I'm wondering if someone can possibly explain to me the significance of the purple numbers in diamonds in Stockhausen's sketch of the Töne von LICHT. Obviously they are sums of numbers of notes, but I'm not quite clear on why those particular collections of notes are added together as their seems (to me anyway) no consistency as to when these numbers of notes are added together (sometimes it is across limbs, sometimes not, sometimes within them, sometimes not, etc). There is probably something really obvious that I am missing.

I think I have attached a jpeg of the sketch to which I am referring.

I think they are summed nuclear tones which are part of the 5 "limbs" of each formula. Each formula has 5 limbs which are sub-phrases within the full formula. The upside down v's are the "pauses" in between each limb. The "day" markers look like they sometimes delineate the limbs also. That's my guess... Oh I just noticed that the EVE Friday section has a LUCIFER G# note stuck in there - that sneaky bastard!

Thanks Ed. I kind of understand what you're saying, but I'm still confused. The actual sketch indicates different numbers of limbs (Glieder) for each row (5 for Michael, 7 for Eva and 6 for Luzifer). I can see how those limbs are formed and counted. And then there are the seven days for each row. And then there's also three-way division, not shown on the sketch but which Jerry has explained in his article (1: Monday-Tuesday; 2:Wednesday-Thursday-Friday; 3: Saturday-Sunday) which gives you twelve pitches total in each division (counting all of the pitches in each of the three formula sections in each division). But I don't understand the five 'limbs', and when and how they are used, and how they relate (a) to the seven days and to the three 12-tone divisions and (b) to the 5, 7 and 6 limbs noted by Stockhausen respectively for Michael, Eva and Luzifer.

And where's the G# in Eva's Friday section? I only see a B flat (bracketed) and a B natural.

And, just to add to my confusion, I don't understand the significance of the squiggly as opposed to straight bar lines either.

Well that'll teach me to try to analyze KS past midnite on a weekend! Yeah that's a Bb. I guess that's just an "echo" from Thursday EVE.
And I forgot that glieder means limb, so I guess it's not limbs that are 5. OK so I "got nuthin'", as they say.

Personally I have never noticed the effect of the diamond head notes and their effect across more than one day. The squiggly lines puzzle me as well. Hmmm...hope someone can answer these questions!

:) This is an issue that I struggled to address coherently in one paragraph in my dissertation, and as Jerry can attest, my first attempt was a hash. I'm not even sure the final version makes it clear. But the bottom line is that Stockhausen is showing that the added numbers are constituted of numerals that have primal significance for Licht.

For instance, in Michael's Sunday limb, the added number in the diamond is 6, but Stockhausen is careful to show that 6 is the result of 3 + 3, a numeral of fundamental importance to Licht.

In Eve's formula, she has two added numbers: 1+3=4 and 1+2=3. Again, the numbers being added are the primal numbers of Licht: 1, 2, and 3.

Lucifer also has one added number: 1+3=4, and bully for him.

At this point, it is reasonable to ask, "But Michael has a 4 in his Thursday limb. Why doesn't Stockhausen add those notes up too?" Here, the sum total of the entire line gives away the answer. Michael is the only character to exceed 12 notes. He gets to 13 because he is pluperfect (or however you want to interpret it). His 4 needs no addition because it signifies his superiority to the other two beings in this matrix.

The straight lines indicate a major division, the squiggly lines indicate a much more permeable barrier. In my view, there's not much need to overthink those things, because you must always bear in mind that the cycle has no real order. Eve bridges the Day of War and the Day of Cooperation in this sketch, but happens if you reversed those two operas? Then that all goes out the window.

I didn't quite follow what you were asking in your subsequent post, but again, I would caution against overthinking things. The most important thing with the formula is not to decode it, because there are LOTS of ways to do that. What's more important is to understand the ways in which Stockhausen used it. Jerry's written great articles about that, such as "Time and Light" (Contemporary Music Review, 7:2, 203 - 219) where he uses Stockhausen's sketches to almost give us a glimpse into Stockhausen's thought process as he works with the formula.

The most helpful quote on this matter comes from Richard Toop, who said, "One could almost regard the Super Formula as a basis for negotiation between the Stockhausen of 1977 (when it began to be formulated), and the many Stockhausens of the next 25 years. And one can scarcely imagine any of these later Stockhausens saying: ‘Well, I mustn’t do that, because I hadn’t thought of it in 1977’." (Lectures, p. 104). What we are looking at in that sketch is not a hard and fast rule for composition, or some kind of unbreakable template. Stockhausen could always return to it and "discover" a new way of using it.

Thanks for this Joe. That all makes a lot of sense.

While I agree about avoiding 'over-analysis', and resisting decoding the formula, it's nevertheless interesting to at least try to understand the things that are there and to ask why - hence my question about the numbers in the diamonds. It seems Stockhausen rarely did anything without some reason, and so I just find it interesting to discover, when I can, what those reasons were. But, of course, as you point out, that doesn't necessarily translate into rules that then govern the 29 hours of LICHT. Maybe at times even some of those ideas went no further than ideas. The quote from Richard Toop that you mentioned is one that certainly resonated with me when I first read it only a few weeks ago.

The article that I have found most helpful on the structure of the Superformula (and of the nuclear formula) is 'Into the Middleground: Formula Syntax in Stockhausen's Licht' (Perspectives in New Music, 28:2, Summer 1990, 262-291). That reveals a plethora of richness in the structure of the formulas that I would surely never have discovered myself.

Jerry's work, once again! :) It's invaluable.

