Yahoo Tuning Groups Ultimate Backup tuning Re: as dumb as a unison

Oz wrote:

> I am the daring author of the argument "you cannot invert a unison, since
> it's not an interval".
>
> Inversion is a very simple concept though. It just applies to intervals, not
> unison.

Dear Oz,

one thing that is confusing some people is the multiple meaning of
inversion. The first meaning, e.g. in "inverting a melody", is when
you take a tune and play it "upside down", so ascending major thirds,
for example, become descending major thirds. This kind of inversion
requires an axis of symmetry, when it deals with pitches rather than
pitch-classes. Then you can also invert chords. A C-major triad in
root position, inverts into a root-position F-minor triad when
inverted around its root. Invert it around its fifth instead, and it
becomes a G-minor triad. Invert it around an axis of inversion halfway
between Eb and E, and it becomes a C minor triad.

If you're working in a universe of pitch-_classes_ rather than
pitches, then no literal axis of inversion need appear. C-E-G can
invert into C#-F#-A, for example. In this sense any minor triad, in
any spacing and position, is the inversion of any major triad.

The second meaning of inversion, which you are talking about, requires
more than one voice or instrumental part combined in a structure.
Inverting the structure means changing the order of its voices--it
would be better if it were called "permuting", but it's not. If the
three voices form a structure of C-E-G, then inverting them can lead
to another position of the chord, e.g. E-C-G. Here we have an
inversion, in the second sense of the word, of a C-major triad.

"Inversion at the octave" is an operation that is applied to a
contrapuntal structure of several parts. If there are two parts, call
them A above B (they are both melodic strands in a multipart
composition), and some time later in the composition we get B above
A--i.e. the structure has been permuted--then we have "invertible
counterpoint". We call it "invertible counterpoint at the octave" if
the difference in the intervals of transposition is an octave. For
example, if A stays put and B moves up an octave; or B stays put and A
moves down an octave; or A moves down a 3rd and B up a 6th; or A moves
up a second and B moves up a ninth - etc. etc. etc. All of those are
called "inversion at the octave" in traditional music theory.

Of course, if A and B are ever separated by more than an octave in
their original appearance, then when you perform the operation on the
contrapuntal structure known as "inversion at the octave" then they
won't have actually changed positions! In this sense of inversion, the
"inversion" of a ninth is a second, and the inversion of an octave is
indeed a unison.

Inversion at the octave is not the only possible inversion. There is
also inversion at the twelfth, where the displacements sum to a
twelfth. Zarlino also shows several examples of inversion at the tenth
(this is quite hard to write so both versions come out sounding ok).
Sometimes, in a 3-voice structure, the relative position of only two
of the three voices is affected. A very common thing in 3-voice
renaissance counterpoint is a structure ABC at one point in the piece,
followed by ACB later on, where A is unchanged but B and C have been
inverted at the twelfth.

I don't know if any of that will help, but quite apart from the
mathematical arguments you have been given (which I agree with), there
is a history of usage of the term "inversion" that is much wider than
maybe you knew.

Cordially, in friendship --Jeremy Targett

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Jon Szanto <jszanto@cox.net>

Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
> What really corresponds most
> closely to how musicians use the word?

I use the word as Jeremy has indicated. However, do note that most
*performing* musicians might be expected to occasionally talk about a
chord inversion - certainly early music/baroque people, and to some
extent jazz players; on the other hand, not many will go into analysis
mode on a regular basis and examine a melodic line to see if it had
been inverted. I certainly did a lot of that during theory and
composition days in school, much less so now (well, I may _think_ it,
just not talk about it!).

I haven't, for the life of me, been able to see how this is such a
compelling docu-drama. The earliest study of chord inversion in
Western music theory teachings - which, I am assuming, is where _most_
of our correspondents cut their teeth - explains the inversion of
intervals in a chord to form chord inversions. C major triad in root
position has C on bottom; transpose that C up one octave, leaving the
third on the bottom, is a 1st inversion chord. The only thing that
moved was the C, so it formed an inversion at the octave. Had two
voices been singing a C, and one jumps up an octave, they have gone
from the relationship of a unison to one of an octave. If the octave
is an interval, and all the other notes of the chord are referred to
(as relationships) as intervals, it seems that utilizing the term of
"an interval of a unison" just follows the logic of the analysis.

