Survival of the fittest: A reevaluation of traditional scale and arpeggio fingerings
Fingering holds a foundational place in fine piano technique. In the words of Carl Philipp Emanuel Bach, "More is lost through poor fingering than can be replaced by all conceivable artistry and good taste."1 For many pianists, traditional fingerings for scales and arpeggios are chiseled in stone. Codified in C.P.E. Bach's Essay on the True Art of Playing Keyboard Instruments and Hanon's The Virtuoso Pianist, and then passed down from teacher to student for generations, these fingering approaches have stood the test of time and for good reason: they feature consistent and memorable patterns that transfer between keys and capitalize on different finger lengths. For convenience, standard fingering patterns tend to begin and end with outer fingers in each hand whenever possible. However, due to the asymmetrical configurations of major and minor scales and arpeggios, favoring outer fingers can sometimes result in different fingering patterns for each hand. If evaluated from the principle that identical patterns of black and white keys merit identical fingerings, many standard fingerings miss the mark. An exploration of symmetrical fingerings provides compelling alternatives to standard approaches, allowing pianists to move beyond tradition to include healthier and more efficient options in their playing.
In order to examine the symmetry of fingering patterns, we must understand how symmetrical inversion takes place at the piano. As shown in figure 1, the keyboard organizes symmetrically around two axes: D and A-flat. Any figure written for one hand can reflect across these axes to produce its mirror image in the other hand. Because human hands are symmetrical, this inversion around D or A-flat results in the same arrangement of black and white keys under the fingers. Since major and minor tonalities are asymmetrical, these inversions produce different pitch collections from the original figure: major triads invert to minor and minor triads to major. Figure 2 shows how an E-major triad in the right hand reflects across D to form an F-minor triad in the LH. For this symmetrical key pair, the physical layout of the keys under the hand remains constant, but the pitch collection differs. See table 1 for a complete list of symmetrical key pairs.
With this framework in place, we can determine which traditional scale and arpeggio fingerings are symmetrical. Table 3 illustrates the standard fingerings for all arpeggios in both hands organized in symmetrical pairs.2 Each distinct fingering pattern is highlighted in a different color. In order to provide exact inversions, the right-hand fingerings ascend one octave degree 1 from to degree 1, and left-hand fingerings descend one octave from degree 5 to degree 5. Figure 4 represents two of these arpeggios visually on the keyboard.
Nearly all arpeggios that begin on black keys have symmetrical fingerings. The remaining, asymmetrical pairs predominately follow the pattern exemplified by the a/C pair shown in figure 5.3 These left-hand fingerings assign the perfect fourth interval to fingers 2 and 1, and the right-hand fingerings use the perfect fourth to shift hand positions through turning under the thumb or crossing finger 3. This fingering discrepancy occurs in most white key arpeggios.4 The remaining pair—b/B-flat—stands alone: the left-hand b-minor fingering conforms to the most common white key pattern (2-4-1-2), but B-flat major, its inverted counterpart, places fingers 2 and 4 on the perfect fourth, similar to the right-hand fingering of an E-flat-major arpeggio. Thus standard arpeggio fingerings fall into two large categories: key pairs with symmetrical fingerings and key pairs that assign different fingerings to the perfect fourth.
Scales present a wider set of possibilities. The traditional approach prescribes different fingerings for identical pitch patterns in some relative major and minor scales. Table 4 shows standard fingerings for major scales, their relative minor scales, and their inversions: four distinct ways of understanding the same collection of keys. Scales that contain greater numbers of black keys have more fixed fingering traditions, with uniform fingerings across all four approaches to each pitch collection. Conversely, the scales that use the fewest black keys present unique fingerings for each of the four approaches (see figure 6).
Variation in scale fingering most frequently stems from the prevailing tendency to place the thumb on degree 1 if it is a white key. This often forces the thumb to turn under finger 3 or 4 as it plays a white key, though a nearby black key may have provided an easier place to shift. In contrast, when degree 1 is a black key, traditional fingerings place the thumb on the first available white key. This wide variety of fingering choices for the same collection of pitches—up to four distinct approaches—reveals that pianists already experience a fair amount of flexibility in fingering but may also suggest that some patterns are more effective than others.
As demonstrated above, standard scale and arpeggio fingerings present a considerable degree of asymmetry between the hands, largely due to consistently placing the thumb on the tonic pitch in both hands. This is doubtless an intuitive and convenient approach. However, in some contexts, adapting scale and arpeggio fingerings to conform to symmetrical fingerings may prove a desirable alternative. These alternatives depart from prioritizing placement of the thumb on degree 1 in order to facilitate more comfortable position shifts.
