Introduction

When Bruno Monsaingeon asked Glenn Gould about his 1981 recording of Bach’s “Goldberg Variations,” BWV 998, Gould replied, “it occurred to me […], that [the 1955 recording] was very nice, but that it was perhaps a little bit like thirty very interesting but somewhat independent-minded pieces going their own way.” Therefore, in his new recording, he intended to find an “arithmetical correspondence between the theme and the subsequent variations, so that there would be some sort of temporal relationship – […] there would be at least a rhythmic design that was continuous and a sense of pulse that went through it.”[2] This could be read as an aim to perform the Variations as one cyclic work (as opposed to a mere collection of thirty-two individual pieces), bringing out the interconnections between the various pieces of the cycle more consciously and clearly. While “arithmetical correspondence” and “temporal relationship” play a significant role in the 1981 recording, this aim is also reflected in a careful choice of repeats: in contrast to his 1955 interpretation, which does not contain any repeats, Gould opts for repeats in thirteen variations – the nine canonic variations as well as the four variations set in strict four-part counterpoint (variations 4, 10, 22, and 30), elucidating their mutual connection and common structural qualities.[3]

Beyond any question, the “Goldberg Variations” do constitute a cycle, primarily for the very fact that any succession of variations on a preceding theme provides a strongly interlinked cyclic form, but also, and more importantly, because they follow an elaborate compositional structure, interlocking character, virtuoso, and canonic variations, implementing musical progression and symmetry as paramount formal principles.[4] As stated by Martin Zenck, “the melody of the chaconne bass and its harmonic pillars create the outer shape […] in which the thirty variations are cast like into a mold.” [5]

The cyclic shape of the work is not identifiable only through the architectonic disposition of the three types of variations mentioned above; there are further traits that link the pieces together, such as dance suite movements which relate to each other in different ways (e.g., passepied, gigue, French ouverture, sarabande, menuet, polonaise), genres (e.g., fantasia, sinfonia, fugetta, quodlibet), time signatures, (minor) key, number of voices, etc.[6] Any variation may be classified according to a number of parameters, simultaneously belonging to several networks of pieces.[7] In this context, the question arises to what extent its performers realize the cyclic potential of the “Goldberg Variations” inherent to its overall structure and to the individual pieces, i.e., how they handle parameters such as tempo, duration, dynamics, repeats, and harpsichord registration. The interpretation of such elements may add further layers to the correlational structures present in the cycle. In the foreword to his 1934 edition of the Variations, Ralph Kirkpatrick writes of the formative employment of such parameters:

It should be noticed that this registration is intended to enhance the symmetrical arrangement of the variations, in that the same 8’ register is employed for all the canons and the same combination for the two-manual arabesques, whereas the greatest possible variety is brought to the other assorted forms, in accordance with their character.
Changes of registration within the variations are quite uncalled-for; they only bring a kind of disturbing restlessness to the expression and utterly destroy the architectural symmetry. Each movement has its own tone-color […].
This same character should be preserved in a performance on the piano, by […] employing the resources of nuance only in smaller degrees in order to enhance the declamation of individual phrases. […] However, each variation should be given a distinctly characterized color or dynamic level.[8]

This article focuses on the question: To what extent has the cyclic form of the “Goldberg Variations” been reflected in its performance history? It centers on the temporal relations between its individual pieces as performed in selected recordings. For this purpose, using tempo measurements as the main analytical criterion seems to be particularly suitable, since durational values would be of limited significance, due to the differences between performers concerning the (non-)execution of repeats.[9] Additionally, the duration of a piece is also strongly influenced by different degrees of rubato and ritardando as well as by durational choices for fermatas, and thus unsuitable for establishing relations between the main tempi of the cycle’s pieces.[10]

This study is based on a database of tempo measurements created during the research project Performing, Experiencing and Theorizing Augmented Listening (PETAL), which focused on performance strategies in cyclic works (piano and lied literature). This database compiles tempo data from seventy-six selected recordings spanning the years 1928 to 2020 (Appendix, Tab. 7), measured using Sonic Visualiser.[11] The calculation of the main tempi for both Arias and variations is based on the first musical phrase formed by the first eight bass notes (G–F–E–D–B–C–D–G, mm. 1–8 or 1–4 respectively).[12] For the purpose of the present article, these measurements have been adjusted to exclude the first and last bass notes of the chosen sections, in order to avoid the tempo deviations occurring in rubato and ritardando passages. Following the principle described by Alf Gabrielsson,[13] the “main tempo” of each piece has thus been derived from the average tempo of the measures featuring the second to seventh bass notes (mm. 2–7 or 1.5–3.5 respectively; see Appendix, Tab. 8). The vast amount of data (2428 measured audio files) yielded by these measurements provides a multitude of possibilities for quantitative corpus analysis. Some conspicuous results discernible in the overview shall briefly be covered below (Tab. 1).

Table 1: Bach, “Goldberg Variations”: mean tempo, maximum and minimum tempo, relative standard deviation, and relative range in seventy-six selected recordings from 1928 (Rudolf Serkin) to 2020b (Lang Lang)

Table 1 depicts the fastest and slowest recordings for each piece, their relative standard deviation (SD) from the respective mean tempo value and the relative range (difference between maximum and minimum tempo values, expressed as a percentage of the main tempo). Variation 14 (10.8%), variation 10 (12.0%), variation 16a (12.2%), and variation 4 (12.4%) stand out for having the lowest SD values, while variation 16a (64.7%), variation 26 (69.4%), and variation 14 (74.0%) have the lowest range; variation 14 and variation 16a therefore stand out concerning both parameters. The recording of variation 14 made by Angela Hewitt 1999 (Audio Ex. 1a:  = 93.2 bpm) matches the mean value (93.0 bpm) most closely; the fastest and slowest recordings of this variation are performed by Rudolf Serkin 1928[14] (Audio Ex. 1b: 127.0 bpm) and Daniel Barenboim 1992 (Audio Ex. 1c: 73.0 bpm) respectively. The low SD values combined with the low range show that ostensibly, most performers seem to agree on a similar tempo choice for this variation. Rolf Dammann describes variation 14, the second to last variation of the first half of the cycle, as a pezzo di bravura (although not to be played “excessively fast”), focusing on mobilità, agilità and prontezza;[15] possible reasons for the relatively low mean tempo value might be found in the technical challenges of this piece, such as hand-crossing, left-hand arpeggi, trills, etc. For variation 16a, the recording matching the mean tempo value (33.9 bpm) most closely is the interpretation of Ton Koopman 1987 (Audio Ex. 2a: = exactly 33.9 bpm). The fastest and slowest recordings are by Rudolf Serkin 1928 (Audio Ex. 2b: 44.8 bpm), and Jörg Demus 1953 (Audio Ex. 2c: 27.2 bpm) and Barenboim (27.6 bpm) respectively. Here, one cause for the strong overlap in tempo choices among the selected recordings might be the dignified French Ouverture character, as well as the fact that variation 16a serves as a festive opening piece to the second half of the cycle.[16]

Audio Example 1a: Bach, “Goldberg Variations,” Hewitt 1999, variation 14 (mm. 1–8)
(Bach, “Goldberg Variations,” Angela Hewitt, CD Hyperion Records Limited, ℗&© 2000, Track 15)

Audio Example 1b: Bach, “Goldberg Variations,” R. Serkin 1928, variation 14 (mm. 1–8)
(Bach, “Goldberg Variations,” Rudolf Serkin, CD Archiphon ARC-105, ℗&© 1992, Track 1)

Audio Example 1c: Bach, “Goldberg Variations,” Barenboim 1992, variation 14 (mm. 1–8)
(Bach, “Goldberg Variations,” Daniel Barenboim, DVD EuroArts, © 2012)

Audio Example 2a: Bach, “Goldberg Variations,” Koopman 1987, variation 16a (mm. 1–8)
(Bach, “Goldberg Variations,” Ton Koopman, CD Erato Disques S.A, ℗ 1988, Track 17)

Audio Example 2b: Bach, “Goldberg Variations,” R. Serkin 1928, variation 16a (mm. 1–8)
(Bach, “Goldberg Variations,” Rudolf Serkin, CD Archiphon ARC-105, ℗&© 1992, Track 2)

Audio Example 2c: Bach, “Goldberg Variations,” Demus 1953, variation 16a (mm. 1–8)
(Bach, “Goldberg Variations,” Jörg Demus, LP Westminster WL-5241, ℗&© 1992, Side A)

On the other end of the spectrum, variation 6 (24.0%), variation 12 (23.4%), variation 19 (21.8%), and variation 21 (21.5%) strike the eye as having the highest SD values, whereas variation 25 sees an exceptionally high range (202.2%). For variation 6, the interpretation matching the mean tempo value (47.4 bpm) most closely is the recording by Alexis Weissenberg 1981 (Audio Ex. 3a:  = 47.3 bpm). The fastest tempo choices appear in Gould 1958 (Audio Ex. 3b: 81.5 bpm) and Andrei Gavrilov 1993 (79.9 bpm), the slowest in Rosalyn Tureck 1957 (Audio Ex. 3c: 31.0 bpm) and Demus 1953 (31.1 bpm). These stark differences could be rooted in different approaches to the notation of the piece. While Gould and Gavrilov both take the whole bar (3/8) as the main pulse, essentially turning the variation into a passepied, Tureck and Demus both play it as a menuet, the pulse here being the eighth note. The two other 3/8 variations also raise this question: while variation 19 also features a relatively high SD value (21.8%), variation 4 appears to show almost no conceptual “disagreements” (12.4%), confirming a shared interpretation as a passepied.[17]

The mean tempo for variation 25 which exhibits the highest range value (202.2%) is 49.2 bpm, matched most closely by Igor Levit 2015 (Audio Ex. 4a:  = 49.0 bpm); its fastest recording is presented by Yūji Takahashi 2004 (Audio Ex. 4b: 77.4 bpm), its slowest by Tatiana Nikolayeva 1992 (Audio Ex. 4c: 25.6 bpm). A comparison of the mean tempo values between pianists and harpsichordists shows that the pianists (mean tempo: 46.9 bpm) tend to play this variation considerably slower (by 11.7%) than the harpsichordists (mean tempo: 53.1 bpm; Fig. 1). On the whole, the interpretations of variation 25 made by the harpsichordists rarely (and after 1985, never) undercut the overall mean tempo value (Fig. 2). The trend lines show that there is a tendency among the harpsichordists to increase the tempo of variation 25 over the course of time, while the data for the pianists shows a trend in the opposite direction.

