1. Upright Petrouchka, Proper Scales, and Sideways Neapolitans
Department of Mathematics
Saint Joseph’s University
Rachel Wells Hall
Dmitri Tymoczko
Jason Yust
Department of Music
Princeton University
School of Music
University of Alabama, Tuscaloosa

2. The “Petrouchka Chord,” Rotated Voice Leadings, and Polytonality

3. Petrouchka chord First appearance of the “Petrouchka chord”
(Second Tableau, r. 49, Clarinets)
--Two superimposed major triads, arpeggiated in different inversions
--Harmonic intervals are not all exactly the same, but have a small range of sizes 3

4. Neapolitan voice leadings
The most efficient “neapolitan” voice leadings
Voice leadings between major triads a tritone apart
Efficient voice leadings change the position of the triad
The other possible voice leading is a simple transposition by tritone 4

5. Neapolitan voice leadings
This links the “harmonic consistency” generated by Stravinsky’s procedure to a familiar musical problem, the problem of finding efficient voice leadings between transpositions of the same or similar chords.
The Petrouchka chord is a “90º rotation” of a neapolitan voice leading (i.e., melodic intervals become harmonic intervals). 5

6. In this example, “Petrouchka chord” simultaneous arpeggiations in the
Vertical
Intervals:
A variation on the “Petrouchka chord” (r.49, mm. 11–12)
Note that there is no suggestion of octatonicism in this example.
In this example, “Petrouchka chord” simultaneous arpeggiations in the
clarinets alternate with those in the piano. Stravinsky juxtaposes different
major triads in each case, but always maintaining approximately a fourth
between the voices.

7. etc
Petrouchka chord 3
–1 –3 – –1 –3 – etc
Another pattern derived from the “Petrouchka chord,”
(Fourth Tableau, r.78: Strings, doubled first by bassoon then clarinet)
The first part of this example, different major triads are juxtaposed to
produce vertical intervals consistently in the vicinity of a major second.
The second part of the example juxtaposes the same major triads in
different ways, so that the intervals swap directions.

8. Rite of Spring: Jeux des Cités Rivales r.57 m. 3–4, Horns
ritual of the rivals 8

9. Rite of Spring: Jeux des Cités Rivales r.57 m. 3–4, Horns
ritual of the rivals
This “polytonal” passage can be thought of as a rotated voice leading
between diatonic scales. 9

10. D major
Bb major
concerto
D melodic minor
These passages from Stravinsky’s Concerto show the same kind of harmonic consistency as the example from the Rite of Spring, but juxtapose two different scale types (diatonic and acoustic). 10

11. In the foregoing examples:
• Stravinsky “rotates” familiar voice leadings
so that melodic intervals appear vertically
and harmonic intervals appear horizontally.
• The vertical intervals are all similar in size
giving the passages a palpable sense of
consistency that is difficult to explain in
traditional theoretical terms.
How can we understand this process?

12. Scalar Interval Matrices

13. Scalar interval matrix for a major triad
Intervals between adjacent
notes in the triad.
Intervals between non-
adjacent notes in the triad.
scalar int. matrices
Scalar interval matrix for a major triad 13

14. Scalar interval matrix for the dominant seventh
scalar int. matrices
Scalar interval matrix for the dominant seventh 14

15. Rite of Spring: Augures Printaniers r.28 m. 5–10, Trumpets
Augurs (trumpet)
Rite of Spring: Augures Printaniers r.28 m. 5–10, Trumpets
Parallel motion in a given scale is a way to explore the scales interval content 15

16. Thirds: 
Fourths: 
Thirds: 
Augurs (trumpet)
Rite of Spring: Augures Printaniers r.28 m. 5–10, Trumpets 4
Stravinsky’s procedure systematically explores the interval content of the diatonic.
5 6
Scalar interval matrix for the diatonic 16

17. Rite of Spring: Augures Printaniers r.31 m. 17–18, Strings
Augurs (strings)
Rite of Spring: Augures Printaniers r.31 m. 17–18, Strings 17

18. Fifths:
Fourths:
Augurs (strings)
Scalar interval matrix for the acoustic scale (melodic minor) 18

19. Rotational arrays and scalar interval matrices (a fortuitous connection)

20. rotational arrays
Rotational array for Stravinsky’s Movements

21. rotational arrays Rotational array for Stravinsky’s Movements
All Combinatorial
Hexachord
rotational arrays
Rotational array for Stravinsky’s Movements

22. rotational arrays Rotational array for Stravinsky’s Movements
All Combinatorial
Hexachord
rotational arrays
Intervals between successive elements
of the hexachord
Rotational array for Stravinsky’s Movements

23. Intervals between elements two places
All Combinatorial
Hexachord
rotational arrays
The scalar interval matrix is a rotational array rotated by 90 degrees.
The elements of a scalar interval matrix are put in strictly ascending order, whereas the order in a rotational array is determined by a serial design.
Stravinsky treats the numbers in the rot. array as pitch class numbers, not as intervals per se.
Intervals between elements two places
apart in the hexachord
Rotational array for Stravinsky’s Movements

