In this section we discuss several of Stockhausen's examples in greater detail, including one which Adriaan Fokker has given some corrections for. First, we discuss Stockhausen's examples on his page 11 summarized here in Table 1, on his page 13 and 16 summarized here in Table 2, and on his page 21 summarized here in our Table 3. Basically, Stockhausen shows various ways one might try to derive a complete scale of equally spaced durations, in correspondence with the equal tempered chromatic pitch class scale; he goes through several unsatisfactory derivations.
After that, we discuss his further examples, in which he does successfully establish a chromatic scale of durations, but at a costly price: his abandonment of serial principles of total quantitative organization of musical parameters, and his acceptance of variable factors out of the composer's control. This revelation leads his music out of the contradiction inherent in the music of total serialism. The final example we consider seems, at least to Fokker, to reinforce the impression that Stockhausen is a musician and not a scientist.
The first example begins with a consideration of whether it is feasible to construct scales of durations, analogous to the scales we construct in the sphere of pitch. Taking a unit duration of one second as the fundamental, Stockhausen refers to traditional notation of duration, in which a smallest unit is given as an eighth-notehead, and other durations are whole-number multiples, i.e. quarter note, half note, whole note. The smallest unit either remains indefinite, or else is metronomically defined (i.e., MM 1/8th note = 60, meaning 1/60 minute, or 1 second).
"It is quite clear that a metronome does not determine durations, but a tempo. It presupposes a series of repeating beats. Therefore it does not define a duration, but a frequency, in cycles per minute. The notation should not be M = 60; but: frequency = 60 cycles per minute (f = 60/M), What is the origin of this muddling of concepts: phase, period, frequency?" ([Fokker1962, p. 69)
In the light of Fokker's suggestion, to clarify the issues as much as possible, we will refer to frequency or duration as appropriate, using MM metronome markings only when explicitly needed in the musical context. Therefore, even though Stockhausen writes MM = 60, we consider "duration = 1 second" when it makes things clearer.
Table 1
| Column 0 | Column 1 | Column 2 | Column 3 |
| From p.11 | From p.11 | From p. 11 | |
| Chromatic number (pitch name) | Mult. by unit duration | Division of Largest unit | Constant 2/1 proportion |
| 13 (A octave) | 13 sec. | 1/13 sec. | 8192 units |
| 12 (Gsharp) | 12 sec. | 1/12 sec. | 4096 units |
| 11 (G) | 11 sec. | 1/11 sec. | 2048 units |
| 10 (Fsharp) | 10 sec. | 1/10 sec. | 1024 units |
| 9 (F) | 9 sec. | 1/9 sec. | 512 units |
| 8 (E) | 8 sec. | 1/8 sec. | 256 units |
| 7 (Dsharp) | 7 sec. | 1/7 sec. | 128 units |
| 6 (D) | 6 sec. | 1/6 sec. | 64 units |
| 5 (Csharp) | 5 sec. | 1/5 sec. | 32 units |
| 4 (C) | 4 sec. | 1/4 sec. | 16 units |
| 3 (B) | 3 sec. | 1/3 sec. | 8 units |
| 2 (Asharp) | 2 sec. | 1/2 sec. | 4 units |
| 1 (A) | 1 sec. | 1 sec. | 2 units |
Column 1 of Table 1 shows the actual durations that result, which form a harmonic overtone series. Note that this method is equivalent to multiplication of the chromatic scale number by the fundamental unit duration. That is, the fifth scale degree has the duration of five seconds. Although he doesn't specifically mention it, the ratio of successive scale degrees is not constant. The ratio of the second to the first scale degree is 2:1, the ratio of the third to the second scale degree is 3:2, and so on (he mentions, in the context of the second example immediately following this one, that a non-constant ratio is unsatisfactory).
His second example, yielding the results in Column 2 of Table 1, is derived by taking a largest duration and sub-dividing it for each successive partial harmonic. This would correspond to a metronome marking assigning MM whole note = 60, or one second for a whole note, with a half note = 1/2 second, quarter note = 1/4 second, etc. Note that this corresponds to multiplying by the reciprocal of the chromatic note number, i.e. the fifth scale degree has the duration of 1/5 second. Also note that the durations get shorter as the scale degree goes higher. This also results in a series of durations with unequal ratios between the successive pairs, which he doesn't like, because his serial principles need equal proportions between the elements of his scale. The ratio between the second and first scale degrees is (1/2):1, which is the same as 1:2. The ratio between the third and second scale degrees is (1/3):(1/2) which is the same as 2:3, different than the first ratio, for example. The ratio between the 12th and 11th scale degrees is (1/12):(1/11) which is the same as 11/12, only slightly less than unity. The ratios between successive scale degrees form the sequence [1:2, 2:3, 3:4, 4:5, ...], which approaches but never quite reaches the limit of unity as the scale degree increases infinitely high.
Next, he tries making a scale of durations which will meet his needs and exhibit constant proportions, i.e. equal ratios between all pairs of successive scale degrees, by using logarithmic relationships. Thus a constant proportional ratio (he uses 2:1 for this example) must be used to derive each next duration. The fundamental is given by him as 2 unspecified units. We don't know why he starts at
21 = 2 units instead of at 20 = 1 unit, which would be more natural. Each successive scale degree is formed by taking the exponential power of 2 raised to the scale degree. Column 3 of Table 1 shows the result, which differs greatly from the duration series of Columns 1 and 2 in its form. The highest value shown, the next "octave", is a whopping 8192 units.
