Dominant 7th 4-5-6-7-8 ?

Daniel Wolf wrote:

> The most striking characteristics of a dominant seventh chord within an
> otherwise triadic environment are - for me - the density of the chord (a
> triad contains three interval classes, a tetrad six)

But a great deal of tonal music is *not* expressed just with triads, so
I cannot agree with this statement. A major seventh or ninth or even
thirteenth can function perfectly well as a stable tonic chord in tonal
music. This proves that "dominant function" is not a result simply of
the "density" of the chord.

> and the appearance of a new (even ''xenharmonic'') interval.

Well this interval is only perceived to be "xenharmonic" if it is played
as 7/4, and I don't believe that "xenharmonicism" is a necessary
condition for the dominant seventh to function as it does.


> These features are enhancements of
> the dominant/tonic voice leading relationship: root movement by fifth, and
> stepwise ''resolution'' in the remaining voices.

Root movement by a fifth is certainly a powerful and satisfying
resolution of the dominant 7th chord, but it is not the only acceptable
resolution. V7 resolves well to vi (minor) as well (i.e. up a major
second), and the augmented sixth chord (e.g. g - b - d - e#) - which is
another chord that I think would be incorrectly represented by a 4-5-6-7
tuning - resolves most effectively to a major or minor chord a *minor
second* below.

The stepwise resolution of the intervals making up the tritone in both
these chords is, I believe, the *critical* factor which determines their
resolution.


> When a dominant seventh chord is
> suspended for a classical cadenza, it suffers little when tuned as 4:5:6:7
> and indeed, the stability of the chord enhances its contrast with the tonic
> triad. On the other hand, the suspension of a 16/9 (or 9/5) as the seventh
> introduces a tone dissonant to the rest of the chord, but not foreign to
> the scale.

If a dominant 7th chord is suspended for a while, then I think a case
can be made for a 4-5-6-7 tuning just to soften the dissonance, but then
how does the soloist tune his fourth scale note? Does he tune it as
21/16 or as 4/3 which has a much more simple and conventional
relationship to the tonic? Can a singer, for instance, really be
expected to pitch the former interval accurately when the latter is so
common and so well-defined. In my experience 21/16 expressed as a
melodic interval sounds like an out-of-tune 4/3 (in the context of
common-practice tonal music).

> One often-raised objection to the septimal tuning is the narrow semitone
> (21/20) voice leading from the seventh to the third of the tonic. I find
> this line of argumentation more compelling, but the interval itself is not
> difficult to sing - and if melodically ''strange'', it does maximize the
> contrast function.

"Strangely" enough :) I wouldn't cite this as an argument against using
4-5-6-7.

I believe that it is the very complex ratio made between the third and
seventh of the dominant seventh (45/32) and the third and augmented
sixth of the augmented sixth chord (25/18) that makes the two notes
making up these intervals so "restless". Substituting 4-5-6-7 gives the
much simpler, and consonant, ratio of 7/5 for both.

For these reasons, and the arguments above, I really think that using
4-5-6-7 in common-practice tonal music is not appropriate.

Using this ratio in a xenharmonic music is of course a different matter!


> Readers might have some fun investigating the traditional rule of thumb for
> the relative density of the chords in V7 - I cadences: In four voices, if
> the voice leading is correct, then one of the two chords has to be
> incomplete (that is with a doubled pc). (Why this should be so I will leave
> as a kind of ''Puzzler'').

Unless, of course, the Dominant 7th is in first or third inversion.

Andrew Milne
Islington
London

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From: PAULE

John C. wrote:

>Doubly postive systems resembling 22-tet are also possible. I
>will post some such tunings at a later date.

What is the definition of "doubly positive?" Do you mean, two positive
systems separated by exactly one-half octave? If so, I hope you don't give
away any more of my secrets, at least until XH 17 comes out, which is, when?
Or did someone else come up with the idea of two positive systems separated
by exactly one-half octave before I did? (My paper includes, in addition to
new ideas on what 22-tet does, two optimized "doubly positive (?)" systems
for doing what 22-tet does. 22-equal actually falls in between the
unweighted and weighted solutions, which no other equal tuning without
vastly more notes can claim.)

Also worth considering are doubly negative systems resembling 26-tet and
triply positive systems resembling 27-tet, if I am using this language
correctly.

-Paul E.

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On Wed, 12 Mar 1997, PAULE wrote:
> John C. wrote:
> >Doubly postive systems resembling 22-tet are also possible. I
> >will post some such tunings at a later date.
>
> What is the definition of "doubly positive?" Do you mean, two positive
> systems separated by exactly one-half octave?

I suspect what John means by "doubly positive" is that the Pythagorean
comma (12 fifths - 7 octaves) is represented by two positive scale
steps.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "'Jever take'n try to give an ironclad leave to
-\-\-- o yourself from a three-rail billiard shot?"


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