Re: How important are overtones?



Steve Latham wrote:
>
> "Joey Goldstein" <nospam@xxxxxxxxxxx> wrote in message
> news:42F3BFA2.DAA243A@xxxxxxxxxxxxxx
>
> >
> > Aeolian songs that attempt to make use of elaborate progressions of
> > chords will likely lead the ear to a major tonic triad on the minor
> > mediant of the original minor key at some point.
>
> Yes, there is always this problem of "drifting" to the realtive major.

Ah. Glad you agree. Thanks for not arguing the point because I'm not
sure I could really back it up! <g>

> > In or order for the tonal center to remain fixed on a minor tonic triad,
> > music in Aeolian needs, generally, to be very simple as far as harmonic
> > progression away from and back to the tonic triad is concerned. In my
> > experience.
>
> Yes, like i - VII - VI - VII (and the VI of course helps out a lot, modally
> speaking)
>
> > Musica ficta just helps to seal the deal and helps keep things focused
> > when elaborate harmonic progression is desired in minor keys. IMO.
>
> I guess it depends on how elaborate it is.

The way I said that was sort of misleading I guess.
Simple progressions that utilize musica ficta like
i V7 i
"seal the deal" too.
The trouble comes in when elaborate progressions that do not use musica
ficta are attempted in minor keys.

> [snip]
> >>
> >> Joey, you've mentioned CEA having a more "major" or stable sound before.
> >> I
> >> would think that that might put CEG and CEA as being closest, and CEbG
> >> being further (note, i.e. the first inversion of a minor triad, EbGC
> >> transposed to C to yield CEA. Interesting how CEbAb fits into that
> >> scheme.
> >> SO are they closest by intervallic content - you said above the frequency
> >> ratios are far more complex in the minor triad - are they more or less so
> >> (even though the intervallic content is different) in ctructures like
> >> CEA -
> >> or do you know?
> >
> > Not sure I understand the question. But...
> >
> > C-E-A does not necessarily retain the same acoustical root upon
> > inversion.
>
> So just to be clear, the acoustical root of CEG EGC and GCE are all C right?

Right.

> ACE is A,

Right.

> CEA is likely C or A,

Usually C. Acoustically C, but just by a hair. Sometimes A according to
context and/or style. Follwoing E7 it will likely sound like Am/C.
Following G7 it will likely sound like C6.

> and EAC is probably A - am I close?

E A C
3 4 5* with A as ac root.
5 7* 8 with C as ac root.
8 11* 13 (or 16 21 25) with E as ac root.

The lower partial numbers win. So, yes. A is the ac root of E A C. But
context (eg. following a G7 chord) could make it feel as if it is a C
chord too.