Figure 1: Bach, “Goldberg Variations,” mean tempo differences (in %) as displayed in the recordings made by pianists vs. harpsichordists

Figure 2: Tempo trend lines for variation 25, pianists vs. harpsichordists

This gradual decrease suggests a romanticized approach to the interpretation of this variation by the pianists. Wanda Landowska calls this variation, the third and last variation in G minor, “the supreme pearl of this necklace – the black pearl”:

In its somber shimmerings, all the restlessness of the romantics may be already discerned. This richly ornamented adagio is overwhelming with the poignancy of its feverish chromaticism. Is not this nostalgic and plaintive curve toward the sixth the same as that later to be rediscovered by Chopin and the Wagner of Tristan?[18]

Similarly, Jörg Demus states that Bach “surprises […] us in the adagio variation No. 25, the crown of the work, with an egregiously audacious, expressive – I’d almost say: romantic – chromaticism, not to be found until later with Franck and Reger.”[19] In the same vein, Glenn Gould describes variation 25 to be evocative of “the languorous atmosphere of an almost Chopinesque mood-piece,” its “wistful, weary cantilena a master-stroke of psychology.”[20] Quite unlike these three viewpoints, Rolf Dammann traces the affectus dolorosus displayed in this variation to narrow and diminished intervals frowned upon in the seventeenth century, the repeated pianto motif in the bass, and the exclamatio depicted by the soprano.[21]

Audio Example 3a: Bach, “Goldberg Variations,” Weissenberg 1981, variation 6 (mm. 1–16)
(Bach, “Goldberg Variations,” Alexis Weissenberg, EMI Classics 5 75952 2, ℗ 1982 © 2001, Track 7)

Audio Example 3b: Bach, “Goldberg Variations,” Gould 1958, variation 6 (mm. 1–16)
(Bach, “Goldberg Variations,” Glenn Gould, CD West Hill Radio Archives WHRA-6038, ℗&© 2011, Track 7)

Audio Example 3c: Bach, “Goldberg Variations,” Tureck 1957, variation 6 (mm. 1–16)
(Bach, “Goldberg Variations,” Rosalyn Tureck, CD EMI Classics 09647-2, ℗ 1958 © 2008, CD 1, Track 7)

Audio Example 4a: Bach, “Goldberg Variations,” Levit 2015, variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Igor Levit, CD Sony Music Entertaiment Inc. 88875140142, ℗ 2015 © 2016, Track 26)

Audio Example 4b: Bach, “Goldberg Variations,” Takahashi 2004, variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Yūji Takahashi, CD Avex Classics AVCL-84069, ℗&© 2014, Track 26)

Audio Example 4c: Bach, “Goldberg Variations,” Nikolayeva 1992, variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Tatiana Nikolayeva, CD Hyperion Records Limited A66589, ℗&© 1992, Track 26)

Methodology

Based on the information obtained through the database, specific variations which stand out through extreme values have been examined. Cyclic interpretation, however, is rooted in the relations between a cycle’s parts. There is a variety of criteria that might be singled out when comparing such parts; for this article, three different dimensions have been selected in order to examine different possibilities of cyclic interpretation in the “Goldberg Variations.” Each of these models focuses on a small group of variations exhibiting specific criteria which establish possible linear and non-linear tempo relations:

1. a progression-based analysis focuses on a constellation of variations based on their progression from one to the next, i.e., as a sequence of pieces (variations 24 to 26)

2. a structure-based analysis examines a constellation of pieces that share a certain interlinking criterion, in this case the minor key (variations 15, 21, and 25)

3. a position-based analysis investigates the pieces framing the cycle (Arias 1 and 2 and their adjacent variations 1 and 30), based on their functional relevance to cyclic symmetry.

As argued by David Epstein, the concept of proportional tempo suggests that in works of multiple movements, “all tempos are intrinsically linked via a common pulse”:

The relationship arises from the organization of the work as a unified and coherent whole in which all movements, all ideas, stem from underlying formative concepts of shape. […] These relationships of tempo can be concisely expressed by whole-number (integral) ratios.[22]

Notably, these ratios are generally integral and of a low order (1:1, 1:2, 2:3, 3:4, or the inverse, occasionally 1:3) and do not seem to exceed 3:5, 5:6, or their inverse; Epstein explains that this principle results from phrase synchrony in proportionally related tempi.[23] As the remaining discussion will demonstrate, this argument is confirmed by the tempo measurements of the PETAL database. Tempo relations for each set of pieces (examined in pairs) are depicted in color-coded tables indicating percent values (Tab. 2). The percentage value representing a tempo relation between two variations indicates a specific mathematical proportion which can also be understood as a relationship between note values. A proportion between tempo A and tempo B (A:B) is calculated by dividing the main tempo value B by the main tempo value A (B/A, expressed as a percentage). Thus, a tempo A of 120 bpm and a tempo B of 60 bpm (proportion 2:1) is expressed by a percentage of (–)50% (60/120 = 0.5). This allows us to understand all tempo proportions in their progressional context: we see how tempo B relates to tempo A or how tempo A evolves into tempo B. To facilitate interpretation of the tables, all negative percentage values are rendered as absolute (positive) numbers.

Table 2: Color-coded tempo relations and their respective percentage values

The second and third columns in Table 2 indicate the tolerance range for each relation. This range has been defined around (rounded) mathematical “core” values: 1:1 = 0%, 2:1 = (–)50%, 1:2 = 100%, 3:1 = (–)67%, 1:3 = 200%, 3:2 = (–)33%, 2:3 = 50%, and 4:3 = (–)25%. The percentages denote the proportional increase/decrease from tempo A to tempo B. Further proportions (such as 3:4, 33%) have not been taken into account, as they do not suggest musically sensible relations between time signatures in this cycle. With the exception of proportion 1:1 (tolerance range: 20%, ±10%), a range of 10% (±5%) has been defined for all relations of the second column (negative percentage values; in all these cases tempo B is slower than tempo A), while the third column (positive values, indicating that tempo B is faster than tempo A) includes a range of 20% (±10%) for the 2:3 relation and a 40% (±20%) range for those two relations which center around a “core” value of 100% or higher (1:2, 1:3). Consequently, a tempo relation of 2:1 between a tempo A of 60 bpm and a tempo B of 30 bpm (half tempo) would also apply if tempo B falls within the range of 27 bpm ([–]55%) and 33 bpm ([–]45%). In the same manner, a tempo relation of 1:3 between a tempo A of 60 bpm and a tempo B of 180 bpm would also be identified if tempo B falls within the range of 168 bpm (+180%) and 192 bpm (+220%). These color-coded percentage values and their respective musical ratios as indicated in Table 2 will be key elements in the comprehension of the following analyses.

In his historically informed study about composed tempo relationships in the “Goldberg Variations,” Ulrich Siegele develops a theory where all tempi within the cycle are related to the tempo of the first Aria through a “principal value” p, defined as 57.6 bpm, and its multiples/fractions.[24] While a detailed discussion of this theory cannot be carried out in this article, the tempo relations calculated by Siegele have been included in the tables below in order to enable a comparison of actually performed tempo relations to this theoretical system.

Cyclic Interpretation I: Progression-based Analysis (Variations 24–26)

The sequence of variations chosen for this analytical model (Ex. 1) stands out through a specific combination of SD and range values: variation 25 is highlighted by having the highest range value (202.2%), whereas variation 26 displays the second lowest range value (69.4%) and a fairly low SD value (13.1%); in addition, variation 24 and variation 25 exhibit a relatively high SD value. Such a unique progression from two variations with a high SD value to a variation with both SD and range values at the lower margin (see the SD pattern red/red/green in Table 1), can only be found in one other instance: variations 12–14. Therefore, the “solo movements”[25] variation 13 and variation 25 have not only their 3/4 time signature and sarabande tendre characteristic[26] in common, but are also linked through a unique pattern in the database. Interestingly, performers tend to make very heterogenous tempo choices for variations 12–13/24–25 but clearly agree on a main tempo as far as variation 14 and variation 26 are concerned.

Example 1: Bach, “Goldberg Variations,” variations 24, 25, and 26 (mm. 1–4)

At the same time, variation 25, as the center of the chosen sequence, also holds an interesting position within the cycle in terms of its compositional genesis. Werner Breig speculates that the cycle might originally have been planned to comprise only twenty-four variations (and therefore no further canons after variation 24, the canon at the octave); Karol Berger remarks that “when Bach oversteps the limits of the octave, he suggests that the series could go on forever.”[27]

Table 3: Bach, “Goldberg Variations,” tempo relations between variations 24, 25, and 26

Tempo Relations between Variations 24 and 25

Tempo Relation 1:1 (0–10%): 24 () = 25 () (green)

As shown in Table 3, eleven recordings display the same tempo or same basic pulse in both variations. The recordings of Demus 1953 (Audio Ex. 5a: 24 = 55.8 bpm; 25=55.6 bpm) and József Gát 1963 (24 = 65.4 bpm; 25 = 65.0 bpm) exhibit the lowest difference in this category (0% difference).

Audio Example 5a: Bach, “Goldberg Variations,” Demus 1953, variation 24 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Jörg Demus, LP Westminster WL-5241, ℗&© 1992, Side B)

Tempo Relation 2:1 (45–55%): 24 () = 25 () (orange)

The orange cells indicate fourteen recordings where the pulse (dotted quarter note) of variation 24 equals the sixteenth notes of variation 25. The recording of Vladimir Feltsman 1991 (Audio Ex. 5b: 24 = 88.0 bpm; 25 = 44.0 bpm) matches this proportion most closely (50%).

Audio Example 5b: Bach, “Goldberg Variations,” Feltsman 1991, variation 24 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Vladimir Feltsman, CD Music Masters, Inc., ℗&© 1992, Tracks 25, 26)

Tempo Relation 3:2 (28–38%): 24 (++) = 25 () (blue)

In eighteen recordings, a whole bar of variation 24 equals a quarter note of variation 25. András Schiff 2001 (Audio Ex. 5c: 24 = 90.1 bpm; 25 = 60.5 bpm) matches this proportion exactly (33%). In two cases, the calculated percentage classifies recordings into both the blue (3:2) and grey (4:3) ranges: both Koopman 1987 and Stefan Vladar 1996 fit right between the two relations but are minimally closer to blue.

