augmented 6th' chords and their septimal cousins

I realize that the context of this discussion has to do with meantone tuning
(a topic in which I am no expert) and I have not followed it from the
beginning. However, while reading this current exchange, it seemed to me
that there might be some value in viewing the augmented-sixth chord in a
somewhat fresh and more practical (I think) light.

Paul Erlich:

>> The Bb7 and Eb7 that Ken refers to are not really dominant
>> seventh chords; since his tuning includes C# and G# but not
>> Db or Ab, these are actually augmented sixth chords. They
>> are "septimal" since, as Ken points out, they very closely
>> approximate 4:5:6:7.

Nowhere in traditional common practice does "spelling" count more than in
the augmented sixth chord. The difference between Ab-C-Eb-Gb on one hand and
Ab-C-D#-F# is critical to aural perception of the "musical action" in that
the C-Gb tritone is significantly smaller (5:7) than the C-F# tritone
(7:10). (35:49 vs. 35:50)

It is very easy to hear the difference when a well tuned dominant seventh
chord and it enharmonic "equivalent" augmented sixth chord are heard in
isolation (without context). When the augmented sixth chord is tuned
properly, one can anticipate its direction even before the resolution chord
is heard.

I guess what I'm saying here is that if a meantone augmented sixth chord
sounds very much like 4:5:6:7, it will not likely be very convincing in
terms of harmonic "direction" and musical expressiveness.

Joe Monzo contributes:
>
> Paul is very careful in his last response here to put
> quotation marks around 'septimal' (they did not appear
> in his original description) and to note that this meantone
> chord *approximates* 4:5:6:7. I'm glad that he added those
> two things, and I think that there's a very important point
> hidden in here that deserves further elaboration.
>
>
> First off, the 'augmented 6th' under consideration actually
> has a more specific name in regular music-theory, actually two,
> which appear in 4-part harmony as follows:
>
> - 'Italian 6th', which is a IV 6+
> i.e., first inversion of the subdominant chord where
> the '3rd', which is in the bass, is flattened by a 1/2-step,
> the 'root' (in an upper voice) is sharpened by a 1/2-step,
> and the '5th' is doubled.
>
> - 'German 6th', which is a IV 6+ 5 3
> i.e., the same as above except that instead of doubling
> the '5th', one voice has the '5th' and the other the '7th'
> flattened by a 1/2-step.

Interestingly, Kostka & Payne, in their theory textbook "Tonal Harmony,"
actually drop the use of "phony" Roman numeral roots in favor of simply
using the names "Italian," German" and "French."
>
> In 12-EDO, both of these have the same interval structure
> as the 'dominant 7th'; the 'Italian 6th' lacks what would
> be the '5th' in the 'dominant 7th' chord, and the 'German 6th'
> contains it.
>
> The point of this is that I think we should be careful
> to refer to the distinction between the two in tuning
> theory when it is relevant to the structure of the chords,
> altho I suppose 'augmented 6th' is good enough in most cases.

Yes, I agree. I think common practice usage is seen more clearly when the
three "constant" members (b6, 1 and #4) are viewed as a secondary dominant
(V7 of V) with a lowered fifth. The difference between "Italian," "German"
and "French" then is simply a matter of the placement of the "remaining"
member. The Roman numerals simply get in the way and confuse the function.

This usage is even clearer when the augmented sixth chord is used in a
simple dominant function (Db-F-G-B or Db-F-Ab-B including the resulting
parallel fifths) directly to tonic resolution (C-E-G-C).

Monz continues:

> Note that there are also two other 'augmented 6th' chords:
>
> - the 'French 6th', which is II 6+ 4 3
> i.e., the same as the 'German' exceptthat the 'flat (or minor)
> 7th' is lowered a further 1/2-step to a '6th', which would be
> a 'flat 5th' in the 'dominant 7th' interpretation.

Or more functionally, the "remaining" pitch in question serves as a
functional root (D) in the V7 of V analysis I mentioned above (Ab-C-D-F# as
opposed to a rather irrational Ab-C-Ebb-F#). (F# should be the LAST of the
four pitches one would consider to be a root in any functional sense.)
>
> - one without a specific name, which is +II 6+ 4++ 3
> i.e., the same as the 'German', but spelled as a sharpened
> '6th' rather than a flattened '7th'.

Spelled Ab-C-D#-F# implying a resolution to the unstable G-C-E-G (tonic
six-four) on the way to the more stable destination G-B-D-G.
>
> (see _The New Harvard Dictionary of Music_, p 752)

Also see Kostka & Payne, pp. 384-391.

For what it's worth,

Jerry