Analysis: Bartók, Music for Strings, Percussion, and Celeste. Jordan Smith
Béla Bartók’s Music for Strings, Percussion, and Celeste highlights Bartók's mastery of orchestration, and innovation with rhythm. However, the opening movement perhaps least exemplifies these features (relative to the other movements). The first movement of the work instead showcases his mastery of counterpoint with a particularly praiseworthy example. Lendvai writes that “If the symbol of the circle means unity, closedness [sic] and fullness, then concerning the concentration, the fugue movement at hand perhaps compares only with the symmetries of the Kunst der Fuge” [emphases in the original].1 While Lendvai’s position may leave room for scholars to debate, this fugue is doubtless a demonstrably taut, elegant and balanced work of art. It is a piece of music that forms a cohesive and coherent whole without aid of conventional tonality or metric regularity. Bartók instead uses techniques and tools creates unity and direction through constructing a subject that is internally self-‐referential in contour, and form, and by achieving formal balance at the micro and macro level through careful structure of subject entrances as well as placement and pacing of dynamics and contrapuntal variation techniques to create a coherent arch form.
Subject: Segmentation and Affinity. Self-‐reference. Score
As Smith notes, “The most distinguishable part of any fugue has to be its first Smith subject entrance.”2 The structure of the fugue subject alone deserves J. considerable attention and comment.
8 12 8 & 8 j j # œ n œ j ‰ j 8 j # œj œj b œj n œ j ‰ # œj n œj 8 nœ œ bœ nœ bœ œ bœ
8 & 8 # œj n œj œj # œj j ‰ œ
j j œ bœ
7 8 œ
j j # œ œj n œ b œ
Figure 1. Fugue subject: mm. 1-‐4
1 Lendvai, Ernő. Bartók's Style. Edited by Katalin Fittler. Translated by Judit Pokoly. (Budapest: Akkord Music Publishers, 1999), 115. 2 Smith, Jordan. "Comparative Analysis: Fugues by Bach, Händel, and Mozart." Unpublished. (2013), 1.
2 First and perhaps most immediately noticeable is the ambiguity created through shifting meter: each measure is constructed using a different time signature than the bar preceding or following it. The first and third measures both make use of 8/8, but this does not violate that principal. Each musical segment consists of one Score measure initiated by its anacrusis; the first two segments use one eighth note anacrusis (both of which are performed by the violas at A3, hereafter referred to as J. Smith 9 using pitch class notation) and the latter two use two eight notes. An eighth rest separates each of the four segments.
8 12 8 j ‰ & 8 j j# œ n œ j ‰ j 8 j 8 b œ n œ œ # œ n œ # œ b œ n œ b œ n œ b œ œ œ a1:
9
T
1
0
8 & 8 # œ n œ Score œ #œ b1: 3
2
0
1
E a2: 9
‰
œ
E
T
nœ bœ
b2: 0
3
1
2 3
7 8 œ 2
0
E
#œ
1
T
1
4
j œ nœ bœ
E
0
J. Smith
Figure 2. Fugue subject segment and pitch class annotation.
Second, and with closer attention to pitch organization (see fig. 2), it becomes clear that the first two segments begin identically with ordered pitch-‐class (PC) set 8 8 j ‰ j 12 ‰ 8 j be j # œcalled j 8 j b œ n œ (9T1); this could & ordinarily a motive. The latter two segments vary #œ nœ nœ nœ œ # œ œ n œ n œ b œ # œ 8 both b œ b œ œ with regard to rhythmic and pitch-‐class content, yet show significant affinity. Indeed the fourth segment perfectly transposes the third segment down a half step and then modifies the 4 rhythmic frame. In this way, the segments are hereafter called a1, 7 a2, b1 and b2 respectively. ! ! ! ! ! j
7 b œ œ #œ œ nœ 8 ‰
12 8
& 8 #œ nœ b œ segments a1 and a2 have many other layers While beginning œwith iœdentically,
of symmetry. First, a2 can be interpreted as merely an interpolated version of a1. This is best visualized when comparing the measures vertically (see fig. 3) 11
&
17
&
j a1 œ
88
‰ j 78 ! j nœ nœ œ
j bœ #œ
!
78 j ‰ j 12 8 j bœ #œ œ bœ nœ nœ bœ #œ nœ a2 œ
Figure 3. Vertical analysis: segments a1 and a2.
Alternatively, these segments can be viewed from the perspective of ordered pitch-‐class intervals. This approach reveals additional layers of quasi-‐motivic3 unity and affinity between each segment. 3 The term “motivic” is consciously avoided in an outright sense in order to avoid confusion. While each segment can be easily broken down into smaller cells, the cells ©
! !