Audio Example 5c: Bach, “Goldberg Variations,” Schiff 2001, variation 24 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” András Schiff, CD ECM Records ECM 1825, ℗&© 2003)

Tempo Relation 3:1 (62–72%): 24 (++) = 25 () (purple)

The five purple cells indicate recordings in which a whole bar of variation 24 equals an eighth note in variation 25. The recordings by Gavrilov 1993 (24 = 99.8 bpm; 25 = 34.7 bpm) and Gould 1955 (Audio Ex. 5d: 24 = 107.5 bpm; 25 = 33.0 bpm) match this proportion most closely.

Audio Example 5d: Bach, “Goldberg Variations,” Gould 1955, variation 24 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Glenn Gould, CD Sony Classical, Inc., ℗ 1956/57, Tracks 25, 26)

Tempo Relation 4:3 (20–30%): 24 () = 25 () (grey)

In eleven recordings, an eighth note in variation 24 equals a thirty-second note in variation 25. The recordings by Claudio Arrau 1942 (24 = 63.0 bpm; 25 = 47.7 bpm) and Edith Picht-Axenfeld 1966 (Audio Ex. 5e: 24 = 65.3 bpm; 25 = 49.0 bpm) both match this proportion exactly (25%). The recording by Isolde Ahlgrimm 1954 (24 = 52.7 bpm; 25 = 65.0 bpm) mathematically yields a percentage which fits into this range, but does not display a 4:3 relation. Since Ahlgrimm is the only performer who plays variation 25 more than 10% faster than variation 24 (resulting in a positive percentage value), the mathematical result cannot be applied here. In this case, the inverted calculation (24:25) shows a (musically irrelevant) 5:6 tempo proportion.

Audio Example 5e: Bach, “Goldberg Variations,” Picht-Axenfeld 1966, variation 24 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Edith Picht-Axenfeld, LP Erato E1036, ℗ 1979 Editions Costallat France, Side B)

Tempo Relations between Variations 25 and 26

Tempo Relation 1:1 (0–10%): 24 () = 25 () (green)

The interpretation by Takahashi 2004 (Audio Ex. 6a: 25 = 77.4 bpm; 26 = 78.5 bpm) is the only recording (single green cell: 1%) expressing this proportion, with variation 25 and variation 26 following the same tempo or same basic pulse.

Audio Example 6a: Bach, “Goldberg Variations,” Takahashi 2004, variation 25 (mm. 1–4), variation 26 (mm. 1–4
(Bach, “Goldberg Variations,” Yūji Takahashi, CD Avex Classics AVCL-84069, ℗&© 2014, Tracks 26, 27)

Tempo Relation 1:2 (80–120%): 25 () = 26 () (orange)

In twenty-one recordings, a sixteenth note in variation 25 equal a quarter note in variation 26. Arrau 1942 (101%) matches this proportion most closely (Audio Ex. 6b: 25 = 47.4 bpm; 26 = 95.2 bpm).

Audio Example 6b: Bach, “Goldberg Variations,” Arrau 1942, variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Claudio Arrau, CD BMG Classics 74321 845 932, ℗ 1988, Tracks 26, 27)

Tempo Relation 2:3 (40–60%): 25 () = 26 (++) (blue)

The blue cells indicate fifteen recordings in which a quarter note in variation 25 equals a whole bar of variation 26. The recording by Richard Egarr 2005 (Audio Ex. 6c: 25 = 49.4 bpm; 26 = 73.9 bpm) matches this proportion exactly (50%).

Audio Example 6c: Bach, “Goldberg Variations,” Egarr 2005, variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Richard Egarr, CD Harmonia Mundi USA – HMU 907425.26, ℗&© 2006, CD 2, Tracks 10, 11)

Tempo Relation 1:3 (180–220%): 25 () = 26 (++) (purple)

In six recordings, an eighth note of variation 25 equals a whole bar of variation 26. The recording by Peter Serkin 1994 (Audio Ex. 6d: 25 = 36.3 bpm; 26 = 108.8 bpm) matches this proportion exactly (200%).

Audio Example 6d: Bach, “Goldberg Variations,” Peter Serkin 1994, variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Peter Serkin, CD RCA Victor Red Seal – BMG Classics 09026 68188 2, ℗&© 1996, Tracks 29, 30)

Tempo Relations between Variations 24 and 26

Tempo Relation 1:1 (0–10%): 24 () = 26 () (green)

The green cells indicate seventeen recordings in which variation 24 and variation 26 follow the same tempo or same basic pulse, e.g., Koopman 1987 (Audio Ex. 7a: 24 = 76.0 bpm; 26 = 76.1 bpm).

Audio Example 7a: Bach, “Goldberg Variations,” Koopman 1987, variation 24 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Ton Koopman, CD Erato Disques S.A, ℗ 1988, Tracks 25, 27)

Tempo Relation 2:3 (40–60%): 24 () = 26 () (blue)

In fifteen recordings, an eighth note in variation 24 equals an eighth note in variation 26, e.g., Ketil Haugsand 2001 (51%) (Audio Example 7b: 24 = 62.5 bpm;26 = 94.2 bpm).

Audio Example 7b: Bach, “Goldberg Variations,” Haugsand 2001, variation 24 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Ketil Haugsand, CD Simax PSC 1192, ℗&© 2002, Tracks 25, 27)

Tempo Relation 1:2 (80–120%): 24 () = 26 (triplet ) (orange)

The recordings by Rudolf Serkin 1928 and Charles Rosen 1967 (Audio Ex. 7c: 24 = 57.5 bpm; 26 = 116.6 bpm) equate a sixteenth note of variation 24 to a triplet eighth note (if viewed in 3/4 time signature, i.e., an eighth note if viewed in 18/16 time signature) of variation 26.

Audio Example 7c: Bach, “Goldberg Variations,” Rosen 1967, variation 24 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Charles Rosen, CD Sony Music Entertainment Inc. – SBK 48173, ℗ 1969, Tracks 25, 27)

Tempo Relations between Variations 24, 25, and 26

As depicted in Table 3, performers use various strategies to create a tempo relation between the variations of a group: some only establish a relation between one pair of variations (thirty-one of seventy-six: 41%), some between two pairs of variations (twenty of seventy-six: 26%), or between all three variations of this group (twenty-one of seventy-six: 28%); the latter case shall be examined more closely below. Only four of seventy-six examined recordings (5%) do not follow any specific relation.

In the recordings by Arrau 1942, Kirkpatrick 1958, and Gustav Leonhardt 1965, the constellation of tempo relations 4:3 (24–25), 1:2 (25–26), and 2:3 (24–26) (grey/orange/blue pattern, as depicted in Table 3) implies the following relations: An eighth note in variation 24 equals a thirty-second note in variation 25; a sixteenth note in variation 25 equals a quarter note in variation 26; and an eighth note in variation 24 equals an eighth note in variation 26. In other words, an eighth note in variation 24 equals a thirty-second note in variation 25, which in turn equals an eighth note in variation 26: 24 () = 25 () = 26 (). In these three recordings, the performers thus choose the eighth note in variation 24 as the common denominator for this three-variation sequence (Audio Ex. 8a: Leonhardt 1965: 24 = 58.3 bpm; 25 = 45.0 bpm; 26 = 88.9 bpm).

Audio Example 8a: Bach, “Goldberg Variations,” Leonhardt 1965, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Gustav Leonhardt, CD Teldec 8.43632, ℗ 1965 © 1987, Tracks 25, 26, 27)

By contrast, Takahashi 2004 chooses the same tempo for the main pulse (as denoted by the time signatures) of all three variations: 24 () = 25 () = 26 (). While the mean tempi of all seventy-six recordings (Fig. 3; see Tab. 1) for these three variations suggests a triangular pattern fast–slow–fast (74.0–49.2–95.1 bpm), Takahashi’s tempo choices implement a sort of linear tempo design, unique among all examined recordings. This horizontal balance allows for the presumption that the exceptionally high tempo of variation 25 (highest tempo value among all selected recordings) has been consciously chosen to approximate the (nearly identical) fast tempi of variations 24 and 26 (24 = 81.6 bpm; 5 = 77.4 bpm; 26 = 78.5 bpm; green/green/green pattern in Table 3, Audio Ex. 8b). This creates an interpretation which reduces the contrast between the three variations to a minimum (resulting in tempo proportions 24:25:26 = 1:1:1), in spite of variation 25 being a somber piece located between two serene variations. Takahashi thus avoids the tempo-related contrast effect which is heard most strongly in Nikolayeva 1992 (between variation 24 and variation 25) and Gould 1955 (between variation 25 and variation 26). That is to say, not unlike Arrau, Kirkpatrick, and Leonhardt, Takahashi lets the notated pulse () of variation 24 determine the tempo of this three-variation sequence, but his execution differs in the proportional alignment of variation 25.

Figure 3: Tempo graph for variations 24, 25, and 26 (grey/orange/blue and green/green/green patterns)

Audio Example 8b: Bach, “Goldberg Variations,” Takahashi 2004, variation 24 (mm. 1–4), variation 25 (mm. 1–4, variation)
(Bach, “Goldberg Variations,” Yūji Takahashi, CD Avex Classics AVCL-84069, ℗&© 2014, Tracks 25, 26)

In Takahashi’s recording of 1976 all three variations are equally interconnected through tempo relations, but unlike the 2004 recording, the tempo for variation 25 is considerably slower (blue/blue/green pattern in Table 3, Audio Ex. 8c: 24 = 101.6 bpm; 25 = 63.7 bpm; 26 = 100.4 bpm). This allows variation 24 and variation 26 (both played pointedly faster than in 2004) to act as a framework for the minor variation. Unlike the near-horizontal progression of the later recording, this choice results in a symmetrical pattern (inverted triangle; 3:2:3, Fig. 4), with the proportions between 24–25 and 25–26 mirroring each other: 24 (++) = 25 () = 26 (++).