One would have to fill in a metronomical duration value for the unspecified units, and he warns that this value cannot be made very small because of the limits of our perceptual facility to distinguish between durations which differ by the ratio of 15:16. He points out that in the sphere of pitch, an equal-tempered semitone corresponds roughly to a pure frequency ratio (interval) of 15:16 Although his remarks are confusing in the context of the example given, we interpret them as pointing out that we can't make the fundamental duration too small a metronomical unit, because any groupings of multiple quantities of these small units will be too hard to distinguish, due to their small overall length. That is, if we made the fundamental unit 1/100 of a second, we couldn't distinguish a grouping of two units (2/100 second) from a grouping of three units (3/100 second) very easily. This is true regardless of how the duration scale itself is chosen, proportionally, which is why his comments are confusing.
On the other hand, the metronomical unit cannot be chosen too large, because of the enormous ratio between the upper scale degrees and the fundamental. Presumably, if they are to actually be used as note durations in a real piece of music, there are practical limits to the longest acceptable duration. For example, if the metronomical unit of 1 second were chosen for the fundamental, then the first "octave", the 13th scale degree, would have a duration of 8192 seconds, which is more than two and 1/4 hours! Even the 6th scale degree would have a duration of 64 seconds, more than a minute. Clearly the fundamental duration of this scale must be chosen very small, and then we run into the perceptual limit mentioned in the previous paragraph at the low end of the time scale. So the vast range of durations in this scale, the way it grows geometrically, is a problem.
Table 2
| Column 0 | Column 1 | Column 2 |
| From p. 13 | From p. 16 | |
| Chromatic no. (pitch name) | Smaller mult. by unit duration | Smaller division of largest unit |
| 13 (A octave) | 1/154 sec. | 1/1300 sec. |
| 12 (Gsharp) | 1/167 (166?) | 1/1200 sec. |
| 11 (G) | 1/182 sec. | 1/1100 sec. |
| 10 (Fsharp) | 1/200 sec. | 1/1000 sec. |
| 9 (F) | 1/222 sec. | 1/900 sec. |
| 8 (E) | 1/250 sec. | 1/800 sec. |
| 7 (Dsharp) | 1/286 sec. | 1/700 sec. |
| 6 (D) | 1/333 sec. | 1/600 sec. |
| 5 (Csharp) | 1/400 sec. | 1/500 sec. |
| 4 (C) | 1/500 sec. | 1/400 sec. |
| 3 (B) | 1/666 sec. | 1/300 sec. |
| 2 (Asharp) | 1/1000 sec. | 1/200 sec. |
| 1 (A) | 1/2000 sec. | 1/100 sec. |
Next (see Column 1 of Table 2), he tries multiplying the chromatic scale degree by a smallest durational unit again. Apparently this is what was done previously in serial music when scales were established for durations, in what he now considers a naive attempt to mirror, in the sphere of duration, the scales of equal-tempered frequencies in the sphere of pitch. This time, his fundamental unit duration is a small fraction (1/2000 second) instead of the 1 second he used last time. And this fraction, when multiplied by the chromatic number of each successive chromatic step, yields durations which approach closer and closer to the upper limit of unity (1 second), unlike those in in Column 1 which will keep increasing unboundedly forever. Thus the overall range of durations is limited, which avoids the problem in the previous example (Column 3 of Table 1). Moreover, the resultant frequencies, assuming that the durations are each one cycle of continuous sound-waves, are in the audio range, producing the sensation of pitch rather than rhythm.
This resultant series in Column 1 of Table 2 is what Stockhausen now calls a "sub-harmonic series of proportions" (the term which, as we already pointed out in the previous section, physicist Backus particularly singled out as unnecessary jargon). The ratios between successive scale degrees is still the same as it was in Column 1, in spite of the change in fundamental from 1 second to 1/200 second. The ratios still go [2:1, 3:2, 4:3, 5:4, ...].
He plots the successive durations as pitches on a treble clef (our durational notation makes it easier to see than his metronomical units, because the fundamental frequency is 2000 cycles per second, the first partial "undertone" is 1000 cycles per second, the third is 666 cycles per second, and so on, with the 13th partial, or the first "octave", being 154 cycles per second). He finds that the pitches form a mode, rather than a chromatic scale, in which not all pitches are present. He comments that this (durations arranged in a mode) was used by Messiaen in his piano study (Modes de valeurs et d'intensites).
He points to the mistaken use of such a mode of durational values, as if it corresponded to a chromatic scale of pitches when it actually corresponds to a mode, and remarks that this led to major contradictions in previous serial music. The extremely long durations, when occurring with equal frequency as the extremely short durations, tended to predominate overwhelmingly in the absolute length of the piece. Plus, the proportions were uneven, and "hierarchically preordained" which is unacceptable in his serial system.
We already commented that this example is much more constrained in its range of values (from 1/2000 second to 1/154 second in its first "octave", converging towards unity as a limit) than the previous example (which ranged on up to 8192 units already in the first octave, with no upper limit). Note that as long as the metronomical unit assigned to the fundamental duration is less than one second, all the chromatic scale degrees will also be less than one second, and will tend towards unity (one second) as their upper limit.
A larger choice of metronomical unit for the fundamental duration, bringing it out of the audio frequency range and into the sphere of rhythm (i.e. with a fundamental larger than 1/16 second) will actually compress the range of values, since they are limited above by unity. Too much range compression would lead us again to the problem where distinctions would be too small to perceive.
We do not fully understand Stockhausen's misgivings about this "subharmonic scale of proportions", especially since his objections, based on his serial principles, to the uneven proportions and hierarchical pre-ordainment of successive durations, are hard to appreciate, some 35 years later in a time with different compositional aesthetics.
