Scriabin's Prelude, Op 74 No. 3   
A Model for Recurring Chords and Nonharmonic Tones in Nontonal Music

copyright 2003 by Larry J Solomon

Theory

Set theory has been developed specifically for the analysis (and composition) of nontonal music. It may be regarded as a kind of General Relativity Theory for music; i.e., it is not a rejection of traditional theory, nor is it completely separate from the latter. Set theory may be regarded as a generalization of the principles of traditional theory. It embraces traditional theory and expands upon it, so that the basic tenets of the older theory still hold. Major and minor chords are sets, as are all the other traditional triadic chords. However, set theory (ST) does not restrict itself to these chords. It embraces all note combinations as possible chords. These are called sets, or pitch sets, or pitch-class sets, in order to distinguish the theory of sets. So, with ST, it becomes possible to identify any possible chord combination that can occur in the twelve-tone equal-tempered system. It became necessary to develop ST because much modern music uses chords that cannot be explained with traditional theory. For example, if one wants to understand the harmonic organization of the late music of Scriabin or Schoenberg, traditional theory just doesn't work. One important limitation of ST is its non-discrimination of enharmonics; e.g., F# and Gb are the same in ST. However, the powerful results of ST more that make up for this limitation, and it is possible to use it in tonal contexts as long as this limitation is taken into account.

The advent of complex mixed-interval chords poses many new problems, only some of which have been resolved. We now have a complete list of all possible tone combinations in the twelve-tone scale, each of which has been given a unique name, or rather a pair of numbers; e.g., the minor triad is 3-11. The first number designates the number of distinct pitch-classes contained in the set, i.e., its cardinality. The number after the dash is a catalog number in alphanumeric order within each cardinality. Because the major triad is also designated 3-11, it has become necessary to distinguish minor from major with a B suffix on the less compact set in the pair. The prime form of the minor triad is 037, while the major triad is 047. These primes may also be used to identify sets. Here, I am adopting a more streamlined system that drops the initial zero, since it is understood to occur as the reference in every prime. Thus, 037 becomes simply 37.

Allen Forte has reduced both major and minor triads to 37, or 3-11, thus making it impossible to distinguish major from minor. By slightly expanding the designations to distinguish the normal forms 37 and 47 and making them both primes, major is distinguished from minor. The set names are then are distinguished by 3-11 for minor (most compact form), and 3-11B for major. I proposed this revision in 1982 in the Netherlands journal Interface. I had hoped that this would resolve some of the concerns raised by theorists, but the momentum of Forte's theory continues to prevail and the resulting unfortunate controversy continues.


Partitioning

Even if one accepts the prevailing theory, locating and isolating sets within real music (scores) remains a controversial problem. This is called partitioning, i.e., dividing the score into harmonic units. Where does a harmony begin? Where does it end? The notes within these boundaries will here be called the harmonic unit, or HU. Are all the notes within a HU equally important? Can a HU contain nonharmonic notes?

In much of our traditional music, the harmonic rhythm is paced at a regular interval, usually inversely proportional to the tempo; i.e., usually the slower the tempo the faster the harmonic rhythm. But, this is a guideline, not a rule. There are some cases where a slow tempo is allied with a slow harmonic rhythm, as in the prelude to Das Rheingold by Richard Wagner. The Bach chorales are mostly uniform in having one chord per beat. Baroque music often has a faster harmonic rhythm than Classical music. In Beethoven's moderately paced pieces, the harmony often changes at about one chord per measure, and in the faster pieces at about one chord for two to four measures. Later Romantic music sometimes has an even slower harmonic rhythm at comparable tempi. However, harmonic rhythm must be supported by a consistent use of a harmonic vocabulary. The chord is the fundamental unit of harmonic rhythm, and, for our purposes a chord will be synonymous with an unordered pitch-class (pc) set.