The passage you referenced in your second post, where Jerry discusses the hidden "magic square" in the super formula is a great example of independent analysis decoding what's there. As I said earlier, there's more than one way to decode the formula. (That's why it was such an effective compositional device for Stockhausen over the four decades that he used it.) That doesn't mean that Jerry's decoding is unassailable, soundly reasoned though it is.

For instance, he says that the "last note of Lucifer's row is transferred to the end of Michael's formula". That sounds reasonable at first, but who's to say that the D which is missing from Lucifer's formula was to come at the end? In the superformula, Lucifer's D can be found in the third bar. There's no indication it would come at the end of his formula except in Jerry's analysis, which provides a great metaphorical way of thinking about the formula complex. I say all this not to assail Jerry's work, but simply to demonstrate that everything is subject to debate.

To revisit my previous answer to this question, it should be clear that, though my statements are declarative (as are Jerry's), they are merely my own analysis. They aren't definitive, and they aren't the only take I have on those numbers. For instance, notice that the characters do not have 1 diamond note on their subject days. It would make a lot of sense if they did. You could argue that it signified the primacy of that particular day for that character. Instead, the 1's tend to come on days when they play important subsidiary roles. Lucifer has 1 on Montag, where he is frenetically working as an antagonist. Eve's 1 comes on Donnerstag, where she is Michael's invaluable counterpart. Michael's 1 comes in Samstag, where he stands (or lies) in protest of Lucifer's nihilism. And again, Lucifer gets a 1 on Sonntag, the day where he has naught to do.

But arguments along those lines should always be taken with a grain of salt, because nothing Stockhausen does with the formulas is immutable. After all, Michael has 3 notes in the Montag limb but he does not appear in the opera. The super formula seems to indicate Michael should have a lot to do on Montag, but Stockhausen made a practical concession to reality after Markus opted out.

Lastly, it is true that Stockhausen often had reasons for his choices, it is not true that he "rarely did anything without some reason". He was more than happy to simply leave it at, "I saw it in a dream" or "I did it intuitively". I suppose those are reasons, but I don't think they are the kind of reasons you have in mind. There is an important balance between rational and irrational thought in Stockhausen's work.

Thanks again Joe!

To briefly respond to the last point first - actually those were exactly the sorts of reasons I had in mind. In fact I was going to semi-facetiously suggest that maybe the reason for having squiggly lines alternating with straight lines was just that Stockhausen thought it looked pretty. I am as interested in discovering the reasons, whatever they are. Not that that then dictates my own understanding of things - I just like to know what was going on in Stockhausen's thinking (insofar as that can ever be understood with anyone) at the time he did the things he did. It helps inform how I then might understand things, but it but no means dictates this. Let us not forget the intentional fallacy!

An interesting point about the missing D from Lucifer. It had never occurred to me that it could have come from anywhere within the formula other than the end - but of course you are right. I have a vague recollection of somewhere seeing the formulas written with the D in Lucifer, written at the end, and only one D in Michael, but I may be making that up or, even if I saw it, it might not have been Stockhausen who wrote it.

It is certainly difficult (but important) to find the right balance when it comes to the issue of 'decoding'. There are so many possibilities there, so many connections and meanings to be found by anyone who wants to spend time looking for them. It obviously can't be the case that all those meanings are equally valid, but neither can it be the case that none of them are. I don't believe there is any definitive balance between those two extremes - but there is a general "space", I suspect, within which different decodings can be more or less justified. But in a work that itself breaks its own "rules" from time to time, the greyness and uncertainty of that space will always be debated. It's another thing I love about my current venture - thinking through those debates, often in my own head as much as anywhere else.

But I am reminded, too, of those people who pore endlessly over old texts, like the Bible, looking for coded messages. I remember once seeing a documentary where someone did the same thing with the phonebook, and got just as many interesting results.

Oh and one other point I forgot to make in my previous post (I'm always remembering things afterwards): I wouldn't quite agree, Joe, that Michael does not appear in MONTAG, although I of course know the sense in which you mean this - the character does not appear on stage. But he is very present in his formula, which is sometimes stated very prominently, such as in the Befruchtung scene in Act 2. There seem to be many places in LICHT where characters are present in the music, even when they are not physically on stage.

I forgot who I was talking to! :) Of course, intuition and dreams are reason enough, and I look forward to seeing what you do with that.

There are obviously degrees of importance here, and they will be different for each interpreter. Certainly, Stockhausen placed great importance on his dreams. But he also used them as a cop out when he wanted to. Someone could say, "What does that mummy signify at the end of Orchester-Finalisten?" and he could shrug and say, "Meh...I saw it in a dream..."

That's essentially a zero point for the analyst. You can make any argument you want to once the composer has absolved himself of any responsibility for his own content.
------

You may recall that I reject this idea that the presence of a trumpet or Michael's formula means that the character himself must be present. It places an unsustainable burden on the operas. The drama would become incoherent if every time that Lucifer's formula sounds we are meant to think he is present and accounted for.

To your specific example, what do you hear when Geburtstags-Formel sounds? Is it Michael's formula? The dominant architecture there seems to be Lucifer. More importantly, the budgerigar arrived on the scene courtesy of a flying piano, which is the instrument Lucifer attempts to play in Samstag. When a budgerigar pianist drives the instrument into Eve's vagina, it's hard to see how we are supposed to associate that with any character other than Lucifer, especially when the music is as overrun by his formula as Piano Piece XIV is.

It's not that I don't buy the idea that the protagonists can be represented by proxy in their formula, but this just doesn't strike me as a good example of that practice. Oktophonie, maybe, but again, I think that piece is meant to underscore the onstage action, not manifest it.