Audio Example 8c: Bach, “Goldberg Variations,” Takahashi 1976, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Yūji Takahashi, CD Denon – Columbia Music Entertainment COCQ-84162, ℗ 1977 © 2006, Tracks 25, 26, 27)

Apart from Takahashi 1976, five other recordings constituting a blue/blue/green pattern (Anthony Newman 1971, Schiff 1982, Koopman 1987, Schiff 2001, and Schiff 2015), two recordings with an orange/orange/green pattern (Feltsman 1991 and Evgeni Koroliov 1999), and two recordings with a purple/purple/green pattern (Nikolayeva 1979 and 1992) equate the main pulse of variation 24 to the main pulse of variation 26, resulting in a green third column: 24 () = 26 (). At the same time, the relations between adjacent variations (24–25 and 25–26) match as well: While the blue pattern equates a quarter note in variation 25 to a whole bar of both variation 24 and variation 26, the orange constellation equates a sixteenth note in variation 25 to the main pulse of both variation 24 () and variation 26 (): 24 () = 25 () = 26 (); the purple constellation finally equates an eighth note of variation 25 to a whole bar of both variation 24 and variation 26: 24 (++) = 25 () = 26 (++). Therefore, these three types of constellations again result in an inverted triangle pattern (Fig. 4, Audio Ex. 8d: Feltsman 1991: 88.0–44.0–93.6 bpm).

Figure 4: Tempo graph for variations 24, 25, and 26 (blue/blue/green, orange/orange/green, and purple/purple/green patterns)

Audio Example 8d: Bach, “Goldberg Variations,” Feltsman 1991, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Vladimir Feltsman, CD Music Masters, Inc., ℗&© 1992, Tracks 25, 26, 27)

Interestingly, all three recordings by Schiff (1982, 2001, and 2015) follow this concept. While in general, the tempi of variation 24–26 become slower with each recording (1981: 96.3–65.9–93.9 bpm; 2001: 90.1–60.5–86.6 bpm; 2015: 84.8–57.8–84.1 bpm), Schiff maintains their tempo proportions, asserting the relations between the pieces (blue/blue/green pattern) (Audio Ex. 8e: Schiff 1982: 96.3–65.9–93.9 bpm).

Audio Example 8e: Bach, “Goldberg Variations,” Schiff 1982, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” András Schiff, CD Decca 417 116-2, ℗ 1983 © 1986, Tracks 5, 6)

Nikolayeva’s 1992 recording displays the slowest tempo for variation 25 among all examined recordings (25.6 bpm); her 1979 recording features the second slowest tempo (30.6 bpm; Audio Ex. 8f). Both recordings fall into the purple/purple/green pattern. Therefore, Nikolayeva’s “very slow” tempo choice for variation 25 (in both recordings) is not simply “romantically influenced”: it ostensibly follows an interpretative concept which creates a striking proportion between the variations. The purple pattern strongly highlights the minor variation in its position and further sharpens the triangular shape introduced by the blue pattern (purple: 24:25:26 = 3:1:3 versus blue: 24:25:26 = 3:2:3 versus orange: 24:25:26 = 2:1:2). (Audio Ex. 8f: Nikolayeva 1979: 86.0–30.6–88.6 bpm)

Audio Example 8f: Bach, “Goldberg Variations,” Nikolayeva 1979, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Tatiana Nikolayeva, CD JVC Victor Japan VICC40126/7, ℗&© 1992, Tracks 25, 26, 27)

Apart from the aforementioned patterns which establish a direct link between all three variations by continuously equating the note value of a variation to its successor (e.g., Arrau 1942: 24 () = 25 () = 26 ()), there are patterns which apply tempo relations between all three pieces by changing the determining note value with the progression from variation 25 to variation 26. This is the case with Bob von Asperen 1991 and Pieter-Jan Belder 1999, whose interpretations create a green/blue/blue pattern (2:2:3, Fig. 5): 24 (+) = 25 (+) = 26 (++). Here, the two relations depend on a reframing of the note value of variation 25 (  ), shifting the proportion in creating the link to variation 26 (Audio Ex. 8g: Belder 1999: 56.4–54.7–80.4). The green/blue/blue pattern reflects Siegele’s suggestions (2:2:3);[28] conversely, the mean tempo values create a blue/orange/[none] pattern, suggesting a proportion of 24:25 = 3:2 and 25:26 = 1:2 (i.e., 24:25:26 = 3:2:4, see below).

Audio Example 8g: Bach, “Goldberg Variations,” Belder 1999, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Pieter-Jan Belder, CD Brilliant Classics 92284, © 2006, CD 2, Tracks 25, 26, 27)

Such reframing processes also take place in three other sets of recordings. In recordings of a green/orange/orange pattern (Fig. 5), e.g. in Rudolf Serkin 1928 and Rosen 1967, the note value of variation 25 is shifted from an eighth note to a sixteenth note: 24 () = 25 ()  25 () = 26 (); 24 () = 26 (triplet ), resulting in the proportion 1:1:2: 24 () = 25 () = 26 (+). As a result of the 25:24 proportion being 1:1 (green) the 24–26 tempo relation matches the 25–26 tempo relation. Similar to the green/blue/blue pattern, variations 24 and 25 follow an identical pulse, but the green/orange/orange pattern results in a higher contrast between 25–26 and 24–26, since it follows a sharper proportion (1:1:2) than green/blue/blue (2:2:3) (Audio Ex. 8h, Rosen 1967: 57.5–52.5–115.6).

Figure 5: Tempo graph for variations 24, 25, and 26 (green/blue/blue and green/orange/orange patterns)

Audio Example 8h: Bach, “Goldberg Variations,” Rosen 1967, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Charles Rosen, CD Sony Music Entertainment Inc. – SBK 48173, ℗ 1969, Tracks 25, 26, 27)

Depending on the distribution of the 26:25 tempo proportion within its percentage range, the 26:24 proportion may fall into a pattern replicating the 26:25 ratio (or not), and vice versa. For example, in Thomas Schultz 1998, the 26:25 percentage (81%) is located at the margin of the designated range (orange: 80–120%), causing the 26:24 proportion (63%) to remain outside of the blue range (40–60%). Such a situation emerges in Tureck 1998 as well: the percentage of the 26:24 proportion (41%) lies near the lower margin of the chosen range (blue: 40–60%), resulting in a “near miss” for the 26:25 proportion (with 39% located just below the blue range). Arguably, since these deviations are a result of the deliberately chosen percentage ranges, these two recordings nonetheless match the aforementioned relation patterns (Schultz: green/orange/[orange], Tureck: green/[blue]/blue).

The blue/orange/blue pattern reframes the note value of variation 25 from a quarter note to a sixteenth note: 24 (++) = 25 ()  25 () = 26 (); 24 () = 26 (), creating a 3:2:4 proportion: 24 (++) = 25 () = 26 (+++), e.g., in Eunice Norton 1942 and Hewitt 1999. One recording, Ji-Yong Kim 2018, forms an orange/purple/blue pattern, shifting the note value of variation 25 from a sixteenth note to an eighth note: 24 () = 25 ()  25 () = 26 (++); 24 () = 26 (). This generates a 2:1:3 proportion: 24 (+) = 25 () = 26 (++).

As mentioned above, the mean tempo values (74.0–49.2–95.1 bpm) suggest a pattern which relates variation 24 to variation 25 in a 3:2 tempo relation, and variation 25 to variation 26 in a 1:2 relation, with the proportion 26:24 not following any discernible (integer) ratio[29] – creating a pattern which displays only two connections (blue/orange/[none]; 24: 34%; 25: 94%; 26: 29%). Strikingly, six performers among the examined recordings match this binary pattern: as reflected by Grete Sultan 1959, Trevor Pinnock 1980, Weissenberg 1981, Pi-hsien Chen 1985, Andreas Staier 2009, and Jeremy Denk 2013. These interpretations equate a whole bar of variation 24 to a quarter note in variation 25, and a sixteenth note in variation 25 to a quarter note in variation 26; Pinnock’s interpretation matches the percentages of the mean tempo values most closely (Audio Ex. 8i: 24: 32%; 25: 93%; 26: 32%; 67,8–46,2–89,2 bpm). The recordings by Norton 1942 and Hewitt 1999 correspond to this pattern as well, but relate variation 24 to variation 26 in a 2:3 proportion (as described above).

Audio Example 8i: Bach, “Goldberg Variations,” Pinnock 1980, variation 24 (mm. 1–4), variation 25 (mm. 1–4), variation 26 (mm. 1–4)
(Bach, “Goldberg Variations,” Trevor Pinnock, CD Archiv Produktion 415 130- 2, ℗ 1980 Polydor International, CD 2. Tracks 25, 26, 27)

Cyclic Interpretation II: Structure-based Analysis (Variations in G minor: Variations 15, 21, and 25)

Evidently, the main reason for choosing to analyze variations 15, 21, and 25 as a group (Ex. 2) is the fact that they are the only minor variations in the cycle. As Rolf Dammann illustrates, the strict three-part canons of variations 15 and 21 could be seen as pieces of the prima pratica. In contrast, the expressive and highly rhetorical “solo” melody of variation 25, set to a two-part accompaniment, constitutes an instrumental paradigm of the seconda pratica.[30] But in addition to that, these variations also feature significant SD and range values: variation 15 exhibits the lowest mean tempo value of all pieces (29.6 bpm; followed by variation 16a: 33.9 bpm; both are located at the center of the cycle); variation 21 has one of the highest SD values and a relatively high range; variation 25, as mentioned previously, possesses a very high SD and the highest range value (most likely also due to its inherent expressivity).

Example 2: Bach, “Goldberg Variations,” variations 15 (mm. 1–4), 21 (mm. 1–2), and 25 (mm. 1–4)

Moreover, variations 15 and 25 are the only pieces of the cycle with added tempo indications (15: Andante and 25: Adagio).[31] This suggests that it was a necessity for Bach to provide tempo indications only in those cases where a tempo could not be derived from common meter and musical structure. Conceding that Italian tempo markings have also been interpreted as denoting mood, Bernard Sherman argues that

for Bach the term Andante slowed the tempo somewhat compared to what it would be without the marking. […] [The] Andante marking may be considered cautionary, warning musicians not to play it as quickly as a movement in common time would normally go. […] [It] could refer to a steadiness or evenness of execution; Bach often uses it in movements with at least one line, usually the bass, moving in continuous quavers (or semiquavers).[32]

While variation 21 is, on average, played slightly faster (by 2.5%) by those performers who are pianists (mean tempo, pianists: 45.5 bpm; mean tempo, harpsichordists: 44.4 bpm; see Fig. 1), both variations 15 and 25 are interpreted more slowly by pianists than by harpsichordists. This is only the case with seven pieces in total; on the whole, the pieces of the cycle have been performed 7.2% faster by pianists. Whereas variation 15 shows only a slight difference (3.5%; mean tempo, pianists: 29.2; mean tempo, harpsichordists: 30.2); variation 25 exhibits a significant discrepancy (as outlined above): it is performed 11.7% slower by pianists (mean tempo, pianists: 46.9; mean tempo, harpsichordists: 53.1). This may imply that harpsichordists tend to interpret the inscriptions Andante and Adagio differently than pianists, possibly due to their inherent experience with and practice of early music, while a reason for the pianists’ slower approach could reflect a tendency to interpret tempo indications in a more “modern” way.