But what do I know - I'm just a musician! :)

Don't know if this shed any light, but it beats working on my taxes...

Cheers,
Jon

🔗klaus schmirler <KSchmir@online.de>

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Jeremy Targett" <jeremy.targett@...>
> wrote:
> >> one thing that is confusing some people is the multiple meaning of
>> inversion. The first meaning, e.g. in "inverting a melody", is when
>> you take a tune and play it "upside down", so ascending major thirds,
>> for example, become descending major thirds. This kind of inversion
>> requires an axis of symmetry, when it deals with pitches rather than
>> pitch-classes. > > I just finished saying that musicians don't use this definition of
> inversion. Am I wrong?

It is used in Renaissance polyphony in a diatonic context, and of course in serial music (along with transposition and retrograde). Ben Johnston's second string quartet does the same in JI.

> > Then you can also invert chords. A C-major triad in
>> root position, inverts into a root-position F-minor triad when
>> inverted around its root. Invert it around its fifth instead, and it
>> becomes a G-minor triad. Invert it around an axis of inversion halfway
>> between Eb and E, and it becomes a C minor triad.
> > I also finished saying that this notion can be converted to an
> inversive operation on classes of chords. What really corresponds most
> closely to how musicians use the word?

In counterpoint, fugue subjects et. al.? Octave complementation?

klaus

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Jeremy Targett <jeremy.targett@gmail.com>

Gene wrote (quoting me):

> --- In tuning@yahoogroups.com, "Jeremy Targett" <jeremy.targett@...>
> wrote:
>
> > one thing that is confusing some people is the multiple meaning of
> > inversion. The first meaning, e.g. in "inverting a melody", is when
> > you take a tune and play it "upside down", so ascending major thirds,
> > for example, become descending major thirds. This kind of inversion
> > requires an axis of symmetry, when it deals with pitches rather than
> > pitch-classes.
>
> 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.

>
> Then you can also invert chords. A C-major triad in
> > root position, inverts into a root-position F-minor triad when
> > inverted around its root. Invert it around its fifth instead, and it
> > becomes a G-minor triad. Invert it around an axis of inversion halfway
> > between Eb and E, and it becomes a C minor triad.
>
> I also finished saying that this notion can be converted to an
> inversive operation on classes of chords. What really corresponds most
> closely to how musicians use the word?

As I said, in multiple ways, which you sort out through context. Most
people mistakenly associate the notion of the "inversion of an
interval" with the "turning something upside down" idea because it
uses the same name, but in fact the "inversion of an interval" is
really a permutation of its elements, same as the inversion of a chord
from root-position to first inversion. (I prefer the use of "position"
for this sense when not modified by an ordinal e.g. "what position is
the seventh chord in?" "Third inversion".) And both of these have the
same underlying sense as the kind of inversion in invertible
counterpoint--a permutation of the voices.

In "inversion at the octave", you invert (in the sense of permute) the
two-voice structure, but given the specific interval mentioned, the
operation might lead to some intervals in the resulting counterpoint
not exchanging positions, but rather being brought an octave closer
together (a tenth might become a third or vice-versa, if the parts
were crossed to begin with). A tenth isn't what Ozan would consider
"the inversion of a third" either, but it IS a result of the
contrapuntal operation that we call "inversion at the octave".

Same goes for other intervals of inversion: when you're performing
"invertible counterpoint at the twelfth", an octave inverts into a
fifth. So in that case it's true to say "the inversion of an octave is
a fifth".

