Does a C major triad have missing fundamentals at D, F and A?

Parncutt and Hair think it does. See the middle of page 3 in
"Consonance and Dissonance in Theory, Practice and Science"
Richard Parncutt and Graham Hair
Conference on Interdisciplinary Musicology, 30.8.-3.9.2011, Glasgow
http://www.uni-graz.at/~parncutt/publications/PaHa11_CIM.pdf

They write:
------------------------------------
"The theory of pitch commonality aims to explain the origin of harmonic relationships between successive tones and chords. For example, the chords C major and D minor have moderate pitch commonality but no common notes. A C major triad has missing fundamentals at D, F and A, while a D minor triad has missing fundamentals at G and Bb. So the two chords have pitches D, E, F, G and A in common.

The chord CEG implies A, because E corresponds the 3rd harmonic of A, and G corresponds to the 7th. CEG also implies F, of which C is the 3rd harmonic and G is the 9th. CEG also implies D, of which C is the 7th harmonic and E is the 9th. Repeating the same procedure for D minor, DFA implies Bb, of which F is the 3rd harmonic and D is the 5th. It implies also G, of which D is the 3rd harmonic, F is the 7th, and A is the 9th.

In this process, the auditory system tolerates quite large mistunings between partials and fundamentals."
------------------------------------

On Tue, Dec 11, 2012 at 9:00 PM, dkeenanuqnetau <d.keenan@...>
wrote:
>
> The chord CEG implies A, because E corresponds the 3rd harmonic of A, and
> G corresponds to the 7th. CEG also implies F, of which C is the 3rd harmonic
> and G is the 9th. CEG also implies D, of which C is the 7th harmonic and E
> is the 9th. Repeating the same procedure for D minor, DFA implies Bb, of
> which F is the 3rd harmonic and D is the 5th. It implies also G, of which D
> is the 3rd harmonic, F is the 7th, and A is the 9th.

Yeah well, that all sounds good, but unless he has some data where
they actually measure people hearing some missing phantom F or A in
the bass when the 12-EDO notes C-E-G are played, it doesn't matter
what he's imagining corresponds to what.

Parncutt is on and off for me, and this is one of those times he's
off. Some of his work with virtual pitch integration and music seems
solid enough, but stuff like this strikes me as fluffier. I'm open to
changing my opinion if anyone has data to suggest that things work the
way he's describing here.

-Mike

It seems to me to be better ultilized as a guide to chord substitution under the proper musical circumstaces.

$$$$$

The chord CEG implies A, because E corresponds the 3rd harmonic of A, and> G corresponds to the 7th. CEG also implies F, of which C is the 3rd harmonic> and G is the 9th. CEG also implies D, of which C is the 7th harmonic and E> is the 9th. Repeating the same procedure
*

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Dec 11, 2012 at 9:00 PM, dkeenanuqnetau <d.keenan@...>
> wrote:
> >
> > The chord CEG implies A, because E corresponds the 3rd harmonic of A, and
> > G corresponds to the 7th. CEG also implies F, of which C is the 3rd harmonic
> > and G is the 9th. CEG also implies D, of which C is the 7th harmonic and E
> > is the 9th. Repeating the same procedure for D minor, DFA implies Bb, of
> > which F is the 3rd harmonic and D is the 5th. It implies also G, of which D
> > is the 3rd harmonic, F is the 7th, and A is the 9th.
>
> Yeah well, that all sounds good, but unless he has some data where
> they actually measure people hearing some missing phantom F or A in
> the bass when the 12-EDO notes C-E-G are played, it doesn't matter
> what he's imagining corresponds to what.
>
> Parncutt is on and off for me, and this is one of those times he's
> off. Some of his work with virtual pitch integration and music seems
> solid enough, but stuff like this strikes me as fluffier. I'm open to
> changing my opinion if anyone has data to suggest that things work the
> way he's describing here.
>
> -Mike
>

If he were talking about looking at patterns that happen when the sound waves interfere and reinforce each other, I think that it's true that these notes might be found as repetitive patterns - but that doesn't mean that they mean anything to the ear. Also, if this is what he refers to, I'm sure that you all know that he references the 7-8-9 triad to justify his D, but the C-E diad in a major triad is not that.