Stockhausen writes that composers tried to work around the shortcomings of such a system by such tactics as transposing the mode up and down when necessary (changing the underying metronome speed) to reduce the range between largest and smallest units, but he feels that this violated one of the stipulations of the serial system, that the absolute tempo of the shortest duration should be fixed. This is because the proportions, his all-important unchanging "light" in which the ever-changing "objects" are to be viewed in ([Cott1973], p. 225]), would be obscured by such changes.
The serial composers who tried to work with this subharmonic series of proportion also had to resort to writing different "parts", each in its own layer of durational series, which are all then superimposed on one another. Stockhausen criticizes this writing of "parts" as being inappropriate in serial music, and criticizes polyrhythms in general too.
Stockhausen tries once again to derive a truly chromatic scale of durations (Column 2 of Table 2) this time by dividing a largest time unit again, like he did in our Column 2 of Table 1, but with a fundamental unit of 1/100 second this time, to bring this series into the sphere of pitch like the last example. This time he produces a harmonic series of durations, which again is seen to be another mode when he displays it on the musical staff as pitches, going up in pitch instead of down like the last one.
Now we come to a crucial turning point in his presentation. The notes which he composes with, in the sphere of pitch, are each composed roughly of harmonic overtone series built on fundamental frequencies just like Column 2 of Table 2. Yet these harmonic pitches can be treated as elements of a chromatic series, in spite of their internal harmonic nature. He calls this a "contradiction" between the harmonic and chromatic scales of perception. Thus there is an inner and an outer nature to the notes, which must be treated separately and not confused. His serial system has to operate on objects at some level, but those objects can then have an internal structure which may also be built up serially, or harmonically.
At the end of our previous section, we quoted H.Vaggione ([Vaggione1990]), who spoke of digital sound-objects being "open" to internal timbral composition, or "closed" under some name, made available to note-based composition. This is basically the 1990's digital object-oriented restatement of Stockhausen's key concept here. To achieve the flexibility he desires in a compositional system, he must have access to both the open and the closed representations of a musical object, in the domains Stockhausen refers to as the sphere of pitch and the sphere of rhythm, which in the 1990's presentation are, respectively, "quantum" and "fractal" in nature ([Laske1990]).
It appears that Stockhausen's discovery of the use of individual impulses to open up the micro-time structure of sound-objects, unlocking some of the puzzles that limited previous searches for musical parameters to serialize, has led to the limit, hence the breakdown ,of his former serial system of totally organized composition. He expresses the breakdown here not only in the form of this contradiction between the "harmonic scale of perception" and the "chromatic scale of perception", but more importantly in the other contradiction he mentions twice later on, between "material and method", between the actual nature of sonic objects and his cherished serial principles! And in order to continue composing, he suggests later, one must come to terms with his new concept of musical time (i.e. the second one presented in his article! - the first was the notion that pitch and rhythm are reflections of the same phenomena, but on different time-scales), in which the variability of the human performer becomes an explicit element, a resource of new life in the music that had grown sterile with the limits ot total serial control.
Now Stockhausen explains that the key to deriving a chromatic series of durations is to apply this contradiction (between the harmonic and chromatic scales of perception), discovered in the sphere of pitch, to the sphere of duration. He does so, and this would seem to be his moment of triumph.
But in reaching this goal, where he can now apply the principle of total serial organization to duration, he finds that other considerations appear, new ones resulting from performance variability and "statistical", or mass-structure, which he finds interesting, perhaps more attractive than his initial goal, now that it has been reached. And these new considerations are examined in the remainder of his article.
So his compositional system reaches new levels and matures. And as a result, he also begins to compromise and change his serial system, to retreat from the absolute principles of total quantitative organization, that so dominated his and other composers' thinking in the previous six or seven years, to a new principle of qualitative structure, which would eventually evolve so far as to allow him to claim that he still followed "serial thinking" even in his improvisational and intuitive space music of the 1960's and 1970's - refer to the quote from our section on serialism, ([Cott1973], p. 100
We now discuss how he finally obtains a true chromatic scale of durations. Columns 1 and 2 of Table 3 show the result when he takes a fundamental unit, a frequency of 1 cycle per second which he denotes by the metronome marking MM = 60, and multiplies it by the 12th root of two, the same way one multiplies a fundamental frequency to derive a 12-note equal-tempered chromatic pitch scale.
In our Column 1 of Table 3, we have followed our convention of notating durations, and in Column 2 of Table 3, we give the corresponding MM metronome markings. We use the numerical approximation 1.05946 to the irrational number which is the 12th root of 2. Each successive scale degree is obtained through multiplication by this factor, in Column 1, or through multiplication by its reciprocal, i.e. division by this factor, in Column 2 (since, as Fokker pointed out, frequency is the reciprocal of the metronome marking, subject to scaling by time-units in seconds instead of minutes).
Table 3
| Column 0 | Column 1 | Column 2 |
| From p. 21 | From p. 21 | |
| Chromatic number | Chromatic
duration
(x 12th root of 2) |
MM markings for Column 6 |
| 13 (A octave) | 0.500 sec. | MM = 120 |
| 12 (Gsharp) | 0.530 sec. | MM = 113.3 |
| 11 (G) | 0.561 sec. | MM =106.9 |
| 10 (Fsharp) | 0.597 sec. | MM= 100.9 |
| 9 (F) | 0.630 sec. | MM = 95.2 |
| 8 (E) | 0.667 sec. | MM = 89.9 |
| 7 (Dsharp) | 0.707 sec. | MM = 84.9 |
| 6 (D) | 0.749 sec. | MM =80.1 |
| 5 (Csharp) | 0.794 sec. | MM =75.6 |
| 4 (C) | 0.841 sec. | MM = 71.4 |
| 3 (B) | 0.891 sec. | MM = 67.3 (67.4?) |
| 2 (Asharp) | 0.944 sec. | MM = 63.6 |
| 1 (A) | 1 sec. | MM = 60 |
For the case of the third scale degree in Column 2 of Table 3, our calculations (60 x 1.059463) give a result of MM = 67.3 when rounded off to one significant decimal place, but Stockhausen's example listed the value MM = 67.4. All our other numbers agree with his, except for this one.
