Consonance and spiral of fifths/fourths

Hello, there.

Recently people have been raising some interesting questions about
consonance and the spiral of fifths or fourths. It's easy to show that
the location of an interval on this spiral does not always determine
its perceived degree of concord or discord, and somewhat more delicate
a matter to delve into the intricacies of an issue such as 81:64
vis-a-vis 5:4.

For example, one interesting facet of Western European harmonic
evolution not so often considered is that 81:64 (the Pythagorean
ditone equal to four fifths minus two octaves, about 408 cents) is
ranked as relatively concordant by most 13th-14th century theorists,
but as "dissonant" by some 16th-17th century theorists. I suggest a
possible explanation below which seems intuitively appealing at first
blush.

While judgments of this kind may differ from listener to listener, and
certainly from era to era or culture to culture, it's easy to show (as
has already been done) that degrees of consonance do not necessarily
follow the sequence of the spiral of fifths or fourths. For example,
let's consider the ranking schemes common among 13th-century European
theorists:

Stable Relatively Relatively Strong
concord concordant tense discord

1 fifth 3:2 x

2 M2 9:8 x

3. M6 27:16 x

4. M3 81:64 x

5. M7 128:81 x

6. A4 729:512 x

It's generally agreed among 13th-century theorists that M2 and M6,
which come before M3 on the spiral of fifths, are nevertheless _less_
rather than more concordant, although they have some "compatibility"
also, and might be placed in a middle region of "unstable but
compatible."

While moving further along this spiral of harmonic intervals requires
entering the realm of accidental alterations and augmented and
diminished spellings, the results might be even more striking.

Within European and related traditions of composition, most people now
as in the 13th century would agree that a major seventh (fifth #5) is
a strong discord in itself. Yet fifth #10 gives us a Pythagorean
augmented second (19683:16384, about 318 cents), a more complex ratio
and also a much more concordant interval to most modern listeners,
being only about 2 cents from an m3 at a pure 6:5.

Similarly, a Pythagorean comma (fifth #12) at 531441:524288 (about
23.46 cents) would be strongly dissonant to just about any listener in
this tradition. Yet a doubly diminished fourth (fifth #16, e.g. d#-gb)
of about 271 cents is only about 3.8 cents from a pure 7:6, which I
suspect many people on this list would regard as more concordant than
a comma (fifth #12) or even a major seventh (fifth #5).[1]

As for 81:64 and 5:4, I'd say that it's a choice a bit like carob
vis-a-vis a very sweet chocolate. Both of them are relatively "sweet"
or "concordant," but the flavor you aren't expecting can seem
remarkably "bitter" (or "out of tune").

In a Gothic setting, 81:64 is treated as relatively concordant in
practice and theory, although in practice, of course, intonation would
vary except on fixed-pitch instruments. It's like carob, with somewhat
more tense intervals like M2 and m7 serving as "spice," and the fifth
and fourth providing rich and stable concord. This raises some curious
questions: might be there be any correlation between liking carob (one
of my favorites) and liking medieval polyphony, or do these thirds
have a higher concentration of complex carbohydrates ?

In a Renaissance setting, however, 5:4 is the new staple: sometimes I
get the "meantone munchies," and can't resist the chocolate-like
flavor of those pure thirds.

To Renaissance musicians for whom 5:4 was the essence of sweetness,
it's easy to imagine how the carob flavor of an 81:64 may have seemed
positively bitter (or "dissonant").

Another way of looking at it is that an M3 at 81:64 or an m7 at 9:5
represents a sonority in motion, while a 5:4 or a 7:4 is more likely
to serve as a point of rest. To my ears, either form of M3 seems
relatively blending while either form of m7 seems relatively tense,
but the microtonal shading can make a difference (along with the
style, of course).

It seems to me that at least one tenable modern approach is to enjoy
each of these intervals in their stylistic settings, and maybe invent
some new ones (intervals and styles alike).

Most respectfully,

Margo Schulter
mschulter@value.net