Changing Series in Walzer Opus 23

Emmanuel Rens
Geneva, April 2009




To examine phenomenological soundness of the Opposite mapping classification of dodecaphonic rows, I reproduced one of the first dodecaphonic works of Schoenberg into a part for computer software, in order to propose various versions where the original notes are replaced by series among the different subsets. The notes in the original piece are simply replaced by notes in the same octavial slice each time.    

Of course I don't pretend that these renderings share artistic value with the original piece. Certainly Schoenberg would have treated another row differently. The process is artificial but not absurd for our purpose since we cannot exclude that Schoenberg in another possible world has chosen to treat another row the way he treated the series in Walzer Opus 23.

I selected this music because it presents several assets. It is quite simple: the row is repeated 66 times in different rythmic organisations. There is only one occurence of the Reverse, and none of the other two instances, this way,  if the Opposite has any influence on music it will not  show directly, but as we hope, structurally. It is also, in my opinion, typically the kind of music piece that will not be appreciated straitforwardly by someone not experienced with dodecaphonic harmonies, unlike some posterior works as string trios, that provided in my sense a smoother passage from the realm of classical music.

Voluntarily breaking the similitudes between the series by choosing different first notes and orders among intervals,Table 1S is probably best designed to make vertical common features of Opposite mapping classification more obvious. The line entitled "special rows" (symetric of various kinds) is the only line that does not take its series from a randomly chosen interval set. Rows that come from another piece of music present a kind of pedagogical interest apart from the experiment itself. The palindromic row from Webern's Chamber Symphony for instance, sounds quite jazzy in the Walzer rythmic context.

What we are looking for, the goal of this experiment, is a vertical similitude among series. If such analogy exists, it could be different from what we are expecting. It could be more related to the type than the degree, that's why we are approaching the various series in 2 sections by column: the whole column is dedicated to one of the 6 degrees of mapped opposites in base 12, but each section contains series of a unique type. There are 77 types in base 12 and 23 degrees, but only 19 types and 6 degrees for mapped opposites..

I don't know yet how to decide when an interval set will have a sample row in every Mapped opposite subset. In these tables some interval sets are not represented within a given degree, and some types aren't either.

The additional renderings of Table 2S are suited for analytic purposes. Its scale being most recognisable, the chromatic version shows clearly the treatment applied by Schoenberg to the row. The harmonic version reproduces without repetitions the order of harmonics in the real world and its semitone-up version reveals the kind of permutational changes implied by transposition. The row called "self-mapped opposite" carries a high degree of discordance in my opinion. It is one of the very rare 4 rows that are the mapped opposites of themselves. The last line of rows contains mapped opposites that are not unmapped, each of them is the first mapped opposite in its degree set.

Table 1B initiates a new step in the experiment. After having heard referential changes in dodecaphonic music, we listen to a new 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: 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, we would have missed something not exemplifying such a referential change in a classical piece. Table 2B offers the same analytic tool as Table 2S, though with less benefit. Strangely, the self-mapped opposite that sounded so bad with the Walzer fits well the idea I can figure out of the Praeludium.

For each version there is a link to a data file that is simply the output of a piece of code (dode.pl) that can be downloaded below. This program facilitates the permutationnal and interval content analysis of the series. Another file explains how to read the special data provided for each row.

In all tables the columns labels remain the same, and identical row names refer to identical data and renderings while identical line names mean that the series are chosen from the same interval sets. The music files have been created with open software Csound and Blue. There are files to downolad and a small bibliography at the end. External links open up in new windows except when they are followed by the sign '(↓)'.

Table 1S. Walzer Opus 23: Quasi(1) random rows in random(2) extended interval sets

MOT = Type of the Mapped Opposite, MOD = Degree of the Mapped Opposite
Sets of Series →
Extended Sets of Intervals (2785 sets) ↓
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
0231510 - ord 899
(442944 rows)
40153B78A692
(MOT=2)
Walzer Op.23 (MOT=4)
42370B1596A8
 
(MOT=7)
listen - data
7962A354801B
(MOT=11)
listen - data
314B762A0895
(MOT=18)

listen - data
no such row
(with given interval set)


Special rows (symetric,
derived, ..)
Schoenberg, Serenade op.24 (Self Transposed
Opposite
) (MOT=2)

listen - data
017B268A5493 (MOT=4)
listen - data
All intervals (MOT=7)
listen - data
Harmonic
(MOT=11)
listen - data
Webern, Quartet  op. 28 (Derived Set)
(MOT=18)
listen - data
Webern, Chamber Symph. op. 21 (Palindromic Row)
(MOT=23)

listen - data
0122133 ord 1523
(535680 rows)