The problem with the resultant scale of durations is that most of the values have to be rounded off to approximate decimal numbers, and can't be represented well on a real-life metronome, but he is willing to put up with some imprecision.
Once he has fixed this chromatic scale of durations, he wants to apply his serial principles to it, first choosing a serial row, then fixing the register of each value, etc., but freely translating between the sphere of pitch and the sphere of duration, so he is trying to maintain an abstract level of proportions, which are instantiated on the two levels of pitch-perception (higher frequency) and duration-perception (much lower frequency).
We now consider his examples 8 - 12, on his pp. 22 - 23, summarized and clarified in our Table 4.
Table 4
| Column 0 | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 | Column 6 |
| No.inSeries (read up) | Chosen Pitch Class | Steps to Next Element | MM No. | Register Choices | Octave Trans. | MM Unit (based on Half Note) |
| 12 | G5 | -6 | 106.9 | G5 | +0 | Half note |
| 11 | Dsharp5 | +4 | 84.9 | Dsharp4 | -1 | Whole note |
| 10 | Gsharp5 | -5 | 113.3 | Gsharp4 | -1 | Whole note |
| 9 | F5 | +3 | 95.2 | F5 | +0 | Half note |
| 8 | E5 | +1 | 89.9 | E4 | -1 | Whole note |
| 7 | Fsharp5 | -2 | 100.9 | Fsharp2 | -3 | Whole X 4 |
| 6 | C5 | +6 | 71.4 | C3 | -2 | Whole X 2 |
| 5 | Asharp4 | +2 | 63.6 | Asharp3 | -1 | Whole note |
| 4 | B4 | -1 | 67.4 | B4 | +0 | 1/2 note |
| 3 | D5 | -3 | 80.1 | D4 | -1 | Whole note |
| 2 | A4 | +5 | 60 | A3 | -1 | Whole note |
| 1 | Csharp5 | -4 | 75.6 | Csharp6 | +1 | 1/4 note |
The first step of his serial procedure is to choose a row-order for the series. He displays his chosen series first in the pitch domain, as note-heads on the musical staff. We list the pitches in Column 1 of Table 4. In his example, he has written them all in the octave starting with A4; that is, from his lowest note head at A4, up to B4, then C5 through his highest note at Gsharp5. We need to keep the octave register numbers straight, since we are using columns of alphanumeric symbols instead of note heads on a musical staff for our representation of his examples. We are confused as to why he does not merely work with octave-less pitch classes at this point, and apply his serial principle to the scale of possible octave registers to devise a series of octave transpositions to be applied to the pitch classes....
He has written in signed numbers indicating, for each element of his series, the number of chromatic half-steps up or down to the next element. We list these in Column 2 of Table 4. Thus from the first element C sharp to the second element A, we must descend by 4 half-steps in the negative direction, and from the second element A to the third element D, we ascend by 5 half-steps in the positive direction.
Note that when we ascend by one half-step in the positive direction, we are actually multiplying a base frequency value by the 12th root of 2 (1.05946) to get a new frequency value, and when we descend by one half-step in the negative direction, we are actually dividing the base frequency by the 12th root of 2. But in the sphere of pitch, we have a convenient shorthand, in the alphabetical note names, for indicating this underlying mathematical procedure.
Next he has indicated (see our Column 3 of Table 4) the corresponding MM metronome tempi, which he previously assigned via logarithmic calculations to each scale degree in his chromatic scale of durations, in the sphere of duration, for each element of the series. Although the correspondence is not spelled out explicitly, we deduce that he is arbitrarily using the alphabetical notation of the pitch class A (denoted on his musical staff by a note head at A4) for the fundamental pitch in his chromatic pitch scale.
The evidence we rely on for this deduction is that the note A is shown to correspond to the tempo MM = 60. And we recall from Column 2 of our previous Table 3, where we listed the tempi he assigned to his chromatic scale of durations that is now being taken as the basis for this choice of duration/tempo elements in series order, that the fundamental duration was assigned MM = 60 there. Therefore, since the note A apparently corresponds to MM = 60, and since MM = 60 was specified as the fundamental duration, we reason that A must be the fundamental pitch in his pitch scale.
The "steps to next element" listed in our column 2 of Table 4 now apply, in the sphere of duration as well as in the sphere of pitch. They still correspond to multiplication of a base value by the 12th root of 2, but since we have no alphabetical note names as a shorthand for this procedure, we must write down the exact metronome tempi instead, (rounded off to one significant decimal place) in Column 3 of Table 4. The first element of the series, in the sphere of duration, is the tempo MM = 75.6.
We can refer back to Columns 0 and 2 of our Table 3, to verify that his original construction of a chromatic scale of durations assigned the tempo 75.6 to the 5th degree of the scale, which has the alphabetical note name Csharp if one starts from degree 1 as the note name A. And we actually do use Columns 0 and 2 of our Table 3 in this fashion, along with Column 1 of Table 4, to produce the entire Column 3 of Table 4, finding the tempo which corresponds to the scale degree which in turn corresponds to the alphabetical note name of the next element in Stockhausen's chosen series.