Best --Jeremy

🔗Jon Szanto <jszanto@cox.net>

🔗klaus schmirler <KSchmir@online.de>

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@...> wrote:
>> Gene Ward Smith wrote:
> >>> I just finished saying that musicians don't use this definition of
>>> inversion. Am I wrong?
>> It is used in Renaissance polyphony in a diatonic context, and of
> course >> in serial music (along with transposition and retrograde). Ben >> Johnston's second string quartet does the same in JI.
> > In a diatonic context, doesn't a major diatonic melody generally
> invert to a major, not a minor, melody? Intervals or melody? If you invert just a melody, I guess its harmonization is a different, independent matter that you can resolve any way you like. As for intervals, an interval class stays the same, but changes direction and often its tone/semitone content. For the sake of keeping the tonality, it is OK to mix different concepts of inversion like the "octave complement".

In the Third Counterpoint of Bach's Art of the Fugue for instance, the head d-a-f-d-c# appears as d-a-c-e-f. The opening fifth is turned into a fourth (its complement), the descending major and minor thirds become ascending minor and major thirds, the half step marks a turning point (besides being diatonical in the inversion) and is unchanged. A more cursory description that would describe a musician's thinking better could simply say that the inversion is also a change from authentic to plagal. Probably nothing that can be done with a mapping.

In serial music, aren't we
> nearly always talking about pitch classes, and not pitches? If so,
> neither would be an example of what I mean. In serial music, a major third stays a major third, it just changes direction (this is what Jeremy described). The axis pitch stays the same, and its tritone maps onto itself. But what do you mean (I'm guessing hard, but it's still only guessing when you introduce them groups with them funny arrows)?

> > I think "mirror inversion" is the term for strict, mathematical
> inversion. The matter can be looked at group theoretically, giving a
> group of order four generated by mirror inversion and major/minor
> exchange.

The problem with these chord inversions is that the final note would also be inverted to be the fifth instead of the root. Common practice has no use for that, as far as I know.

klaus

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:

> > ... you don't use first, second, third inversion, but rather
> > say that an F minor triad in close root position whose fifth is middle
> > Cis an inverted C major triad in close root position whose root is C?
>
> What you have just written makes absolutely no sense to me, and is not
> the way I described it. An inversion of a triad never changes the
> pitches, only the order in which they are stacked. And of primary
> importance is which pitch is on the bottom. Pretty standard stuff.

It *is* standard stuff, but now you contradict what you just got
through saying, which is that you are saying the same thing as Jeremy.

> > It seems to me that when musicians do that, they don't necessarily or
> > generally mean an inversion so strict that major becomes minor. Is
> > that wrong?
>
> Wrong. Chord inversion never changes the notes of the chord, only the
> order of notes from bottom to top.

I claimed that inverting a chord does not normally entail changing
major to minor, and you reply this is wrong because it *never* entails
changing major to minor. It seems to me people aren't even reading my
questions before answering them, and the whole question is more
confusing now than when the explanations began.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗klaus schmirler <KSchmir@online.de>

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@> wrote:
> >
> > > In serial music, aren't we nearly always talking
> > > about pitch classes, and not pitches? If so,
> > > neither would be an example of what I mean.
> >
> > In serial music, a major third stays a major third,
> > it just changes direction (this is what Jeremy described).
>
> Which says, to octave equivalency, that a major third
> has become a minor sixth. In other words, major has
> become minor.
>
> > The axis pitch stays the same, and its tritone maps
> > onto itself. But what do you mean (I'm guessing hard,
> > but it's still only guessing when you introduce them
> > groups with them funny arrows)?
>
> The arrows just say what maps to what--in other words,
> they define a function.
>
> > The problem with these chord inversions is that the
> > final note would also be inverted to be the fifth
> > instead of the root. Common practice has no use for
> > that, as far as I know.
>
> Which agrees with what I was claiming--that "inversion"
> does not ordinarily mean strict mirror inversion.

I haven't been following this thread too closely, but
i do see that one source of confusion is that there are
indeed several different definitions of "inversion" being
invoked.

