Analysis by Larry J. Solomon, copyright © 1997
The four preludes of Scriabin's Op. 74 were written in 1914, shortly before the composer died. They are among the most daring harmonic conceptions in his complete work. A study of the preludes from early to late reveals a gradual development from the Chopin-like pieces of Op.2-11 to the highly chromatic preludes of Op. 35, and the tertian extensions of Op. 51, to the nearly pantonal preludes of Op. 74. This literature shows, perhaps more than any other, the gradual and logical development of an pantonal vocabulary.
1. Why is there a conspicuous gap in the triangle at chord 21?
I first performed the Dover edition of this Prelude in the 1970s and noticed something peculiar about it. Measure 21 sounded wrong to my ears. I had a hunch that there were wrong notes in the edition I was using. It is usually easy to determine incorrect notes in traditional triadic music, but how is one to know in a piece with as complex harmonies as this? First, I looked at other editions. Burkhart's Anthology for Musical Analysis, third edition, contained this piece, so I looked at it. It had the same notes as the Dover edition. Since the two editions agreed, I became somewhat sceptical about solving the problem. Then, I looked at the Peters edition, which showed different notes in measure 21. The sharp on the G was on the D instead in the Peters edition. I was sure that the Dover edition was wrong, and the Peters edition was correct, but I couldn't support that without examining the original manuscripts, which were not accessible to me. It wasn't until years later, after I had created the Music Analysis System that I was able to corroborate my suspicion.
The harmonic rhythm in this prelude is consistently one chord per measure, so that each measure can represent a set. After inputting the sets, the relations triangle showed a prominent gap of relation symbols both horizontally and vertically at set number 21, which is from measure 21 of the Dover edition. In both editions, the same chord occurs in measures 21 and 22. On the triangle, set 22 represents the chord in the Peters edition.
= is an equivalence S is a most significant subset (one in which the cardinality of the subset is at least half of superset) + is less significant subset (the subset has less than half the cardinality of the superset) Z means the sets have the same IV and may be inversionally related (They are inversions if they have the same set-name except for a "B" at the end of one of them). X is Forte's maximum similarity relation with interchange feature O is Forte's maximum similarity relation without interchange feature R is Solomon's maximum similarity relation (see "Solomon's Glossary of Technical Terms" on its Webpage, for R-relation) (also, INTERFACE, Journal of New Music Research, vol.11/2, 1982)
It is clear from the density of relations exhibited by all the other sets and the lack of any relationship to set 21, that chord 21 is illogical and out of place. It has no relationship to the other chords. Thus, the analysis shows that it is not logical, corroborating the ear. When the analysis was redone replacing the maverick chord, thereby conforming to the Peters edition, the second chart was the result.
2. What does it mean to have such a density of relations displayed?
The extreme density of relations confirms a tight harmonic organization. Harmonically, it couldn't be more consistent. A relation occurs everywhere on the triangle.
3. Do all sets display the subset relation?
Yes. This is easily determined from the statistics table which shows 100% S (subset) relations? Notice that all of these are S (most significant) rather than + (less significant). See the key below the first triangle for further explanation.
4. What are the most and least significant chord relations?
Again, this is determined from the statistics table. The only significant relationship that can occur between sets of differing cardinality is the S relation, of which there are 100%. The relations between between sets of equal cardinality can be =,R,X,O, or Z (see the above key for explanation). Of these, the equivalence relation is ubiquitous (100%), and is the most significant. Maximal similarity X is next in line with 64%. Insignificant are the +, O and Z relations, of which there are none.
5. Why are there no O, Z, or + relations?
It seems that the composer choose to avoid these types of relations.
6. What percentage of the chords are mirrors?
The table shows 69%. This is quite a significant majority, showing that Scriabin favored this type of sonority.
7. What is the parent set?
The parent set is 8-28. It is the only set with eight pcs, the largest of all, and from the triangle we can see that all the other sets are subsets of 8-28. Notice, particularly the last two lines on the triangle.