The music of the "Second Viennese School" is a part of the Western tradition. Thus, the music of Schoenberg, Berg, and Webern, which may at first seem to have little in common with the music of Beethoven or Wagner, is actually a descendant and logical continuation of it. Schoenberg and his students, themselves, often commented about the evolutionary, rather than revolutionary nature of their music. Therefore, it should not be surprising that many of the same principles apply, like voice leading and part writing. Although we may not find triads and tertian structures anymore (they were actually avoided for reasons that I will not go into here), there is till a harmonic and melodic consistency . Thus, even the twelve-tone works may contain a small number of repeating harmonic units, or sets. But, how are these to be determined?

Another way to view harmonic rhythm is as a boundary, enabling divisions between chords, like a territorial boundary. If we can know the harmonic rhythm, then we can partition the sets. So, the question becomes "How do you determine where the boundaries are?" This task may actually easier in twelve-tones works than in non-twelve tone atonal music, because the basic set can be analyzed first for its internal structures, such as source sets, combinatoriality, symmetry, and interval repetitions. If the basic set has such special structures, they are likely to be employed in the music and this can then be used as a way of hypothesizing chords. If these are found, their boundaries (HUs) can help to establish the harmonic rhythm.

Partitioning is one of the most difficult things to assess in the analysis of nontonal music. This is because there is no standardization of chords, such as triads, in nontonal music. However, the more standard tonal model may be studied as a possible guide for nontonal contexts. Partitioning of tonal music is more or less taken for granted; i.e., the systematic process, although relatively simple, has not been described, to my knowledge. Simply put, how does one identify chords (compound harmonic units) in tonal music and separate them from nonharmonic tones? And how does one determine when the harmonies change?

In more complex music, basic sets are groups of notes associated by virtue of their proximity. But, what separates one group, or set, from another? One factor is harmonic rhythm, or the time involved for a chord change. Another is the possible, and likely, infiltration of the harmonies by nonharmonic notes. To simplify partitioning here, we will restrict our inquiry to music that is harmonically consistent, i.e., music that repeats a small group of sets, specifically chords (or scales), consisting of at least three pcs. Partitioning should be divided into two steps: 1. dividing the music into HUs, and 2. determining which notes are nonharmonic within each unit.

One goal is to identify the chords that recur. Another is to determine any relationships that the chords may have. If unsure about the harmonic rhythm, one may start with some hypothetical unit, perhaps the measure. This unit is not entirely arbitrary. It is one of the common units of harmonic rhythm, and it will not be a serious error if it turns out to be incorrect. At this stage we are searching for the proper unit, which is not "cast in stone", but a hypothesis. An analysis of an atonal work may not account for some notes being more harmonically significant than others; i.e., some may even be nonharmonic. [A nonharmonic tone or note is defined as one that is not a part of the current harmonic unit.] Traditional nonharmonic notes include passing (p), neighbors (n), suspensions (s), appoggiaturas (ap), pedals (ped), anticipations (a), cambiatas (cam), and escape (e) notes. (For more on nonharmonic tones, see my Theory of Nonharmonic Tones. All involve step motion or no motion, and these are together called minimal motion, or MM. Step motion is linear motion by semitone or tone, but no more. Leaps are anything larger. Pedals simply repeat or are held through a chord change, and therefore have no motion. Some, like the escape tones, appoggiaturas, and cambiatas, use step motion in combination with a leap. (Leaps are larger than a whole step.) They are less common. Additionally, we know that most nonharmonic notes occur off the beat, rather than on the beat.

Melodies, i.e., linear motion, outline chords and scales. From Schenker we know that leaps outline chords, whereas step progressions follow scale patterns. However, even step motion can be shown to have a harmonic background by considering rhythmic/metric emphasis.


Scriabin's Prelude Op 74, No. 3

Scriabin's music is a part of the tradition of Western music. So, although his late works explore new harmonic terrain, we would reasonably expect a similar organization of the harmony.We will hypothesize a harmonic rhythm of one chord per measure in Scriabin's Prelude. Therefore, there will be a total of 26 chords (or pc-sets), equal to the total number of measures. Each HU will occupy a unit of one measure.

