Circle of fifths

In the introduction I alluded to the possibility of the quartets being a whole tonal work. Elizabeth Wilson supplies the evidence for this. She recalls that when asked how many quartets he intended to write, Shostakovich replied, twenty-four:

Haven't you noticed that I never repeat a key? I'll write twenty-four quartets, so as to have a complete cycle (Shostakovich: A Life Remembered, p 389).

If he intended to write twenty-four and never repeat a key then it is pertinent to enquire in which order he intended to compose them. As we shall see this, in the phraseology of the mathematician, is a non-trivial question.

The seminal work for a complete tonal cycle is Bach's 'Das Wohltemperierte Klavier'. This consists of two collections, BWV 846 to 869 and BWV 870 to 893, each composed of twenty-four preludes and fugues. In each collection Bach used the same simple system, based on semitones, to progress from one scale to the next throughout the cycles. He started with C major then selected, as the tonic for the next scale, the next note to the right on the keyboard thus placing flats before naturals, and naturals before sharps. Before proceeding to the next major scale he composed a piece in the same minor scale. This arrangement is shown in Table 1 where the right-hand column shows the number of flats or sharps in each of the scales. This system produces an "alphabetic" arrangement rather one than based on an orderly progression of flats or sharps.

Table 1:
The tonality arrangement in Bach's 'Das Wohl- temperierte Klavier'
No.keyno. of ♯ or ♭
1C major0
2C minor3
3C sharp major7
4C sharp minor4
5D major2
6D minor1
7E flat major3
8E flat minor6
9E major4
10E minor1
11F major1
12F minor4
13F sharp major6
14F sharp minor3
15G major1
16G minor2
17A flat major4
18G sharp minor5
19A major3
20A minor0
21B flat major2
22B flat minor5
23B major5
24B minor2


Between October 1950 and February 1951, Shostakovich composed his opus 87 as a tribute to Bach. Like Bach this work is a cycle of twenty-four preludes and fugues for piano, each piece being written in a different key. But he chose not to follow Bach's method for progressing from one key to the next, deciding instead on a slightly different arrangement. Shostakovitch also arranged the pieces in pairs but now each couple consisted of a major and its relative minor (that minor scale with exactly the same notes as the major). The first two pieces were in C major and A minor respectively; both of these scales have neither sharps nor flats. The third and fourth pieces were in G major and E minor, each of which has one sharp in its scale (it is actually C sharp). The two scales with two sharps were then employed for the following two pieces and this system, of adding one sharp, was continued until the thirteenth piece in F sharp major; a scale with six sharps. The next prelude and fugue was in E flat minor which has six flats. From then on scales were employed with flats but the number of flats was progressively decreased by one until the final two pieces were reached. These, written in F major and D minor, have only one flat (it being B flat).

This arrangement is generated not by raising each tonic by a semitone, as Bach did, but by using the dominant  (the fifth note in the rising scale) as the tonic for the next scale. Doing this will generate a major scale with one sharp more than those in the previous major scale,  and then, after G flat major (which is equivalent to F sharp major) has been reached it, will decrease by one the number of flats incrementally.  Finally Shostakovich, like Bach, placed the major scale before the minor.

 This arrangement, is shown in Table 2. In this table the first column shows the order of the piece Prelude and Fugues as it appears in the opus 87. Column 2 shows the scale in which the work is written whilst the next column shows the number of sharps or flats in that scale. Next comes an explanation how the next major scale is derived from its predecessor (remember the following minor scale is just the relative one of the major) , whilst the final column explains the change that has occurred in the scale from that of the previous pair. As can be seen Shostakovich's method of arranging the works is just as methodical as that used by Bach, and although it might at first sight appear rather miraculous it is based on elementary musical theory. The reason for giving such a detailed explanation of the tonal arrangement of the opus 87 will be understood when the scheme used in the fifteen string quartets is analysed.

Table 2:
Tonality Arrangement in Shostakovich's 24 Preludes and Fugues Op.87
No.Keyno. of ♯ or ♭Reason for choice of scaleNotes in scale
1C major0 notes are all naturals
2A minor0  
3G major1 G being the dominant of C major  as in C major but leading note F raised to F
4E minor1  
5D major2 D being the dominant of G major  as in G major but leading note C raised to C
6B minor2  
7A major3 A being the dominant of D major  as in D major but leading note G raised to G
8F sharp minor3  
9E major4 E being the dominant of A major  as in A major but leading note D raised to D
10C sharp minor4  
11B major5 B being the dominant of E major  as in E major but leading note A raised to A
12G sharp minor5  
13F sharp major 6 F sharp being the dominant of B major as in B major but leading note E raised to E
 D sharp minor6 notes identical to E flat minor (no.14)
 C sharp major7 notes identical to D flat major (no.15)
 A sharp minor7 notes identical to B flat minor (no.16)
 C flat major7 notes identical to B major
(no.11)
 A flat minor7 notes identical to G sharp minor (no.12)
 G flat major6 notes identical to F sharp major (no.13)
14E flat minor6  
15D flat major5 D flat being the dominant of G flat major  as in G flat major but leading note lowered to C
16B flat minor5  
17A flat major4 A flat being the dominant of D flat major as in D flat major but leading note lowered to G
18F minor4  
19E flat major3 E flat being the dominant of A flat major as in A flat major but leading note lowered to D
20C minor3  
21B flat major2 B flat being the dominant of E flat major  as in E flat major but leading note lowered to A
22G minor2  
23F major1 F being the dominant of B flat major  as in B flat major but leading note lowered to E
24D minor1  


For the quartets Shostakovich appears to have selected a system similar to that used in the opus 87. As we have seen, in the cycle for piano the dominant note of the major scale was used to generate scales with an increasing number of sharps and then a decreasing number of flats, each of which he then selected for a composition. For the string quartets he used the submediant note in the scale instead of the dominant note for this purpose. The submediant is the sixth note in the rising scale, and in any major scale the submediant note defines the tonic of the major scale's relative minor, that is the minor scale which uses exactly the same notes (naturals, sharps and flats) that the major uses. For example the scale of C major has no flats or sharps and thus uses only the white notes on the piano keyboard. The sixth note in this scale is A,  and the scale of A minor (the relative minor to C major) has exactly the same notes in its scale as C major (the difference being, of course, that the scale starts on A and is played as a minor!)  By generating scales using the submediant a cycle would contain pieces which first contain an increasing number of flats and then a decreasing  number of sharps as is shown in Table 3.