We use Column 2 of Table 4 to verify that the tempo numbers are correct -- we carry out the actual multiplication of the previous tempo value in the series by the 12th root of 2, raised to the power given by the next highest element of Column 2 of Table 4(the number of steps to the next element), which then yields the next highest element of Column 3 of Table 4.
The next step (see Columns 4 and 5 of Table 4) is to place the pitches of the series in octave registers, and to carry out the analogous operation in the sphere of durations. This is a series of octave transpositions, one for each element of the series. The register choices in the sphere of pitch are given in our Column 4 of Table 4 as alphanumeric note names, and the signed magnitude of the octave transpositions from the original portrayal of the pitch series is given in Column 5 of Table 4.
We wonder how the sequence of octave transpositions (Column 5 of Table 4) was chosen; why did he not discuss this in the context of his serial method?
We would expect, for the next step of this procedure, to apply the octave transpositions of Column 5, element-by-element, to the series of metronomic tempi in Column 3. We expect to put the results in Column 6 of Table 4. Stockhausen does something unexpected here though.
Octave transposition corresponds to multiplying by 2 for each octave up, or dividing by 2 for each octave down. Thus, for the first series element MM = 75.6 to be transposed +1 octave, we would expect a new tempo of MM = 151.2 to be the result. But instead, Stockhausen carries out the octave transpositions by using a different note-head to denote the fundamental durational unit of each series element in its transposed form, without changing the tempo number itself. He starts with a basic unit of one half note as the original assignment.
Thus for the first series element MM = 75.6 transposed +1 octave, he keeps the MM tempo of 75.6, but assigns the note value of one quarter note to it, since a quarter note is twice as short in duration as the original half note. For the second series element of MM = 60 transposed -1 octave, he writes MM whole note = 60. And so on, for all the elements of the series.
Column 6 of Table 4 lists the note head values. Each one corresponds to the octave transposition register in Column 5 of Table 4 beside it, with the origin, corresponding to +0 transposition, denoted as a half note.
Table 5
| Column 0 | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 |
| No. in Series (read up) | Rounded Metrical Series | 2nd Rounded Metrical Marks | Stockhausen "harmonic proportions" | Fokker corrected proportions | Fundamental durations |
| 12 | 1/2@Q= 107 | H=103 | (?? not given) | (8:11) | 5/9 sec. |
| 11 | 2/2@H= 85 | W=87.3 | 4:10 | 7:5 | 1-4/11 sec. |
| 10 | 2/2@H= 113 | W=110.4 | 12:9 | 18:7 | 1-1/29 ?sec. |
| 9 | 1/2@H= 95 | H=97.7 | 5:3 | 3:4 | 3/5 sec. |
| 8 | 2/2@H= 90 | W=89.6 | 6:13 | 7:12 | 1-1/3 sec. |
| 7 | 2/2@H= 101 | 4xW.=102.4 | 2:7 | 15:7 | 4-11/16 sec. |
| 6 | 4/2@H= 72? (71) | 2xW.=70.3 | 11:8 | 25:7 | 3-3/8 ? sec. |
| 5 | 2/2@H= 64 | W=63.3 | 9:5 | 7:10 | 1-7/8 sec. |
| 4 | 1/2@H= 67 | H=63.5 | 13:6 | 4:7 | 7/8 sec. |
| 3 | 2/2@H= 80 | W=80 | 7:12 | (7?) 9:15 | 1-1/2 sec. |
| 2 | 2/2@H= 60 | W=60 | 3:4 | 4:3 | 2 sec. |
| 1 | 1/4@Q= 76 | Q=75 | 10:2 | 5:1 | 2/5 sec. |
Next comes a transformation which we do not understand, which results in Column 1 of Table 5. Stockhausen's remark ([p. 22]) is "If this were barred normally, the result would be the following metrical series..." and then he gives a set of new specifications for each series element. Each element has a note head unit value and a metronomical tempo, but it also has an indication of the measure.
Thus the first series element is a measure of 1/4, at MM quarter note = 76. Every other series element after the first one has a measure with 2 in the denominator instead of the 4 that this first measure has in its denominator, and we don't understand why the denominator is varied only for this first measure.
Perhaps because it is the only one which involved an octave transposition of +1, in contrast to the rest of the series elements which all have either zero transposition or else have negative octave transpositions, in magnitudes ranging from -1 to -3. Perhaps there is an unstated principle here that the denominator of the measure indicator should only change when the underlying unit has to get smaller than the fundamental half note; the other series elements just involve larger multiples of the fundamental half note so that the constant unit of half note can be carried through them all, just by changing the numerators of each measure indicator to match the magnitude of the octave transposition.
But this cannot explain why the quadruple-whole-note series element (#7) and the double-whole-note series element (#6) get a measure of 2/2 time, which is the same as the whole-note gets! It seems that information is being discarded here and we don't understand the principle involved. Only the half-notes get a new measure numerator, making them a single 1/2 time measure of MM half note = 67 for series element #4, MM = 95 for element #9, and MM = 107 for element #12.
And the metronome tempi have been rounded off to the nearest whole number, except for the case of element #6, original MM = 71.4, new value MM = 72 (which should be 71 to be consistent with the rest of the series, which has followed the standard rule of rounding up for a decimal value greater than .5 and rounding down for a decimal value less than .5).
