Bach, Bagels, and Bilateralism

[Kristofer Emerig]

The other day, while chewing my morning bagel and listening to some JS Bach, the Crab Canon, no less, my thoughts drifted from my toroidal breakfast morsel to the lemniscate of Bernoulli specifically and the concept of symmetries nested within symmetric isometries in general, and its ramifications within the discipline of counterpoint.

This was not a novel chain of association for me, as the thought has been crowding my mind with ever growing urgency and frequency for the last two years, and served as a central principle of design and impetus in much of the music I've produced in that span. Nick Capocci recently remarked to me, and I quote without permission,

"It [symmetry] is a powerful intellectual and philosophical concept, and, naturally, finds a ready medium in free counterpoint".

I quote the above because I can not find more apt words than Nick's.

So what of it? Forays into the exploration of symmetric form within counterpoint are nothing new, and imitative counterpoint could be viewed as fundamentally grounded in that pursuit. The methods of textural inversion and imitation (transformational symmetry, mathematically speaking) have been well established and documented for four centuries.

There is nevertheless a glaring hole, both theoretical and practical, where melodic inversion and retrogradation is concerned, particularly of entire polyphonic textures. The smattering of extant works left by Bach in the mirror fugues of Kunst der Fuge and the Crab Cannon are hardly a large enough body of work from which to derive a rigorous contrapuntal methodology for mirror techniques.

Until I experienced a recent epiphany, just a year or two old now, it was apparent to me that such mirror forms in counterpoint could only be improvised haphazardly case by case through a blend of intuition and trial and error alone. Viewed in light of the realization that music, as an abstract contrapuntal construct at least, is an isometry of a frequency axis (v) against a time axis (h), it became apparent to me that it is indeed possible to derive from the conventional (or any) protocols of voice leading a consequent set of protocols for melodic invertibility and retrogradibility.

The epiphanic realization was in understanding that melodic invertibility and retrogradability are integrally related, in fact, for phrases which are mutually symmetric across (v) and (h), their results are precisely identical. Moreover, simultaneous melodic inversion and retrogradation of such a phrase is identical to the original phrase itself. And so a methodical analysis of any sample of music with respect to melodic invertibility and retrogradability must begin with parsing that phrase into the largest elements which are symmetric to either or both axes. The inherent guarantee in this approach is that the smallest of possible elements would be merely two consecutive notes, a configuration which will invariably be mutually symmetric to both (v) and (h).

Well, I'm still assembling the specific details of my methodology for writing melodically invertible and retrogradable counterpoint, a system which might be viewed as an extension and modification of the principles outlined by Fux, with focused scrutiny on considerations of symmetry, yet there's a great deal to be learned by practice and sometimes failure. I hope the Mass Mysteria to be a proving ground for the application of my theoretical observations.

http://www.box.net/shared/j3jt4l07b5

http://www.box.net/shared/msjy0u95nl

Edit 9/30/10: Still working on the mass (in between four other projects) which despite being a sincerely devotional and religious effort, shares also the purpose of a proving ground for the aforementioned techniques. Upoaded to my page are two motets "In Diebus Vocis Septimi Angeli", I and II, which are strict vertical mirror images of eachother. I suppose it doesn't really matter which is designated as "Inversus" and "Rectus".

[Kristofer Emerig]

To answer your question directly Fredrick, yes, but it's not a case of making those necessary compromises of aesthetic and form in isolated cases, but in a continuous and discreet manner. The ideal is to discover with unflinching stubbornness those rare solutions for which strict forms of symmetry (cf "dictates of numbers") satisfies the aesthetic expectations sublimely, with the fewest and least significant sacrifices. In drafting even the simplest fugue or canon, one will undoubtedly stumble upon many moments where an alteration of the strict subject will yield more beautiful musical expressions. The key is balance, as many of the finest examples show an occasional deviation for the purpose of making the music work, but only sparingly and in good judgment.

To answer indirectly, I don't necessarily see symmetry as equatable to mathematics (depending on one's definition of mathematics) to which music is intrusively subjected. I see symmetry rather as a powerful archetypal concept with which music, mathematics, and all disciplines of thought are occasionally imbued. In fact, the entire process alluded to here can be viewed entirely in terms of musical, rather than mathematical terms.

[Fredrick zinos]

Math and Music are cousins, but not exactly bedfellows.

Well said: and to carry the anaology one step further, there is plenty of evidence that when first cousins ARE bedfellows the offspring, with alarming regularity, are idiots. So too with allowing the regulations of arithmetic to dictate, wihtout exception, the notes on the page. Form and structure are what separate 'composition" from notational drooling, but is there not a point at which the dictates of numbers have to be broken to provide a more satisfying musical experince?

