Hello all!

Initially, I thought I had already asked about this issue on this forum, but it doesn't seem that I have, although I am sure I have had the discussion with some of you individually, perhaps via email, and I think some years ago. But still I am curious and would be interested to revisit the question of enharmonic notation throughout LICHT.

There are, of course, many cases of it, and many places where we might ask why Stockhausen chooses to notate a pitch as, for example, G sharp rather than A flat, and sometimes the reasons for this will be more evident than other times - but the one that I find most intriguing is the different ways of notating Lucifer's leap at the opening of his formula. In the famous graphic of the nuclear pitches, it is notated as G to F sharp (a Major 7th), but then in the Superformula it is notated as G to G flat (a diminished octave). Throughout the score it seems to shift from one to the other without, as far as I have been able to decipher, any consistency.

There are, of course, musicologists and composers who will argue very strongly that there is a definite difference between an F sharp and a G flat (and also plenty of musicians - especially string players - who would play the note differently), and surely Stockhausen would have been aware of these arguments. But were they relevant to him in his own decisions of how to notate these pitches, and why the shifts in different appearances of the same formula?

Does anyone have any thoughts on why these, or other, pitches are notated with these enharmonic differences?

I would be intrigued to know anyone's thoughts on this (and sorry if this means asking any of you to repeat thoughts that you have already shared with me, but which, in my over-crowded head, now seem to have become somehow lost)!

Many thanks,
Ian

Why are my ears burning? Maybe because this question seems so familiar, that I am sure I have discussed it with you, Ian.

I think it is important to remember that Stockhausen was trained on the piano and had an acute sense of absolute pitch. For him (as also for many other musicians) it was a matter of indifference whether a note was written as A-sharp or B-flat. The choice in most cases comes down to which is the most easily read. In the published score of Zeitmasze, for example, he experimented with the idea of using only sharps and naturals since, in a thoroughly chromatic context, indiscriminately mixing flats and sharps is needlessly complicated. The manuscripts were originally notated with the usual preference for flats on B and E, and sharps on F and C, with context determining the rest. In the long term, he evidently found this method unsatisfactory, since he only used it in a few subsequent scores (Gruppen, I think, and some of the Klavierstücke).

When talking about the difference between his formula technique, beginning in the 1970s, he said that the only real difference from what he did before was to bring the notes closer together in register, in order to make the lines more singable. This is something of an oversimplification because, when you do this, it makes traditional voice-leading harder to avoid. In some compositions, in fact, he succumbed wholesale not only to strong leading-tone resolution tendencies, but even to a modal-diatonic sound world (admittedly, usually of the type that Bartók called "polymodal chromaticism", with rapid shifting from one diatonic mode to another). This in turn leads to the problem you raise, which is especially easy to see in the opening of the Lucifer formula: is the second note a major seventh above the first note, or a diminished octave? As I said above, for Stockhausen, the notation is as matter of indifference, but the way the interval is used in context is another matter. In some compositional contexts, Stockhausen's use of this interval suggests a more or less traditional major-seventh context, while in others it suggests a diminished ocatave, resolving downward a semitone to the minor seventh above the initial, central tone. Sometimes, the choice of enharmonic notation may reflect this; other times, not.

I would be happy to continue this discussion, but this should be enough to be getting on with.

Thank you Jerry. Yes, think we have discussed this earlier, but I could find where. Some of your answers here are triggering some memories. It is comforting to know that the reason I often cannot find a reason for this notation rather than that might simply be that there is none!

Alternatively, there may be a reason for notating a note one way or the other that has nothing to do with what you are looking for. For example, in the superformula's basic form, the Lucifer formula begins on G. Notating the second note as F-sharp is more "readable" than writing G-flat. If the formula is transposed down a semitone, I think most musicians would find it easier to read F-sharp up to F-natural, rather than to E-sharp. Stockhausen's practice usually is to make the performer's job of reading the notes as easy as possible, and this may not be the same thing as clarifying structural aspects for the analyst. There are well-known examples of the same thing in the piano sonatas of Beethoven, where a musicologist working with pencil and paper will complain that Beethoven should have written an A-sharp to spell his chord correctly, but a pianist will counter that it is easier to read when written as a B-flat. If I recall correctly, Beethoven is not very highly regarded as a musicologist, but he did have a fairly good pedigree as a pianist.

