A Phenomenological Experiment
With Arnold Schoenberg

Emmanuel Rens
Geneva, April 2009 - Revised in June 2012


Unless we wanted - as prescribed by the Swiss conductor and phenomenologist Ernest Ansermet - to reduce willingly our artistic curiosity for theological reasons, there will always be - and has always been since the beggining of the 20th century - a moment in which we'll experience a music that doesn't sound like music. The initiator of this cultural experience is named Arnold Schoenberg and the experience itself dodecaphonism (cf. note 1).

The main stream of music, the music that is heard, remains devoluted from the point of view of harmony. Contemporary music has made a large use of Schoenberg's discoveries but has also specialized in official and upper class happenings. Despite Adorno and perhaps because complexity can be seen as oppressive, dodecaphonism has only been received among composers like a liberatory form of art.

In terms of complexity, this presentation will stay hardly introductory, far below the level of state-of-the-art discourse in this field. It is only an attempt to address one specific problem: the fact that music theory turns out poor in integrating a musical practice as ordinary as transposition. Indeed consonance theory claims that it is the order of heard harmonics that decide of our feeling about an interval: the sooner the better. The chord or interval will sound harmonious when the same interval appears sooner in the harmonic scale (which is a fixed and theorically infinite compound of sounds that is part of almost every sonic event). In algebra it is peculiar to permutations to be mainly concerned by the position of elements in a row. On the other hand the row (or series) was the main tool invented by Schoenberg to produce music. It was a justified consequence that musical theorist Milton Babbitt finally described the schoenbergian and post-schoenbergian serial systems as permutational: privileging certain remote intervals they fake another order in the harmonic scale, they permute its values. This act gave a new impulse to musical creativity, till then confined to the beggining of the harmonic scale and its related circle of fifths.

What are permutations?

We can think of a permutation as an operation like multiplication, which does something with two things to produce a third. The permutation itself looks like a number but it's mostly a word. In base 12 the permutation contains cyphers that do not exist in base 10 because we are talking of permutations of distinct elements, so we need two more cyphers, A for 10 and B for 11. "01234456789AB" for example is a permutation of 12 elements, and each element has two values: its meaning and its place in the row. So let's write this permutation like ((0,0), (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7), (8,8), (9,9), (A,A), (B,B)). It shows that this permutation is not the best example since every cypher is exactly where it means. At least for this particular permutation - call it the chromatic scale - we should be able to skip the parenthesis notation. But we said earlier that a permutation is like an operation with two arguments and it is true to say that "0123456789AB" is the permutation of itself by itself. It will put each element of "0123456789AB" at the place it is already, so nothing will change actually and it will be like multiplying 1 by 1. As any number can be thought of as a multiplication of itself by one, every permutation can be thought of as permuting the chromatic scale (corresponding to its base, here 12). As 4 = 1*4, "19B786A24305" = "0123456789AB" P "19B786A24305" , where "P" is the operation of permuting and means strictly "permuted by". We see that a permutation can be described by saying what has been permuted in the chromatic scale and what has not. Taking fewer elements, in base 4, for the sake of clarity, we say that "0213" has permuted "2" and "1" but left untouched "0" and "3", and we will write this simply by putting "2" and "1" together when each one of the other signs will stay alone: (12,0,3), here the order has no importance, as long as what remains of the chromatic scale is isolated, the description will be good, so (3,21,0) is ok for the same "0213". This description is called here the "partition" of the permutation. Each permutation has several partitions that are equivalent and describe only their specific ordering of signs. In our previous example "19B786A24305" was not any random permutation, but the first row that was used in serial music by Arnold Schoenberg when he composed his Walzer Opus 23, in 1923. All permutations however are not permutations of the chromatic scale. For example you can permute the elements of "19B786A24305" a second time with "19B786A24305" and that will produce another row. If you repeat this operation enough times you will fatally come back to the chromatic scale whichever was the permutation you took in the first place, but depending on the permutation the return to the well ordering of cyphers will take more or less time. Permutations can differ by this feature. For example permuting "19B786A24305" by itself 10 times will produce the chromatic scale which only needs 1 permutation to reproduce itself when "0213" requires 2 to yield its analogon of the chromatic scale in base 4, and "9A0346578B12" - which is a row in the Serenade opus 24 - requires 4 times the same operation to get to the same chromatic scale. We can see this feature as the power, or degree, of a permutation, since it is a bit, mutatis mutandis, like what happens when we say that 4 is 2^2 or 8 the third power of 2.

