Re: Superimposed fifths and fourths in 3-limit, etc.

Hello, there, and I'd like to respond to some very interesting remarks
by Christopher Stembridge <stembridge@iol.it> in Tuning Digest 213.
The topic of sonorities combining fifths and fourths with major
seconds or ninths or minor sevenths is also a subject that Paul Erlich
and I have been discussing in another thread.

> The other day I heard Michel P�r�s 'Ensemble Organum' singing in a
> church in Cremona; I was glad to have read Margo's pythagorean piece
> and appreciated being able to listen to chords built up of perfect
> 4ths - that to us would be discords with clashing major 2nds - in
> 4-part P�rotin (conductus 'Deus misertus'). However, that piece was
> fortunately near the beginning; tuning later suffered.

Your comment raises the point that listeners may hear the major 2nd in
this kind of quintal/quartal sonority either as a "clash" or as what
I'd describe as an "energetic blend or fusion."

From one 20th-century point of view, sonorities of this kind
are described as "fourth chords" (e.g. G3-C4-F4) or "fifth chords"
(e.g. G3-D4-A4), and often considered relatively concordant or even
potentially stable. Likewise the combination of a fifth and fourth
above the lowest tone (e.g. G3-C4-D4) is used as an independent
concord by various 20th-century composers.

A striking example occurs at the opening of Debussy's _Et la lune
descend sur le temple qui fut_, which begins with a series of parallel
chords having the fifth and fourth above the bass. Here I would
describe the effect as a spacious and opulent concord.

However, these sonorities have a rather restricted role in tertian
styles of the Renaissance-Romantic eras, serving mainly as suspensions
or the like.

It might be interesting to see how different people on this Tuning
List, coming from various stylistic and intonational perspectives,
perceive these sonorities.

Most appreciatively,

Margo Schulter
mschulter@value.net

Margo Schulter wrote:

>It might be interesting to see how different people on this Tuning
>List, coming from various stylistic and intonational perspectives,
>perceive these sonorities.

As I already mentioned, the principle by which these chain-of-fifths chords
are formed in 3-limit music is the same as that by which major and minor
seventh and ninth chords (also minor elevenths and major ninth sharp
elevenths) are formed in 5-limit music -- notice how the latter are
considered consonant in jazz harmony (partly because they contain no
tritones) which can be considered the dissonant "endgame" of 5-limit music,
much as medieval music using the analogous 3-limit constructs was reaching
an endgame by embracing more and more dissonance in the late 14th and early
15th century.

The triads by fourths or fifths are comparable to major and minor triads in
dissonance (especially when voiced without close-position major seconds);
the former have two 3-limit intervals and one 9-limit interval while the
latter have one 3-limit interval and two 5-limit intervals. But once the
5-limit was entrenched, superimposed fifth/fourth chords occured only as
suspensions. The 5-limit being the standard of consonance meant that without
5-limit intervals, a chord would sound incomplete, but with 9-limit
intervals, a chord would sound dissonant. So in common-practice harmony,
triads of the type Margo describes have to resolve to triads (or to seventh
chords that then resolve to triads).

There is an interesting parallel between the Pythagorean thirds, with their
out-of-tune 5-limit implications (I'll call them quasi-5-limit), becoming
more and more accepted in the late medieval period, and the preponderance of
tritones in jazz, whose 7-limit implications are unavoidable. What (if any)
style will take the 7-limit as the tuning norm and rejects the over-extended
5-limit and quasi-7-limit chords of the jazz style, the way Renaissance
music took 5-limit as the tuning norm and rejected the over-extended 3-limit
and quasi-5-limit chords of the late medieval style? That remains to be
seen, but a possible mechanism for doing so is described by my paper
(http://www-math.cudenver.edu/~jstarret/22ALL.pdf). Meanwhile, some jazz
musicians such as McCoy Tyner have often abandoned the 5-limit and
concentrated a good deal on extended 3-limit chords (the ones Margo was
taking about, often with five or six different notes) probably because
12-tET has such good fifths!