Stockhausen mentions a simplification which can be made in series elements #3 and #11, because they follow in a simple harmonic ratio from the preceding series element. Thus #3 which is notated as a 2/2 measure of half note = 80 follows a measure of half note = 60, and this forms a ratio of 80:60 = 4:3, so he rewrites series element #3 as a measure of 3/4 time at the same tempo as the previous measure (i.e. half note = 60).
Similarly, he shows series element measure #10 is a measure of 2/2 time at half note = 113, and measure #11 is a measure of 2/2 time at half note = 85, and somehow he must be rounding off measure #11's tempo to MM = 84.75 which puts the two elements in a ratio of 113:84.75 = 3:2, so he rewrites series element #10 as a measure of 3/4 time at its previous MM = 85, followed by measure element #11 as a measure of 2/2 time at this same tempo. He seems to be rounding up and down at will to make things approximately fit a tempo scheme that will be easier to count and conduct. (we have not made these two simplifications in our Table 5, however).
Now he addresses our question concerning the octave register placements selected to transform Column 1 of Table 4 into Column 4 of Table 4 in the sphere of pitch, and Column 3 of Table 4 in the sphere of duration into Column 1 of Table 5. We asked whether the choice of octave registers was made arbitrarily, and why it was not treated according to his serial principles.
He mentions that "One is misled into doing things in this order" ([p. 23]) because the chromatic series with its unit interval of the 12th root of 2 between successive scale elements is assumed. And then he says something which shows a significant breakdown of his previously strict serial principles - he says that it is not as interesting, when listening to series, to observe that all the chromatic steps should appear, as it is to observe which proportions are chosen between the elements of the series; that is, to see how the series elements are composed in relation to one another (even if some of them are left out?) He seems to be violating a cardinal principle of his previous serial system, by allowing the possibility of using a "mode", even though he spoke out against modes and their inappropriateness in the serial method, earlier in this same article.
Following this drastic announcement, he now rewrites the series in terms of the all-important ratios, or intervals, between successive elements. Thus, instead of deriving the "steps to next element" which we wrote in Column 2 of Table 4 previously, from the elements themselves, he derives the elements themselves from the ratios. He first specifies a series of ratios, and a given first element of the derived pitch/duration value series. Then all the elements of the pitch and duration series follow from these generators
And he writes pure small-number ratios, admitting that the resultant actual elements will deviate from the chromatic equal-tempered values. This then violates the second of the cardinal principles of his previous serial system, because the elements of the derived pitch and durations scales which result do not completely cover the spectrum in equal-valued steps!
It seems that he has had to remake his serial system in order for it to survive the contradictions he has pointed out in his article - the contradiction between the harmonic and chromatic "scales of perception" on the one hand, which is leading him to accept deviations from his equal-stepped chromatic scales, including the possible omitting of certain elements, resulting in the "modes" which he spoke out against, and the contradiction between the "material and the method" on the other hand, (his serial system and the human interpreters) which is leading him to simplify, through arbitrary and sometimes inaccurate rounding-off, the tempo numbers his system is generating, presumably in the interest of easier performance by human musicians and conductors.
He writes out a series of proportions which, he indicates, approximately express the ratios of the successive elements of the fundamental durations/pitches in his previously selected element-series. And he lists a new set of metronome markings, which we are asked to compare with the previous set (listed as the right side of the @-signs in our Column 1 of Table 5, i.e. disregarding the measure time signatures to the left of the @-signs).
We list these new metronome markings as Column 2 of Table 5, and we list these "harmonic proportions" ([p. 23]) in our Column 3 of Table 5. These metronome markings presumably arise from multiplying a fundamental tempo by the successive ratios of this new harmonic proportion series, but we are not quite sure of the details, since there used to be a fundamental tempo of MM half note = 60 in the 2nd element of the series, but it has now been transformed into a whole note = 60 through some means we do not understand.
At this point, we turn to physicist Adriaan Fokker, who has been trying to follow Stockhausen's complex procedure in an attempt to help explain it more simply, because we do not understand this next step at all ourselves!.
And Fokker indicates that Stockhausen's series of "harmonic proportions" contains some glaring errors which seem to be deliberate distortions. We list Fokker's corrected proportions in Column 4 of Table 5, alongside Stockhausen's proportions.
Fokker questions why Stockhausen has chosen more complex ratios, with higher integers in them, and has even chosen inaccurate ones, like the 7:12 for a major sixth interval which should be 3:5 (Fokker is "startled" by this "high-handed procedure", [Fokker1962] p. 72). Unfortunately, the text of Fokker's article shows 7:15 as his correction for Stockhausen's 7:12 in the ratio of the 8th scale element to its upper neighbor, so we are confused; perhaps it is a typographical error and should be 9:15 = 3:5, and we have written it so in our Column 4 of Table 5.
Fokker lists other problems with Stockhausen's interval series: Asharp through C to Fsharp is written as 5/9 x 11/8 = 99/40, but Fokker says it should be 100/40 = 5/2, giving a much simpler ratio. C through Fsharp to E should be 2/5 = 22/55 but Stockhausen has written it as 11/8 x 2/7 = 22/56. E to Gsharp should be 4:5, but Stockhausen has written 6/13 x 5/3 = 10/13 = 20/26 instead of 20/25 which reduces to 4/5. Stockhausen correctly writes 5/3 for F to Aflat, but also uses the same ratio for F to Gsharp, which is wrong (the correct ratio is 12/7). Stockhausen hs written 7/12 for the major sixth from D to B, and is hence off by a factor of 35/36. And the interval C to Fsharp, which Stockhausen wrote as 11/8, should be 10/7.
