Real and tonal, diatonic and exact answers and inversions (was: [tuning] Digest Number 3943 and: as dumb as a unison)

Hi Daniel, Gene, and all,

On Wed, 08 Mar 2006, Daniel Wolf wrote:

[snip]
> There are (unfortunately) two usages of "inversion" in music theory. The
> first, harmonic use, which you mention, refers to the voicings or
> positions of a chord: for example "root" or 1st position = tonic in the
> basss (5/3), 2nd position = third in the bass (6/3), 3rd position =
> fifth in the bass (6/4). The second, equally legitimate usage, is more
> associated with melodic uses, and original with answers in imitative
> counterpoint. Answers are distinguished between real (the intervals are
> inverted exactly) or tonal (the contour is generally maintained, but
> intervals are adjusted to accommodate the tonal environment). Real
> answer becomes especially important in atonal and 12-tone contexts
> (where interchangeability between melodic and chordal material is also
> essential), but there are plenty of examples of real imitation to be
> heard in earlier repertoire as well: not only Partch was fond of chordal
> inversion at the 1/1 (To hell with it, I'm going to walk! 1/1 O7 - 1/1
U7).
>
> DJW

Spot on, Daniel! This is exactly the way I learned it
(in Australia) in the 1950s and 1960s. "Real" and
"tonal" [melodic] answers lead naturally to "real"
and "tonal" [melodic] inversions. This is the basic
language in (British) English of counterpoint. The
assumption, before Schoenberg, was that the answers
and inversions would usually be diatonic (preserving
key or mode), and after him, that it might be exact
(preserving the chromatic intervals).

And yes, we also say that chords are in root position,
first inversion or second inversion. Personally, I much
prefer to call these instead the first, second and third
positions as you have done, since doing so means that
"inversion" is entirely in the melodic realm and "position"
is entirely in the harmonic realm. Although I know what
first inversion is, I think of it as a transposition of one
chord member to another position, not as an inversion.
Of course, when scribbling down harmonies, it's often
much quicker to use the figured bass notation for the
different positions, eg of the chord of the subdominant:

..... IV ....... IV 6 ............ IV 6
..................................................... 4

where the default 5s and 3s of first (5 over 3) and
second (6 over 3) positions are not shown; full version:

..... IV 5 ....... IV 6 ............ IV 6
............. 3............... 3 .................... 4

Gene Ward Smith replied to Daniel:

> > Answers are distinguished between real (the intervals are
> > inverted exactly) or tonal (the contour is generally maintained, but
> > intervals are adjusted to accommodate the tonal environment).
>
> Is the vocabulary of "real" and "tonal" common in English? I've heard
> "mirror inversion" for "real".

Yes, Gene, that's been British music practice for
a very long time. I have an English book on music
theory from the 19th Century, and some Australian
ones from mid-20th century, with the same usage.

Jeremy Targett replied:
>
> Gene wrote:
> > --- In tuning@yahoogroups.com, "Jeremy Targett" <jeremy.targett@...>
> > wrote:
> >
> > > > I just finished saying that musicians don't use this definition of
> > > > inversion. Am I wrong?
> > >
> > > Yes, you'd be wrong about that. It's the standard term, from Bach
> > > fugue subjects to Schoenberg's rows.
> >
> > Since Schoenberg's rows involve pitch classes, not pitches, I wonder
> > if you understood what I'm asking. When people talk about Bach
> > inverting a fugue subject, do they always require a strict mirror
> > inversion, sending major to minor and vice versa?
>
> An inversion of a Bach fugue subject is usually diatonic, not exact.
> So some kind of ascending third gets mapped to some kind of descending
> third, major or minor. This doesn't matter in that it is still a
> *reflective* inversion rather than a *permutation*.
>
> In 12-tone practice the reflective inversion is exact, because you are
> operating in a universe of 12 tones rather than 7 diatonic tones. You
> could think of the diatonic inversion in Bach as being exact within
> the realm of the seven diatonic notes. Here is an example, from WTCI:
>
> original subject: Eb-Bb-Cb-Bb-Ab-Gb-Ab-Bb-Eb-Ab-Gb-F-Eb
> inversion, which first appears in the bass about halfway through the
piece:
> Bb-Eb-D-Eb-F-Gb-F-Eb-Bb-Eb-F-Gn-Ab
>
> contour is preserved (up to reflection), i.e. this is a reflection in
> pitch, not just pitch-class space. It is an exact inversion in
> diatonic space, and a less strict inversion in chromatic space. Some
> theorists refer to these as "generically exact" and "specifically
> inexact".
>
> The distinction between "tonal" and "real" is a different one in
> traditional counterpoint theory. A "real" answer to the subject above
> would start:
>
> Bb-F-Gb-F-Eb-Db-Eb-F...
>
> but a "tonal" answer, which Bach uses in this piece and roughly half
> the time in WTC, goes like this:
>
> Bb-Eb-Gb-F-Eb-Db-Eb-F...
>
> i.e. you are permitted to answer "1-5" by "5-1", exchanging 4ths for
> 5ths. Bach does this when he is still in the tonic at the entrance of
> the second voice. If he has modulated to V for the entrance of the 2nd
> voice, he will use a real answer. Here he is still in Eb minor at the
> entrance of the 2nd voice, and Bb-F wouldn't fit.
>
> In Schoenberg's practice, you can be assured of a reflective inversion
> in 12-tone pitch-*class* space, but not in pitch-space--the contour is
> almost never preserved.
>
> Summing up, we have:
>
> inversion can mean reflection or permutation.
>
> Reflection can be in pitch space or pitch-class space.
>
> Reflection can be exact (chromatic), as in Schoenberg, or diatonic, as in
Bach.
>
> In specific baroque practices, a subject can be imitated (whether in
> inversion or "right way up") *tonally* which is even looser than
> diatonic, in that 4ths can replace 5ths and vice-versa, in order to
> preserve key. "Real" vs "tonal" usually refer to a "right-way-up"
> imitation, but could also be used to refer to imitation by inversion
> when diatonic interval-classes are not preserved exactly.

Jeremy,

Thanks for a useful summary and example!

I'd heard all these varieties before except for
"generically exact" and "specifically inexact".
And that was no great loss, as they don't help
much to clarify what's happening.

For my own use, I'd like to slightly revise your
summary as follows:

* Inversion can mean melodic reflection or
chord-tone permutation.

* Reflection can be in pitch space or pitch-class
space.

* Reflection can be exact (chromatic), as in
Schoenberg, or diatonic, as in Bach.

* Diatonic reflection can be real (preserving
diatonic intervals), or tonal (preserving key or
mode). This parallels real and tonal answers, which
are unreflected, transposed repetitions.

Regards,
Yahya