Yes, and that makes total sense to me. My initial bewilderment on this all came from the example I cited earlier - that the Lucifer leap at the beginning of his formula is notated as a a Major 7th to F sharp in the 'Töne von LICHT' sketch, but as a diminished octave to G flat in the actual Superformula. As both of these are more abstract than performative, I couldn't see a reason for the difference, but then noticed many other examples throughout the scores. Your explanation, Jerry, that sometimes it just will seem more 'readable' for the musician explains many of these. But I guess the initial example is still best explained by your earlier comment that sometimes there is simply no reason at all.

As an aside, there is somewhere in one of Morton Feldman's pieces (I can't remember which one) where a violin part has a note in two enharmonic notations side by side. I was told that performers typically change their fingering at that point and, with this, comes a minute change in pitch. But of course these different notations can be more significant for a string player than for some other instruments such as the piano which, as you pointed out Jerry, was the instrument on which Stockhausen mainly trained.

I see your point about the abstract nature of the Töne sketch and the "actual" Superformula. Stockhausen's habit (no different from the vast majority of musicians today) of thinking in terms of twelve-equal temperament of course means there is no difference in sounding pitch between a G-flat and an F-sharp in the same register.

Stockhausen and I once had a conversation about microtones in Licht, and he told me that, for him, they were mainly attractive for the variations in tone colour they produce on woodwind instruments. Their use was much less interesting to him on strings or brass. Obviously, there are exceptions, in particular the quarter-tone flugelhorn in Pietà from Dienstag, but this is still a matter of equal temperament, only with 24 tones to the octave instead of twelve. At the time of our conversation, Pietà had already been composed so, when he said he believed that the twelve-note chromatic scale would continue to be the norm for many years to come, he must have been thinking of such pieces as relative novelties. The electronic pieces from the early Studies down to Cosmic Pulses are full of sub-chromatic intervals of various kinds.

Getting back to enharmonic notation, there are of course some of Stockhausen's earlier pieces where he deviates from equal temperament. Sternklang, for example, uses a sequence of five justly intoned chords, all sharing one constant note (E = 330 Hz) as the fifth, sixth, seventh, eighth, and ninth partials of different overtone series. Stimmung also sticks to the just intervals of the overtone series, though with less deliberate use of microtonal byproducts, and there is the isolated example of a septimal seventh in Gruppen, produced as the natural seventh harmonic on the double bass.

The example you cite from Feldman is interesting because, from your description, it does not sound to me that Feldman specifies which of the enharmonic notes is higher and which is lower. Depending on the tuning adopted, the sharp note may be higher than the enharmonic flat (Pythagorean tuning, for example), or the reverse (in most forms of just intonation, for example), and in the latter case the difference is scarcely "minute"—it amounts to nearly an equal-tempered quarter tone. The inference, then, would be that Feldman did not care exactly what pitches are used, but only that there be a small pitch wobble of some sort. The nearest thing to this attitude that I can think of in Stockhausen's music are the indeterminate microtones in pieces like Xi and Ypsilon, where it is up to the performer to determine how small the intervals of the scale can be. Of course, there is no question here of enharmonic notation, the differences of which go back to various musical traditions.

Without wanting to get too far away from Stockhausen (and especially not too close to Feldman!) I did manage to find at least one example of what I was referring to - near the beginning of the second string quartet there is, on page 3 of the score, in the viola part, a bar beginning with a C sharp followed by an F flat then a dotted 16th note rest, followed in the next bar by another dotted 16th note rest and then an E natural followed by an E flat. Then in the bar after that there is again the rest followed now by an F flat and then an E flat. I notice, actually that that score is full of unexpected notations. Even the opening bar, where the first violin begins with a D double sharp (which it then repeats over and over again for the entire first page (and more)), while the second violin plays an E flat, repeating it in the same way. I checked one of his piano scores (Triadic Memories) and there doesn't seem to be any of these oddities there, so it is presumably somehow related to the fingering and more nuanced intonation he was seeking in the strings, as you suggest.