This set of three variations in a minor key (variation 15: 2/4; variation 21: 4/4; variation 25: 3/4; all three in a binary structure) only allows for two meaningful ratios: 1:1 (green) and 2:1 (orange), as depicted in Table 4.

Table 4: Bach, “Goldberg Variations,” tempo relations between variations 15, 21, and 25

Tempo Relations between Variations 15 and 21

Tempo Relation 1:1 (0–10%): 15 () = 21 () (green)

In three recordings, a quarter note in variation 15 equates a quarter note in variation 21, i.e., both variations share a main pulse, e.g., in Wanda Landowska 1945, Pinnock 1980, and Schultz 1998. In the recording by Landowska 1945 (27.7–27.2 bpm), the tempi of both variations match almost exactly, making her tempo of variation 21 the slowest among all examined recordings (Audio Ex. 9a).

Audio Example 9a: Bach, “Goldberg Variations,” Landowska 1945, variation 15 (mm. 1–4), variation 21 (mm. 1–2)
(Bach, “Goldberg Variations,” Wanda Landowska, CD RCA Victor Gold Seal – BMG Classics, ℗&© 1992, CD 1, Tracks 16, 22)

Tempo Relation 1:2 (80–120%): 15 () = 21 () (orange)

In thirteen recordings, an eighth note in variation 15 equals a quarter note in variation 21, e.g., in Norton 1942 (26.8–52.2 bpm; 95%) and Cecilia Li 1996 (Audio Ex. 9b: 22.7–46.6 bpm; 105% ).

Audio Example 9b: Bach, “Goldberg Variations,” Li 1996, variation 15 (mm. 1–4), variation 21 (mm. 1–2)
(Bach, “Goldberg Variations,” Cecilia Li, CD Bayer Records/Amati 9602/1, ℗&© 1996, Tracks 16, 22)

Regarding the question whether the main pulse in variation 15 is to be derived from an eighth note or a quarter note, the performers’ choices concerning the tempo relations between variation 15 and variation 21 might offer some insight into their view of the matter. Those applying a 1:1 relation have assessed the quarter note of variation 15 as the main pulse (putting their tempo choices for variation 21 among the slowest of all recordings); those implementing a 2:1 relation have chosen the eighth note as the main pulse of variation 15.

Tempo Relations between Variations 21 and 25

Tempo Relation 1:1 (0–10%): 21 () = 25 () (green)

The twenty-six green cells indicate recordings where a quarter note in variation 21 equals an eighth note in variation 25, as it is the case in the recordings by Nikolayeva 1979 (30.6–30.6 bpm) and Egarr 2005 (Audio Ex. 10a: 49.4–49.4 bpm), both with exactly 0% deviation.

Audio Example 10a: Bach, “Goldberg Variations,” Egarr 2005, variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Richard Egarr, CD Harmonia Mundi USA – HMU 907425.26, ℗&© 2006, CD 2, Tracks 6, 10)

Tempo Relation 1:2 (80–120%): 21 () = 25 () (orange)

One recording, Landowska 1945, equates a quarter note in variation 21 to a quarter note in variation 25, using the same main pulse for both pieces (Audio Ex. 10b: 27.2–49.5 bpm). All other interpretations which feature a relation for 21–25 choose a 1:1 ratio (as reflected in the mean tempi 45.1–49.2).

Audio Example 10b: Bach, “Goldberg Variations,” Landowska 1945, variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Wanda Landowska, CD RCA Victor Gold Seal – BMG Classics, ℗&© 1992, CD 1, Tracks 22, 26)

Tempo Relations between Variations 15 and 25

Tempo Relation 1:1 (0–10%): 15 () = 25 () (green)

The two green cells indicate recordings where a quarter note in variation 15 equals an eighth note in variation 25, as in the recordings by Gould 1955 (31.0–33.0 bpm) and Nikolayeva 1992 (Audio Ex. 11a: 24.8–25.6 bpm).

Audio Example 11a: Bach, “Goldberg Variations,” Nikolayeva 1992, variation 15 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Tatiana Nikolayeva, CD Hyperion Records Limited A66589, ℗&© 1992, Tracks 16, 26)

Tempo Relation 1:2 (80–120%): 15 () = 25 () (orange)

In twenty recordings, an eighth note in variation 15 equals an eighth note in variation 25, e.g., Lang 2020a (Audio Ex. 11b: 19.9–39.1 bpm).

Audio Example 11b: Bach, “Goldberg Variations,” Lang 2020a, variation 15 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Lang Lang, CD Deutsche Grammophon B089TWSBDT, ℗&© 2020, Tracks 16, 26)

Tempo Relations between Variations 15, 21, and 25

As depicted in Table 4, there are seventeen recordings (23% of the examined recordings) that do not suggest any specific tempo relations, whereas in fifty-four recordings (71%) only one relation, in four recordings (5%) two relations, and in only one recording (1%) three tempo relations can be observed. The recording by Norton 1942, as the only recording showing three relations, follows a symmetric pattern (1:2:2): 15 () = 21 () = 25 (). Not only is the eighth pulse of variation 15 a determining factor for the tempo choice for the other minor key variations, its character and atmosphere have been applied to the other two: the Andante is interpreted in a very stable tempo and without any substantial rubato. The same flow is discernible in variation 21 and variation 25 (Audio Ex. 12a: Norton 1942: 26.8–52.2–52.9 bpm –).

Audio Example 12a: Bach, “Goldberg Variations,” Norton 1942, variation 15 (mm. 1–4), variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Eunice Norton, CD Norvard Recordings 0005-2, ℗&© 2000, Tracks 16, 22, 26)

Out of the four recordings showing relations between two variations, two follow the same pattern (orange/green/[none]): Chen 1985 and Vladar 1996. These two performers match an eighth note in variation 15 to a quarter note in variation 21, and a quarter note in variation 21 to an eighth note in variation 25. Although this relation would suggest that the eighth note in variation 25 equals the eighth note in variation 15 (tracing a 1:2:2 tempo pattern), the distribution of values near the margins of the percentage ranges for the 21:15 and 25:21 proportions does not allow 25:15 to match the orange pattern (Chen: 21:15: 80% – 25:21: 9% – 25:15: 63%; Vladar: 21:15: 89% – 25:21: 7% – 25:15: 76%), so that the proportion applies only in a chronological direction: 15 ()  21 ()  25 () (Audio Ex. 12b: Vladar 1996: 31.5–59.8–55.4 bpm).

Audio Example 12b: Bach, “Goldberg Variations,” Vladar 1996, variation 15 (mm. 1–4), variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Stefan Vladar, CD Preiser Records 90771, © 2009, Tracks 16, 22, 26)

Karl Richter’s 1970 recording also features two tempo relations but forms a different pattern: [none]/green/orange. Here, a quarter note of variation 21 equals an eighth note of variation 25, and the eighth notes of variation 15 and variation 25 match: 21 () = 25 (); 15 () = 25 (). Consequentially, one would assume that the use of this relation would automatically equate an eighth note in variation 15 to a quarter note in variation 21, resulting in a 1:2:2 tempo proportion, as implied in Chen and Vladar. But since the green percentage value for 25:21 is 10% and therefore at the margin of the corresponding percentage range, the expected (orange) ratio for 21:15 doesn’t apply (74%). Richter’s 1956 recording matches the 1:1 percentage bracket for 25:21 somewhat more closely (7%) but features a “near miss” for an orange 25:15 proportion (79%) (Audio Ex. 12c: 30.5–53.1–58.4 bpm).

Audio Example 12c: Bach, “Goldberg Variations,” Richter 1970, variation 15 (mm. 1–4), variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Karl Richter, CD Deutsche Grammophon 445 057-2, ℗ 1972, Tracks 16, 22, 26)

The proportion for 15–25 displayed in Landowska 1945 (79%) puts her on the threshold to a 1:2 tempo relation, which, given a higher tolerance regarding the 80–120% range, would result in a linear set tying together all three variations (as approached in Norton 1942). In contrast to Norton’s tempo relations (15 () = 21 () = 25 (); orange/green/orange, 1:2:2), Landowska however equates the quarter notes of all three variations (15 () = 21 () = 25 (); green/orange/[orange], 1:1:2). This pattern conforms to the one suggested by Siegele.[33] Consequently, Landowska’s 1945 interpretation is the only one among all examined recordings which chooses a uniform tempo for the pulse of all three pieces (referring to a quarter note in all three cases, as denoted by the time signature of each variation: 2/4 – 4/4 – 3/4). This choice results in her remarkably slow tempo for variation 21, the slowest among all seventy-six recordings (Audio Ex. 12d: 27.7–27.2–49.5 bpm).

Audio Example 12d: Bach, “Goldberg Variations,” Landowska 1945, variation 15 (mm. 1–4), variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Wanda Landowska, CD RCA Victor Gold Seal – BMG Classics, ℗&© 1992, CD 1, Tracks 16, 22, 26)

A comparison between variation 15 and variation 25 reveals a seemingly contradictory aspect: As mentioned earlier, these are the only two variations with a tempo indication: Andante for variation 15 and Adagio for variation 25. These inscriptions would suggest Andante to be played at a palpably faster tempo than Adagio, yet, contrary to expectation, twenty of seventy-six recordings equate the notated pulses (quarter notes) of both variations (see Table 4, 25:15, orange), with Norton 1942 and Lang 2020a approximating this proportion (100%) most closely (Norton: 98%, Lang: 97%). These two performances – while audibly basing both variations on virtually the same pulse – show a remarkably dissimilar approach to character and atmosphere, resulting in two very different expressions of the same ratio: Norton maintains the dignified pace and flow from the Andante in variation 25, essentially cutting both pieces from the same “fabric” (Audio Ex. 12e: 26.8–52.9 bpm). Lang executes variation 15 at a steadily walking pace before creating a strong contrast in moods by incorporating a fair amount of rubato in the Adagio, shaping time in a more liberal manner to achieve a different level of expression (Audio Ex. 12f: 19.9–39.1 bpm).[34]

Audio Example 12e: Bach, “Goldberg Variations,” Norton 1942, variation 15 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Eunice Norton, CD Norvard Recordings 0005-2, ℗ 2000, Tracks 16, 26)

Audio Example 12f: Bach, “Goldberg Variations,” Lang 2020a, variation 15 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Lang Lang, CD Deutsche Grammophon B089TWSBDT, ℗&© 2020, Tracks 16, 26)

Another significant observation concerning variation 25 is the strikingly low number of 1:1 relations between variation 15 and variation 25: Gould 1955 and Nikolayeva 1992 are the only performers to equate an eighth note in variation 25 to a quarter note in variation 15, with both interpretations choosing conspicuously low tempi for variation 25, Nikolayeva’s 1992 choice being the slowest, Gould’s 1955 choice the third slowest tempo. While in her 1979 recording, Nikolayeva matches the similarly slow tempo of variation 25 (which is the second slowest choice for this variation among all examined recordings: 30.6 bpm) to that of variation 21 (Audio Ex. 12g: 30.6 bpm), her 1992 tempo for variation 25 (25.6 bpm, slowest recording) equals that of variation 15 (Audio Ex. 12h: 24.8 bpm). This is also the case for Gould: in his 1955 recording, the tempo choice for variation 25 (33.0 bpm) matches that of variation 15 (Audio Ex. 12i: 31.0 bpm); in his 1959 recording, it equals the tempo of variation 21 (49.7 bpm), which results in a much more flowing tempo for variation 25 (Audio Ex. 12j: 48.1 bpm).

