'augmented 6th' chords and their septimal cousins

>>>>> [Paul Erlich, TD 493.20, re: Bach/Busoni Chaconne]
>>>>> I'm still interested in the possible septimal augmented
>>>>> sixth chords as in Ken Wauchope's rendition.

>>>> [John deLaubenfels, TD 497.5]
>>>> I missed the details of Ken Wauchope's version - when
>>>> did that go by?

>>> [Paul Erlich, TD 497.12:]
>>> Are you serious? On 11/24/99, Ken Wauchope wrote,
>>> [extended quote...]

>> [John deLaubenfels, TD 499.9]
>> Well, I DID see those posts, but I don't find the phrase
>> "septimal augmented sixth" in them; thus my question.

> [Paul Erlich, TD 499.18]
> 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.

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.

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.

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.

- 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'.

(see _The New Harvard Dictionary of Music_, p 752)

The qualifier which Paul put in quotations refers to the
prime-limit of the chord or interval. The ones in common
use are:

prime Latin numeral
limit name

3 Pythagorean
5 just
7 septimal
11 undecimal
13 tridecimal
17 septadecimal
19 nondecimal

They can be extended in the same way with the Latin
numerals for higher primes.

I think that we should distinguish clearly between
those chords/intervals that *are* septimal, undecimal, etc.,
and those that *imply* septimal, undecimal, etc.

The proportions 4:5:6:7 by definition describe a 'harmonic 7th'
chord, a tetrad (= 4 notes) to be exact.

The meantone 'augmented 6th' chord sounds very much like
4:5:6:7, but it has proportions which include irrational
ones and which would look far different from this. So it
should be called an 'implied harmonic 7th' or 'implied
septimal dominant 7th' chord.

I suppose that 'implied septimal augmented 6th' would
convey well the information that Paul was transmitting,
but I find it somewhat less than adequate because the
meantone 'augmented 6th' has its basis in the 5-limit
'just augmented 6th', which likewise also sounds very
much like the 'septimal harmonic 7th', but whose ratios
only imply but do not contain 7.

In fact, the 5-limit 'augmented 6th' is a good example
of a chord that contains a ratio that may be analyzed as
'extended reference'. For example, the 'German 6th' is
('octave'-specific notation):

45/32
F#
/
/
1/1---(3/2)---(9/8)
C ( G ) ( D )
/ \
/ \
4/5---6/5
Ab Eb

The interval between 4/5 and 45/32 is:
(45/32) / (4/5)
= (45/32) * (5/4)
= 225/128
= ~976.54 cents

This is very close to the 7/4 [= 968.83 cents], and this,
coupled with the 4:5:6 proportion of the other three notes,
gives it a strong resemblance to 4:5:6:7.

But analyzed as extended reference, the 45/32 is actually
the 4:5 'major 3rd' above 9/8.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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On Wed, 26 Jan 2000, Joe Monzo wrote:
> 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+ [snip]
> - 'German 6th', which is a IV 6+ 5 3 [snip]
>
> Note that there are also two other 'augmented 6th' chords:
>
> - the 'French 6th', which is II 6+ 4 3 [snip]
> - one without a specific name, which is +II 6+ 4++ 3 [snip]

I agree with Daniel about not worrying too much about the spelling. The
important thing is what the pitches are (in 12TET) and how they resolve.
So I wouldn't consider the fourth something separate from the regular
German.

OTOH, don't forget the "Tristan chord", bane of mid-level theory
students everywhere. The pitches sounding look like a 1/2-diminished
7th chord, but it doesn't resolve that way; it resolves like an
augmented 6th chord. One could consider it a variant of the French 6th,
with a minor 3rd above the bass instead of major.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
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