Well actually, when I hear Klavierstück XIV, I tend to hear all the formulas at different points - but when it appears in MONTAG it is, for me, Michael's formula that dominates that scene because of the girls' choir singing it so markedly. So I don't quite see there the associations with Luzifer as strongly as you do, although they are obviously there. Admittedly the piano is Luzifer's instrument in the first scene of SAMSTAG, but do you also associate it with him him EXAMEN in DONNERSTAG? Maybe this is another example of the multiple interpretations being possible of which you spoke earlier in another context.

As for your point about the presence of a character's formula not necessarily meaning that we should construe the character being present also - I do take that point although, on the other hand, it doesn't strike me as that incredible to think that perhaps all three characters are indeed present for much of the time at some level. They are archetypal characters, after all, and are arguably all just different facets of the one being - but that is, I guess, getting into a psychoanalytical interpretation (which is certainly something I am interested in). But I do think, if we see the characters as symbols beyond their various roles in the narrative, then the concept of their presence in places where they are not physically on stage is much less problematic.

The superformula clearly shows that all three characters deserve the occasional rest. Stockhausen knew how tedious it would be to have all three present at all times. His whole concept of Licht was to examine the three of them in different combinations. One could argue that the impact of each character's personality can be traced in every scene, but that is quite different than a concrete appearance.

The argument that Michael, Eve, and Lucifer are three sides of the same personality is certainly fruitful, but only to a point. Reading all of Licht from that perspective is a little artificial. It starts to get unwieldy. So too with the idea that an appearance by a formula = an appearance by a character. It's an interpretation that bears fruit, but not for long.

Speaking of bearing fruit...

In Befruchtung, when the girls playfully sing Michael's formula, have they suddenly transformed into Michael? Has Michael "appeared" in the scene, simply because his formula sounds? That strikes me as an incoherent reading of the scene and of Stockhausen's theater. "Formula" has a magical connotation, as Stockhausen defines it, but it's not some kind of conjurer's trick which invariably summons the characters to the stage every time it sounds.

The better question to ask is why the girls are singing Michael's formula at that moment. Are they hoping that Michael's spirit will inseminate Eve, despite all evidence to the contrary? If further proof were needed that Lucifer is the dominant force here, look at the pitch center that the girls are using: Not Michael's D, or Eve's C, but Lucifer's G.

As you know, I argue that there are degrees of appearances in Licht. Some are quite clear and others are so ambiguous that even Stockhausen was confused by them. I'm not totally opposed to the idea that the formula can sometimes conjure one of the protagonists into a scene, but in this case, Michael's formula seems to be a bit of wishful thinking on the girls' part, not an actual appearance by the archangel.

There are few hard and fast rules in Licht. For example, I would never argue that the piano always signifies Lucifer. But when a piano flies (which it does not in Examen), it's hard to dismiss the association with Lucifer, whose piano floats in the clouds in Samstag. We cannot be meant to ignore the connection between the flying piano and the budgerigar in the previous Act and the piano-playing budgerigar who shows up to inseminate Eve during Befruchtung.

Some interesting food for thought there Joe, as always! I wouldn't quite agree, though, that linking of the formulas to appearances is not fruitful for long: I'm actually finding it very fruitful, although it's important to be clear what I mean by "characters" and "appearance". I mean both of these in what I suspect is a rather more symbolic sense than you might be using them and was really, after all, just responding to your initial comment that Michael's formula is quite active in the Montag limb and yet he, as you put it, does not appear in the opera. I think he does, just in a different and less literal way. I am finding it interesting, whenever a portion of a character's formula appears in the music, and particularly when it appears prominently, to think what sort of presence, of what aspects of that character's symbolism, we can decipher in that.

While I do not for a moment think that the girls are transformed into Michael when they sing his formula in Befruchtung, nor am I too convinced by your suggestion that it's indicative of some "wishful thinking" on their part. I'm not sure that any of it is, or can be, as concrete or as literally defined as that. It is perhaps a little analogous to the "presence" of Wotan in Götterdämmerung ß he never actually appears on stage at all in the opera (other than at the very end where we see him engulfed in the flames of Valhalla), but his presence is often very strong in the music, like in the powerful interlude between the Gibichung scene and the Waltraute scene in Act 1, where Wotan's spear motive blasts so ominously and apocalyptically through the orchestra. There is no "conjurer's trick" summonsing him to the stage, but he is there, somewhere more ineffable, in the music. That's what I sense here in Befruchtung too. It's a presence, if not an appearance.

And I do agree with you that the superformula clearly shows that all three characters deserve the occasional rest. That's why there are rests! I'm interested in what's happening when there are not rests, and yet when their presence is less straightforward. Sometimes, of course, it might not mean anything much. But it's always worth thinking, and wondering and seeing what can be found beneath the surface.

But, like you say, there are no hard and fast rules.

It's great that you are thinking through definitions, because that's where the defense of your argument has to begin. You haven't defined for us what you mean by an appearance, which is fine (it's just a message board, after all!). However, it's hard to argue with a privately held notion of what constitutes an appearance.

