Theory : what exactly is "Invertible Counterpoint" ?

[m1469]

Definition from Harvard Concise :

"Counterpoint so designed that it can be performed with the lower part transposed (usually by an octave) to lie above the upper part or with the upper part transposed to lie below the lower part."

I don't quite understand what makes counterpoint invertible and what makes it not invertible.  For example, couldn't a person take any of the voices of a fugue or invention and play them in a different octave ?  I am confused.  I have been thinking about this for a few years now and I just don't understand still exactly what it is.  I must just be missing something simple. 

Anybody willing to help out ?

Thanks a lot,
m1469

[rhapsody in orange]

Hmm here is what I have:

To retain the original form of the melodic line exactly, throughout such transposition (melodies first heard in the upper voice appear later in the lower voices), the parts have to be written in invertible counterpoint.
The amount of transposition is equal to the sum of the intervals of transposition in the opposite directions of both voices. (therefore you'd have inversion at the octave, 12th, 15th etc).
Take for example the inversion at the octave. For it to be 'invertible', the voices must not be more than an octave apart, otherwise the octave transposition will not be sufficient to place the lower part above the other.

Hope that helps..

[m1469]

hmmmm... 

okay, if I take Bach's invention no 1 in C major and play the soprano voice with my LH and the bass voice with my RH, is that invertible counterpoint ? -I kinda don't think so, but I just don't understand

I think my basic concept is just plain wrong.  -just be blunt with me

Thanks for your response rhapsody in orange


m1469

[cadenz]

what i think rhapsody in orange is saying is that.... if for instance the lower part is like if the lower part is about two octaves below the upper part for instance, raising it by one octave, it will still be below the upper part and the harmony of the piece wouldn't have been dramatically altered. but you probably understood that bit.
but my take on the idea of counter point being invertable or not is a little deeper is whether the intervals and harmonies still work once the upper part has been placed below the lower part or vice versa. for instance if it creates problems with discords or destroys the point of the harmony, then it wouldn't be invertable. and thus is a little more open to interpretation.

[ted]

I wouldn't have a clue either, m1469. Tell you what though, there was a good trick I used to do with files of Bach and other contrapuntal stuff on the Amiga years ago. Write a programme to operate on the file and invert all the notes mirror fashion about some central note. I preferred many of the inverted pieces to their originals ! Inversion preserves degree of dissonance and location within a key (usually not the same key as the original of course) and if the piece is roughly vertically symmetrical - most Bach WTC pieces are more or less - the results sound pretty good.

[Floristan]

Most of Invention #9 in f minor fits this description, whereas Invention #1 doesn't.  In #9 this is literally what Bach does for most of the piece.  (I have never heard this term before, BTW.)

[Bob]

Counterpoint has rules for how two lines has to move, mainly the intervals.

You can invert two lines by placing the lower one above the upper one -- that's inversion, "invertible counterpoint."

It doesn't work for all counterpoint.  The rules aren't always followed when you invert the lines -- some counterpoint will work, some won't.

If you invert a third and get the sixth, there's probably no problem.  If you have a fifth and invert it to a fourth, then you might have problems and it might not work.


Dang... I'm starting to forget this stuff.

I think there's also double interverted counterpoint -- two counterpoints that can be inverted.   AB with CD inverted to CD with AB.  (What was this "Bach" guy smoking?)

[m1469]

Thank you for all of the responses.  I am sorry to say that I just don't get it still  .  What am I missing ?

In #9 this is literally what Bach does for most of the piece.

I don't understand what "this" is.  I am looking at invention no 9 right this minute and I just don't understand.

Is it simply placing one line on top of the other at the octave ?  Why can't this be done with any counterpoint ?

Is this one of those things that defies an exact and clear definition ?  Ambiguous ?

argh... 

I wouldn't have a clue either, m1469. Tell you what though, there was a good trick I used to do with files of Bach and other contrapuntal stuff on the Amiga years ago. Write a programme to operate on the file and invert all the notes mirror fashion about some central note. I preferred many of the inverted pieces to their originals ! Inversion preserves degree of dissonance and location within a key (usually not the same key as the original of course) and if the piece is roughly vertically symmetrical - most Bach WTC pieces are more or less - the results sound pretty good.

Do you mean, if I were to have two notes, "middle C" and "G 3", I could use "C" as my central note and then move the G up an octave changing the interval in relation to the central note from a P4 to a P5 ?  So, would the central note be necessary as a reference in order for the inversion to exist (I mean, I guess it would but, I am still trying to figure out how this all fits together for the "invertible counterpoint" situation)?

Thanks everyone,
m1469

[ted]

My comment was just an interesting aside which has probably caused needless confusion. I just meant inversion in the crude arithmetical sense. If everything were inverted about, say, middle C then a note x semitones above middle C would become the note x semitones below middle C. Es become Abs , Fs become Gs , F#s and Cs would become themselves etc.