As was pointed out already, one definition is very loose,
as employed in "chord inversion", which really refers more
to permutation and to the concept of inverting in a strict
sense.

But there are also two versions of the stricter definition:

* one as used to describe Bach fugue subject, which is a
looser one where, say, a major-3rd upward may transform
into a minor-3rd downward, or in fact, it might even become
a perfect-4th downward;

* the other one, as used in the serial method invented
by Schoenberg, is totally strict: a major-3rd upward
always becomes a major-3rd downward.

And yes, since both the Bach and Schoenberg versions
generally invoke octave-equivalence, when looked at
that way, you get the "major turns into minor" effect
you're talking about, but the intervals also become
totally different, i.e., when a major-3rd upward becomes
a major-3rd downward, that's the same as a minor-6th upward.

That's my 2 cents.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

🔗Jeremy Targett <jeremy.targett@gmail.com>

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

🔗Jeremy Targett <jeremy.targett@gmail.com>

By the way, I should add that the Bach fugue I quoted (no.8, book 1 of
WTC) appears in many editions in D# minor rather than Eb minor. I
think this is Bach's original notation--he wrote the prelude in Eb
minor but the fugue in D# minor. I always thought Ebm was a much
friendlier choice, as so many of the chromatic alterations are towards
the sharp side, and I'd rather read naturals in a context of 6 flats,
than double-sharps in a context of 6 sharps!

I just read Jon Sz's message about not recognising the use of
inversion to mean reflection in pitch-class space, i.e. C-E-G can be
inverted into C#-F#-A. It's true that inversion (in the mirror
reflection sense) has been more often talked about as a melodic
phenomenon, in pitch-space, rather than having to do with chords. But
in atonal theory, *anything* can be reflected. Rather than a
pitch-axis of reflection, you talk about an index of reflection, like
this:

0..4..7
1..9..6
-------
1..1..1

(0,4,7 represents C,E,G; 1,9,6 represents C#,A,F#; each column sums to
1 mod 12--so 1 is the index of inversion. There doesn't have to be an
axis of inversion. This is all standard atonal theory, see Rahn,
Straus or Forte for example.)

We write the index of inversion as a subscript to the inversion
operator I, like this: I_1({C,E,G}) = {C#,F#,A}. I_2(CM) would be Gm;
I_0(CM) would be Fm, and so forth.

I_n is equivalent to T_n(I_0). But maybe this stuff should be on tuning math...

Another "by the way": all the harmonic dualists in the 19th century
made a big deal of the fact the minor triad is the inversion of the
major triad. So the idea of inverting a chord by mirroring it dates
from much further back than atonal theory.

The common usage of "inversion" to mean the different positions of a
triad comes from the French "renversement" by the way, which I believe
was introduced by Rameau sometime in the first half of the 1700s. In
French, as in English, the only important tone for determining which
"renversement" a chord is in is the bass tone. So (reading upwards)
E-G-C and E-C-G have always been considered the same "renversement".
With that in mind maybe "permutation" wouldn't have been a good name
after all.

Jeremy

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Jeremy Targett <jeremy.targett@gmail.com>

Gene wrote:

> --- In tuning@yahoogroups.com, "Jeremy Targett" <jeremy.targett@...>
> wrote:
>
> > But
> > in atonal theory, *anything* can be reflected. Rather than a
> > pitch-axis of reflection, you talk about an index of reflection, like
> > this:
> >
> > 0..4..7
> > 1..9..6
> > -------
> > 1..1..1
> >
> > (0,4,7 represents C,E,G; 1,9,6 represents C#,A,F#; each column sums to
> > 1 mod 12--so 1 is the index of inversion. There doesn't have to be an
> > axis of inversion.
>
> This represents x |--> 1-x (mod 12), so the axis of inversion would be
> 1/2, since 1-1/2 = 1/2.

Well, if you did want an axis of inversion you could also choose 13/2
(thirteen halves i.e. 6.5). My point in saying "there doesn't have to
be an axis of inversion" is that there doesn't have to be (and there
usually isn't) a pitch that things are disposed around symmetrically,
unlike the case for reflection in pitch-space.