[Simon Godden]

Sylvester, are you telling me that Marcel Dupré in his "Prelude and Fugue in G minor" - op. 7 no. 3 pulled off a situation whereby he precisely inverted and retrograded a subject into PURE symmetry and was able to pass it off as a listenable piece of music (when I say 'listenable', I mean in the accessible, generally consonant, tonal sense).

If so, I'll need to listen to it. Because to me, it sounds like an immensely difficult exercise. Even Bach had escape clauses such as false subject entries, rests, bridge passages etc.

[Sylvester Wager]

"...methodology for writing melodically invertible and retrogradable counterpoint, a system which might be viewed as an extension and modification of the principles outlined by Fux"
First and foremost! Lose the love of Fux! Fux was a fuddy-duddy. What great works did he leave us?

Bach knew how to do it, but his examples of inversion/retrograde are well-pondered ones. The trick is always this: the vertical and the horizontal must meet to make some kind of sense. And the problem for 100 years is that most people want to take the easy way out and use rhythm as the cohesive tool, instead of harmony.

It is posssible, very possible to pull off a fugue, consider Marcel Dupré (1886-1971) in his
Prelude and Fugue in G minor op. 7 no. 3.


This man pulled it off. It is a rare thing to manage. Math and Music are cousins, but not exactly bedfellows.
You need a flexible tune. That is the thing.

[Kristofer Emerig]

True, but as I've indicated before, it (unresolved dissonance) works for many reasons in your music which are far out of the scope to enumerate here. Trying to tack that onto my music without the proper harmonic preparation would give rise to some truly horrendous sounds. Wait a minute, Crucifixus?

[Simon Godden]

I thoroughly agree with you Kris. It was just an observation. However, I will say this. Schoenberg's system, although sounding like a string orchestra trying to tune up with no consideration for the neighbours, with a wind section trying to blow the corks out of there instruments, did provide a platform for 20th century composers to extend, such as the incidental music to horror movies etc, plus it's combination with tonality does provide more freedom for composers of today, such as me. Unresolved dissonance if disciplined can be emotionally mind blowing.

[Kristofer Emerig]

Precisely. Schoenberg is symmetry in a symmetric system which eschews the asymmetric considerations of tonal language. It just doesn't seem that intriguing to me when the result is not expected to sound like music (subjective, I know). It frankly strikes me as some form of pointless mathematical masturbation. The invertible fugues from Kunst, now that impresses me, but wouldn't so much if they sounded like crap in inverted form. Not much of a trick in the absence of aesthetic constraints, is it? I'm much more intrigued by the pursuit of symmetry bound by an asymmetric tonal system, ie, sounds like music (subjective, I know).

[Simon Godden]

Can you send them over as .mus files. Then I can read them.

Incidentally, I am not hinting that your method will sound anything like Schoenberg's. I'm just saying that your method of precise symmetry is similar. Obviously the differences such as tone equality and absence of key is bound to make the music sound different.

[Kristofer Emerig]

Simon, I can provide you with some MP3 excerpts from the 8 part Kyrie if you wish to listen to them (Pacem was built on the same principles, but much more conservative). They'd mocked up poorly in Finale, but it'll give you some idea of my present direction. Besides, I'd like to hear your opinion. I'll stick my neck out and predict that you'll find the material to sound nothing like Schoenberg, and not too much like Fux. Incidentally, the concept of full resolution of dissonance by step is still preserved. This is less about redefining conventional contrapuntal methods and more about extrapolating from those principles similar ones more narrowly focused upon invertibility (melodic) and retrogradability. The crucial realization that makes the application of such a set of principles feasible is that the phrase must be spliced into the largest possible fragments which are, as isolated objects, symmetric to one or both axes.

[Simon Godden]

Whilst most composers compose from the heart, as they should, it is never too late to use mathematically inspired guidelines to provide a basis upon which to create good (as opposed to 'nice') music. What you have done here Kris (whether intentionally or unintentionally) is mould the rules of both Fux and Schoenberg, and I commend you for it. However, both these composers, especially the latter, had an advantage in that their applications were a lot easier. Fux's because he provided an escape clause through the use of 'false subject entries' and 'bridge passages', and Schoenberg's because his contemporaries were already exploring the avenues of unresolved dissonance.

All I can say is that you have one hell of a wrestling match in front of you. This is not a put down of course by any means. Indeed, I sincerely look forward to your first successful composition using this method.