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ipar1306 Offline



Posts: 199

Thu Nov 08, 2018 1:06 pm
#11 RE: Enharmonic notation in LICHT (especially Lucifer) reply

Sorry for the very late reply to this Jerry - I meant to do so earlier but got sidetracked. As usual, you raise many helpful and insightful points.

The issue of dissonance and consonance is one that I have been giving a lot of thought to lately. Interestingly, Stockhausen described Lucifer's formula as a dissonant one - and it was pretty clear that he meant this as melodic dissonance. I see your point about context and, of course, in twelve-tone music issues of dissonance can be especially fraught. What I find interesting, though, is that there are so many consonant intervals in Michael's and Eve's formulas - I guess intervals that we traditionally associate with harmonic consonance more than melodic (the fourths of Micheal and the thirds of Eve) - whereas Stockhausen points to the 'several tritones (dissonant intervals)' in Lucifer's formula. All of these references seem to be to harmonic dissonance, but he is describing them as melodic progressions in the formulas, so maybe he is not making the distinction. But, be that as it may, I think it is important that, at least for Stockhausen, he considered there to be a dissonance in Lucifer's formula that he found remarkable and that he did not identify in Michael's and Eve's formulas. This accords very much with how I 'hear' them, although I know that that is a very subjective thing.

Another thing - and now only tangentially related to the original topic - I have always been of the impression, without ever having actually checked, and I think because I am sure I read it once somewhere and just assumed it was correct, that all of the possible intervals of a twelve-tone octave scale are included in the nuclear formula tones: and (and I am ashamed to have to admit this) I noticed only today that there is neither a Major 6th nor a minor 7th in there. Everything else (or is enharmonic equivalent!) is. I am a little staggered that I never noticed this until today.

Jerry Offline



Posts: 173

Fri Nov 09, 2018 1:10 am
#12 RE: Enharmonic notation in LICHT (especially Lucifer) reply

This is a very complicated business, depending in part on how you measure intervals and in part of which version of the superformula you examine: the core tones, the core (or nuclear) formula, or the full formula. I think we have been speaking mostly about the full formula.

Lucifer's formula is clearly dissonant according to several criteria at the beginning. The leap of a major seventh or diminished octave is both harmonically and melodically dissonant by the usual measures. But is a diminished-octave leap more or less dissonant than one of a major seventh? I think I have been arguing that it is more dissonant, but this cannot be so because of the actual distances of the two intervals which, in equal-tempered terms, are identical. The distinction rests on a diatonic, as opposed to a chromatic reading. Whether or not this is a valid interpretation is, to say the least, debatable. Lucifer has four (or five) tritone leaps in the full formula, but only one in the core formula. His "extra" tritones are a result of repeating the first one (F down to B in bar 4) twice in the immediately following bar, and a G to C# between bars 7 and 8. This one is particularly interesting because it is a repeat of the framing interval of the chromatic rising scale, and occurs both descending and ascending. What is more, if you allow the scale figure to define a dissonant tritone between is starting and ending notes (as would be the case in a traditional diatonic context), then this tritone is actually stated three times. This is not just dubious on stylistic grounds: it opens up identification of a whole host of possible non-adjacent tritones in all three formulas (the framing G-sharp to D in the first beat of Michael's second bar, for example).

Michael and Eve have one tritone leap each in the full formula (Eve right away in the first bar, Michael in the second-to-last bar, and in both directions (does that mean two tritones instead of one?). Both of these places embed the tritone in a traditional dominant-seventh chord context and, since Stockhausen not only shows no interest in resolving these chords, but actually treats them as stable, restful sounds (the last one in particular is the melancholy closing chord of the superformula, manifested plainly as a closing sonority in the Donnerstags-Abschied and at the end of Michaelion, for example) they take on more of the function of a "tonic chord" (or, at least, suspension) than of something urgently requiring onward movement. This is what I meant about Stockhausen not necessarily treating the tritone as a dissonant interval at all, either harmonically or melodically.