I've spent a lot of time on this particular issue, and I don't come by the "incoherent" label lightly. When I was devising my representation scale, I originally started with 7 steps because of Stockhausen's own sketches. On the low end of such a scale would be the appearance of the formula or an instrument like the trumpet. So, okay, there's a trumpet in Orchestra Finalists, that's a very low grade representation of Michael, because the trumpet is his instrument. Well then, what do you do with the 2 trumpets onstage in Michaels Reise? Are the clowns in that scene making fun of themselves, since they must also be Eve because they play her instruments?

The same kind of gibberish results when one holds that every appearance of the formula constitutes an appearance of a character, or is symbolic of that character. It simply does not. Stockhausen used the formulas to generate material. The mathematical, not the magical, definition is the more important one. Here again, Jerry's work is really helpful, in particular his dissertation. It really clarifies how Stockhausen manipulated material. Hermann Conen's book is particularly helpful on the technique of formula composition.

A friend had once gone through several Licht scores and highlighted every fragment of the formulas he could find in blue, red, and green. They were beautiful. They looked a lot like the Helicopter String Quartet score, but he had gained no real insight about the narrative from the work he'd done. He'd learned a lot about how Stockhausen constructed those scores, but nada about the dramatic significance of the music.

I'm not saying that there's none to be had in tracing the formulas, of course. I'm just saying that the formulas have a dual function, one of which is purely musical. A common mistake in post-tonal analysis is to assign too much weight to what are unavoidable coincidences. There are only so many ways to structure a tetrachord or a hexachord. When you get down to trichords, you're really in the soup. Seeing dramatic symbolism in every iteration of a formula in Licht strikes me as the same type of mistake.

Taking your operational premise of examining what sort of presence or what aspects of a character's symbolism is contained in every appearance of their formula, I think you'll find as you progress that this method is indeed quite unproductive after a while. Take for instance the bass solo in Scents-Signs. Most of his material is Eve's major third. Now, if we are operating with your hypothesis, we must look for how Eve is embodied in this interval, though it is sung by the bass. How does Eve's symbolic nature come through in the bass solo, since he is singing her primary interval?

Well, clearly, it doesn't. The bass is singing about death, anger, and prison. The only way Eve figures into his solo is his sneering contempt for her. I'd imagine you would say, "Well, that's not nothing!" True enough, but it's a tangential point that oversells the symbolism of the formulas.

As always, the best guide is Stockhausen himself, who freely admitted that the presence of a formula did not necessarily have any real meaning. When discussing Sonntag, he said that he did his best to keep Lucifer's formula out of the opera, but that after spending 26 years with it, the formula was so deeply embedded in him that it showed up in the music anyway. I know that subconscious activity in Stockhausen's mind interests you, which is to your credit. It will be interesting to see what you do with such a rapacious appetite for context.

Backing up a bit to return to the subject of this thread: Joe has questioned whether Lucifer's D must come at the end (and in the same breath has questioned whether my analysis of the nuclear formula is "unassailable"). Naturally, I am obligated to respond. No, there is no reason why Lucifer's formula should have ended with a D. That D could have come anywhere (assuming a twelve-tone row is somehow compulsory). What matters is that Lucifer has only eleven tones, and one missing is a D (pace Joe, who pretends, presumably for comic effect, that the pitches of the ornamental additions in the fully elaborated superformula work on the same level as the core tones), whereas Michael's "extra" tone is the D that completes the third aggregate of the 36 total core pitches.

If you want to "assail" elements of my article, Joe, you don't have to work so hard, not even when restricting your critical gaze to what I say about the core pitches. Mustafa Bor, at the time a graduate student at the University of British Columbia, took me to task for that table showing prominent trichords in the array of nuclear pitches. This was in a paper titled "Pitch and Duration Contours in Karlheinz Stockhausen's Super-Formula for Licht", presented at a conference in 2007 and published the next year as chapter 6 of Musical Currents from the Left Coast, edited by Jack Boss and Bruce Quaglia (Newcastle: Cambridge Scholars Publishing, 2008), pages 129–54.

What Mr Bor objected to in my analysis was what he perceived as the arbitrary forming of set types from "distant pcs, freely intermingled among the three protagonists". He offered an alternative analysis, "which avoids voice-crossings between the characters and focuses on adjacent pc segments" (p. 133). Fair enough, though his purpose and mine were quite different. Mr Bor appears to have been seeking a more rigorous, objective analysis of the pitch relationships, whereas my purpose was to illustrate many relationships which I had already found by analysis of the scores in Licht—by ear as much as by eye. Admittedly, at the time I wrote my article, I had only three—more like two-and-a-half—of the operas to actually work from (Donnerstag, Samstag, and portions of Montag: I had only heard Klavierstück XIV and Evas Lied in concert performances in Montpellier, and none of the Montag scores were yet published). I nevertheless do find Mr Bor's insistence on ignoring the pitch relationships between the three lines of the superformula rather odd, since after all those three lines are often made to sound together in performance, and the principle of three consecutive aggregates (together with three simultaneous horizontal ones) suggests that Stockhausen was not thinking solely in linear, melodic terms.