! !
Score J. Smith 3
8 12 j ‰ n œ 88 & 8 j j# œ n œ j ‰ j 8 j b œ œ nœ nœ bœ #œ nœ œ bœ #œ œ bœ 8 & 8 #œ nœ D1
U1
D2
U3 D1 D1
œ #œ U1
D2
œ
(D2)
‰ (U1)
U1
nœ bœ U3
U3 U1 U1 D3 D1
7 8 œ
D1
D1
#œ
D2
D1 (U3)
U3 D1
j œ nœ bœ U1
D2
Figure 4. Ordered pitch-‐class interval analysis (with direction)
Looking at only measures 1 and 2 in tabular form, further conclusions may be drawn. As previously noted, a1 and a2 begin with ordered PC set (9T1), ergo they begin with identical intervals. They also conclude with two identical downward half steps (labeled D1). However the remaining interior intervallic content of the longer segment a2 can also be analyzed against that of a1. a1 a2
U1
U3
U1
U3
D1
D1
U1
U1
D3
D1
D1
Figure 5. Interior interval class comparison. U=up, D=down, 1=one half-‐step, 3=three half-‐steps.
As shown in fig. 5, the interior of a2 is itself a transposed retrograde of a1. Therefore, every interval of a1 can be said to have an intelligible form of affinity with the slightly lengthier a2. This type of work can be repeated in detail across the following two segments. Of b1 and b2 (whose interval contours are identical), the most noticeable affinity can be found by comparing a1 with its anacruses omitted. With that omission it is easy to see that all three segments relate very cleanly to the head. Segments b1 and b2 thereby represent a form of musical ”pig latin”. These overlaps and affinities can also thought of as a form of self-‐reference, as will be discussed.
Subject: Pitch Content:
Upon still closer analysis, the subject yields a clear sense of the harmonic structure to come. Interestingly, the subject does make use of eight pitches, just as would any diatonic major or minor scale (including the necessary resolving octave). However, the eight pitches used are eight consecutive chromatic neighbors, those © being A3 through E4. As traditional tonality has, therefore, given way to what many
may or may not represent a degree of salience and consistency that is often ascribed to that term. While no one aspect of the term is violated with severity, the term as a whole seems to have inappropriate connotations in view of the existing relationships between segments in the piece, or so I would argue.
4 post-‐tonal analysts refer to as centricity, one may need to explore “the entire spectrum of centric effects”. 4 Indeed, alternative analyses are required here. Pitch Class
Frequency Only
Duration Included
9
2
2
T
4
5
E
4
5
0
5
7
1
5
6
2
3
4
3
2
2
4
1
1
rest
(3)
3
Total
27
35
Figure 6. Pitch Salience: with and without considering duration
Under the lens of pitch class salience, the fugue subject yields additional data. Taking the eight pitch classes in a tabular form (see fig. 6), pitch content can be used statistically. The “frequency only” column lists the number of occurrences of each pitch class while the “duration included column awards “additional points” for quarter notes (double the duration=double the points). In the chart in fig. 7 demonstrates these relationships using a bar graph. In this light, PC0 is clearly the dominant class, and also happens to be roughly in the center between PC9 and PC4. This is puzzling, at least in part, because the subject entrance (both for segment a1 and a2) is PC9, yet these are the only two occurrences of that PC. Historically, entrances of fugue subjects are tracked using the first pitch of each entrance, even in myriad cases where the first pitch is not the same as the tonic. This ambiguity may lead one to ask whether PC9 is meant to be heard as central. This question will resurface at the end, both of the movement, and of this document.
Figure 7. Subject Pitch Class Content. Black=”Frequency Only”; Grey=”Frequency+Furation”
4 Straus, Joseph N. Introduction to Post-‐Tonal Theory. ed. Sarah Touborg. (Upper Saddle River, NJ: Pearson, 2005), 131-‐133.