I don't think I have ever heard the claim that the superformula contains all possible intervals in the twelve-octave scale but, as to the minor seventh, I see one in the Lucifer formula, bar 5, just before the end of the first 11-toplet group. This is the embellished resolution of the diminished octave presented across the first four bars of the superformula, now arpeggiated between adjacent notes. The three-note pattern (G up to G-flat, down to F) also occurs in the "scale" embellishment in Lucifer's third bar, anticipating the core-tone resolution between bar 2 and 4. I agree with you about the major sixth; I can't find one, either, but there is at least one melodic leap that exceeds the span of an octave, in Michael's formula, between bars 7 and 8: a leap of two octaves plus a major third. If size of leap is a measure of melodic dissonance, then surely this is the most dissonant interval in the entire superformula.

ipar1306 Offline



Posts: 199

Fri Nov 09, 2018 2:29 pm
#13 RE: Enharmonic notation in LICHT (especially Lucifer) reply

I think the point that I found most interesting on this issue of dissonance in Lucifer's formula was not so much whether one can (let alone should) argue that it has more dissonance than the formulas of either Michael or Eve (and I am talking here particularly of the full formulas, rather than their nuclear formulas), but that this was how Stockhausen described it himself and so, presumably, what he was intending to create when he composed the Superformula. The reference he made along these lines is in an interview he gave in 1997 and is in TEXTE Volume 11 pages 285-298. This is where he contrasts what he describes as the 'dissonant intervals' of the 'several tritones' of Lucifer's formulas with the Michael formula that he describes as having 'mainly descending and ascending fourths' and Eve's as being 'predominantly major thirds' (which, of course, he had already described as 'soul fourths' and 'body thirds'in the libretto of EVAs ZAUBER. Obviously he was not making these comments in a rigorous way by counting and comparing just how many of each interval was in each formula and their contexts: but I guess it does give some indication of how he saw the issue of consonance and dissonance applying to the three formulas. This was the aspect of his remark that I found especially interesting.

On the matter of my mistaken thought that all the intervals were contained in the formulas - this was not the Superformula, but the nuclear formulas where I thought I had once read that. Even now I am almost sure that I read it somewhere, but can't for the life of me remember where or by whom or, more importantly, why I didn't think at the time to check. But it certainly wouldn't be the first time that I have been absolutely sure I have read something, but have never ever been able to find it again. This makes me worry just a little about the current state of my memory, which I am sure was once very reliable. But then maybe that's a distorted memory too.

Jerry Offline



Posts: 173

Sun Nov 11, 2018 3:03 am
#14 RE: Enharmonic notation in LICHT (especially Lucifer) reply

Well, there you go, then. If the composers says the Lucifer formula is more dissonant than the other two, and gives his reasoning, then that is all you need to know. Of course, we listeners may find that our senses do not confirm this. On a related matter, Stockhausen says in many places that the Michael formula's first note is its "central tone" (D in the basic transposition), but I find it impossible to hear the opening three notes as anything but a descending major third with an intervening appoggiatura, making B-flat the firm centre at the beginning of Michael's formula. Naturally, because Stockhausen says it should be D, I am open to persuasion, but so far no one has been able to convince me to hear it differently. I am probably just an old stick-in-the-mud, brainwashed by years of listening to "that old tonal muck" (as Prof. von der Vogelweide so eloquently puts in in the Hoffnung Music Festival Concert), but this is one of the problems of not rigorously avoiding "tonal referents" when composing music: the listener is bound to fall back on experience, and who knows where that might lead! I have often noticed at the Stockhausen Courses that the trumpet players frequently warm up by playing major and minor arpeggios, but hardly ever major-minor triads and seventh chords, let alone chords built of fourths or sevenths. Perhaps this is because of some built-in defect of their instruments, but such chords are at least as important to Stockhausen's music as the major and minor triads, so if the instruments resist them, the players ought to be making an effort to overcome the limitations, rather than reinforcing them, don't you think?

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I feel that it could be useful to have a discussion-forum on the music of Stockhausen. There are so many people from all over the world, young and old, learned and eager to get into contact with this musical world: musicologists, composers, musicians, music lovers; people who plan concerts - who write books or have to give lectures and so on. So there should be much stuff, many ideas that we can share. And when we have open questions, there may be people who studied just that and could give a hint or a stimulus. A problem might be the English language, but i feel that is the only possibility that many people who are interested can participate. And we can exercise tolerance to mistakes! Thomas Ulrich
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