Chord inversions: effect of order (triads) - 1 top, 2 bottom, 3 middle

Hello -

I've thought about chords of "5 limit" triadic harmony and effect of chord inversion on the chord's "intensity of just-third-ness (4/5 ratio)". Instinctively I've felt that the 2nd inversion chord 3:4:5 brings out that effect with the greatest intensity (usual cases I'm used to hearing). But that effect is very powerful and noticeable also when the third is the bottom note in the first inversion 5:6:8 chord. The effect is by no means lost in the root position 4:5:6 chord, but I think where the third note is exposed at the top the effect is greater, and when it's at the bottom, I get a sensation of a kind of liberating relaxation when the third drops down from its equal tempered "drawn up" tense position to the more natural just position.

I've not accumulated quite as much listening experience with inversions of three-note chords having some note frequency ratios involving higher odd numbers than 5 - have only just now thought more directly and analytically about the effect of how "exposed" a note is within a chord - but the way chord inversions affect "exposure" of their notes and their prominence in the mind of the hearer adds an additional dimension to the analytic numerical perceptual study of chords and musical harmonies and their tunings.

On another topic, I thought Margo Schulter's improvisations sounded striking and different from any musical phrases I'd heard before - a bit like the "ars nova squared"

Dave Hill, Borrego Springs, CA

Dave Hill wrote:

> Hello -

> I've thought about chords of "5 limit" triadic harmony
> and effect of chord inversion on the chord's "intensity
> of just-third-ness (4/5 ratio)". Instinctively I've felt
> that the 2nd inversion chord 3:4:5 brings out that effect
> with the greatest intensity (usual cases I'm used to
> hearing).

Hi, Dave! Zarlino, interestingly, made a similar observation
in 1558: the ratio 3:4:5 best follows "the order of the
sonorous numbers" 1-2-3-4-5-6, not yet recognized as the
harmonic series, but equivalent to it. In this era there was
considerable caution about a sonority with a fourth above
the bass: Zarlino recognized this, but argued that really
3:4:5 should be treated as fully concordant.

> But that effect is very powerful and noticeable also when
> the third is the bottom note in the first inversion 5:6:8
> chord. The effect is by no means lost in the root position
> 4:5:6 chord, but I think where the third note is exposed
> at the top the effect is greater, and when it's at the
> bottom, I get a sensation of a kind of liberating
> relaxation when the third drops down from its equal
> tempered "drawn up" tense position to the more natural
> just position.

Zarlino takes 4:5:6 as the most usual position, but regards
5:6:8 or 3:4:5 as also a "natural" arrangement following the
ordering of ratios in the series of sonorous numbers
(basically 1-6, plus 8 for some of these voicings). This is
in contrast to 10:12:15, 12:15:20, or 15:20:24, all of which
are "artificial" and less sonorous. So, in modern terms, the
inversion of a "natural" sonority will also be natural,
while that of an artificial sonority will also be
artificial.

But in the 16th and 17th centuries in Europe, when flexible
pitch performances were likely close to 5-limit and meantone
was the rule for keyboards, there was a lot of discussion
and debate about a fourth above the bass, or something like
3:4:5. In practice, the ideal was something like 2:3:4:5,
with Zarlino noting how a major tenth is more harmonious
than a third.

> I've not accumulated quite as much listening experience
> with inversions of three-note chords having some note
> frequency ratios involving higher odd numbers than 5 -
> have only just now thought more directly and analytically
> about the effect of how "exposed" a note is within a
> chord - but the way chord inversions affect "exposure" of
> their notes and their prominence in the mind of the
> hearer adds an additional dimension to the analytic
> numerical perceptual study of chords and musical
> harmonies and their tunings.

Well, you've led me to try comparing some things that I
might not have otherwise -- thank you! For example, I find
that 6:7:9 is very harmonious, and often about the next most
concordant sonority I use after 2:3:4 or 3:4:6. As Zarlino
would say, 6:7:9 follows the order of the sonorous numbers
-- or would, if he had included 7! With 7:9:12, there's a
lot more energy and expansiveness, very effective for a
cadence, but more "JI-buzzy" as 7:12:18, or 14:18:21:24.
And something I really hadn't focused on before is 9:12:14,
very sweet and pleasant.

There are also things like 8:11:13 and 13:11:8 -- this could
be a very extensive discussion, and I'm still reflecting on
some of your valuable observations from back around
1998-2000 or so, based on lots of hands-on experience with
JI and meantone.

> On another topic, I thought Margo Schulter's
> improvisations sounded striking and different from any
> musical phrases I'd heard before - a bit like the "ars
> nova squared"

That's something I'll take as a very happy compliment; one
of my mottoes is _ars intonationis subtilior_ or "a more
subtle art of intonation."

And your questions about inversions and voicings may give
lots of us some artful ideas.

> Dave Hill, Borrego Springs, CA

With many thanks,

Margo

Margo Schulter <mschulter@...> wrote:

> But in the 16th and 17th centuries in Europe, when
> flexible pitch performances were likely close to 5-limit
> and meantone was the rule for keyboards, there was a lot
> of discussion and debate about a fourth above the bass,
> or something like 3:4:5. In practice, the ideal was
> something like 2:3:4:5, with Zarlino noting how a major
> tenth is more harmonious than a third.

Were doubled basses relevant? If you double the bass of
4:5:6, you get 2:4:5:6. From 2:3:4:5, you get 1:2:3:4:5.
But 3:4:5 leads to 3:6:8:10 which is the worst of the bunch.

Graham

Dear Graham,

Your question is very relevant: doubled basses are in
fact very common in 16th-century music, and often are
encouraged by the emphasis on ideally placing intervals
in their "natural order in the series of sonorous numbers."
Thus Zarlino held that a major third, 5/2, was more
harmonious than 5/4, while 5/1 would be ideally harmonious,
1-2-3-4-5.

According to Tomas de Santa Maria, who was focusing on
the art of improvising four voices on keyboard, although
with reference to composition also, three of the most
common sonorities are third-fifth-octave,
fifth-octave-tenth, and octave-tenth-twelfth.

When 3:4:5 occurs as 3:6:8:10 in an unstable role as a
"consonant fourth" sonority, then the tension you note
serves a musical purpose, because this sonority is at
once consonant enough so that it can be treated more
freely than a usual dissonance, but yet also unstable
in a way that typically leads into the more definite
dissonance of a suspension and resolution. For example:

E5 D5 E4
C5 B4 C4
G4 G4
G3 C3

But if the "consonant fourth" were intended as stable,
then your caution would apply! And its possible, as
you suggest, that this might lead to a more general
rule of being cautious about simple 3:4:5 as well as
3:6:8:10.

Curiously, some of the original 14th-15th century
arguments for treating the fourth as less than a
full concord followed the logic that since 8:3 wasn't
regarded by the Pythagoreans as concordant since it's
not superparticular, then caution might be indicated
for a simple 4:3 as well, even though it is, of course,
superparticular! One article I saw in a journal called
_In Theory Only_ during the early 1980's concluded that
in fact 8/3 _is_ acoustically more tense than 4/3. which
may also be in line with your suggestion.

Some of the 16th-century debate was more purely
subjective. Thus Zarlino pointed to the free use
of the fourth in the Greek Orthodox music heard in
Venice, and to the aural reality that two strings
at a pure fourth are fully concordant to the ear.
Thomas Morley argued that the fourth was rightly
treated as a dissonance because it mightily offended
the ear. But it would be an interesting connection
with some debates going back to ancient Greece if
it were the question of 8/3, the eleventh, which
played a role in the more cautious treatment of
the fourth generally.

Best,

Margo