8. What scale is being used?
It is also 8-28*, the alternating-octatonic, or "diminished" scale: F#,G,A,A#,B#,C#,D,E,F,F#. It is called "alternating" because of the repeating semitone, tone sequence throughout the scale. It is also called "diminished" because all triads constructed in this scale are diminished. Tonic and dominant are a tritone apart, which is a most significant interval in this piece. The bass sequence swings incessantly from F# to B#, and later from D# to A. Although the identity of this scale is not shown in the triangle, it is identified in the list of sets generated in the analysis, of which only a part is shown here.
12 b# f# a# e g a c# d# 8-28* 0 1 3 4 6 7 9 10 ........... 448444 Alternating-Octatonic scale 13 b# f# e d# c# a# 6-Z23*..45 0 2 3 5 6 8 ........... 234222 Super-Locrian hexamirror/comb.=I8
9. How many different chords occur?
Only six different sets can be found on the triangle: 6-Z23*, 6-Z50*, 6-Z49*, 6-30, 7-31, and 8-28*.
10. Which chords occur most often?
6-Z23* occurs most often -- eleven (11) times. 7-31 is next, six (6) times. They are tabulated in the next diagram.
11. Do all chords recur?
From the table it is clear that they all recur.
12. Which chord functions as a tonic?
6-Z23*, the Super-Locrian hexamirror, which occurs ten times, much more often than any other set, functions as tonic. This can be corroborated by consulting the score and the rhythmic placement of this set.
13. Is there a repeating harmonic progression?
Yes, there is. The first twelve measures repeats its progression in the last twelve.
____________________________________________ 1___6-Z23*..45 | Number of Different X2___6-Z50*..29 | times each set occurs =X3___6-Z23*..45 | --------------------- SSS4___7-31 | 6-Z23* 11 =X=S5___6-Z23*..45 | 6-Z50* 2 XXXSX6___6-Z49*..28 A1 6-Z49* 3 =X=S=X7___6-Z23*..45 | 6-30 2 =X=S=X=8___6-Z23*..45 | 7-31 6 SSS=SSSS9___7-31 | 8-28 2 SSS=SSSS=10__7-31 | --- XRXSXXXXSS11__6-30 | total---- 26 SSSSSSSSSSS12__8-28*________________________| =X=S=X==SSXS13__6-Z23*..45 | X=XSXXXXSSRSX14__6-Z50*..29 | =X=S=X==SSXS=X15__6-Z23*..45 | SSS=SSSS==SSSSS16__7-31 | =X=S=X==SSXS=X=S17__6-Z23*..45 | XXXSX=XXSSXSXXXSX18__6-Z49*..28 A2 =X=S=X==SSXS=X=S=X19__6-Z23*..45 | =X=S=X==SSXS=X=S=X=20__6-Z23*..45 | SSS=SSSS==SSSSS=SSSS21__7-31 | SSS=SSSS==SSSSS=SSSS=22__7-31 | XRXSXXXXSS=SXRXSXXXXSS23__6-30 | SSSSSSSSSSS=SSSSSSSSSSS24__8-28*____________| =X=S=X==SSXS=X=S=X==SSXS25__6-Z23*..45 (25) XXXSX=XXSSXSXXXSX=XXSSXSX26__6-Z49*..28 (26)
14. What is the form, and how is it articulated?
The form is a symmetrical simple binary, A1 A2. The first twelve measures repeats transposed down a tritone in the second half. This is seen in the repeating harmonic progression above and is corroborated in the score. The extra last two measures serve as a final cadence.
15. Are there "nonharmonic-tones", and if so, what are they?
This is one of the most difficult things to assess due to the complexity of the harmony and a lack of standardized harmonic units, like triads. Basic sets are established by what is called partitioning, which should be divided into two steps: 1. dividing the music into harmonic units, and 2. determining which notes are nonharmonic within each group. The harmonic rhythm here has been determined to be one chord per measure. For an explanation of harmonic rhythm and the determination of nonharmonic notes see: Toward a Theory of Partitioning.
