Counter-arguments are very useful, agreed. Frankly I love to be "illuminated" if the argument holds water (and I often update my blog entries when this happens). This is way off topic, but my beef with some of the theses I've come across is that they sometimes go out of their way to be dismissive of a predecessor's work, or read like the writer has his nose up in the air. But perhaps that is the academic culture and I'm more accustomed to friendly discussion over a beer after a good gig.

Right you are. That's just par for the course, unfortunately. Even when people are not actively sounding an aggressive tone, there is often an implied aggression in their writing, because the nature of peer review requires defense of one's ideas. It's like a defensive crouch from which the writer is ready to spring at any moment in response to an attack. Stockhausen's writing often has this tone.

And of course, a lot of analysts of Stockhausen's music feel like they are in some kind of intellectual competition with him. So they adopt a sort of sneering or aggressive tone in order to bolster their own work. It's a shame, but it's also hard to see how it would ever change.

For instance, if Ian were to proceed with Lacan as a theoretical lens, he will run into unavoidable criticism from anyone who thinks Lacan's work is garbage. Right out of the gate, Ian's got to be prepared to defend his decision to use Lacan (which I'm sure he'll be able to do!). It almost sounds like an abusive relationship, but it's just the nature of academia.

This thread is getting a bit tangled, so I had better make clear that I am now responding to:

Zitat

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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.

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I think Joe explained this fairly well, though I do have a couple of additional thoughts. First of all, I think we are all clear that this is not a "sketch", but rather an analytical diagram made after the fact. These are fairly common in Stockhausen's output. The numbers in diamonds, as Joe pointed out, refer to groupings of pitches in each of the formula lines. I differ from Joe, however, on the issue of the primacy of the numbers 1, 2, and 3, and this is why: Whenever Stockhausen begins to work out his material, he starts looking for ways either of grouping smaller things together (supraordinating), or subdividing larger elements (subordinating), preferably in ways that will produce numerical sets. Certain numbers go together in certain ways and, as Richard Toop has pointed out, Stockhausen was particularly adept at seeing these relationships. We see, for example, the number 28 as the sum of the integers 1 + 2 + 3 + 5 + 5 + 6 + 7 (used in Zeitmasze and Pole, for example), or the number 15 as the sum of 1 + 2 + 3 + 4 + 5 (e.g., in Aus den sieben Tagen). When such basic number sequences don't work for his purposes, he may resort to others, such as an incremental series like 1 + 2 + 4 + 7 + 11 = 25 (the difference between successive members of the series increases: 1, 2, 3, 4), which he uses for example in Mixtur, or the Fibonacci series, 1 + 2 + 3 + 5 + 8 = 19 or, if this is not a useful sum, starting on something other than 1, such as 3 + 5 + 8 + 13 = 29. These number series may also be compounded, for example taking 3 + 4 + 5 + 6 + 7 (= 25) as multipliers, respectively, of the descending Fibonacci numbers 21, 13, 8, 5, 3, yielding 63 + 52 + 40 + 30 + 21 = 206 (in the first section of Harmonien, for example). The use to which such groupings are put depends on Stockhausen's needs in a particular composition.

Mercifully, in the present case there is nothing so complicated. Stockhausen merely uses a four-number sequence (representing groups of notes) in three rotations: 1, 3, 2, 4 (Lucifer), 2, 4, 1, 3 (Eve), and 3, 2, 4, 1 (Michael). He then extends each number series in circular fashion to return to its starting number, giving Lucifer one note, Eve two, and Michael three at the end. This produces what he needed, a total of 11 notes for Lucifer, 12 for Eve, and 13 for Michael. The arrangement is a little forced, since only Michael has a genuine group of four notes, and in order for Eve's fourth group to conform, a 1 + 2 pair must be combined to get 3. Then there is that oscillating group of tones at the end of Michael's formula that actually bring the total number of pitches in this particular diagram to 16 (in others, Stockhausen counts the repetition of the first F# as two notes, and declares Michael's formula as totaling 17).

The six vertical red "barlines" together of course divide the core tones into the seven days of the week (as shown at the bottom), though the tidy pattern of rotating vertical sets (from bottom to top: 1–2–3, 3–1–2) breaks down already with Wednesday, because of the inserted rests in both Michael (who would have had one note) and Lucifer (who would have had two). Lucifer gets "postponed" to Thursday, but Michael conflates his one note with the three he otherwise would have had anyway. Even though Stockhausen does not mark it as such, this four could be therefore be seen as 1 + 3. Naturally, this means that the next vertical grouping (Friday) is also disturbed, since Michael's three notes have already been used in Thursday, so he takes a rest. Saturday at last returns to "normal", with a 3–2–1 vertical set, but in Sunday all hell breaks loose (so to speak) because Lucifer should have had two notes but has "lost" one to Michael, which upsets what logically would have been an uncharacteristic 2–2–2 vertical arrangement, brought about by Lucifer's steady repetition of 1–3–2, 1–3–2 against Michael's permutation 3–2–(1+3)–1–2 and Eve's more irregular groupings.