An analysis of the set relations of these chords without nonharmonic notes is as follows:

1___8-12< (1)
S2___6-Z50*..29 (2)
=S3___8-12< (3)
SSS4___7-31 (4)
=S=S5___8-12< (5)
SXSSS6___6-Z49*..28 (6)
=S=S=S7___8-12< (7)
=S=S=S=8___8-12< (8)
  S   S  S    9___8-28* (9)
  S   S  S    =10__8-28* (10)
SSS=SSSSSS11__7-31 (11)
  S  S   S     ==S12__8-28* (12)
=S=S=S==     S  13__8-12< (13)
S=SSSXSSSSSSS14__6-Z50*..29 (14)
=S=S=S==     S  =S15__8-12< (15)
SSS=SSSSSS=SSSS16__7-31 (16)
=S=S=S==    S   =S=S17__8-12< (17)
SXSSS=SSSSSSSXSSS18__6-Z49*..28 (18)
=S=S=S==     S   =S=S=S19__8-12< (19)
=S=S=S==     S   =S=S=S=20__8-12< (20)
SSSSSSSS     S   SSSSSSSS21__9-8B (21 Dover)
  S  S   S    == S=   S  S  S      22__8-28* (22 Peters)
SSS=SSSSSS=SSSS=SSSSSS23__7-31 (23)
  S  S   S    ==S=  S   S  S       =S24__8-28* (24)
S  SSS   SSSSSSS  SSS SSSSSS25__5-10 (25)
SXSSS=SSSSSSSXSSS=SSSSSS 26__6-Z49*..28 (26)
Total number of sets:                                  26
Number of each cardinality
C5 = 1    C6 = 5    C7 = 4    C8 = 15     C9 = 1
Number of mirror sets:                                10      38 %
Number of sets showing S relation:         26    100 %
Number of sets showing + relation:           0        0 %
Number of sets showing + or S relation:  26     100 %

Relations within equal cardinalities:
Number of sets showing equivalence:    24      92 %
Number of sets showing Z:                        0        0 %
Number of sets showing X+O:                  5       19 %
Number of sets showing R:                        0        0 %
Number of sets with =/R/X/O/Z:              24      92 %

Although this analysis shows an overall consistency of the harmonies, the ear can hear an even greater consistency. How, then, can this consistency be demonstrated? We can assume that not all the notes are equally significant to the harmony. For example, notes on the beat are more important than those off the beat. Notes that leap are harmonically freer than those that must move by step.

Commencing with our hypothesis, we can formulate basic premises for separating harmonic and nonharmonic notes in this piece. Each melodic voice part should be identified and kept separate from the others. Each voice is then examined for the following criteria to determine harmonic and nonharmonic tones.

  1. Two successive notes forming a leap are harmonic.
  2. Nonharmonic notes are surrounded by harmonic notes. Therefore, the notes surrounding a nonharmonic note (immediately preceding and following) are identified as harmonic unless factors indicate otherwise.
  3. Notes are assumed to be nonharmonic if approached stepwise from a harmonic note. However, if they are left by leap within the HU, both notes of the leap are harmonic.
  4. Unaccented notes approached by step and left by leap into the next HU are nonharmonic escape tones.
  5. Notes tied across a barline may be harmonic or nonharmonic in the next HU. If held for a third or less of the next HU, it is nonharmonic.
  6. Scale sets should be identified separately from harmonic sets, or RCs (recurring chords).
  7. Notes occurring on the beat are harmonically more important than those off the beat.

Consider Ex 1, measures 1-3. First, we divide up the score into four voices, soprano and alto in the treble, and tenor and bass in the bass. This is easy to do since the composer has written the music with stems up for the soprano and tenor, and down for the alto and bass voices, as if it were a vocal piece. The bass and tenor voices are always leaping in this Prelude, so all the notes in these voices are harmonic. The alto voice is also completely harmonic for the same reason. (The superfluous leading zero is omitted from the prime forms in these examples.)