But I have seen nothing as perplexing as this in Stockhausen! Of course the microtonal pieces and just intoned pieces you mention are another thing again, and usually seem to suggest to me that Stockhausen was rather clearer in explaining what sorts of nuanced intonation he was seeking, when he was stepping outside the more straightforward semitone (or quarter-tone) scales.

Another example in Stockhausen that I find interesting is in the Eve formula, and the G flat that appears there in the Wednesday limb. That is notated as G flat in both the nuclear and Superformula forms. I find it interesting because it creates an overall interval of a diminished fourth (preceded by a minor third) in the Wednesday limb, whereas the Monday-Tuesday limbs (if we look at the nuclear notes) moves from a Major third (C to E), then down to E flat. Thus, the intervallic envelope of the first three notes become a Major 3rd falling down to a minor 3rd (that is, overall, spanning from the C to the E flat), while the next three notes do the exact reverse, a minor 3rd moving up to a Major 3rd now enhamronically transformed to a diminished 4th (that is, overall, spanning from the D to the G flat). I realise, of course, that viewing the notes in these groupings can be somewhat arbitrary and there is no real reason to compare the first three notes of Eve's nuclear formula with the second three notes, especially as the pitches of the first three do not form any single gesture in the Superformula (where the E and E flat are separated by the downward noise glissando) and where the second three notes of the nuclear formula form a very definite gesture in the Wednesday limb of the Superformula. But, even so, I find these patterns fascinating when I discover them, and I often wonder how much Stockhausen was conscious of these little patterns and riddles that he sometimes created.

The example you cite from the Eve formula is less contradictory than it seems at first sight, if you take into account the larger span of the entire formula. You will recall that I invoked Bartók's "polymodal chromaticism" in connection with some of Stockhausen's formula-based melodies. Well, this is as good an example as any.

The Eve formula begins on, and is at least initially centred on C. The opening arpeggio conforms to the C-Mixolydian scale, followed by a short chromatic connection establishing the major third from C up to E. This is followed by an "improvisation" on these two notes, including the lower octave of the E and a grace-note upper-neighbour ornament of F. Nothing so far has deviated from C-Mixolydian. Then comes the noise-glissando from "on high" down to a noisy A, and then a pitch glissando down to E-flat. Why not D-sharp? Well, clearly E-flat is part of a diatonic scale based on C, whereas D-sharp would be a foreign tone: an augmented second above C. So we have a modal shift from C-Mixolydian to some C scale with a minor third. Given the strong presence of A natural, C-Dorian is the best fit.

Here is where things get interesting. The figure marked "modulation" in the second bar of Tuesday involves the semitone between D and E-flat, so we can continue to regard this as being C-Dorian. This brings us to the Wednesday segment, and the reason for notating G-flat instead of F-sharp. The oscillating E-flat and D become the grace-note pickup to the downbeat of the first Wednesday bar, with a D moving up to the fourth diatonic scale degree, F. In order to continue being diatonic, the next note is written as G-flat, which means that the glissando down to E-flat and back up presents not a raised (Lydian) fourth degree, but a diminished fifth scale degree, therefore inflecting to C-Locrian, except for the continued presence of D-natural, indicating a mixture of Dorian and Locrian. The following B-flat octave in the Thursday segment does nothing to dispel that impression and, after the noise-glissandos, the B-flat octave returns in Friday.