The interrupting silent "days" in Michael and Lucifer reduces their "active" Glieder to 5 and 6, respectively, whereas Eve has core tones in all seven days. This is another way in which Stockhausen partitions his material, into a 5 + 6 + 7 set this time. This leaves only the question of the difference between the straight and wavy barlines, and I must confess I do not see a neat and tidy explanation for these. At first, it looks like straightforward division of aggregates into component hexachords (Monday hexachord + Tuesday hexachord = aggregate), but the next aggregate covers three days (Wednesday, Thursday, Friday), and yet the second straight barline falls between Thursday and Friday. The last wavy barline does separate two hexachords though, so perhaps the second straight barline is a mistake, and should be switched with the next wavy one. I'm not really prepared to accept this solution, though, because it seem too glaring a discrepancy to be merely a careless mistake.

The marking in pink in the lower-right corner, by the way ("13 + 12 + 11 = 36 Töne" with "(3 x 12)" underneath it) I take to refer to the simultaneous horizontal and vertical arrangement of the pitches: three horizontal lines of different numbers, and three equal-sized "vertical" aggregates. It may also prove useful at some stage to observe the dashed vertical lines, which indicate alignment of the pitches, the octave-displaced pitches in Lucifer (Tuesday) and Eve (Sunday), and the six circled notes (two in each horizontal formula), which within the straight barlines are grouped as 3 + 1 + 2 (G–D–F#, Bb, and A–B). Since, as I said, this is an analytic diagram rather than an actual sketch, these circled notes must have some significance, but I cannot recall having seen them discussed anywhere in the literature. Did I overlook this in your dissertation, Joe?

In any case, I think this shows how very dubious any analysis is that seeks merely to find simple patterns in Stockhausen's music. His method was always to probe his initial material for possibilities, then to introduce fresh ideas or rearrange the elements where the original failed to supply what he needed. It is no wonder that he found the formula "so flexible"—it wouldn't dare be anything else!

Thanks for this Jerry. More fascinating connections to think about. I, too, had wondered about the dashed vertical lines.

These observations help me to understand, even more than I did before, Stockhausen's remark that he could continue composing using the superformula forever. It does seem to offer a boundless sea of possibilities for development.

And I have to confess that I actually didn't know that this was not a sketch. That helps me to understand it rather differently.

If you want to see the actual sketch that was redrawn for this diagram, it is reproduced on p. 153 of TEXTE V. It is dated 21 March 1978, and differs in several details from what you see in the "fair copy." For one thing, it is a major third lower in pitch (Michael starts on B-lat instead of D, for example). For another, the table of rotations of the four-element set I described is given explicitly to the right. There are also a lot of further analytic diagrams below, with all sorts of useful information. And I should also say that you may still not want to call it a sketch, technically, because it is labelled "Abschrift" ("copy") at the top. The actual sketch from which this was reconfigured is on p. 152 of TEXTE V, marked "Kyoto" and dated "October 1977".

Such redrafting of sketches is common—and not only for Stockhausen—but there are some interesting details that go right back to that original sketch from October 1977. For one thing, the 13 + 12 + 11 distribution of pitches is there from the start, and for another, those six circled pitches are already so marked. The usual difference between a "true" sketch and a redraft lies in the presence of erasures, overstrikes, and arrows connecting originally separated ideas together. There are often also signs of some material added later than others, such as with a different pencil, and in Stockhausen's case, at least, verbal annotations tend to be less neatly written than in the case of redrafts. Although he often used colored pencils in his sketches, the kind of elaborate color coding you see in that later diagram we have been discussing is more characteristic of the drawings he made for the covers of scores and CD booklets.

There is an even earlier sketch, from 26 July 1977, reproduced on p. 150 of the same volume, which entirely lacks the pitches, but shows that the distributions of notes and melodic contours were decided first. This is faintly reminiscent of the (probably apocryphal) story about Debussy who, when asked one day about how his latest orchestral composition was coming along, replied, "Splendidly. I finished the orchestration this morning. Now all I have to do is put in the notes."

Thanks for the detailed explanation Jerry. So I wonder, how do these diamond notes, solid/squiggly/dashed indications get expressed in the music? I suppose one could find these numerical sets expressed in proportions somewhere? (besides the obvious "trot out the ol' nuclear formula for another turn round the CAROUSEL" trick).

