affinity with the progressions of Gothic
> Europe, I would suggest that the most common Gothic progressions
> involve motion of all voices, including the lowest, by seconds
> or thirds.

Margo, thank you so much for this!

It was really naive of me to equate Pythagoreab *tuning*
with voice-leading exhibiting Pythagorean-base root-movement.
Your clarification is well noted.

What's interesting is that, in jazz voice-leading, while
the bass generally moves by skips of alternating "4ths"
and "5ths" in the ii-V-I cadential progressions that appear
over and over again in different tonicizations throughtout
the piece, all of the *other* voices generally move as
short a distance as possible, either being held as
common-tones, or moving by semitone or whole-tone.

This is of course very similar to traditional "common-practice"
(c. 1600-1900) contrapuntal technique, but a major difference
is that in chord progressions where common-practice *triadic*
(or even tetradic) voice-leading requires a voice to skip in
order to obey a "rule", jazz often does not require the skip
because a held common-tone or a move by semitone or whole-tone
can result in a pitch which would be labeled a "non-harmonic"
tone in common-practice but which fits very well into the
extended sonority of jazz harmony.

(Whew! These sentences are too long...)

As always, your responses to my speculations, even where they
so gently show that I am in error, provoke me to further
interesting thoughts. I'm always grateful for that.

My biggest regret in regard to responding to your contributions
lately is that my mind has simply been very far away,
stylistically and chronologically, from the musical realm
that you so comfortably inhabit. I'll eventually take a
step back in time and engage more frutifully with your work.

-monz
http://www.monz.org
"All roads lead to n^0"

Hello, there, Monz, and thank you for a most friendly response, to which I
would add a correction to a URL in my earlier post, cautioning again that
this article dates back to 1997, and uses symbols such as "8/5" to show
the interval each upper voice forms with the lowest voice (here an octave
and a fifth) rather than a tuning ratio such as 8:5:

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

Your point about jazz progressions often being more conjunct melodically
in the upper parts because of the more complex vertical sonorities deemed
consonant (or at least not requiring special treatment as
"dissonances," in contrast to the "last common practice period" of
Corelli-Wagner or the like) is an interesting one I hadn't considered.

In Gothic progressions, generally, the strongest directed resolutions
involve _all_ voices moving stepwise, or sometimes all voices moving
either stepwise or by a third. This is another side of the general
preference for directed resolutions of unstable intervals involving
stepwise or near-conjunct contrary motion.

By the way, in response to your dialogue with Paul, I tend to regard the
Halberstadt keyboard as a slice of Pythagorean or something similar in a
Gothic/neo-Gothic setting (a not unhistorical view, given its 14th-century
provenance), and as a slice of meantone in a Renaissance setting.

Possibly this suggests that conventions can and should vary depending on
the intonational style and setting. Of course, this can be a complication
when trying to come up with a "standard" notation for tunings open to
various interpretations.

Most appreciatively,

Margo Schulter
mschulter@value.net