> And the acoustical root of this particular voicing is a toss
> > up between C and A, with C being the victor but by a very slight margin.
> > (see below.)
> >
> > In a chord voicing where the individual intervals all have different
> > acoustical roots, the strongest interval (i.e. the interval with the
> > simplest freq ratio) that is found in the lowest range of the chord
> > voicing will often sway the ear that the lower note of that interval is
> > the overall acoustical root of the entire chord. This is why chord
> > voicings that have a P5th on the bottom always feel like the lower note
> > of that 5th is the root of the chord, no matter how weird the notes
> > above that 5th happen to be.
>
> Yes, I agree. Even if you take C-Eb-G, and you put an F below the C, you
> "change" the root from C to F, but going C Eb G D doesn't quite do the same
> thing.
>
> [snip].
> >
> > If we ask whether A is the ac root of the voicing C-E-A, then the
> > partial numbers would be:
> > 5*-6-8
> > "*" indicates a distorted partial.
>
> > If we ask whether C is the ac root of the voicing C-E-A, then the
> > partial numbers would be:
> > 8-10-13
> >
> > [The 13th partial above C is really somewhere in-between what we call A
> > and Ab. So the feeling here of A as the 13th partial is tenuous at best.
> > This is why true 13th chord voicings usually require that the 7th be
> > present in some form as well (b7 = 7th partial, maj 7th = distorted 7th
> > partial).]
>
> Makes good sense.
>
> >
> > So 8-10-13 isn't it, especially when we have 5*-6-8. I.e. Our ears
> > expect look for low partial numbers (simple freq ratios) when we are
> > looking for a root. Lower partial numbers are better. Lower partial
> > numbers without distorted partials are best and give us the least
> > ambiguous sense of root.
> >
> > Or, if we have a strong interval on the bottom of the voicing that is
> > leading our ears to hear C as the root (like C-E) then we might
> > experience the A as a distorted version of 7th partial:
> > 4-5-7*
> >
> > Notice that these partial numbers are even lower than 5*-6-8. So, at ths
> > point it's a toss up between an acroot on A or C, with C being just
> > slightly more persuasive but not by much. Both views involve altered
> > partials and fairly low partial numbers. It's really a close call.
> >
> > If we ask whether E is the ac root of the voicing C-E-A, then the
> > partial numbers would be:
> > 13-16-21
> > Not very harmonically persuasive at all. E is clearly not the ac root of
> > this voicing.
>
> Clearly.
>
> >
> > And, if we added a low A or C bass note below this voicing we will have
> > an even stronger feeling that it is the acoustical root of the entire
> > chord.
>
> Right, this is why in classical music you don't want to move from V to VI6,
> because (in C) the bass moves G-C (implying V-I) and the A note in the VI
> sounds like a mistake (of course the V-I feeling adds to the sense of the
> latter chord being a C chord too).

Yet in jazz, we do this all the time. But we call vi7/6 (a vi7 chord in
1st inv, eg. Am7/C in the key of C) a "I6" chord. The only time we would
see this as Am7/C rather than C6 is if there were some other preceding
harmonic material more clearly rooted on A.
I don't believe in the classical world there exists a maj6 chord does there?
It's always vi7/6?
According to the way I learned through Delamont, the ac root of a min7
chord in 1st inversion is the 3rd of the min7 chord.
So chords can have "constructional" or "functional" roots that are not
the same note as the "acoustical" root.

> > Now there are contexual settings in which C-E-A will be felt more like
> > Am than as C6.
>
> Right. I'd say the same is true, by extension, of ACEG and CEGA. In fact,
> when I play open voicings of an Am7 chord (AECG) on guitar and cycle through
> the various inversions, when AE or EA is on the bottom, it sounds like A,
> and when CG or GC is on the bottom, it sounds like C. I actually did a
> little piece once where a mived from G CE A to ACEG (just the G and A
> swapped) and it's amazing how that one little change makes it sound lik two
> different harmonies (roots).

The most sure-fire way to ensure that A is heard as the root of Am or
Am7 is to put an A in the bass.

Making sure that G is present and that A isn't in the bass can help to
ensure that voicings/inversions of C6 sound like C is the root. With C
in the bass this chord is almost always hard to hear as an Am chord.

E G A C
5 6 7* 8 with C rt
3 7 8 10* with A rt
This voicing is ambiguous but leans just a bit more towards a C root.
Could go either way really, depending on context.
But with G omitted:
E A C
5 7* 8 with C rt
3 4 5* with A rt
The scales are tipped towards an A root.

G A C E
6 7* 8 10 with C rt
7 8 10* 12 with A rt
is ambiguous but leans even more towards a C root.

> As a matter of fact, since the partial numbers are really
> > splitting hairs here, if a composer is not careful it might be heard
> > either way.
> That is a danger theory students often face in part-writing exercises.
>
> One thing is for certain though. It will not be heard as
> > some sort of chord whose root is E.
>
> Agreed.
>
> >
> > If you asked yourself whether A#/Bb, B, D#/Eb, or any other note is the
> > acoustical root of this voicing you won't find any better partial
> > numbers than the ones we've already found.
>
> Surely.
> >
>
> > The next best thing is an ac root on D.
> > 7-9-12
> > C-E-A
> > Here we have no distorted partials. But the ac root of the chord is an
> > omitted tone from the actual voicing of the chord, (what Rameau would have
> > called an "implied" root - like when C# E G Bb is considered a "rootless"
> > Ab9 chord)

You meant C Eb Gb Bb, right?