On a semi-related note, I was just reading the Kramer article in PoNM 36.1 where Felder compares the European-style of analysis with the American style - that is, at Darmstadt the "how was it made" is examined (which is what you've described), but in America, the "how does one listen to it" is analyzed. I personally try to cover both angles (admittedly, in a simplified, hopefully general-audience-friendly way) so I'd be curious to know if the markings have any use as aural listening aids...

The obvious answer is: it depends on how good a listener you are. I cannot personally hear the straight and wavy barlines as such, but I certainly can hear the harmonies that change at those points. Do I hear the hexachord boundaries? Probably not. Does it matter? Can I tell that Michael's formula starts with three notes, and then has two? I think so, if I try really hard. Can I count the six collective notes in all three lines of the opening segment? I imagine so, if I want to, though it doesn't seem to me an especially productive way of hearing (for example) the first act of Montag.

As you say, there is no particular reason to deny the "how it is made" angle, just because you are interested in "how it sounds", and it seems to me that the relationship between the two ought to be worth investigating. I know for a fact that Stockhausen wanted people to listen to his music, rather than compile statistics on the contents of the scores. On the other hand, he also wanted listeners to stretch their capabilities, and learn to hear things they didn't think was possible.

A good example is the form scheme of Donnerstag, which clearly shows how the 4 notes in Michael's Thursday limb shape the entire opera. I'm not sure if that's what you mean by the diamond notes, but it is an instance where one can easily see how Stockhausen expands on the four notes in the "Pitch Sketch".

Whether or not these things are useful as aural listening aids is down to the tastes of the analyst and the listener. You could make hay out of the fact that the climax in Michael's Samstag appearance is a high F, perfectly aligned with the pitch sketch. You could sit someone down with Oktophonie and show them how clearly it flows from every layer of the formula, from the pitch level, to the nuclear level, to the super formula level.

However, as we've been discussing, Stockhausen did not fetter himself to what appears in these formula layers. The pitch sketch contains a rather devilish obfuscation that highlights the gap between what the ear hears and how the music is written. As Jerry says, there's a rotation of 1 3 2 4 in this pitch sketch. Lucifer only has one note in the Monday limb at both the pitch and the nuclear level.

BUT, our ear actually hears his signature "Pro-test!" interval of a Major Seventh/Diminished Octave in the second bar of the super formula, which is still early in the Monday limb. By putting the 2nd note of Lucifer's formula in the Tuesday limb, he was able to sustain this numerical conceit at the pitch and nuclear level, but it comes apart in the finished product. The aural reality of Lucifer's formula puts that Protest interval squarely in the Monday limb.

So, I think it's really up to the end user to decide what's germane.

The way I tend to look at the issue of how much of these structural elements are apparent to the listener is a bit like going into a building. Whether or not I am aware of its structure doesn't alter the effect that the structure has on the building. And if its a good and thoughtful structure, chances are that my entry into the building will be a better experience for me than if it's poorly structured. My understanding the structure might help me to understand my experience, and why it's good or why it's not so good, and it might even enhance my experience. But I don't have to be aware of it to reap its benefits. I reckon many things are like that, and especially music such as Stockhausen's where there is just so much to the structure.

OK that's some good food for thought and I'm glad I got a take from each of you guys. Ian's point about a good structure creating a good experience reminds me of when I first learned about the Fibonacci series and the Golden Ratio used in Bartok's and Debussy's works (and KS' of course). The fact that the structure of his works (Bartok SQrtt 4? 6? I forget...) is based on an additive numerical sequence is not something I need to stopwatch as the piece progresses. It just feels good! Though sonata-allegro form is still probably my favorite form structure...

I remember when I first wrote about MANTRA and followed Richard Toop's analysis (which is superb of course). Also the British Lecture Stockhausen gave on MANTRA, probably the most lucid analysis I've ever seen by him on video (not a lot to choose from obviously). The expansions of the MANTRA formula are a fascinating technique, but when the formula is stretched registrally in proportion like that - well that pretty much makes the melody into an extreme variation far beyond what would normally be considered "variation" on the classical Beethovenian sense. And then with all the poly-layering... Frankly it was a chore to listen to MANTRA like that, and I think someone in the audience asked KS if it was necessary to know all these details of construction. KS replied to the effect that it was not necessary to know astronomy to appreciate the night sky, but by knowing it, it could possibly expand one's consciousness (as Jerry mentioned). Speaking from experience, I've loved parts of the LICHT operas without knowing a thing about the super-formula (or even reading the booklets), but now having dived deeper into it, it makes the listening experience much richer since I can now hear the linear and to some extent, the vertical, thematic material.

However it may be some time before I really appreciate the effects of the temporally-stretched formula layers (not to mention the 4 Michael notes Joe mentioned stretched over Donnerstag), aside from just thinking "oh another drone layer, OK that fills up the timbre-space nicely..." But it is interesting... Now if someone could just explain to me how my iTunes shuffle function works, or how Google formulates its search results...