24. Proper Scales

25. Propriety
The harmonic consistency of all the preceding examples is not an automatic result of vertically superimposing different forms of the same chord or scale. It only works for certain kinds of chord or scales.
The chords and scales that Stravinsky uses--triads, seventh chords, and diatonic and acoustic scales--have the property of propriety.
In a proper scale the range of interval sizes within each row is relatively small, so that there is no overlap between the range of adjacent rows.
In other words, the range between rows is non-negative. (In the case of the dom. 7th it is always positive, which makes this chord strictly proper).
The dominant seventh chord is a proper scale, because there is no
overlap in interval sizes between the rows of the interval matrix. 25

26. Propriety
The harmonic consistency of all the preceding examples is not an automatic result of vertically superimposing different forms of the same chord or scale. It only works for certain kinds of chord or scales.
The chords and scales that Stravinsky uses--triads, seventh chords, and diatonic and acoustic scales--have the property of propriety.
In a proper scale the range of interval sizes within each row is relatively small, so that there is no overlap between the range of adjacent rows.
In other words, the range between rows is non-negative. (In the case of the dom. 7th it is always positive, which makes this chord strictly proper).
The interscalar matrix for any transposition of a scale inherits
the propriety property, since adding a constant does not change
the ranges of interval sizes within rows or the gaps between rows. 26

27. Rite of Spring: Augures Printaniers r.28 m. 5–10, Trumpets
Thirds:  ( )
Fourths: ( )
Augurs (trumpet)
In the diatonic scale, thirds are always smaller than fourths 27

28. Rite of Spring: Jeux des Cités Rivales r.57 m. 3–4, Horns
ritual of the rivals
The vertical intervals are “roughly equal”: this is relative to the size of the set.
For seven-note scales, a range of two interval sizes is small,
but for 3- or 4-note sets, a range of three or four is small.
Different juxtapositions of the diatonic scales in Jeux des Cités Rivales
would produce vertical intervals consistently larger than an octave
or consistently smaller than a major seventh, because
the diatonic scale is proper. 28

29. Region of Proper Scales
Chords at the center of the space tend to be even, so propriety is a kind of evenness.
(i.e., a proper chord cannot be made improper by making it more even).
The proper scales form a compact, convex region at the center of
n-note chord space

30. Interscalar Interval Matrices (for transpositionally related scales)

31. –2 –3 –1
ISI matrix
The interscalar interval matrix for major triads a tritone apart represents all one-to-one crossing-free voice leadings between the chords 31

32. –2 –3 –1
ISI matrix
The interscalar interval matrix can also represent the harmonic intervals resulting from simultaneous arpeggiations in different positions 32

33. SI and ISI matrices 4 3 5 7 8 9 12 12 12 –2 –3 –1 1 2 3 6 6 6 –6
–2 –3 –1 –6
SI and ISI matrices
Scalar interval matrix
for a major triad
Interscalar interval matrix
for tritone-related major triads
The interscalar interval matrix is derived from a scalar interval matrix 33

34. Petrouchka chord –2 –3 –1 1 2 3 6 6 6 Interscalar interval
–2 –3 –1
Interscalar interval
matrix for major triads
a tritone apart
--Two superimposed major triads, arpeggiated in different inversions
--Harmonic intervals are not all exactly the same, but have a small range of sizes
The harmonic intervals of the Petrouchka chord represent a
row of the interscalar interval matrix. 34

35. Vertical
Intervals:
Petrouchka chord 2
ISI matrix
for major
triads at T
ISI matrix
for major
triads at T
Note that in this example, there is no overlap of interval sizes between adjacent rows between as well as within the different matrices. This is possible because of the propriety of the major scale. 1 6
When Stravinsky contrasts forms of the Petrouchka chord juxtaposing
different major triads, the vertical intervals come from different interscalar
matrices for the major triad. They articulate rows of these matrices that
that have similar interval sizes.

36. The first part of this example articulates different rows of the same
etc
Petrouchka chord 3
–1 –3 – –1 –3 – etc
–1 –2 –3
The first part of this example articulates different rows of the same
matrices as the previous example. The latter part of the example articulates
different rows of a single matrix (the one for the original Petrouchka chord)

37. Petrouchka chord 3 Because the major triad is proper,
etc
Petrouchka chord 3
–1 –3 – –1 –3 – etc
–1 –2 –3
Because the major triad is proper,
there is no overlap in interval sizes
between the two rows of the ISI
matrix realized in this example.
This makes Stravinsky’s “direction
flipping” effect possible.