0A7184B56392
(MOT=2)

listen - data
71853B609A42 (MOT=4)
listen - data
24A5B9038176 (MOT=7)
listen - data
562039718B4A
(MOT=11)
listen - data
816079352BA4
(MOT=18)
listen - data
3802B65A4179
(MOT=23)

listen - data
0224121 ord 886
(1505664 rows)
5A4678B29031
(MOT=2)

listen - data
3028957AB416 (MOT=4)
listen - data
B423095617A8
(MOT=7)
listen - data
1693A02B7854
(MOT=11)

listen - data
917A052436B8
(MOT=18)

listen - data
853B69A70142
(MOT=23)

listen - data
0320511 ord 509
(211392 rows)
827B9A651340
(MOT=16)

listen - data
6A84359127B0 (MOT=26)
listen - data
059A4621B378
(MOT=11)

listen - data
A3B789510264
(MOT=25)

listen - data
15AB08264379
(MOT=17)

listen - data
B12A37640895
(MOT=19)
listen - data
0432102 ord 246
(270144 rows)
20689A17543B
(MOT=16)

listen - data
B9A203417865 (MOT=26)
listen - data
0B2A45786931 (MOT=6)
listen - data
421B3587609A
(MOT=25)

listen - data
536209871BA4
(MOT=17)

listen - data
1B0A96458237
(MOT=19)

listen - data
0111342 ord 1672
(548352 rows)
374810629B5A
(MOT=16)

listen - data
5038916A427B
(MOT=26)

listen - data
9082713B6A54
(MOT=6)
listen - data
27B6154093A8
(MOT=25)
listen - data
6170495B238A
(MOT=17)

listen - data
A27B50389164
(MOT=19)

listen - data
0820101 ord 43
(3456 rows)
no such row
(with
MOT=16)
8764359AB012
(MOT=26)

listen - data
56021BA98734
(MOT=6)
listen - data
3456210BA879
(MOT=25)
listen - data
no such row
(with
MOT=17)
7B0A98654321
(MOT=19)

listen - data
   
1). Informatic randomness has been helped to look more random (distribution of rows among the total set).
2). Except symetric rows and some other rows with special historical and musical interest (line 2).

Table 2S: Additional Renderings for Walzer Opus 23

MOT= Type of the Mapped Opposite, MOD = Degree of the Mapped Opposite
Sets of Series →
Other Sets
(if any)↓
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

Chromatic
(MOT=2)

listen data
Walzer Op.23 (MOT=4)
listen - data
1B093A276485,
All Intervals (MOT=7)
listen - data
Harmonic
(MOT=11)

listen - data
Schoenberg, Suite for piano op. 25
(MOT=24)

listen - data
Berg, Lyric suite,
All Intervals

(MOT=23)

listen - data

Circle of Fifths
(MOT=2)

listen - data
1B09473A2865,
All Intervals
(MOT=4)
listen - data

Bach, Praeludium I
Transposed Harmonic
(MOT=11)

listen data
Schoenberg, Ode to Napoleon op. 41,
Derived Set
(MOT=18)

listen - data


Self-mapped Opposite (MOT=2)
listen data






Table 1B. Praeludium I: same rows as Table 1S

MOT = Type of the Mapped Opposite, MOD = Degree of the Mapped Opposite
Sets of Series →
Extended Sets of Intervals (2785 sets) ↓
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
0231510 - ord 899
(442944 rows)
40153B78A692
(MOT=2)
Walzer Op.23 (MOT=4)
listen - data
42370B1596A8
(MOT=7)
listen - data
7962A354801B
(MOT=11)
listen - data
314B762A0895
(MOT=18)

listen - data
no such row
(with given interval set)