Now that we have seen similitudes between permutation and multiplication, let's take a look at a big difference: if "0213" is called S, then "0213" P "0213" can be described as S^2 for instance, if the degree of S is 2, it means that S^2 = "0123", S^3 = S^1 = "0213" = "0123" P "0213" and S^-1 is also "0213" when S^0 = "0123". But don't bother too much! The point here is that whatever positive or negative integer you'll choose as power for your permutation, you will hit a series in its gradual subset (or suite) because unlike multiplication, permutation is cyclic. That is a difference from what happens with multiplication because no number greater than 1 equals the power 7 of itself but some permutations do (it is the case with "0213" for instance, which being of degree 2, equals also all odd positive or negative powers of itself). You might also want to permute two series that are not member of such a cyclic subset, and we will preferably speak of a composition of permutations in this case. This leads to another great difference: multiplying x by y always yields the same product as y by x, but permuting x by y is not the same as permuting y by x. For example "0312" P "3210" = "3021" when "3210" P "0312" = "2130".

It can reveal quite tedious to realise a permutation by hand because it's really fast to mess up with signs and order when you have to put for example the first sign of "0312" into the place "indicated" by the first sign of "3210", that's why I propose below a little utility program in Perl that makes it easy to do such things and others like telling the degree or the elementary partition of a series, or even, what is more useful, its "type" which is a description of the partition, i.e. a description of a description. The type of a permutation is the structure of its partition and it is unique. So if the partition of "3210" is (30,21) its type will be "0200" because there are 2 elements of 2 elements into the partition. You see that we use the order of the sign here as a column name. The type of "0213" is "2100" because we have seen that its partition was (12,0,3) where there are 2 subsets of 1 sign, and one subset of 2 signs: 2 is located in the column 1 and 1 in the column 2, the other columns are left at 0 because there is no subset of 3 or 4 signs in the partition. In base 12, which is the base of western music, we have 77 types for about 480 millions permutations among 23 degrees. There is only one degree by type but there can be several types for one degree. For those who would like to do permutations with a paper and a pen there is however a wonderful method given by Franck Ayres Jr., in his Theory and Problems of Modern Algebra, that can help prevent many errors.

Interval contents and transposition

Back to music we are going to use some similar tricks. For example, a given series will differ from another in terms of intervals. Intervals in the chromatic scale are only semitones. In "19B786A24305" the first interval 1-9 can be considered as the distance from a C# to an A, which represents 8 semitones, or as the distance from an A to the next octave's C#, which represents only 4 semitones. So, for each series we can create a kind of interval type that is made of only 6 columns because no interval is bigger than 6 when we go across octaves. In fact we will use rows of 7 columns to leave a place for 0 semitone intervals if one day we want to treat permutations of non-distinct elements. The interval set of the Walzer Opus 23 is then "0231510". But musicians also defined intervals with reference to a sole octave because with 11 possible intervals in a row you get necessarily repetitions in representing them with 6 sizes only. That is a good reason to use also a 12 signs row for our interval contents. This time the series of Walzer Opus 23 obtains the set "023121002000".

The problem of transposition can be seen now from a better point of view: when I transpose "19B786A24305" of 1 semitone up to "2A0897B35416" for example, the interval set remains "0231510" but the degree and the type change. The permutational nature of this series is modified by the musical operation, so it seems difficult to characterize our musical system as permutational like we felt we had to. There might be a way of overcoming this problem. If we take a look at a piece of music we will see a kind of graph with notes at different heights. These heights represent the frequencies of the sounds. This is what happens in the chromatic scale. But if we wanted to see how harmonious these notes sound to us, following our admitted theory of consonance, we could do it too, replacing the chromatic scale by the order of apparition of harmonics, which is the row "074A268B1359", or one of its transpositions, depending on the note you take as fundamental. In C major, to the G - seventh semitone - we would give the height 2 on the graph, because it's the most harmonious note after the fundamental at height 1, in fact the next in the harmonic scale. Looking at the musical part after having changed the notes this way, we would see a representation of our ability to understand the melody: the higher it would get the worse we would take it, the more we would want to go back on the ground. We would have made a kind of map of our feelings instead of a map of the frequencies. This is what happens in classical or "tonal" music, the game we play when we listen to it, except that we do not write it according to this phenomenological principle, but according to the physical principle of the chromatic scale. Schoenberg wanted to change this, take another referential with each row and ask: - What would it be like if these intervals were the most common ones? How would we feel then?