Audio Example 12g: Bach, “Goldberg Variations,” Nikolayeva 1979, variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Tatiana Nikolayeva, CD JVC Victor Japan VICC40126/7, ℗&© 1992, Tracks 22, 26)

Audio Example 12h: Bach, “Goldberg Variations,” Nikolayeva 1992, variation 15 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Tatiana Nikolayeva, CD Hyperion Records Limited A66589, ℗&© 1992, Tracks 16, 26)

Audio Example 12i: Bach, “Goldberg Variations,” Gould 1955, variation 15 (mm. 1–4), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Glenn Gould, CD Sony Classical, Inc., ℗ 1956/57, Tracks 16, 26)

Audio Example 12j: Bach, “Goldberg Variations,” Gould 1959, variation 21 (mm. 1–2), variation 25 (mm. 1–4)
(Bach, “Goldberg Variations,” Glenn Gould, CD Sony Classical 52685-2, ℗&© 1993, Tracks 22, 26)

Cyclic Interpretation III: Position-based Analysis (Beginning and End)

Undisputedly, the strongly interlinked symmetrical disposition of the “Goldberg Variations” offers a multitude of possible relations to be examined in terms of position-based (cyclic) qualities. For example, the symmetrical structure of 1+15+15+1 pieces invites a detailed examination of the cycle’s middle and the tempo relations across the middle axis (mainly the relations between variations 14 to 17). In this final analysis, however, I will place the focus on the beginning and end: on both Arias and their (adjacent) variations (Ex. 3).

Example 3: Bach, “Goldberg Variations,” Aria (mm. 1–4), variations 1 (mm. 1–4) and 30 (mm. 1–3)

The following tempo relations have been examined: Aria 1–Aria 2, Aria 1–variation 1, variation 30–Aria 2, and variation 1–variation 30 (Table 5). These combinations provide further information on possible arcs between the first and last positions of the cycle, as executed in the selected recordings.

Table 5: Bach, “Goldberg Variations,” tempo relations between Aria 1, variation 1, variation 30, and Aria 2

Tempo Relation Aria–Aria da capo (Aria 1–Aria 2)

Tempo Relation 1:1 (0–10%): Aria 1 = Aria 2 (green)

A collation of Aria 1 and Aria 2 shows that fifty-nine recordings – about 78% of all examined recordings – display the same tempo for both arias; the mean tempo values of both arias strongly resemble each other (Aria 1 = 50.2 bpm; Aria 2 = 48.3 bpm).

The high number of recordings equating the tempi is hardly surprising given the fact that Aria 2 is simply the Aria da Capo è Fine, as indicated in the first print, in which the Aria was only printed once at the very beginning of the “Goldberg Variations,” prompting the performer to return to the first page at the end of a performance.[35] As a consequence, the cycle closes with a “memory of its beginning,” conforming to the Baroque convention of repeating the first piece of a variation cycle at its end.[36]

This repeat of the Aria seems itself to say something about the strange power of great music, for as one hears it a final time, its aura is different. It has changed from a greeting to a farewell, from elegantly promising to sadly concluding. But how can that be, when the notes are the same and even the manner of playing them need not have changed?[37]

This introduces the question which interpretational choices are offered by the remaining 22% of performers who do not play both Arias in the same tempo, i.e., whose interpretations display a tempo deviation above 10% (seventeen recordings): Silver 1957, Gát 1963, and Takahashi 2004 choose a faster tempo for Aria 2 (Fig. 6), with Takahashi displaying a considerable tempo difference: Aria 2 is played 30% faster in relation to Aria 1 (Audio Ex. 13a: 60.0–78.1 bpm). Remarkably enough, Takahashi’s 1976 recording starts out at virtually the same tempo as in 2004 (61.6 bpm) and maintains it in Aria 2 (63.2 bpm), generating a green 1:1 relation.

Figure 6: Tempo diagram for Aria 1 and Aria 2

Audio Example 13a: Bach, “Goldberg Variations,” Takahashi 2004, Aria 1 (mm. 1–4), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Yūji Takahashi, CD Avex Classics AVCL-84069, ℗&© 2014, Tracks 1, 32)

Out of the remaining fourteen recordings which assign a slower tempo to Aria 2, two interpretations stand out: Ernst 2020 and Lang 2020a. These two recordings display the largest difference between both arias (Ernst: 27%, Audio Ex. 13b: 39.2–28.5 bpm; Lang: 24%, Audio Ex. 13c: 46.1–35.2 bpm). Until 2020, the largest tempo difference between Aria 1 and Aria 2 was the one performed by Gould 1981 (17%; Audio Ex. 13d: 33.6–27.9 bpm). Coincidentally, both of these choices constitute the slowest tempi for the respective Arias among all examined recordings (Fig. 7). Interestingly, all of Gould’s earlier recordings (1954, 1955, 1958, 1959) fall into the green 1:1 bracket. This unusual decision – as taken by Gould in 1981 – to play Aria 2 radically slower has not been replicated until Ernst’s and Lang’s recent interpretations, with both performers producing an even more radical contrast than Gould. Lang’s studio recording of the same year (2020b) also displays a tempo reduction in Aria 2 (14%), but to a lesser extent than the concert version (2020a).

Figure 7: Tempo diagram for Aria 1 and Aria 2

Audio Example 13b: Bach, “Goldberg Variations,” Ernst 2020, Aria 1 (mm. 1–4), Aria 2 (mm. 1–4)
(Moritz Ernst, unpublished live recording, Stadtkirche Bayreuth, 23/07/2020, used with kind permission)

Audio Example 13c: Bach, “Goldberg Variations,” Lang 2020a, Aria 1 (mm. 1–4), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Lang Lang, CD Deutsche Grammophon B089TWSBDT, ℗&© 2020, Tracks 1, 32)

Audio Example 13d: Bach, “Goldberg Variations,” Gould 1981, Aria 1 (mm. 1–4), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Glenn Gould, CD Sony Classical 52619-10, ℗ 1982 © 1993, Tracks 1, 32)

Among the recordings marked green in Table 5 regarding their relation between the two Arias, three stand out for displaying the exact same tempo for Aria 1 and Aria 2: Richter 1956 (52.7–52.7), Gavrilov 1993 (42.7– 42.7), and Egarr 2005 (51.3–51.3). This raises the question – as posed by Bruno Gingras during a PETAL conference[38] – whether this precise match between the two Arias might be a result of a technical reproduction of one track, a question which shall be explored here.

In total, there are 14 recordings which exhibit a 0–1% difference[39] between the tempi of Aria 1 and Aria 2. Several other factors beyond main tempo have to be taken into account when examining the relationship between the two Arias in these recordings, such as the setting (live or studio recording), the durations of each part of the Aria (A: mm. 1–16 and B: mm. 17–32) and whether and how repeats (A' and B') were executed, as well as further musical properties perceived through close listening (ornaments, nuances, rubato, etc.).[40] Table 6 offers an overview of the durations of all four sections for those 14 recordings below 1% tempo difference.[41]

Table 6: Durations of Aria 1 and Aria 2 and their repeats (parts A, A', B, and B' respectively)

Three of the interpretations included in Table 6 have been recorded live: Gould 1958, Grigory Sokolov 1982 (Audio Ex. 13e; 57.2–57.4 bpm), and Schiff 2015, which excludes them from this assumption. Their measurements show great control in replicating tempo and duration very closely on stage; this is especially the case for part B in Gould 1958 (0.6 sec difference), part A in Schiff 2015 (0.15 sec difference), and parts A, A', and B in Sokolov 1982 (0.55/0.37/0.18 sec difference).

Audio Example 13e: Bach, “Goldberg Variations,” Sokolov 1982, Aria 1 (mm. 1–32), Aria 2 (mm. 1–32)
(Bach, “Goldberg Variations,” Grigory Sokolov, CD Melodiya/MEL-CD 10-02049, ℗ 2012, Tracks 1, 32)

The durations for Egarr 2005 show that, while the initial tempo for both arias is exactly the same, the durations of the repeated parts differ substantially, excluding the possibility that the track of Aria 1 has been reused for Aria 2. The recordings of Picht-Axenfeld 1966, Schultz 1998, Ingrid Marsoner 2009, Denk 2013, and Levit 2015 show some remarkable similarities in duration but exhibit obvious differences between Arias 1 and 2 concerning the execution of ornaments, rubato, and other nuances. Except for Schultz (who plays both Arias without repeats), all of these recordings contain at least three audibly different versions of parts A and A' (marked blue).

Neither of the remaining five recordings offers an unambiguous explanation for the durational and qualitative similarities of Arias 1 and 2, leaving the question of technical reproduction open to interpretation. In the cases of Richter 1956 and Marlowe 1962, there is virtually no difference in duration between parts A in Arias 1 and 2. Contrary to Egarr, Richter’s 1956 Arias, played at exactly the same (initial) tempo (52.7), show a very negligible difference in duration (0.03 sec). This also applies to Marlowe, whose slightly different tempi result in a very small difference in duration (0.56 sec). Both Richter’s and Marlowe’s recordings, however, display a substantial durational difference for parts B; therefore, it seems safe to assume that they have indeed recorded the Aria twice. Whereas in Kempff 1969, the durations and musical traits are virtually identical, the fact that part A' (which constitutes the sole repeat in both Arias in this recording) uses distinctly different ornaments from A while producing a very similar duration could imply (but not ascertain beyond doubt) that the pianist, in theory, played a second Aria with identical properties.