Special rows (symetric,
derived, ..)
Schoenberg, Serenade op. 24 (Self Transposed Opposite) (MOT=2)
listen - data
017B268A5493 (MOT=4)
listen - data
All intervals
(MOT=7)
listen - data
Praeludium I, (harmonic) (MOT=11)
listen - data
Webern, Quartet  op. 28 (Derived Set)
(MOT=18)

listen - data
Webern, Chamber Symph. op. 21 (Palindromic Row)
(MOT=23)

listen - data
0122133 ord 1523
(535680 rows)
0A7184B56392
(MOT=2)

listen - data
71853B609A42 (MOT=4)
listen - data
24A5B9038176 (MOT=7)
listen - data
562039718B4A
(MOT=11)
listen - data
816079352BA4
(MOT=18)
listen - data
3802B65A4179
(MOT=23)

listen - data
0224121 ord 886
(1505664 rows)
5A4678B29031
(MOT=2)

listen - data
3028957AB416 (MOT=4)
listen - data
B423095617A8
(MOT=7)
listen - data
1693A02B7854
(MOT=11)

listen - data
917A052436B8
(MOT=18)

listen - data
853B69A70142
(MOT=23)

listen - data
0320511 ord 509
(211392 rows)
827B9A651340
(MOT=16)

listen - data
6A84359127B0 (MOT=26)
listen - data
059A4621B378
(MOT=11)

listen - data
A3B789510264
(MOT=25)

listen - data
15AB08264379
(MOT=17)

listen - data
B12A37640895
(MOT=19)
listen - data
0432102 ord 246
(270144 rows)
20689A17543B
(MOT=16)
listen - data
B9A203417865 (MOT=26)
listen - data
0B2A45786931 (MOT=6)
listen - data
421B3587609A
(MOT=25)

listen - data
536209871BA4
(MOT=17)

listen - data
1B0A96458237
(MOT=19)
listen - data
0111342 ord 1672
(548352 rows)
374810629B5A
(MOT=16)

listen - data
5038916A427B
(MOT=26)

listen - data
9082713B6A54
(MOT=6)
listen - data
27B6154093A8
(MOT=25)

listen - data
6170495B238A
(MOT=17)

listen - data
A27B50389164
(MOT=19)
listen - data
0820101 ord 43
(3456 rows)
no such row
(with
MOT=16)
8764359AB012
(MOT=26)
listen - data
56021BA98734
(MOT=6)
listen - data
3456210BA879
(MOT=25)
listen - data
no such row
(with
MOT=17)
7B0A98654321
(MOT=19)
listen - data

Table 2B: Additional Renderings for Praeludium I

MOT = Type of the Mapped Opposite, MOD = Degree of the Mapped Opposite
Sets of Series →
Other Sets
(if any)↓
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

Chromatic.
(MOT=2)

listen data

1B093A276485,
All Intervals Row. (MOT=7)
listen - data

Schoenberg, Suite for piano op. 25.
(MOT=24)

listen - data
Berg, Lyric suite,
All Intervals Row.
(MOT=23)
listen - data

Circle of Fifths.
(MOT=2)

listen - data
1B09473A2865,
All Intervals Row
. (MOT=4)
listen - data


Schoenberg, Ode to Napoleon op. 41,
Derived Set.
(MOT=18)

listen - data


Self-mapped Opposite.
(MOT=2)
listen data


Praeludium I, (harmonic)
(MOT=11)

listen - data



Documents and utilities

Le dodécaphonisme comme outil d'analyse A previous and more precise article on permutational music, co-authored with Jérôme Palfi, in French.
How to read special data (↓) A reminder on the meaning of data formats used here.
Degrees and Types (↓) Distribution of types by degrees in base 12 and type numeration (MOT).
wop23.csd The Walzer Op. 23 part for Csound synthesis open software.
dode.pl A Linux command line utility written in Perl, for the analysis and generation of series (run "./dode.pl --help" in a terminal window to get usage information).
csdmod.pl The Linux command line utility written in Perl for the replacement of one row by another in .csd files (run "./csdmod.pl --help" in a terminal window to get usage information).
Piano-Akai_Steinway_III.sf2 The public domain piano sound used for renderings.
History (↓) Record of the modifications brought to any document in this project.

Bibliography

Franck Ayres, Jr., Theory and Problems of Modern Algebra, McGraw-Hill Inc, New York, 1965.
A good manual.
Kenneth R. Rumery, Twelve tone composition. Detailed examples for different types of rows.
Emmanuel Amiot, La série dodécaphonique et ses symétries. A detailed overview of groups in some symetric series, with theorems and demonstrations (in French).
Milton Babbitt, Collected Essays, Princeton University Press 2003.
Deals with hexachords and groups. As difficult as Babbitt can be.
David J. Hunter and Paul T. von Hippel, How Rare Is Symmetry in Musical 12-Tone Rows?, The Mathematical Association Of America, February 2003.
Still too hard and groupy for me, but row examples are nice.

Link back to the introduction on permutations

© Emmanuel Rens, Geneva, April 2009. home top