Fokker asks, after pointing out all these discrepancies, "It is an amazing puzzle. Why was the author unaware of these discrepancies?" (op.cit.) Stockhausen has made errors of an entire semitone (25/26 and 77/80) in magnitude, which are quite audible in the sphere of pitch intervals.
Finally, Fokker notes that Stockhausen left out one ratio. Substituting a value of 11/8 for the empty spot (at the 12th scale degree in our Column 3 of Table 5), and thereby returning from G to Csharp for the next octave of the series, as one would naturally expect, Fokker notices that Stockhausen would have written out a collection of fractions where all the numbers from 2 through 13 have appeared, both somewhere in the numerators and somewhere in the denominators.
"The completeness would be lost if any of the discrepancies shown were to be amended. That is true. But there is no reason for that. What is the use of such a frivolous play with, or of such regard for, numbers? Is there any music in it? If any of us, physicists or other scientists, were to offer this sort of argument, we should certainly be reminded at once that numbers do not make music. The author [Stockhausen], however, is quite an honourable and serious musician. I therefore feel justified in asking: what is the musical sense of such a dance with numbers, which so obviously falsify actual facts?" (op. cit.)
We have one more column remaining in our Table 5 -- in Column 5, we list the fundamental durations that Stockhausen finally lists, in seconds, for the elements of the series, after all the rounding off has been completed to his satisfaction.
The first element is given as 2/5 seconds duration. Looking back at our previous tables, this element comes from an original pitch-series note Csharp6, and a MM marking of a 1/4 time measure at quarter note = 75.6 (subsequently rounded off first to 76 for Column 1 of Table 5, then rounded off to 75, for Column 2 of Table 5. We should be able to find some fundamental somewhere, probably related to either half note = 60 or whole note = 60, which is in a 5/2 ratio with some aspect of this element #1, presumably with the new rounded metrical mark quarter note = 75. We do note that 75 x (2/5) = 30, which is 1/2 of 60, so there must be some relation among the half note, quarter note, and whole note units which makes this come out right.
Since the 2nd series element, which should still correspond to the original fundamental (it was an A3 in the sphere of pitch, and MM half note = 60 in duration, before the most confusing transformations), is listed as having a duration now of 2 seconds, this would suggest the factor of 2 needed to match 75 x (2/5 seconds) = 30 and somehow relate it to the original MM = 60, but we are at a loss to explain how this exactly works out. It doesn't seem to be that essential to Stockhausen's purpose anyway.
All of this lengthy buildup, from the derivation of a chromatic scale of durations through the establishment of a corresponding series, rounded off somehow, and related to an intervening series of mysterious small-number ratio proportions, has been merely a prelude to the next subject. Now Stockhausen introduces Groups of fundamental durations, where the numerators of the proportional ratios indicate a first group-half and its division into sub-parts, and the denominators of the proportional ratios indicate a second group-half and its division into sub-parts.
Fokker suggests a new terminology to make this idea clearer. He has been denoting time-intervals as "whiles" instead of Stockhausen's misnomer "phases"; more specifically, he calls time-intervals in the sphere of duration "macrowhiles", and calls time-intervals in the sphere of pitch "microwhiles". Then this lengthy procedure of establishing a series whose elements correspond on the one hand to pitches and on the other hand to durations is a process of establishing proportional relations between macrowhiles and microwhiles:
"[Stockhausen] does not carry the idea of placing macrowhiles proportional to microwhiles to its conclusion. He inverts the idea. He takes a number of macrowhiles together in a group. For the sake of definite clarity I propose to call such a group a super-while. I want a special word, because it is not easy to explain the intricate procedure of the author.
"He places two equal super-whiles in succession. Each is divided into a number of macro-whiles. The two numbers should be proportional to the frequencies to the microwhiles concerned in the interval which has to be represented. However, he never uses the word frequency...." ([Fokker1962], p. 73)
Stockhausen is using, as input for high-level construction procedures, the numerator and denominator of each ratio in the series of "harmonic proportions" that he derived from the elements of his pitch/duration built on a permutation of his long-sought chromatic scale of durations. Recalling that Fokker has indicated the erroneous nature of many of these ratios, at least if they are supposed to truly represent the underlying series of duration/pitch elements, we see that Stockhausen is basically using numbers which he has constructed arbitrarily for their own sake. For example, Stockhausen uses 7/12 instead of the 3/5 which Fokker points out as the true ratio representing a major sixth interval; so Stockhausen's "super-while" equal-duration groups of "macrowhiles", which are divided, the first half into 7 equal parts and the second half into 12 equal parts, cannot be taken as a representation of the original frequency ratio in the sphere of duration.
It is evident that Stockhausen has shifted paradigms on us, leaving behind his original goal of establishing correspondence between pitch and duration for the purpose of total serial control, and instead pursuing a new goal in this new context of groups. This new goal is the composition of a work like "Gruppen" where three separate orchestras play different metres and rhythms (Stockhausen's "formant-spectra") in different spatial locations, with variability and uncertainty as an explicit factor in the composition.
The prescribed series of intervals is 10:2, 3:4, 7:12, 13:6, etc. Stockhausen has two equal super-whiles, which he calls groups. The first super-while (obtained from the first element of the original series of durations, i.e. the one involving quarter note = 76 tempo, now mysteriously changed once again into half note = 76) is divided into ten macrowhiles, namely ten quarter notes; the second super-while (from the original fundamental duration, half note = 60), which is the second half of this group-pair, is divided into two macrowhiles, namely two whole notes. These are super-whiles to which, as a pair, Fokker assigns the number I.