This leaves the recordings of Norton 1942 and Gavrilov 1993, both of which exhibit an alarmingly exact durational congruence between both Arias, leading to the assumption that this is the outcome of a technical copying of tracks. While for Norton’s recording, an explanation for this decision might potentially be sought in contemporary technical conditions, such a supposition seems meaningless in a recording made as recently as 1993. Regardless of the exact circumstances of this interpretation, it may be hypothesized that Gavrilov’s intention could have been to reproduce an exact – technical – copy of the Aria to follow the variations, supposedly taking the instruction Aria da capo quite literally: taking the (recorded) Aria from the beginning and placing it at the very end. Other performers (as listed in Table 6) might have executed the same intention to a humanly possible extent (e.g. Sokolov in his live recording).

Tempo Relations Aria 1–Variation 1 and Variation 30–Aria 2

Tempo Relation 1:1 (0–10%): Aria 1 () = Variation 1 () (green)

The comparison between Aria 1 and variation 1 addresses three tempo relations (see Table 5). The single green cell corresponding to Landowska’s 1933 recording (10%) indicates that a quarter note in Aria 1 equals a quarter note in variation 1 (Audio Ex. 14a). This is the only performance using this relation.

Audio Example 14a: Bach, “Goldberg Variations,” Landowska 1933, Aria 1 (mm. 1–4), variation 1 (mm. 1–4)
(Bach, “Goldberg Variations,” Wanda Landowska, CD Naxos Historical 8.110313, ℗&© 2005, Tracks 7, 8)

This interpretation features the slowest tempo for variation 1 (54.1 bpm) among all examined recordings; possibly, it is the direct result of matching it to Aria 1 (49.1 bpm). The second slowest tempo for variation 1 is also performed by Landowska, in her 1945 recordings (63.0 bpm); this second interpretation comes fourth in the table regarding the tempo difference between both pieces (38%), surpassing the earlier recording by reducing the tempo for Aria 1 (to 45.5 bpm) and increasing the tempo for variation 1 (to 63.0 bpm). After Landowska 1933, the second lowest difference between Aria 1 and variation 1 pertains to the recording of Blandine Verlet 1992 (17%; 75.2–88.3 bpm), the third lowest difference appears in Kempff 1969 (18%; 80.8–95.4 bpm).

Audio Example 14b: Bach, “Goldberg Variations,” Verlet 1992, Aria 1 (mm. 1–4), variation 1 (mm. 1–4)
(Bach, “Goldberg Variations,” Blandine Verlet, CD Astrée Auvidis E 87459, ℗&© 1993, Tracks 1, 2)

While both of Landowska’s recordings of variation 1 differ considerably from the mean tempo of all examined recordings for variation 1 (54.1 and 63.0 bpm; mean value 101.3 bpm), her tempi for Aria 1 strongly converge to the mean tempo (49.1 and 45.5 bpm; mean value 50.2 bpm). Conversely, the tempi for variation 1 chosen by Kempff and Verlet approximate the mean tempo for variation 1 (Kempff: 95.4 bpm; Verlet: 88.3 bpm), and might be a result of their fast tempi for Aria 1 (the fastest of all examined recordings; Kempff: 80.8 bpm; Verlet: 75.2 bpm) (Fig. 8).

Figure 8: Tempo diagram for Aria 1 and variation 1

Tempo Relation 1:2 (80–120%): Aria 1 () = Variation 1 () (orange)

The thirty-one orange cells in Table 5 denote recordings which equate an eighth note in Aria 1 to a quarter note in variation 1, e.g., Christine Schornsheim 2016 (Audio Ex. 14c: 52.2–104.1 bpm).

Audio Example 14c: Bach, “Goldberg Variations,” Schornsheim 2016, Aria 1 (mm. 1–4), variation 1 (mm. 1–4)
(Bach, “Goldberg Variations,” Christine Schornsheim, CD Capriccio C-5286, ℗&© 2016, CD 1, Tracks 7, 8)

Tempo Relation 1:3 (180–220%): Aria 1 () = Variation 1 (++) (purple)

The single purple cell reveals that the recording of Ernst 2020 applies a 3:1 tempo relation between Aria 1 (39.2 bpm) and variation 1 (122.3 bpm); this implies that a quarter note in Aria 1 equals a whole bar of variation 1. This makes Ernst’s recording the one with the largest tempo difference between Aria 1 and variation 1, resulting in a sudden startling impulse when the first variation begins (Audio Ex. 14d).

Audio Example 14d: Bach, “Goldberg Variations,” Ernst 2020, Aria 1 (mm. 1–4), variation 1 (mm. 1–4)
(Moritz Ernst, unpublished live recording, Stadtkirche Bayreuth, 23/07/2020, used with kind permission)

The question arising at the end of the cycle is how performers correlate variation 30 with Aria 2, or rather, how they depart variation 30 towards the Aria da capo:

Tempo Relation 1:1 (0–10%): Variation 30 () = Aria 2 () (green)

The green cells indicate recordings where a quarter note in variation 30 equals a quarter note in Aria 2. This is only the case in Chen 1985 (61.5–57.7 bpm) and Verlet 1992 (Audio Example 15a: 64.9–66.4 bpm).

Audio Example 15a: Bach, “Goldberg Variations,” Verlet 1992, variation 30 (mm. 1–2), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Blandine Verlet, CD Astrée Auvidis E 87459, ℗&© 1993, Tracks 31, 32)

Tempo Relation 2:1 (45–55%): Variation 30 () = Aria 2 () (orange)

The fourteen orange cells indicate recordings where a quarter note in variation 30 equals an eighth note in Aria 2, e.g., in Marlowe 1962 (Audio Ex. 15b: 79.5–40.3 bpm).

Audio Example 15b: Bach, “Goldberg Variations,” Marlowe 1962, variation 30 (mm. 1–2); Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Sylvia Marlowe, LP Decca DL 10056, ℗ 1962, Side B)

A comparison between variation 1 and variation 30 yields forty-five relations in total, distributed into three proportions; this variation pair has the second-highest number of tempo relations, following the obvious link between Aria 1 and Aria 2 (fifty-nine tempo relations).

Tempo Relation 1:1 (0–10%): Variation 1 () = Variation 30 () (green)

The twelve green cells indicate recordings where a quarter note in variation 1 equals a quarter note in variation 30, as is the case in Schiff 2015 (Audio Ex. 16a: 100.7–100.8 bpm).

Audio Example 16a: Bach, “Goldberg Variations,” Schiff 2015, variation 1 (1–8), variation 30 (mm. 1–8)
(Bach, “Goldberg Variations,” András Schiff, BBC-Proms, 22.08.2015)

Tempo Relation 2:1 (45–55%): Variation 1 () = Variation 30 () (orange)

The three orange cells indicate interpretations where a quarter note in variation 1 equals an eighth note in variation 30, e.g., Weissenberg 1981 (Audio Ex. 16b: 124.4–62.7 bpm). Landowska’s 1933 recording (54.1–83.2 bpm) yields a percentage value which fits into this bracket but does not display a 2:1 relation. Analogously to Ahlgrimm’s 1954 recording (as described above for the 25:24 ratio), the calculation needs to be inverted and reveals a (in this case, musically irrelevant) 2:3 tempo proportion.

Audio Example 16b: Bach, “Goldberg Variations,” Weissenberg 1981, variation 1 (mm. 1–8), variation 30 (mm. 1–8)
(Bach, “Goldberg Variations,” Alexis Weissenberg, EMI Classics 5 75952 2, ℗ 1982 © 2001, Tracks 2, 31)

Tempo Relation 3:2 (28–38%): Variation 1 () = Variation 30 () (blue)

As indicated by the blue cells, thirty recordings equate a whole bar of variation 1 to a half note in variation 30; i.e., the harmonic rhythm of one variation is equated with the other as in Norton 1942 (Audio Ex. 16c: 118.9–80.9 bpm). In this case, Landowska’s 1945 recording mathematically matches this pattern without displaying this relation; an inverted calculation shows that her tempo values actually match a 3:4 pattern (in this case, musically irrelevant).

Audio Example 16c: Bach, “Goldberg Variations,” Norton 1942, variation 1 (mm. 1–8), variation 30 (mm. 1–8)
(Bach, “Goldberg Variations,” Eunice Norton, CD Norvard Recordings 0005-2, ℗ 2000, Tracks 2, 31)

Tempo Relations Aria 1–Variation 1–Variation 30–Aria 2

As depicted in Table 5, Ulrich Siegele’s equal tempo for the four pieces postulates a 1:1:1:1 relationship (green). Notably, none of the performers complies with this idea, but three interpretations offer proportions for all four pairings: Tureck 1957, Marlowe 1962, and Schiff 2015. They all implement a green/orange/green/orange pattern (Aria 1 : variation 1 : variation 30 : Aria 2 = 1:2:2:1), i.e., a 1:1 relation between Aria 1 and Aria 2 as well as a 1:1 relation between variation 1 and variation 30, suggesting a sort of plot based on the tempo design. Similarly, the mean values for these four pieces (50.2–101.3–76.8–48.3 bpm) create a green/orange/[none]/[none] pattern, only reflecting the first two relations: Aria 1 : variation 1 : Aria 2 = 1:2:1, as realized by Kenneth Gilbert 1986, Keith Jarrett 1989, Murray Perahia 2000, Levit 2015, and Mahan Esfahani 2016. The only recording displaying an Aria 1 : Aria 2 : variation 1 = 1:1:1 pattern (green/green/[none]/[none]) is Landowska 1933, resulting in an exceptionally slow tempo for variation 1 as explained above. This concept of equating (!) the tempo of variation 1 to that of the arias (as suggested by Siegele) has not been realized in any other interpretation.