Fortunately this operation of mapping is well represented in the language of permutations. The mapping we described above for S, the tonal melody, and Q, the harmonic scale, is a composition of permutations given by the formula: (Q) P (S^-1), or in another notation: S^-1(Q).

Schoenbergian instances

When Schoenberg invented dodecaphonism, he gave strict rules (that he did not always respect) : the notes of the row had to be played either in the correct order or as chords. You could use a limited number of series that had to be closely related to the original one, as modified instances of it. Among admitted instances of the row were the original series, and two different kinds of inversion: the horizontal inversion, call it Retrograde or Reverse, as if you played the series reading it backwards, and the vertical inversion of the row, let's call it Inverse, where an interval of say 4 semitones up in the original row is replaced by an interval of 4  semitones down. If you write the row as we did above with two parameters for each note, the Inverse modifies the first, the value, and the Reverse the second, the place. It was not too amazing then that Schoenberg introduced a fourth instance of the row which combined both of the preceding ones, the Reverse of the Inverse of the row, and we call it Opposite. It was not amazing but we will see now that it was a precious intuition in the perspective of theory, since this last instance of the row will offer us a way to solve our transposition problem.

You can map many series in many ways but if you choose to map a row on itself you will obtain the chromatic scale. This is not surprising either, it just shows that you have given to your series the value of your old referential. This is of no use for music. If you map the Reverse onto the original row you get another fixed row with a fixed degree and type. This is more interesting yet because its degree is higher than 1. But as any series produces the same mapping, you can't claim having differentiated your rows in any way. Things become still more interesting when you map the Inverse onto the same original row. This time you get a set of 10395 series each of them showing the same degree and type. When you map the Opposite onto whichever row, you obtain a reverse row out of the preceding set of 10395, but these series are now distributed into 6 subsets of different degrees and 19 subsets of various types. You have gained already a way to differentiate between great families (6) and sub-families (10395) of series. Moreover if you transpose your original row and map its Opposite onto it, you will get the same map as the untransposed series and this is a means of unifying permutationaly the 12 transpositions of a row. Our set of 480 millions series (12!) has reduced to 40 millions (11!) because we don't count transpositions anymore. This new set is divided into 6, 19 and 10395 subsets with permutational features, and so compatible with the theory of consonance that generated this musical system.

Why would we experiment?

Everything should be great then if our perception was to confirm this distinctiveness. Otherwise, what value would have such an artistic system? This is where beggins the interactive and experimental part of this paper. Instead of telling what I think about the proposed splitting of the total serial set, I will let the reader listen to different music samples and make his or her own decision. In fact at the time I'm writing this line, I did not hear enough samples to decide yet.

To conclude I will stress again that if the row has to be considered in the same time transposable and musically referential, there must be a mathematical means capable of ensuring that the transposition will not distort its permutational nature. The purpose of this experiment with real sounds is to test the sensitive accuracy of one such mathematical instrument, it does not mean that the instrument is adequate, nor that it's the only possible one. Actually there might exist already better answers to the problem of transposition and I'm not literate enough to know them. Nevertheless the experiment remains relevant.

Finally to do justice to our Swiss theologian - who was mathematician - we must remind that the beautiful physicalist theory of consonance for which we provided a structural interpretation here, is at its best in explaining why things might sound bad; but under its rule subsists a little mystery that, in C major, is named A.

2012 Update

This new edition doesn't provide the full experiment anymore. As its complicated appearance seems to have discouraged feedback in the first version I will only propose here a small subset of dodecaphonic transcriptions of the Walzer opus 23 specially chosen so as to examplify our concern in this paper ( cf. note 2). It should remain clear that we can't hope to obtain this way an answer to the original question about wether or not our sets of rows, are musically (i.e. phenomenologically) adequate. This new revised version of our experiment only illustrates the mapped opposite based sets. Thus, the six version of this piece of music will be taken from our six main subsets of permutationnaly compatible rows. To add some interest to this musical experience, I have chosen the representative rows not randomly from each subset but with the help of some musicological considerations. So at the same time we will be able to appreciate the musical properties (if any) of each one of our subsets and to hear how it would have sounded if Schoenberg had decided to write his Walzer with other rows, for example the one that he used in his Serenade op. 24, or the row used later by Webern in his Quartet op. 28. The transcriptions achieved on Csound score files are simple: take a new row and for each note of the Walzer op. 23, ask which indice it had in the original row, then replace it by the corresponding note in the new row (cf. note 3). This way only the chosen row changes but the other musical properties of the original score are left untouched. As I am not a musician, it is very likely that the Csound interpretation I propose can be criticized in many ways although I tried to respect the score and compared it to several famous interpretations in order to take decisions at some stages. I only hope the general idea remains close enough to be regarded as a valid interpretation of Walzer op. 23.