The only voice that is truly melodic is the soprano, and therefore it is the only one that must be studied for nonharmonic notes.

Since the first soprano interval is a leap both notes are considered harmonic, i.e., notes within the set. However, the G# comes from the A by semitone, and it is off the beat (a dotted quarter gets the beat here). So it is circled and labeled nonharmonic, passing to a G. The G is harmonic for these reasons: 1) it follows a nonharmonic note, and 2) it leaps to E. The last condition means that both G and E are part of the harmony. The D#, however, is off the beat and comes from the harmonic E by step motion; so, it is nonharmonic in measure 1. Measure 2 has no step motion, so it is all harmonic. Measure 3 contains a transposition of the soprano voice from measure 1; therefore, the nonharmonic notes are in the same relative locations. When the harmonies of these measures are analyzed without the nonharmonic notes, the set names are shown in red. An analysis of the first 13 measures then yields the following sets.

Notice the escape tones at the end of measures 5 and 8. They are approached by step off the beat (nonharmonic) but leap into the next measure. A consistent upper neighbor pattern also recurs. Similar passages are encountered throughout the rest of the Prelude. They are analyzed in a consistent manner. The tied note A# at the end of measure 6 becomes nonharmonic in measure 7 since it is held for only a third of measure 7.

Measure 24 is a scale set, 8-28*, the diminished octatonic scale, sometimes called the alternating octatonic. It is also the "parent set", i.e., the set that contains all the others. Measure 25 contains a subset of the same scale.

The results of analyzing the sets without the nonharmonic tones reveals a consistency of harmony that rivals even the most traditional tonal music.

1___6-Z23*..45 (1)
X2___6-Z50*..29 (2)
=X3___6-Z23*..45 (3)
SSS4___7-31 (4)
=X=S5___6-Z23*..45 (5)
XXXSX6___6-Z49*..28 (6)
=X=S=X7___6-Z23*..45 (7)
=X=S=X=8___6-Z23*..45 (8)
SSS=SSSS9___7-31 (9)
SSS=SSSS=10__7-31 (10)
XRXSXXXXSS11__6-30 (11)
SSSSSSSSSSS12__8-28* (12)
=X=S=X==SSXS13__6-Z23*..45 (13)
X=XSXXXXSSRSX14__6-Z50*..29 (14)
=X=S=X==SSXS=X15__6-Z23*..45 (15)
SSS=SSSS==SSSSS16__7-31 (16)
=X=S=X==SSXS=X=S17__6-Z23*..45 (17)
XXXSX=XXSSXSXXXSX18__6-Z49*..28 (18)
=X=S=X==SSXS=X=S=X19__6-Z23*..45 (19)
=X=S=X==SSXS=X=S=X=20__6-Z23*..45 (20)
SSS=SSSS==SSSSS=SSSS21__7-31 (21)
SSS=SSSS==SSSSS=SSSS=22__7-31 (22)
XRXSXXXXSS=SXRXSXXXXSS23__6-30 (23)
SSSSSSSSSSS=SSSSSSSSSSS24__8-28* (24)
S SSS SSSS SS SSS SSSS S25__5-10 (25)
XXXSX=XXSSXSXXXSX=XXSSXS 26__6-Z49*..28 (26)

The only set that has gaps in the triangle is number 25: 5-10. This is due to the grace note G, which occurs in measure 26, and which should be included in set 25. When it is, the results are even tighter.

This result demonstrates a very high degree of harmonic unity in the Prelude. There are only six different sets, five are harmonic, recurring chords (RCs), and one scale set (8-28*). The scale set 8-28* is also the parent set, containing all the other sets within it. It is the diminished octatonic scale. Only five different RCs occur in the Prelude, and all are closely related. The cardinality of these harmonic sets ranges only from 6 to 7, and most of these are hexachords. Nearly 70% of the sets are symmetric mirrors. The most important RC is 6-Z23* (Table C), which occurs eleven times. The only heptachord, 7-31, contains all the hexachords. All the sets occur at least twice.