This little trick probably has nothing to do with "invertible counterpoint" in the academic sense. I really do not know what the expression means and should no doubt have refrained from complicating matters.

 

[crimper]

Google search revealed the following:

Illustrated confusion is the best!

https://www.teoria.com/articulos/analysis/kdf/03-eng.htm - "The problematic interval in the inversion at the 12th is the 6th interval that becomes a 7th. In other words, a consonant interval (an a very useful one in counterpoint!) becomes a dissonance."

Inversion at the 12th in fugue BWV 885(step-by-step):
https://www.teoria.com/articulos/analysis/BWV885/01-eng.htm

Other pages on that site require a plugin.

[m1469]

My comment was just an interesting aside which has probably caused needless confusion. I just meant inversion in the crude arithmetical sense. If everything were inverted about, say, middle C then a note x semitones above middle C would become the note x semitones below middle C. Es become Abs , Fs become Gs , F#s and Cs would become themselves etc.

This little trick probably has nothing to do with "invertible counterpoint" in the academic sense. I really do not know what the expression means and should no doubt have refrained from complicating matters.

Ted, I don't think your response added confusion.   I was thinking though, that maybe I could understand my question better by understanding your comment.  They did seem related to me.  PLUS, I am always happy to hear from you and do find your perspective to be valuable .

Google search revealed the following:

Illustrated confusion is the best!

https://www.teoria.com/articulos/analysis/kdf/03-eng.htm - "The problematic interval in the inversion at the 12th is the 6th interval that becomes a 7th. In other words, a consonant interval (an a very useful one in counterpoint!) becomes a dissonance."

Inversion at the 12th in fugue BWV 885(step-by-step):
https://www.teoria.com/articulos/analysis/BWV885/01-eng.htm

Other pages on that site require a plugin.

Okay, crimper, I will check these out.  I did google before I asked, but I just did not understand still.  Perhaps these sites will be different.  I had also asked a theory professor a while back which ended in him telling me to try writing some...  :- which did not explain to me what it is and if I have written any, it is purely accidental.  But I will read these.

Thanks,
m1469

[abell88]

From Eric Rollinson's Free and Double Counterpoint:

"When counterpoint is written so that any part will make a good Bass for the other parts, it is said to be invertible.

Quadruple Counterpoint. Any one of the four parts will serve as Bass for the other three above it: the last movement of Mozart's symphony in C, commonly called the Jupiter symphony, is largely constructed of quadruple counterpoint.

Triple Counterpoint is in three voices, and any one of the three voices will serve as Bass for the other two.. .

Double Counterpoint is in two voices. It is classifed as being "At the 15th", "At the 12th", or at any other interval. When the upper part is trasposed down, or the lower part moved up the interval named, the parts will be inverted and the Bass will be good. "

The Quality of the Bass Part

From the above it is clear that all voices of Invertible Counterpoint must be written in such a way as to make good Bass parts....What special qualities are necessary to make a good Bass part? The answer lies in the treatment of Second Inversions of Triads."

 He goes on to remind the student of the acceptable use of 6/4 chords and how they may be acceptably used in invertible counterpoint..."There is no objection whatever to the use of 6/4 chords in Invertible Counterpoint providing the chord can be classified as Cadential 6/4, Appoggiatura 6/4, Auxiliary (or Pedal) 6/4, Passing 6/4, or Arpeggio 6/4 -- but any chord which cannot be classifed as one of those is classifed as "Bad".

[Floristan]

Sorry, m1469.  I just confused the issue for you.  In the 9th Invention, the two voices in the first four bars are strictly inverted in the second four bars.  From there on Bach diverges from strict inversion.  The theme goes from f minor to c minor in the top voice, then is developed, then is restated fully in the left hand in c minor, then there's more development and modulation back to f minor.  I was just thinking of the first eight bars where the inversion is literal.  You don't see that literal inversion in the 1st Invention.

[m1469]

Sorry, m1469.  I just confused the issue for you.  In the 9th Invention, the two voices in the first four bars are strictly inverted in the second four bars.  From there on Bach diverges from strict inversion.  The theme goes from f minor to c minor in the top voice, then is developed, then is restated fully in the left hand in c minor, then there's more development and modulation back to f minor.  I was just thinking of the first eight bars where the inversion is literal.  You don't see that literal inversion in the 1st Invention.

Actually, I think I may slowly be starting to understand now.  OOOOHHHHH, I think I get it ... LOL I was really confused.  Okay, I am going to need to explore this a little more.  I am not sure to find any other examples in the 2-part inventions, though I think there is supposed to be.  Do you know of anymore ?  What about invention no 12, first 6 bars ?

Thank you all for your help(s),
m1469