> The best way to think of that is presumably that the inversion is around the
> 0-1 interval, C to C#.

When the index of inversion is 1, you could think of an axis of
inversion around the interval 0-1, as you mentioned, but also around
the interval 11-2, 6-7, 4-9, or any other dyad whose members sum to 1.
The most general notation used for mirror inversion is I[superscript
m][subscript n], where m and n map into one another. This is redundant
(it gives you 72 notated inversions for 12edo, but we know there are
only 12 distinguishable inversions); but useful if you want to be
sensitive to the conceptual difference between "the inversion that
maps E into A" from "the inversion that maps Bb into Eb". There are a
couple of David Lewin's articles where he writes about when it might
be useful to consider such distinctions.

-Jeremy

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, "Jeremy Targett" <jeremy.targett@...>
wrote:

> Well, if you did want an axis of inversion you could also choose 13/2
> (thirteen halves i.e. 6.5). My point in saying "there doesn't have to
> be an axis of inversion" is that there doesn't have to be (and there
> usually isn't) a pitch that things are disposed around symmetrically,
> unlike the case for reflection in pitch-space.

There is a pitch around which the reflection is an inversion in pitch
space half of the time. This becomes all of the time if you are
willing to extend the pitch space. In the case of pitch classes, it
makes sense to first invert pitches in order to induce an ordering on
pitch classes, in which case that would have an axis, at any rate. The
reflection x |--> 13-x, and the reflecting x |--> 1-x, are different
on pitches and the same on pitch classes, which is why we get these
multiple axes.

> The most general notation used for mirror inversion is I[superscript
> m][subscript n], where m and n map into one another. This is redundant
> (it gives you 72 notated inversions for 12edo, but we know there are
> only 12 distinguishable inversions); but useful if you want to be
> sensitive to the conceptual difference between "the inversion that
> maps E into A" from "the inversion that maps Bb into Eb". There are a
> couple of David Lewin's articles where he writes about when it might
> be useful to consider such distinctions.

It seems overly elaborate. Why not just say I_n, where I_n sends x to
n-x mod 12?

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Oh no you don't. The tonic of melody A seperated from melody B by two
octaves, when inverted by an octave, will meet at the unison, which is not
an inverval at all, but the new tonic.

Is tonic an interval?

SNIP

>
> I haven't, for the life of me, been able to see how this is such a
> compelling docu-drama. The earliest study of chord inversion in
> Western music theory teachings - which, I am assuming, is where _most_
> of our correspondents cut their teeth - explains the inversion of
> intervals in a chord to form chord inversions. C major triad in root
> position has C on bottom; transpose that C up one octave, leaving the
> third on the bottom, is a 1st inversion chord. The only thing that
> moved was the C, so it formed an inversion at the octave. Had two
> voices been singing a C, and one jumps up an octave, they have gone
> from the relationship of a unison to one of an octave. If the octave
> is an interval, and all the other notes of the chord are referred to
> (as relationships) as intervals, it seems that utilizing the term of
> "an interval of a unison" just follows the logic of the analysis.
>
> But what do I know - I'm just a musician! :)
>
> Don't know if this shed any light, but it beats working on my taxes...
>
> Cheers,
> Jon
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Dear Jeremy,

> >
> > Inversion is a very simple concept though. It just applies to intervals,
not
> > unison.
>
> Dear Oz,
>
> one thing that is confusing some people is the multiple meaning of
> inversion. The first meaning, e.g. in "inverting a melody", is when
> you take a tune and play it "upside down", so ascending major thirds,
> for example, become descending major thirds.

Augh. You had to put your finger in it now, didn't you?

This kind of inversion
> requires an axis of symmetry, when it deals with pitches rather than
> pitch-classes.

Diatonical melodic inversion is hardly anything like chord inversions.
Whole-tones and semi-tones may be swapped in an inversion of a theme, so
that the resultant melody is harmonically agreeable.