Nonetheless, my method of analysis is "assailable" in that completely different set types can be arbitrarily found, simply by drawing lines around different groups of notes. Why did I not, for example, consider the initial simultaneity of G/C/D? The answer is simple: because I did not find the 0,2,5 trichord type particularly prominent in the actual compositions of Licht (it does occur, of course), by comparison to the ones featuring interval-class 1 (the semitone), and major and minor thirds. Stockhausen was never content to let the music compose itself, as a blind consequence of some number chart or other. He always made choices based on what he could hear, and this meant bending the formulas to his compositional purposes. Now that we have all of the opera cycle to examine, my opinion has not changed in its essentials. I might now expand my view to take into account Stockhausen's use of tetrachords and pentachords, though of course trying to cram all of this into that analytical diagram would complicate it beyond any usefulness.

I haven't really jumped in on the symbolism and character appearances discussion here since I'm happy to wait to read Ian's thesis to read about that aspect of LICHT (also just not my current focus). However, Jerry, I was hoping you could weigh in on the original question of the "diamond" shaped notes and squiggly lines aspect of the Nuclear tones sketch. Joe touched on it a bit and I was wondering about your thoughts.

Regarding the (un)assailability of your articles, Jerry, I think there's too much assailing going in in academia, everybody's out to find a raison d'etre for their own writing I guess. I have to admit tho, that the Bor-style analysis is closer to my own initial take, and when Stockhausen points to a track and says "here" is the M/E/L (nuclear) formula then it does lead one to read it in that linear fashion. But after reading your texts and correspondence I've definitely been more attuned to "hearing" things as vertical structures. After all, when the pieces were first premiered, no one had a score in front of them with the formulas highlighted - the tones naturally come at us as multi-layered polyphony and harmony. As you said, those lines sound together in performance, and that IS an unassailable fact.

I suppose I could skin a metaphor by saying that knowing the formulas in their linear form is like learning the dialect of LICHT, and knowing them in their vertical form is like learning grammar structure?

Actually, Joe, I think we are more on the same page on this issue than not. First, I was using "appearance" very loosely - I haven't even begun to describe these issues in my actual research yet. I probably mean it in the sense of "having some sort of presence" - but even that is more a vague description than a definition. And I certainly don't mean to go through the whole of Licht and identify every segment of every appearance of every bit of every formula - it's really just the more prominent or significant instances, like the one Befruchtung that we've been discussing, where an immediate reference to a character visible on stage is not there, but where its presence is prominent enough to consider what it might signify. But I totally agree that it might often signify something much more mathematical than anything else.

That said, I do imagine we might have some differing views (at least when mine are more developed than they currently are) about what is significant and what is tangential. Remember, I am tending towards an approach to Licht that is rather different to yours (I'm actually leaning more towards Lacanian psychoanalysis now rather than to Barthes, but I'm still exploring different approaches to see which ones seem to offer the richest possibilities). One of the things I love about trying out new ore different theoretical frameworks for analysis is that they have the potential to yield such different results in terms of what matters and what is tangential. Sorry to keep returning to Wagner - but I think, for example, of George Bernard Shaw's Marxist analysis of the Ring and Robert Donington's Jungian take: both end up focussing on and emphasising details that the other sees as incidental, and yet both are, I think, valid analyses. As long as you're transparent about the approach you're taking, and, importantly, as long as it's one that can yield coherent and consistent results, I think it can have potential. But I don't for a moment pretend that I am at the point yet - but I'm enjoying exploring the potentials, and don't want to dismiss them as fruitless too quickly. But I do think Lacan is looking very promising as a new lens to consider.

Just another point - I'm actually less interested in Stockhausen's unconscious than I was initially. I am now a little more inclined to accept that it would have been active and that that therefore opens up opportunities to explore connections and meanings that might not be explicit or of which he himself was unaware. But I certainly agree that that doesn't mean it's all just open slather. It has to make sense!

But thanks, as always, for making me think and for helping to keep an eye on what can be a tendency to over-zealousness.

We've a surfeit of assailants! I think it was perfectly clear from my invocation of that word that I meant no harm, especially when it comes to work that has been so helpful. It was a reference to the fortress of presumption. If one is so inclined, an assault can be mounted and the flag of a counterargument can be raised o'er the ramparts.

To Ed's point, a lot of that can be petty bullshit, but the fundamental business of argument is a vital activity that advances knowledge. Licht can only benefit from new sets of eyes with different ideas about how to analyze it.

That's precisely the kind of stuff that will be grist for the mill. Just imagine all the arguments that a Lacanian analysis of Licht would prompt. What fun! The other exciting thing about such a theoretical lens is that it is independent of Stockhausen's!

And one more thing in favor of assaults and arguments: it really helps us clarify our work. Nothing firms up an argument like having someone lovingly pick it to pieces!

Yes ... I agree. I hate to think where I would be if no one had ever challenged my views.