It is only here that a chromatic shift is made, to B-natural. However, since we have been thinking all along of the C as the central tone, this is a perfectly familiar shift from the minor-seventh scale degree to the (leading tone) major seventh. At this point, the mixture of modes becomes more intensively chromatic, returning to B-flat plus the Locrian fifth degree G-flat, the fourth degree F, and the mixture of the major and minor thirds (E and E-flat), finally introducing the Phrygian/Locrian D-flat, closing to the central tone C. I don't think I need to give a detailed description of the rest, which eventually establishes the normal dominant degree G but ends, somewhat surprisingly, with the minor sixth A-flat, supported by the brief recurrence of D-flat, but also a wavering between the minor and major sixth degrees.

So, it is not a matter of whimsically deciding to use a diminished fourth in one place and a major third in another but, rather, an effort to maintain a referential C, with as many notes diatonically related to that note as possible. This is not the same thing as writing tonally, since there are several audible shifts of pitch focus in the course of the formula, and the nominal dominant of G fails to function in that capacity, becoming instead a leading-tone to the A-flat at the end. Of the three formulas, Eve's is the easiest to explain in these terms (so I thank you for choosing it), but the same principle can be applied to Michael and, to a lesser degree, Lucifer as well.

Many, many thank for this extremely helpful analysis, Jerry. I did not pursue the modal connections you mentioned earlier, and I can see here how they explain the notation in a way that other explanations (other than whimsicality, if that's a word) could not. Was Stockhausen, to your knowledge, making these modal references consciously, or is it something you have noted yourself?

The closest I was able to come to an alternative explanation in the Eve formula - but it is not one that I was able to sustain throughout the whole formula - was a theory that Stockhausen was reinforcing the primacy of the interval of a 3rd (be it Major or minor) throughout her formula - so the notation to of G flat in the Wednesday limb creates a minor 3rd with the following E flat (but of course, that doesn't quite follow, because it sacrifices the Major 3rd with the preceding D that would have been created through notating it as D sharp instead). We then see a dominance of thirds in the downwards figure of the first bar of the Saturday limb (the beginning of the passage you describe as the more chromatic mix of modes), which you also note (B flat going down to G flat, then down to E flat, then down to C, although I realise I have rather arbitrarily picked that E flat out of the downwards scale, beginning on the F and where you make reference to the Phrygian/Locrian mode with its D flat).

But of course my 3rds theory falls apart completely in the Sunday limb, with its mix of minor seconds and Major 7ths and even the diminished 4th reappearing, but now between the A and the D flat, which then immediately resolves back to a Perfect fourth by returning down to A flat.The thing I do find interesting there though, is that those intervals (minor 2nd and Major 7th) are the intervals I associate more with Lucifer's formula, and so it is odd that they would be entering here so markedly in the Sunday limb of Eve's formula while, in Michael's formula, in the Sunday limb, the wavering between F sharp and F natural, ultimately returning to the D, seems to suggest the Major and Minor 3rds that are more associated with Eve. It makes sense that some of Eve's characteristics would enter Michael's formula in the Sunday limb, given that it is the day of their mystical union, but it is more perplexing that Lucifer would dominate in Eve's formula, notwithstanding the eventual resolution (of sorts) to Michael's perfect 4th, rounded off, however, by a return to the minor 2nd, by going back to the G and then back to the final A flat (where, I guess, the G could be read as a leading note to the A flat, but then that is not consistent with the notion of the formula's centrality on C that you suggest - although, finishing on A flat does, I guess, again restate the Major 3rd in relation to the opening C). But seeing a Major 3rd relationship there, between the opening C and the closing A flat, is, I have to admit, not entirely plausible either, as it is a downwards move, against the overall upwards trajectory of Eve's formula. Maybe that is caused by Lucifer's infiltration - but then that whole explanation would be much more credible if it was happening in the Friday limb, the day of Lucifer tempting Eve, and Eve's subsequent fall. Art any rate, my recollection is that these more detailed 'meanings' for the days were created after the Superformula had been written, but I can't precisely remember the timing of that.

Anyway, the bottom line is that my focus on intervals rather than overall modal structure leaves many things unanswered.