> and the partial
> > numbers are considerably higher than our other two candidates.

5 6 7 9

> And this goes to the Dominant9 idea you were talking about before.

Yes. Acoustically speaking, min7b5 chords (aka half dim 7th chords),
when voiced in root position, are dom9 chord in 1st inv with root
omitted, acoustically speaking.

Oh...
Maybe you meant C# E G Bb as A7b9, right?
Well yes, but these 4 notes, especially in equal temperament, have the
potential to also be C7b9, Eb7b9 and/or F#7b9. But in root position
C#dim7, is acoustically speaking, A7b9/C# with root omitted. b9 is an
altered/distorted version of the 9th partial of A.

C# E G Bb
5 6 7 9*

If C was the root the partial numbers would be higher for this root
particular voicing of C7b9:
Db E G Bb
9* 10 12 14
If Eb was the root:
Db Fb G Bb
7 9* 10 12
If F# was the root:
C# E G A#
6 7 9* 10

So, in root position, C#dim7 has an acoustical root on A.

> > As a matter of fact in a just intonation for a minor key it is my
> > understanding that this is how the tonic min triad is actually tuned
> > (3-7-9 for a root position min triad).
> >
> > It is interesting to examine a root position min triad in this light.
> > Is it:
> > A-C -E (ac root on A)
> > 4-5*-6
> > or
> > 3-7-9 (ac root on D)
> > ??
> > I.e. Is the ac root of A-C-E an A or is it a D?
> > Another close call.
>
> >
> > Clearly though, the way we use Am triads in the key of A minor is not
> > normally in a manner in which D is experienced as the root. So there
> > must be something persuasive about the presence of that P5th between A
> > and E as far as the feeling of root is concerned. And I guess that
> > having an octave double of the fundamental actually sounding in the
> > chord voicing, rather than omitted, helps too.
>
> Sure - also a reason students should avoid doubling the 3rd of a VI chord if
> they can help it - C C E A sounds awful "c like".

Well putting C in the bass is the problem there, not doubling it.
Delamont recommends doubling the 3rd of vi because it is a "tonal note"
(1 4 or 5) in the key. I.e. The preferred doubles in the key of C are C,
F, and G. With exceptions of course, usually involving not doubling a
note that is part of a dissonant interval. F wouldn't be a good double
on Bdim. C is not a good double on Dm7. etc.
But on Am as vi the preferred double is C. The root is always available
as a double too, except on viidim because it is the leading tone. I
believe the preferred double on viidim is the 3rd of the chord.

In the key of A minor though the tonal notes are A D and E.
So on Am as i, although doubling the 3rd is often necessary, the
preferred doubles are A or E.

This may be different from CPP practice though.

> > Now, if we add a G to our C-E-A chord:
> > C-E-G-A
> > 4-5-6-7*
> > Then the case is solidified in favor of an ac root on C, because the
> > interval in that chord that has the strongest harmonic suggestion is
> > C-G, 4-6 in this voicing.
> > If A were the ac root the partial numbers would be:
> > C -E-G-A
> > 5*-6-7-8
> > higher numbers, distorted partial is lower in the series, etc.,...
> > overall, a little bit less persuasive feeling of a root on A.
>
> I'm wondering though, if C and E don't also cause an implied (or inferred is
> better) G - sometimes I swear I hear "ghost" notes like this that seem to be
> a result of overtones or other vibrations. So even CEA might sound like CEGA
> in some situations.

Well in a situation where C-E-A is heard as a C6 chord, the G is there
by implication.
In a setting where C-E-A is heard as Am the G is not implied. Voo-doo.