All the sets are composed of six to eight notes. In triadic music, nonharmonc tones are normally easy to determine, because of the standardization of the triad and tertian chords. They are simply notes that don't occur in these chords. There is no reason that nonharmonc notes shouldn't occur with more complex harmonies. Tertian trichords and tetrachords have no inherent monopoly over nonharmonic notes.
But, how can these be determined in this piece? If we use the traditional model, most nonharmonic tones move stepwise, so this is likely in Scriabin as well. And, most are passing and neighbor notes. If all the notes of every measure are analyzed, the triangle comes out as:
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) SSSSSS7___9-10* (7) =S=S=SS8___8-12< (8) S S SS 9___8-28* (9) S S SS =10__8-28* (10) SSS=SSSSSS11__7-31 (11) S S SS ==S12__8-28* (12) =S=S=SS= S 13__8-12< (13) S=SSSXSSSSSSS14__6-Z50*..29 (14) =S=S=SS= S =S15__8-12< (15) SSS=SSSSSS=SSSS16__7-31 (16) =S=S=SS= S =S=S17__8-12< (17) SXSSS=SSSSSSSXSSS18__6-Z49*..28 (18) SSSSSS=SSSSSSSSSSS19__9-10* (19) =S=S=SS= S =S=S=SS20__8-12< (20) SSSSSS S S SSSSSS S21__9-8B (Dover 21) S S SS ==S= S S SS 22__8-28* (22) SSS=SSSSSS=SSSS=SSSSSS23__7-31 (23) S S SS ==S= S S SS =S24__8-28* (24) S S SS ==S= S S SS =S=25__8-28* (25/26) Total number of sets : 25 Number of each cardinality C. 6 = 4 C. 7 = 4 C. 8 = 14 C. 9 = 3 Number of mirror sets 12 48% Number of sets showing S relation 25 100% Number of sets showing + relation 0 0% Number of sets showing + or S relation 25 100% Relations within equal cardinalities: Number of sets showing equivalence 24 96% Number of sets showing Z 0 0% Number of sets showing X+O 4 16% Number of sets showing R 0 0% Number of sets with =/R/X/O/Z 24 96%
Notice the "holes" in the triangle. These show sets that lack relations to others sets. The set 8-12 in particular creates gaps in logical relations. This set, then, is suspected of containing nonharmonic notes. Sets larger than eight pcs are so complex that they are difficult to impossible to distinguish from tone clusters and from one another, so they are unlikely to be integral to the harmonic structure. (They could be, though, if analyzed as pitch-sets, rather than pc-sets, retaining their registrational positions.) Admittedly, this is an observation based on perception, but because of this I considered the set 9-10 as another set that contains nonharmonic notes.
I know of no better and faster way than using the ear to determine which notes are nonharmonic in those measures containing the sets 8-12 and 9-10. But, by familarization with the harmonies and careful listening, one can single them out, and they are all either passing or neighbors. They are as follows, from each measure:
1 g# d# 14 none 2 none 15 g# d# 3 d a 16 none 4 none 17 d a 5 g# d# 18 none 6 none 19 b f# e 7 e# b# a# 20 g# d# 8 d a 21 e 9 a# 22 e 10 a# 23 g 11 c# 24 none 12 none 25 none 13 d a 26 none
This prelude shows a high degree of harmonic organization using only six chords which recur throughout. These are all non-standard chords, especially for the time of the composition. The six chords range from hexachords to octachords, over half of which are mirrors, a definite preference; inverting each results in the same chord. This prelude is based on the alternating-octatonic scale, with all chords derived from this scale. Tonic is F#, which repeats in the bass and carries a dominant swing-substitution to the tritone B#. Tritones play a prominant role in the music. The bass constantly swings in tritones, and every chord contains at least two tritones, which can be seen from the interval vectors. Every chord contains at least two of every interval-class, and each has at least four minor thirds. The tonic chord is F# B# A# E G A, 6-Z23*, which occurs ten times, about twice as often as any other chord. The form is a binary A1 A2, with the second part being a tritone transposition of the first.
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