An unexpected result of the analysis table is the clear exposition of form: Binary, with A2 repeating the same harmonic progression as A1. Section A2 is a tritone transposition of A1. Although many maximal similarities are shown as Xs on the triangle, significantly no Zs or Os occur. This means that no inverse sets are used, and no different sets with the same interval vectors are used. Finally, no maximally similar sets with non-switched interval pairs occur.


The complete list of sets:

No.| notes
set name         prime form    interval vector  descriptive name

1  f#b#a#ega
6-Z23*..45          2 3 5 6 8................ 234222     Super-Locrian hexatonic/comb.=I8

2  f# b# gc#eac#
6-Z50*..29         1 4 6 7 9................ 224232    comb.=I1

3 f#b#a#ed#c#
6-Z23*..45          2 3 5 6 8................ 234222    Super-Locrian hexatonic/comb.=I8

4  f#b#c#ga#d#a
7-31                    1 3 4 6 7 9.............. 336333    Alternating heptatonic.1

5  f#b#ea#ag
6-Z23*..45          2 3 5 6 8................ 234222    Super-Locrian hexatonic/comb.=I8

6  ad#c#a#gf#
6-Z49*..28         1 3 4 7 9................ 224322    Prometheus Neapolitan scale/comb.=I4

7  ad#c#gf#e
6-Z23*..45         2 3 5 6 8................ 234222    Super-Locrian hexatonic/comb.=I8

8  b#f#ea#c#d#
6-Z23*..45         2 3 5 6 8................ 234222    Super-Locrian hexatonic/comb.=I8

9  d#ac#ecgf#
7-31                    1 3 4 6 7 9.............. 336333    Alternating heptatonic.a

10  d#ac#ecgf#
7-31                   1 3 4 6 7 9.............. 336333    Alternating heptatonic.a

11  b#f#a#ed#a
6-30                  1 3 6 7 9................   224223    minor-bitonal hexachord/comb.=P6/#I5+11

12  b#f#a#egac#d#
8-28*                1 3 4 6 7 9 10..........448444    Alternating-Octatonic/diminished scale

13  b#f#ed#c#a#
6-Z23*..45       2 3 5 6 8................ 234222     Super-Locrian hexatonic/comb.=I8

14  b#f#c#ga#d#
6-Z50*..29       14 6 7 9................ 224232    comb.=I1

15  b#f#ea#ag
6-Z23*..45       2 3 5 6 8................ 234222     Super-Locrian hexatonic/comb.=I8

16  b#f#gc#ead#
7-31                 1 3 4 6 7 9............. 336333    Alternating heptatonic.a

17  b#f#a#ed#c#a#
6-Z23*..45     2 3 5 6 8................ 234222    Super-Locrian hexatonic/comb.=I8

18  d#agc#b#e
6-Z49*..28     1 3 4 7 9................ 224322    Prometheus Neapolitan scale/comb.=I4

19  d#agc#b#a#
6-Z23*..45     2 3 5 6 8............... 234222    Super-Locrian hexatonic/comb.=I8

20  f#b#a#age
6-Z23*..45     2 3 5 6 8............... 234222    Super-Locrian hexatonic/comb.=I8

21  ad#ga#f#c#b#
7-31                1 3 4 6 7 9.............336333    Alternating heptatonic.a

22  ad#ga#f#c#b#
7-31                1 3 4 6 7 9.............336333    Alternating heptatonic.a

23  f#b#ea#f#ad#
6-30                1 3 6 7 9................ 224223    minor-bitonal hexachord/comb.=P6/#I5+11

24  f#b#ea#af#ec#b#agd#
8-28*             1 3 4 6 7 9 A....... 448444    Alternating-Octatonic/diminished scale

25  d#c#b#a#ag
6-Z23*..45    2 3 5 6 8................ 234222    Super-Locrian hexatonic/comb.=I8

26  f#ega#b#d#
6-Z49*..28    1 3 4 7 9................ 224322    Prometheus Neapolitan scale/comb.=I4

 

  
2003 GUEST