But, I don't think interference patterns are what he's talking about. He follows with:

"In this process, the auditory system tolerates quite large mistunings between partials and fundamentals. For example, the difference between the so-called natural seventh with a ratio of 7:4 and the equally tempered minor seventh is about a 1/3 semitone. We know from psychoacoustical experiments that mistunings of this order
have little effect on the salience of derived virtual pitches"

In essence, I have no idea what he means by 'implies'.

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:
>
> Parncutt and Hair think it does. See the middle of page 3 in
> "Consonance and Dissonance in Theory, Practice and Science"
> Richard Parncutt and Graham Hair
> Conference on Interdisciplinary Musicology, 30.8.-3.9.2011, Glasgow
> http://www.uni-graz.at/~parncutt/publications/PaHa11_CIM.pdf
>
> They write:
> ------------------------------------
> "The theory of pitch commonality aims to explain the origin of harmonic relationships between successive tones and chords. For example, the chords C major and D minor have moderate pitch commonality but no common notes. A C major triad has missing fundamentals at D, F and A, while a D minor triad has missing fundamentals at G and Bb. So the two chords have pitches D, E, F, G and A in common.
>
> The chord CEG implies A, because E corresponds the 3rd harmonic of A, and G corresponds to the 7th. CEG also implies F, of which C is the 3rd harmonic and G is the 9th. CEG also implies D, of which C is the 7th harmonic and E is the 9th. Repeating the same procedure for D minor, DFA implies Bb, of which F is the 3rd harmonic and D is the 5th. It implies also G, of which D is the 3rd harmonic, F is the 7th, and A is the 9th.

I can get CEG to generate a D if I tune it to 14:18:21 and distort it like crazy, and I can get it to generate an A if I tune it to 19:24:28 with the same distortion. It's actually pretty neat that the difference tone changes so clearly when I move the C by 57/56. I can't get it to generate an F at all though, no matter how I try to cheat. I'm not sure what these guys were thinking, because if every harmonic is a multiple of 3 then it's simply a harmonic series on C and has nothing to do with F. There is no "period tripling" that would be needed to produce F.

If there's no distortion I can't hear any of these things, only imagine them.

The two base notes for the minor triad are of course very clear with modest distortion, but again pretty much absent without it.

Keenan

On Wed, Dec 12, 2012 at 12:39 PM, bigAndrewM <bigandrewm@...> wrote:
>
> "In this process, the auditory system tolerates quite large mistunings
> between partials and fundamentals. For example, the difference between the
> so-called natural seventh with a ratio of 7:4 and the equally tempered minor
> seventh is about a 1/3 semitone. We know from psychoacoustical experiments
> that mistunings of this order
> have little effect on the salience of derived virtual pitches"

I'd like to see some of those experiments.

To be totally honest, I don't hear any VFs at all when 12-EDO dyads
are played, save for perhaps a bit on the 3/2's.

These days, I think that actual true blood VFs are most directly
important when you play larger chords, like 2:3:4:5 or something, with
the obvious VF often being played in the bass. I think that exposure
to these sorts of sounds can carry over later to situations when
there's no VF at all, such as in an arpeggiated chord - you still
remember the sound lurking in the background that you know would be
there if you just played all of the notes at once. To a lesser extent
I think the same applies to things like dyads, where save for stuff
like 3/2, I find that unless my level of consciousness is somewhere up
there with "Buddhist monk," I don't really hear any overt VF at all.

This is just my experience though, YMMV. But FWIW, I find that some
principle like the above models very well how my hearing has changed
as I've played more in 19-EDO and 22-EDO. 9/7 used to sound woefully
discordant to me, but once I started playing 4:7:9 and 9-limit harmony
in general a lot, it started sounding more like an abstract "piece" of
that chord. The same applies to 11/8; it still has that same sort of
semi-discordant flavor to it, but it's hard for me to play that dyad
and not immediately think of a full 8:9:10:11:12 lurking somewhere in
the vicinity after playing around in porcupine temperament so much,
and this covers up the discordance somewhat.

-Mike