For the next proportion 3:4, one needs a super-while with three macro-whiles. The last of the now extant superwhiles I contains only 2 macrowhiles (the two whole notes), so a third macrowhile is added, extending a super-while I (the 2nd one) to a super-while II, containing 3 macrowhiles. Then an equal super-whileII follows, divided into 4 equal macrowhiles. For the next proportion 7:12, we have to extend this last super-while II by adding 3 more equal macro-whiles to give a new superwhile III with 7 equal macro-whiles. An equal super-while III follows, divided into 12 equal macro-whiles. This procedure is continued, adding a new super-while after extending the last one to get the right number of divisions for the numerator of the next proportion. So the interval 10/2 is served by super-whiles I, the 2nd proportion 3/4 is served by super-whiles II, ...., the proportion 11:8 is served by super-whiles VI, 2/7 is served by super-whiles VII, etc...
The posterior super-while VI contains 8 macro-whiles. The anterior super-while VII must contain no more than 2 macro-whiles. After these the posterior super-while VII enters. The anterior part of the first super-while VII overlaps with the posterior part of the second super-while VI. More overlappings occur elsewhere in the serial structure.
Stockhausen then says
"... in this way different numbers of equal duration (macrowhiles) are united in groups (super-whiles): these are equal in length from one group to the next (the first ten macro-whiles are equal in length to the next 2 macro-whiles)....Every group (super-while), however, with the exception of the first and the last, is ambiguous. It is a second member of a first interval (10:2) and a first member of a next interval (3/4).... From this ambiguity the result is either rests or temporal superpositions" ([p. 24]).
But Fokker says this is wrong, there is no ambiguity. The previous super-while is not actually a first member of the next interval, because of the rests that must be added. The new super-while, which consists of the old super-while plus new rests, is a different entity from the old one, so there is no ambiguity at all.
So in the first super-while II the so called 'group' contains two macro-whiles of the second super-while I, then also contains 1 rest, in order to make a complete super-while II of 3 macro-whiles. There are two "groups", with some constituent macro-whiles serving as members of both. The "temporal superposition" that Stockhausen mentions, according to Fokker, relates to the super-while VI (for the proportion 11/8) being longer (8 is larger than 2) than the superwhile VII (for the proportion 2/7) and this super-while is again longer (7 larger than 6, in the next super-while VIII for the proportion 6/13).
Stockhausen's diagram shows that sometimes, four super-whiles overlap. We are in agreement with Fokker; there is no ambiguity, and there is no "temporal superposition", there are merely multiple hierarchical levels, with shared elements playing different roles in the different entities at higher levels. We recall, however, that one of Stockhausen's previously expressed serial principles (if it isn't one he has abandoned along with the need for comprehensive equally-spaced intervals and for exact precise values with no rounding off, that is) is the inappropriateness of hierarchies in serial music. Perhaps Stockhausen is reluctant to look at this situation in the most natural way, in terms of multiply overlapping hierarchies, for this lingering ideological reason.
"Here again, if some scientist happened to hatch a temporal construction like that, intrigued by what looks so interesting on paper, everybody would ignore it. The author, however, is an honourable leading composer. We must presume that there are other composers who are working with such things and who can find use for them in their music." ([Fokker1962], p. 75)
"...These observations have been critical but I must emphasise that I want to support these endeavours at innovations in a constructive mood. I am not too old to understand that there are great problems which have to be tackled in the face of new possibilities. If someone attempts to present a scientific account of his aims and methods, we have to consider his endeavour with sympathy and with a realisation of its promise. But he ought not to invest scientific terms with other meanings....
"Pseudo-science is no better than pseudo-music. The explanations should be given in the simplest manner in the most familiar words. Where I have been denouncing failures, these denouncements belong to the feedbacks (in the sense of cybernetics) which have to be activated whenever a great task is attempted." ([ibid.] p. 79)
And those are Fokker's last words on the subject. We end this example, and this section of the paper, with one final thoughtful quote from Gottfried Michael Koenig's article which concluded the literary dialogue engendered by Stockhausen's paper.
"When studying theoretical articles, which the composers themselves must provide today, one must clearly distinguish between musical facts and resources of presentation frequently borrowed from mathematics. A widely-spread superstition is that composers of serial music are really mathematicians or at least arithmeticians; in the case of electronic music that they are really technicians or sound-engineers. Unfortunately the people who are taken in by such rumours do not bother to convince themselves of the actual state of affairs by glancing at the scores....Anybody in need of a plumber is also happy if the latter can also get a bent lock open. But woe betide the composer who knows how to work with logarithm tables, or who puts his nose in a book in order to find about the physiological processes of hearing. He who has contact with music has his own ideas, however they may be, about it; it is not always his fault if they are wrong. Only intellectual theft is spoken of; but the removal of mistaknness and untruth is at least as strictly punished. The confusion of music with non-musical assistance keeps on penetrating the music itself. Since serial music has been being composed, the opposition has not died out; now that some composers are beginning to transcend serialism too, the lie-a-beds and reactionaries are getting up and wanting to preserve serialism at least. Once one is used to opposing, one gets angry when the resistance disappears. There are reasons for assuming that it is evaporating even in composing. Especially as serialism -- as a coming-to-itself of all rationalisation in music -- favours the interception of all laterals in systematically placed nets. A sort of auto-motion is imputed to the musical material, the composer passing on the pressure from above....."
"...Music could be said to be the practical superstructure of theory. But this would cede to the latter a priority which it does not possess. By penetrating the form, what is heard inwardly assumes something of its constraint. It easily degenerates to an agreeable arrangement, to applied art." ([Koenig1962], p. 98)
Next section (Comparison of Henry Cowell's Rhythmic Concept with Stockhausen's, part 6)
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