I may have overplayed the modal aspect of the Eve formula just a little. The emphasis should be on explaining the reasons for the choice of notation for certain intervals. The opening of the formula firmly establishes C-Mixolydian, but the continued reference to C is more a matter of notational convenience than compositional focus on that particular note. To my ear, for example, the Wednesday segment is quite clearly focussed on D, with both a major and minor third above it, never mind the notation.

You will recall that Stockhausen mentions, in "... How Time Passes ...", that some rhythms may be written in different ways, and the performer's uncertainty increases in proportion to the complexity of the chosen method of notation. Well, the same applies to notation of pitches. If I were to write an ascending scale as D, F-flat, E-sharp, G, A, A-sharp, D-flat, D, I doubt that many musicians would find it as easy to sight-read as D, E, F, G, A, B-flat, C#, D, and I could make things far worse by the liberal application of double flats and sharps.

It is really the fault of our notation system, where sooner or later we are going to come up against that awkward split between "soft" and "hard" B (B-flat and B natural, or one of its transpositions; in German, of course, one says B and H). Stockhausen makes this divide on the third scale degree of C, where the E and E-flat can be quickly recognised as the major and minor third. Once having established this break at E/E-flat, Stockhausen continues "in one key" for as far as he can, until it becomes necessary again at G-flat/G-natural where, fortunately, the notes do not have to occur in direct succession.

Turning to Lucifer's formula, the opening G to G-flat is really just the conventional way of notating a descending chromatic scale, continuing down to F, except that the first note is taken down an octave. It may be worth noting that the very first sketch of the core pitches of the superformula is written a major third lower, so Lucifer begins on E-flat and leaps not to an E-double-flat, but to D. Compositionally speaking, when Stockhausen wants to focus on a shorter segment, where the two notes may be thought of as part of a single harmony rather than a scale segment, it makes sense to write the interval as a major seventh instead of a diminished octave. The combination of notes in all three strands in the superformula's first two bars can then be understood as a major-seventh chord on G with a minor third (Michael's second note being an upward-resolving appoggiatura to B-flat), and added fourth and sixth in the Eve layer.

This brings me to another observation, which is that many of our conventional (textbook) definitions of consonance and dissonance rely on musical contexts of bygone eras. In much twentieth-century music (and, especially, in jazz) simultaneous intervals that would have been sharply dissonant in, say, a motet by Palestrina or a string quartet by Haydn are treated as consonant. The major-seventh chord is a cliche ending in many a jazz standard, and the dominant seventh is treated as a perfectly restful Mixolydian tonic seventh chord, given the correct context and voicing. Voicing of chords is also an important factor in the Licht superformula and, by extension, in the elaborations of it into the component parts of the opera cycle.

Melodic dissonance is a separate category, but by no means any less dependent on style. The traditional definition roughly equates size of interval between two successive notes with the degree of dissonance, one important exception being the tritone, which is regarded as much more dissonant than the (larger) perfect fifth or minor sixth. However, it is also more dissonant as an augmented fourth than as a diminished fifth, because of the possibilities for resolution once the leap has been made. This depends entirely on a solid diatonic context, however, and to a large degree on a monophonic or contrapuntal texture free of consideration of arpeggiating chords. In other words, the style of the 15th and 16th centuries. By contrast, think about the beginning of Burt Bacharach's "(They Want to Be) Close to You". Assuming a key of C minor, it begins with two upward leaps, C to E-flat, and then on to B-flat, a minor seventh above the opening note. Not what you would call a singer-friendly melodic line, unless you consider a minor-seventh chord as a workable framework. Bacharach certainly does; Artusi would have had a fit, in part because he wouldn't even know what a "chord" or an "arpeggio" are, let alone a seventh chord.

So, is Lucifer's initial leap a dissonance? It depends on the context. As a part of that major-seventh chord with a minor third, the harmonic context can make it seem a consonance; with a more essentially melodic focus, and in light of the third note of the Lucifer formula, it may be heard as the sharpest of dissonances, resolving downward by semitone to the relative consonance of a minor seventh.