One such alternative fingering applies to white key arpeggios. As mentioned previously, the standard fingering for right-hand white-key arpeggios (1-2-3-1) places the hand position shift over the perfect fourth. The left-hand approach to these same patterns takes the perfect fourth comfortably with fingers 1 and 2 and shifts over a third. The right-hand can adopt this approach by beginning with finger 2 on tonic as shown in figure 7. This alternative fingering is gaining wider acceptance, having recently been advocated by Rami Bar-Niv and Carol Leone.5 Many pianists may find that they already use this approach in their own playing. An excerpt from the finale of Schubert's Sonata in A Major, D. 664, illustrates a passage well suited to this fingering (figure 8). The following right-hand arpeggios could benefit from this alternative approach: F, C, G, D, A, E, f, c, g, d, a, and e.
If right-hand arpeggios can benefit from copying left-hand fingerings, the opposite may be true for scales. Right-hand scale fingerings consistently coordinate the turning under of the thumb to follow finger 3 or 4 playing a black key. Left-hand fingerings more frequently require the thumb to turn under a white key, even when black keys are present. For example, in a D-major scale, the right hand thumb turns under when finger 3 plays F-sharp and finger 4 plays C-sharp. In the left-hand descent, the thumb turns under 3 on B and 4 on E. However, if the left-hand adopts the approach of B-flat major, its symmetrical counterpart, the thumb will then turn under C-sharp and F-sharp as shown in figure 9.6 Another excerpt from Schubert's sonata shows this alternate fingering in action (figure 10). The following left-hand scales could benefit from this approach: G, D, A, f, c, and g.
Even for those pianists who do not choose to modify traditional fingerings, an understanding of mirror relationships between scales can help identify differences in the physical requirements for each hand. Playing scales and their inversions simultaneously promotes this awareness. Pianists have long practiced scales in contrary motion, but beyond C major, the pattern of black and white keys differs in each hand when scales are played this way. To get the truest sensation of playing a mirror image in each hand, pianists should perform contrary motion scales with the key pairs outlined above. For example, to practice the mirror image of a right-hand G-major scale, the left hand plays an F-major scale beginning on A as shown in figure 11. Conversely, to mirror a left-hand G major scale, the right hand plays an F-major scale beginning on A. Beyond increasing awareness of keyboard topography, mirroring the hands builds technical skill. Many pianists will find that one hand has developed a greater mastery of a particular scale pattern. By practicing with the hands mirrored, the less-skilled hand can piggyback on the other's strength and copy its physical approach, making meaningful improvements that will extend beyond this exercise. Thus, even if pianists stick with traditional fingerings, through contrary motion practice they can maximize technical efficiency for these patterns.
Traditional fingering approaches have served pianists well for centuries, but they are not the only solutions to fingering questions. As shown above, tradition often presents more than one fingering option for identical patterns of black and white keys depending on which hand is playing. When these fingerings differ for each hand, pianists can choose the approach they deem most suitable and mirror it in the other hand. As Chopin put it, in piano playing "everything is a matter of knowing good fingering." Those with a thorough understanding of the keyboard's symmetry can make the most efficient and effective fingering choices.7
1Carl Philipp Emanuel Bach, Essay on the True Art of Playing Keyboard Instruments, trans. William J. Mitchell (New York: W. W. Norton, 1949), 41.
2These fingerings are derived from Charles-Louis Hanon, The Virtuoso Pianist, ed. Theodore Baker (New York: G. Schirmer, 1936).
3Key pairs are notated (LH/RH) with uppercase letters for major triads and lowercase for minor. Thus, a/C refers to an A-minor triad in the left hand and a C-major triad in the right.
4The left-hand arpeggio fingerings for F-sharp, E, D, and A substitute finger 3 for finger 4 but maintain the placement of fingers 2 and 1 over the perfect fourth.
5Rami Bar-Niv, The Art of Piano Fingering: Traditional Advanced and Innovative (Tel Aviv: AndreA, 2012), 40; Carol Leone, "Putting a Strong and Versatile Technique Within Your Students' Grasp."
6Carol Leone suggests a similar approach in "Putting a Strong and Versatile Technique Within Your Students' Grasp."
7From Chopin's incomplete Projet de Méthode, reprinted in Jean-Jacques Eigeldinger, Chopin: Pianist and Teacher As Seen by His Pupils, trans. Naomi Shohet, ed. Roy Howat (Cambridge: Cambridge University Press, 1986), 195.