What's impossible to prove, I suppose, is that any of this is really
going on in a listener's ear, all by itself, without the listener's
knowledge or participation, and without the listener necessarily having
any musical training. I'm the last one to want to grasp on to
metaphysical things, but this does seem to be an almost mystical
process, one that can not really be proved. I know that since being
exposed to this theory *I* certainly hear things this way and the theory
helps to guide *me* in my own voicing choices. If and when other trained
musicians say they hear things that way too that helps to "prove" the
theory. But as to why untrained listeners respond to harmony as well is
a mystery.

> > But if the same 4 tones are voiced as follows:
> > A-C-E-G
> > then the strongest interval is A-E putting the ear squarely on a root at
> > A.
> > If C were the root of this voicing then the partial numbers would be:
> > 7*-8-10-12
> > or
> > 13-16-20-24 (pretty unlikely)
> >
> > Unlike the major triad and the dom7 chord, the voicings and inversions
> > of these other chords (and of *all* other chords) do not necessarily
> > retain the same acoustical root. This is what makes the maj triad and
> > the dom7th chord unique.
> >
> > I.e. Any inversion or voicing of a maj triad will always have the same
> > acoustical root. This is true for dom7 chords as well. This is *not
> > true* for any other possible chord type.
>
> What about diminished and augmented?

Come to think of it, dim triads retain their ac root in all inversions too.
B D F
5 6 7
F B D
7 10 12
D F B
6 7 10
This 1st inv Bdim chord can also be heard as and function as a Dm6 with
5th omitted
4 5* 7*
at least by jazz players it can.
The case is weak though with 2 altered partials so low in the series.
But the same could be said for the min(maj7) chord:
C Eb G B
4 5* 6 7*
The addition of an A to the 1st inv Bdim chord helps to put the ear on D
as the ac root, somewhat more, again having a P5th present in the lower
voices, especially including the bass voice helps in this process:
D F A B
4 5* 6 7*
[Now, *these* 4 notes can have the root jump around in the ear between
G, B, and D depending on the inversion and/or context.]

The omission of the G on Cm(maj7) is something to ponder. Don't have
time now to really explore it but I suppose it would begin to lean a
little bit towards a B7b9 (in 4th inv with a bunch of omissions),
acoustically speaking, which means that it is approaching the limit of
what it means for a chord to actually have an acoustical root. Certainly
having the G present helps the ear to find the root.

There's a lesson there to jazz players used to habitually omitting the
5ths from our voicings. Sometimes keeping the 5th is really helpful too.

Now aug triads, especially in 12 tet, have a different acoustical root
in each inversion.
The lowest note in the voicing will always be the acoustical root.

C E G#
4 5 6* with C as rt
6* 8 10 with E as rt
5 6* 8 with G#/Ab as rt

E G# B#
4 5 6* with E as rt
6* 8 10 with G#/Ab as rt
5 6* 8 with C as rt

Ab C E
4 5 6* with Ab as rt
6* 8 10 with C as rt
5 6* 8 with E as rt

> >> [snip]
> >>
> >> > The ruling interval in a tonic triad, whether it be minor or major, is
> >> > the P5th, not the 3rd.
> >>
> >> I think that's what makes it "stable", and the deviation from the
> >> "natural"
> >> overtone makes the m3 sound "sad" as we typically describe it.
> >
> > Yes. I see it that way too. Im is poetically like an imperfect home,
> > just like in real life. I guess that only in death do we truly reach the
> > I major chord, and only if we've been good. <g> If we've been bad then
> > we are doomed to an eternity on a dim or aug triad!
>
> That sounds like Beethoven's 9th - Dm to DM (I think the fifth is Cm to CM
> too).
>
> Thanks Joey - that is what I was asking - anything lacking I'll throw at you
> in other discussions :-)
>
> Steve

--
Joey Goldstein
http://www.joeygoldstein.com
joegold AT sympatico DOT ca
.

Relevant Pages

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