Table 1. Walzer Opus 23

MOT = Type of the Mapped Opposite, MOD = Degree of the Mapped Opposite
Set 1
191692800 rows.
MOD= 12
MOT= 2, 16
data
Set 2
106168320 rows.
MOD= 10
MOT= 4, 26
data
Set 3
66355200 rows.
MOD= 8
MOT= 6, 7
data
Set 4
66355200 rows.
MOD= 6
MOT= 10,11,12,22,25
data
Set 5
44697600 rows.
MOD= 4
MOT= 9,15,17,18,24
data
Set 6
3732480 rows.
MOD= 2
MOT= 3,19,23
data
Row: 9A0346578B12
Schoenberg, Serenade op.24 (Self Transposed Opposite, MOT=2)

listen
Row: 19B786A24305
Walzer Op.23 (MOT=4)

listen
Row: 2A5390B18476
All intervals (MOT=7)

listen
Row: 074A268B1359
Harmonic (MOT=11)

listen
Row: 769801AB3254
Webern, Quartet op. 28 (Derived Set, MOT=18)

listen
Row: 5876A934012B
Webern, Chamber Symph. op. 21 (Palindromic Row, MOT=23)

listen

Table 2 initiates a new step in the experiment. After having heard referential changes in dodecaphonic music, we listen to the same set of referential changes but in a tonal piece, the Praeludium I of Johan Sebastian Bach. This extension might be more confusing than helping the pursuit of our objective, since the main difference between dodecaphonism and classical music- after the row - lies in the prohibition of repetition: strict dodecaphonic music doesn't repeat notes except in their serial turn. By this means the composer avoids to fall back into an avatar of tonal music, giving privilege to some intervals in the serial set. Thus should be the referentials more apparent in the Walzer versions than in the Praeludium ones, which do favour some intervals in their sets. However, I think that we would have missed something not exemplifying such a referential change in a classical piece too.

Table 2. Praeludium I

MOT = Type of the Mapped Opposite, MOD = Degree of the Mapped Opposite
Set 1
191692800 rows.
MOD= 12
MOT= 2, 16
data
Set 2
106168320 rows.
MOD= 10
MOT= 4, 26
data
Set 3
66355200 rows.
MOD= 8
MOT= 6, 7
data
Set 4
66355200 rows.
MOD= 6
MOT= 10,11,12,22,25
data
Set 5
44697600 rows.
MOD= 4
MOT= 9,15,17,18,24
data
Set 6
3732480 rows.
MOD= 2
MOT= 3,19,23
data
Row: 9A0346578B12
Schoenberg, Serenade op. 24 (Self Transposed Opposite, MOT=2)

listen
Row: 19B786A24305
Walzer Op.23 (MOT=4)

listen
Row: 2A5390B18476
All intervals (MOT=7)

listen
Row: 074A268B1359
Bach, Praeludium I, (harmonic, MOT=11)

listen
Row: 769801AB3254
Webern, Quartet op. 28 (Derived Set, MOT=18)

listen
Row: 5876A934012B
Webern, Chamber Symph. op. 21 (Palindromic Row, MOT=23)

listen

Notes:

note 1: Real atonality has to be called dodecaphonism. It is one of Schoenberg's discoveries that to remain truly atonal it had to become serial. Of course there are nowadays many other musical experiences with sounds, however they must be declared musical to gain this statute, otherwise they can also serve other ends, e.g. informative or incentive, this is not the case with dodecaphonism which shows its musicality immediately by the use of classical musical intruments.

note 2: The original text of the experiment (with html links disabled) can be found here.

note 3: Csound is a free software for sound processing. In this experiment we used it as a sequencer for pre-recorded piano samples. The score we used with some comments on interpretation choices can be found here.


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