Table 3: Tonality arrangement using submediant of previous scale
Basic TonalityNumber of sharps / flatsSubmediant of Scale
C major0A
A minor0F
F major1D
D minor1B flat
B flat major2G
G minor2E flat
E flat major3C
C minor3A flat
A flat major4F
F minor4D flat
D flat major5B flat
B flat minor5G flat
G flat major6E flat
E flat minor6B
B major5G sharp
G sharp minor5E
E major4C sharp
C sharp minor4A
A major3F sharp
F sharp minor3D
D major2B
B minor2G
G major1E
E minor1C


However Shostakovich then added cryptic complications into this scheme. Whether or not this was his aim, the first complication disguises his plan for the tonality of the quartets. What he did was that instead of using the submediant note to generate the next scale in an alternating succession of major and minor as is shown in Table 3, he first chose to use only major scales and only when they had been used did he start to compose in the minor scale. This first modification is shown in Table 4. In that table the first three columns reproduce the scheme already shown in table 3, whilst the fifth shows his scheme of using the key of column 3  first  as the tonic of a major scale before it is employed in the minor.

Table 4: Submediant tonality arrangement and Shostakovich's scheme for the Quartets
Basic Tonality Number of  sharps/ flatsSubmediant of ScaleQuartet NumberShostakovich's basic tonality scheme for the quartets
 Q1C major
C major0AQ2A major
A minor0FQ3F major
F major1DQ4D major
D minor1B flatQ5B flat major
B flat major2GQ6G major
G minor2E flatQ7E flat major
E flat major3CQ8C minor (C major already used)
C minor3A flatQ9A flat major
A flat major4FQ10F minor (F major already used)
F minor4D flatQ11D flat major (C sharp major)
D flat major5B flatQ12B flat minor
B flat minor5G flatQ13G flat major (F sharp major)
G flat major6E flatQ14E flat minor (E flat major already used)
E flat minor6BQ15B major
B major5G sharpQ16G sharp minor (A flat major already used)
G sharp minor5EQ17E major
E major4C sharpQ18C sharp minor (C sharp major already used)
C sharp minor4AQ19A minor
A major3F sharpQ20F sharp minor
F sharp minor3DQ21D minor
D major2BQ22B minor
B minor2GQ23G minor
G major1EQ24E minor
E minor1C C major


But in 1960 he introduced a second complication when he wrote his seventh quartet. Table 5  compares the tonality scheme derived in the previous table with those that he actually employed. As can be seen there is complete agreement up to, and including, quartet number six, but then two further modifications occur. The tonality of F sharp minor instead of E flat major is suddenly used for quartet number seven and then after correctly proceeding to C minor, E flat major follows instead of A flat major.

Table 5:  Shostakovich's scheme for the Quartets and the actual tonality used
Quartet  Shostakovich's
planned tonalities
(Table 4, Column 5)
Actual tonalities
Q1C majorC major
Q2A majorA major
Q3F majorF major
Q4D majorD major
Q5B flat majorB flat major
Q6G majorG major
Q7E flat majorF sharp minor
Q8C minorC minor
Q9A flat majorE flat major
Q10F minorA flat major
Q11D flat majorF minor
Q12B flat minorD flat major
Q13F sharp majorB flat minor
Q14E flat minorF sharp major
Q15B majorE flat minor
Q16G sharp minor 
Q17E major 
Q18C sharp minor  
Q19A minor 
Q20F sharp minor 
Q21D minor 
Q22B minor 
Q23G minor 
Q24E minor 
 C major 


Why this divergence? Both these alterations occur in the quartets dedicated to his wives. As can be seen in Table 5 these modifications however cause only a transient disturbance to the guiding scheme which Shostakovich seemed to have set himself when he composed his first quartet in 1938. From the tenth to his final fifteenth quartet he returned to, and faithfully followed,  his original scheme. However because of the modifications started in 1960 the tonality of these last six quartets would, due to the insertion of the F sharp minor in quartet seven, always be one quartet  in arrears compared to his original scheme. Can his deviation from his first concept be explained by certain tonalities associations with his first and third wives, or could it be more prosaic?

Ian Strachan1 has proposed that the insertion of F sharp minor and the rotation of E flat major and C minor were done so that quartet number nine would be written in E flat major and quartet number sixteen in B major. (By doing so Shostakovich would ensure that his initials (DSCH) were used as the keys in quartets whose number are a perfect square (D major: quartet number four or 2 squared; S, in the German notation or E flat major in the English: quartet number nine or 3 squared; C major: quartet number one or 1 squared; and H or B major in the English notation as quartet number sixteen, or 4 squared).

So it seems that Shostakovich, a tonal composer who delighted in keeping detailed numerical records of football scores2, indulged in numerical as well as musical ciphers.








Footnotes:

  1. [1]. Ian Strachan, 'Shostakovich's DSCH Signature in the String Quartets' DSCH Journal, 10, (Winter 1998), 48-49. back

  2. [2]. Laurel E. Fay, Shostakovich: A Life (Oxford University Press, 2000), p.111. back