Then you can also invert chords. A C-major triad in
> root position, inverts into a root-position F-minor triad when
> inverted around its root.

What? Which base pitch did you transpose where to come to F-minor? The only
inversions of the C-major triad are the ones I have already given.

Do you mean inverting by a seventh down?

G
E E
C C C
A A
F

But this is modulation, not inversion, and it does not yield a minor triad
on F.

Oo, you mean a diatonical mirror operation of a theme:

C D E F G
C Bb Ab G F

You are simply changing the direction of intervals with C as 1/1.

Invert it around its fifth instead, and it
> becomes a G-minor triad.

You mean:

G A B C D
G F Eb D C

Invert it around an axis of inversion halfway
> between Eb and E, and it becomes a C minor triad.
>

You mean:

E F G
Eb D C

> If you're working in a universe of pitch-_classes_ rather than
> pitches, then no literal axis of inversion need appear. C-E-G can
> invert into C#-F#-A, for example. In this sense any minor triad, in
> any spacing and position, is the inversion of any major triad.
>

This only serves to complicate matters even more. One must not confuse
interval inversions with chord inversions.

And I cannot imagine an inversion without an axis of symmetry.

> The second meaning of inversion, which you are talking about, requires
> more than one voice or instrumental part combined in a structure.
> Inverting the structure means changing the order of its voices--it
> would be better if it were called "permuting", but it's not.

Permuting... That's a great word. We must say chord permutations then. Let's
establish this as a rule in this list.

If the
> three voices form a structure of C-E-G, then inverting them can lead
> to another position of the chord, e.g. E-C-G. Here we have an
> inversion, in the second sense of the word, of a C-major triad.
>
> "Inversion at the octave" is an operation that is applied to a
> contrapuntal structure of several parts.

That's invertible counterpoint, double, triple or quadruple, etc...

If there are two parts, call
> them A above B (they are both melodic strands in a multipart
> composition), and some time later in the composition we get B above
> A--i.e. the structure has been permuted--then we have "invertible
> counterpoint".

But you used permutation here again, as in, changing the order of. The order
of pitches in a melody have not been changed, so this should be called
inversion proper.

We call it "invertible counterpoint at the octave" if
> the difference in the intervals of transposition is an octave. For
> example, if A stays put and B moves up an octave; or B stays put and A
> moves down an octave; or A moves down a 3rd and B up a 6th; or A moves
> up a second and B moves up a ninth - etc. etc. etc. All of those are
> called "inversion at the octave" in traditional music theory.
>

Agreed.

> Of course, if A and B are ever separated by more than an octave in
> their original appearance, then when you perform the operation on the
> contrapuntal structure known as "inversion at the octave" then they
> won't have actually changed positions! In this sense of inversion, the
> "inversion" of a ninth is a second, and the inversion of an octave is
> indeed a unison.

As long as we reserve the word permutation for what was known previously as
`chord inversions`. Now the matter is resolved! However, say instead
inversion by ..., not inversion of ...

>
> Inversion at the octave is not the only possible inversion. There is
> also inversion at the twelfth, where the displacements sum to a
> twelfth. Zarlino also shows several examples of inversion at the tenth
> (this is quite hard to write so both versions come out sounding ok).
> Sometimes, in a 3-voice structure, the relative position of only two
> of the three voices is affected. A very common thing in 3-voice
> renaissance counterpoint is a structure ABC at one point in the piece,
> followed by ACB later on, where A is unchanged but B and C have been
> inverted at the twelfth.

This was most refreshing. It's been quite a while since I pondered upon
these.

>
> I don't know if any of that will help, but quite apart from the
> mathematical arguments you have been given (which I agree with), there
> is a history of usage of the term "inversion" that is much wider than
> maybe you knew.

I simply ignored this part in my obsession with chord inversions. You have
been most helpful.

>
> Cordially, in friendship --Jeremy Targett
>
>

Gratefully,
Oz.