5
Fugue Structure
The fugue structure itself is a rich tapestry of design that is best illustrated by a chart showing the subject entrance pitch plan, adapted from Landvai. (see fig. 8) The chart demonstrates a number of points The fugue structure itself is a rich tapestry of contrapuntal virtuosity in service of the larger, carefully calibrated double arch structure. This structure is shown below in the the subject entrance pitch plan, adapted from Landvai. (see fig. 8) The chart demonstrates a number of points. Central to the organizational structure is the progression through the complete circle of fifths simultaneously in both directions, a double helix of sorts. They meet at roughly the midpoint of the movement's 88 measures: at the anacrusis to m. 45 in the most distant entrance at the tritone (Eb/PC3). However, even as the movement begins to return back through each pitch center, another more powerful peak built using dynamics continue through to measure 56. This is approximately the “golden section” of the piece. p (con sordino)
(senza sordino)
mp
mp
g# e
f#
b
f
c
canon
p
(con sordino)
c
5
9
13
17
g
d
golden mean
a db
f eb
f#
b
e
ab
stretto
stretto
canon
rectus 1
p - ppp
bb f
bb exposition
f
eb
centre
g
fff
c#
a d
ff
recapitulation
inversus 27
34
45
56
58
65
rectus+inversus 69
73
Figure 8. Pitch Center/Structure/Dynamics Diagram5
A basic understanding of the “golden section” is necessary to authentically discuss the music of Bartók. His well-‐documented fascination and often the explicit formal design of his compositions stem from this special ratio, which is roughly 1.618. Like pi, this constant has an associated Greek letter, phi. It is the recursive and self-‐referential nature of this ratio that allows it to create interesting fractal like patterns in the music. For instance, the golden mean of the subject is found at its 21st eighth note. This coincides with the highest pitch class in the subject. At the larger scale, as seen above, the golden section is also the peak point of impact dynamically. Also note that Bartók does not yet begin to return through the keys nor does he begin to further develop his fugue technically until this peak is reached. Worth noting is Bartók’s deference to his own intuition. The actual golden section would occur at m. 54 on beat 4. That calculates, using the formula in fig. 9a , both when calculating either the number of measures or the number of beats (using 5 Lendvai, p. 115
78
88
6 the eighth note as constant throughout). In the case of the fugue, it would suggest that the length of the beginning to the fff at m. 56 is to m. 56 to the end, as the entire fugue is to the beginning to 56. Further, if this rough equivalency can be accepted, then it is worth noting that the “recapitulation” is roughly in proportion to the “inversus” (which is inaugurated by the entrance of the celeste) as the whole of m. 58 to the end is in relationship to the “recapitulation”.
Figure 9a. Mathematical Notation of Phi6
Figure9b. Visualization of Golden Ratio7
Lastly, a note on the discussion regarding the disparity between the importances accorded the first pitch in a subject and its salience in Bartók. This disparity does indeed find its resolution. “A” (PC9), which was on the far low end of the pitch frequency chart, does indeed establish its primacy as the soli unison pitch on which the entire final measure is pinned. This also completes, with a sense of finality, the double helix pitch center scheme, as well as the locally formed double helixes shapes created by homorhythmic presentation of both the fugue subject and its inverted form.
Conclusion
The first movement of Bartók’s Music for Strings, Percussion, and Celesta, makes excellent use of all of the best techniques well known and appreciated by contrapuntists over the past 300 years. However, the fugue is so rich with invention, there is virtually no need to simply approach the work. Through his tightly integrated fugue subject, novel approach to dodecaphony via two countervailing cycles of fifths, and innovative use of the recursive golden mean, Bartók tightly knits together a work of immense complexity that is indeed worthy, if not to stand alone next to Bach’s Art of Fugue, then among only a small number of worthy peers. Further, Bartók’s surface accomplishments in orchestration and rhythmic invention only serves to enhance and augment his ability to create deep structure and finely tuned formal balance in his compositions. 6 Wikimedia Foundation. Golden ratio. http://en.wikipedia.org/wiki/Golden_ratio (accessed December 8, 2013). 7 Wolfram Alfa. http://www.wolframalpha.com/input/?i=golden+section&a=*C.golden+section-‐_*MathWorld-‐ (accessed December 8, 2013).
7
Bibliography
Lendvai, Ernő. Bartók's Style. Edited by Katalin Fittler. Translated by Judit Pokoly. Budapest: Akkord Music Publishers, 1999. Smith, Jordan. "Comparative Analysis: Fugues by Bach, Händel, and Mozart." Unpublished, 2013: 1. Straus, Joseph N. Introduction to Post-‐Tonal Theory. Edited by Sarah Touborg. Upper Saddle River, NJ: Pearson, 2005. Wikimedia Foundation. Golden ratio. http://en.wikipedia.org/wiki/Golden_ratio (accessed December 8, 2013). Wolfram Alfa. http://www.wolframalpha.com/input/?i=golden+section&a=*C.golden+section-‐ _*MathWorld-‐ (accessed December 8, 2013).