38. Interscalar Interval Matrices (for scales not related by transposition)

39. ritual of the rivals + 8 Scalar interval matrix
. . .
+ 8
Scalar interval matrix
for the diatonic
Interscalar interval matrix for
diatonic scales a major third apart 39

40. petrouchka chord 4 A variant of the Petrouchka chord
(Second tableau, r.60, ostinato in piano and strings)
petrouchka chord 4
Note that this version is not octatonic. The octatonic scale is not an inherent feature of the petrouchka chord.
Stravinsky’s procedure of harmonic juxtaposition is not limited to
transpositionally related chords or scales . . . 40

41. petrouchka chord 4 # A variant of the Petrouchka chord
(Second tableau, r.60, ostinato in piano and strings)
petrouchka chord 4 –1
Interscalar interval matrix
for D minor and F major triads #
. . . but neither are interscalar interval matrices. 41

42. petrouchka chord 4 # 1 1 –1 4 5 4 8 8 9 Interscalar interval matrix–1
Interscalar interval matrix
for D minor and F major triads # 42

43. ISI matrix, diff. chords
An interscalar interval matrix for different set types can be constructed
from a scalar interval matrix and a voice leading between the set types 43

44. ISI matrix, diff. chords
Interscalar interval matrices (whether for the same set type or different
set types) catalogue the possible voice-leadings between chords, or the
vertical intervals that result from juxtaposing different rotations of them. 44

45. Co-Proper Scales

46. concerto come from transpositionally related ISI matrices.
The two matrices are related by a constant because they both relate diatonic and acoustic scales, just at different transpositions.
Vertical intervals in the polytonal scalar passages from Stravinsky’s piano
concerto come from transpositionally related ISI matrices. 46

47. (Gaps between the range of interval sizes in adjacent rows)
} Gap = 0
} Gap = 1
concerto
(Gaps between the range of interval sizes in adjacent rows)
When interscalar interval matrices for distinct transpositional set classes
are proper, the set classes are co-proper. Diatonic and acoustic scales are
co-proper. 47

48.
0 – –1 –1
} Gap = 1
} Gap = 0
} Gap = 0
} Gap = 1
concerto
Adding a constant or rotating rows obviously does not effect the ranges
within or between rows, so co-proper scales are co-proper regardless
of the transpositions of the individual scales. 48

49. copropriety } Gap = 0 } Gap = –1 } Gap = 1 } Gap = 0 } Gap = 1
The minor minor
ninth is not co-
proper with the
dominant minor
ninth
copropriety
} Gap = 0
} Gap = 1
The minor minor
ninth is co-proper
with the pentatonic
scale.
A scale can be co-proper with some relatively even scales, but not co-proper with other (less even) scales, even if it itself is not proper (as is the case with the minor minor ninth) 49

50. Scalar interval matrix for Scalar interval matrix for
minor minor ninth
Scalar interval matrix for
pentatonic scale
copropriety
Co-propriety can be determined from scalar interval matices alone.
For two scales to be co-proper it is necessary and sufficient that there is no overlap between adjacent rows of the two different scalar interval matrices. (This is a non-trivial result). 50

51. Stravinsky’s T-chaining Technique, exploiting copropriety and the interscalar interval matrix

52. octet
Lower bassoon arpeggiates F major
Upper bassoon arpeggiates a D# diminished seventh, which can be thought of as two interlocking diminished triads: D# dim. and F# dim.
Going from D# dim to F# dim., the pattern moves up a row while adding a constant of 3. (It also moves ahead by one column).
When coming back to D# dim., the constant of 3 is canceled, but we remain a row higher.
Since the scales are co-proper, the last three intervals are all smaller (or at least no larger) than the first three.
Another, more sophisticated, procedure we find in Stravinsky explores multiple rows of interscalar interval matrices at different transpositional levels by T-chaining two forms of one set against a single form of another. In this passage from the Octet, Stravinsky does this with major and diminished triads, which are co-proper. 52

53. petrouchka tetrachords
–2 –1 –
One more variant of the Petrouchka chord
Second tableau, r. 59 m.7 , Piano
This variant on the Petrouchka chord juxtaposes
two different four-note sets:
A half-dim. seventh ( ) and a fifths-generated chord, ( ).
These sets are co-proper. 53

54. petrouchka tetrachords
–2 –1 –1 1
petrouchka tetrachords
Stravinsky uses the T-chaining procedure in this example, T-chaining
transpositions of ( ) to present three rows of the interscalar matrix. 54

55. Conclusions

56. • Voice leading is a phenomenon that can occur in many guises: ordinary chordal voice leading, modulations between scales, and also vertically in the kind of polytonal passages we have been considering.
• Scalar and interscalar interval matrices are useful for understanding voice leadings between scales.
• Scalar interval matrices are closely related to the rotational arrays that appear in Stravinsky’s music.
• The concept of “scalar propriety” was originally introduced by David Rothenberg to describe the gap between the rows of a scalar interval matrix. This concept can be extended to interscalar interval matrices.
• Co-proper scales provide a wealth of opportunities for the sorts of simultaneous polytonal arpeggios and scales found in Stravinsky’s music.
•This fact might be useful to contemporary composers, many of whom are returning to the polytonal ideas found in Stravinsky’s early music.

57. Upright Petrouchka, Proper Scales, and Sideways Neapolitans
Department of Mathematics
Saint Joseph’s University
Rachel Wells Hall
Dmitri Tymoczko
Jason Yust
Department of Music
Princeton University
School of Music
University of Alabama, Tuscaloosa 57

58. propriety
blank
Scales can have equal values between rows and still be proper.
The dominant minor ninth has three or four interval sizes in
each row, but is still proper. 58

59. The minor minor ninth is improper as a five note scale.
propriety
blank
The minor minor ninth is improper as a five note scale. 59