Tureck 1957 presents a uniquely slow tempo for Aria 1 (34.0 bpm; only paralleled by Gould’s 33.6 bpm in 1981), breathing life into a very held-back, melodious piece played in long arcs. The ending of the first Aria culminates in a prolonged penultima before it fades away in a drawn-out ritardando. The subsequent variation 1 stands in stark contrast to this opening, both in dynamics and articulation, conjuring “a vision of a great archway to the experience awaiting us.”[42] Similarly, variation 30 paints a lively and serene picture, then retracts twice during B' before celebrating a solemn and prolonged ending (essentially turning the fermata into a lunga). Mirroring the beginning, the following return of the Aria brings back the same melodious, unobtrusive calm (Audio Ex. 17a: 34.0–74.2–66.8–35.6 bpm). Again, the Aria displays long and delicate arcs, this time played without repeats and progressively petering out, virtually coming to a standstill:

The actual ending of the 30 Variations is given to the return of the Aria. This return to the beginning, following the unfolding of the Aria’s potential in the experiencing of a cavalcade of multifaceted ideas and expression, completes the life cycle. This return is not a repeat; it is a return to the source. The very return to the beginning carries with it a fundamental sense of renewal and, as such, reveals yet a new meaning. The form is not circular, therefore, but cyclical, moving to a new plane of vision and perception. This return to the beginning, this end-beginning, is one of the most sublime moments in all music.[43]

Audio Example 17a: Bach, “Goldberg Variations,” Tureck 1957, Aria 1 (mm. 1–8), variation 1 (mm. 1–8), variation 30 (mm. 1–4); Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” Rosalyn Tureck, CD EMI Classics 09647-2., ℗ 1958 © 2008, CD 1, Tracks 1, 2, CD 2, Tracks 5, 6)

In a similar way, Marlowe 1962 opens with a calm and steady Aria 1, using a gentle registration and slowing down only slightly at the end. Variation 1 emerges in clear distinction, vigorous and with a mellow registration. Likewise, variation 30 is well-registered and celebratory, with the first (and only) repeat offering another opportunity to relish the full color. Its ending is shaped by a strong ritardando which creates a ceremonious and stately finish, letting the last note linger until it fades away. This festive piece is followed by the return of the serene and steady Aria and its tranquil registration; a distinction from Aria 1 is barely to be made, save for the slight difference in tone (Audio Ex. 17b: 40.7–75.6–79.5–40.3 bpm). Marlowe’s use of color and registration certainly stands out in this selection of pieces, especially in variation 30. In a 1971 interview, she criticizes the contemporary fashion of playing without the added tonal dimension of registration:

The new wave of harpsichordists go for absolutely no registration at all, which I believe is a mistake. […] There are certain harpsichordists, very well known and very well thought of, who use no registration. They play everything on one eight foot. […] The idea of playing in a very boring way, without and change in color or tempo or anything is… well… boring. I think that any music has to come alive. I don’t think one should exaggerate on one hand by making dramatic changes all the time, but I think it is necessary to use different colors. I see nothing wrong with it. Besides, the instrument has these possibilities which should be exploited.[44]

Audio Example 17b: Bach, “Goldberg Variations,” Marlowe 1962, Aria 1 (mm. 1–8), variation 1 (mm. 1–8), variation 30 (mm. 1–4), Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” Sylvia Marlowe, LP Decca DL 10056, ℗ 1962, Sides A, B)

Schiff’s 2015 (live) recording starts out with a gently flowing, thoughtfully phrased Aria, played with a narrative gesture and lingering shortly on the penultimate note. In the Aria, as throughout the whole cycle, both repeats are religiously executed, and differentiations are mostly made through varying ornamentation.[45] The vibrant first variation enters as if woven from the same fabric, differing in texture mainly through articulation and touch. Analogously, a very spirited and deciso variation 30 leads through a diverse interlacing of playful articulation and melodic phrases reminiscent of a cantus firmus, slowing down only somewhat at the end. With Aria 2, a contemplative gentle flow returns; especially in comparison to the preceding variation, it evokes a sense of calm. As in Aria 1, the motion hardly slows down, essentially simply ending with only a slight ritardando (Audio Ex. 17c: 55.8–100.7–100.8–55.4 bpm). There are no salient differences from its first appearance to be perceived: “After the last variation the opening Aria returns, unchanged. However we are hearing it with new ears because of the experiences of the past 70 minutes.”[46]

Audio Example 17c: Bach, “Goldberg Variations,” Schiff 2015, Aria 1 (mm. 1–8), variation 1 (mm. 1–8), variation 30 (mm. 1–4), Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” András Schiff, BBC-Proms, 22/08/2015; https://youtu.be/sbOwhF1hFcg)

Traces of the same proportional concept can be detected in both of Schiff’s earlier “Goldberg” recordings as well. The relations presented in 2015 (55.8–100.7–100.8–55.4 bpm) are met very closely in his 1982 recording (58.1–104.5–102.0–60.2 bpm), missing the arbitrary percentage bracket for the Aria 2: variation 30 proportion by a neglectable 1%. The same thing can be said about the 2001 recording (61.5–104.1–107.0–60.2 bpm) which again barely misses the bracket for the orange Aria 2: variation 30 proportion (by 3%), falls out of the margin for the variation 1 : Aria 1 proportion (by 11%), but unerringly displays the two green (1:1) relations between the two Arias as well as the two variations. Unsurprisingly, this similarity seems to be rooted in Schiff’s very consistent tempo choices which show only a minimal divergence.

Going by the pattern, the common denominator for Schiff’s recordings is the green relation both in the first and the third column of Table 5. Such a setup (naturally also applicable to Tureck 1957 and Marlowe 1962) also makes an appearance in five other interpretations (Richter 1956, Leonhardt 1965, Richter 1970, Peter Serkin 1994, and Egarr 2005), indicating a concept which “double–frames” the cycle: connecting both the first and last, as well as the second and penultimate elements (but neither of the adjacent pieces!). This seems especially apposite for those recordings which feature no other relation beside these two (Richter 1956, Leonhardt 1965, Richter 1970, Schiff 2001, and Egarr 2005; Audio Ex. 17d: 51.3–77.1–78.9–51.3 bpm), thereby highlighting the double arc.

Audio Example 17d: Bach, “Goldberg Variations,” Egarr 2005, Aria 1 (mm. 1–8), variation 1 (mm. 1–8), variation 30 (mm. 1–4); Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” Richard Egarr, CD Harmonia Mundi USA – HMU 907425.26, ℗&© 2006, CD 1, Tracks 1, 2, CD 2, Tracks 15, 16)

It could be argued that such a construction points to a structural design which isolates the two Arias from the variations, juxtaposing a static arc (Aria 1–Aria 2) with a dynamic arc (spanning from variation 1 to variation 30).[47] Ostensibly, this isolation of the Arias from the rest of the work could imply an interpretation of Aria 1 and Aria 2 as a sort of prologue and epilogue, not fully integrated into the cycle but framing the plot – which leads from variation 1 to variation 30 – from outside.

By contrast, those recordings which only connect the two Arias (1:1) – without any evident relationship between variations 1 and 30 or those two variations with their adjacent Arias – might indicate a differently imagined framework: it suggests a layout which draws the same arc as above but does not explicitly isolate Arias from variation(s), thus integrating the Arias into the cycle as its actual beginning and ending (Kirkpatrick 1958, John Gibbons 1979, Schultz 1998 (Audio Ex. 17e: 44.7–105.3–63.2–44.8 bpm), Haugsand 2001, and Cédric Pescia 2004).

Audio Example 17e: Bach, “Goldberg Variations,” Schultz 1998, Aria 1 (mm. 1–8), variation 1 (mm. 1–8), variation 30 (mm. 1–4); Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” Thomas Schultz, CD Wooden Fish Recordings, ℗ 2003, CD 2, Tracks 1, 2, 31, 32)

This leads to the discussion about how the performers interpret the end of the cycle. As shown in Table 5, the relation Aria 2: variation 30 shows the lowest number of proportions performed, leaving these two pieces unconnected (in terms of tempo relations) in around 80% of all examined recordings. Therefore, the question arises whether the artists assign the role of “ending” to the last variation or the Aria da capo.

Variation 30, inscribed Quodlibet (“what you please”), structurally appears in lieu of a canon at the tenth; “[even] the little upbeat announces that something different is happening here – it is the first and only one.”[48] The variation presents a “festive character,” set in an “exorbitant” four-part stretto, combining two melodies (both well-known in Bach’s time) in the tightest of spaces.[49] The source for the melody first appearing in the tenor is a traditional song set to the lyrics “Ich bin so lang nicht bei dir g[e]west” (I have been so long away from you),[50] which, in the seventeenth and eighteenth centuries, was commonly danced as a Kehraus, i.e., the last (round) dance of a dance evening or a wedding celebration, to send the party home in high spirits.[51] The second melody which follows immediately in the alto most likely stems from a song whose most commonly known version bears the following lyrics: “Kraut und Rüben haben mich vertrieben / hätt’ die Mutter Fleisch gekocht, so wär’ ich länger blieben”[52] (Cabbage and turnips have driven me away / if mother had cooked meat, I would have stayed longer).

Rolf Dammann reasons that the coherence between these melodies becomes evident if one proceeds on the assumption that Bach chose to include them to symbolize goodbye.[53] Apparently, the impression of parting was so strong that artists used to finish their concert performances without repeating the aria; Otto Baensch reports this as a wide-spread “misunderstanding” in 1934, explaining that the Quodlibet is not meant to be a final crowning moment (and therefore not to be played in a thunderous fortissimo) but a good-humored transition leading back to the Aria da capo. In his interpretation, it is the Aria that speaks to the ground bass (“I have been so long away from you”), having been ousted (by “cabbage and turnips”) and announcing its imminent return.[54] Aptly, Bach ends this last variation with the very melody corresponding to “wär’ ich länger blieben” (would have stayed longer), additionally illustrating “blieben” with the subsequent fermata. On the other hand, Dammann argues that the finale of the cycle is not to be found in the “civil idyll” of the Quodlibet but in the “courtly” Aria. He attributes a melancholy longing and nostalgia to the last variation, characterizing it as a fading-out of the cycle, and precisely not as a blithe dance[55] (or one of the popular, playful quodlibets purportedly sung by Bach’s family),[56] or, as Glenn Gould calls it, a “boisterous exhibition of Deutsche Freundlichkeit.”[57]

A closer look at the recordings reveals several distinct dramaturgical decisions with regard to tempo, connecting variation 30 and Aria 2 in different ways (Fig. 9). The only interpretations which take a faster tempo for Aria 2 than variation 30 are Kempff 1969 (19% difference) and Takahashi 2004 (18%), both making astoundingly similar tempo choices. Yet, whereas Kempff replicates the fast tempo of Aria 1 with staggering precision (0% deviation), Takahashi’s Aria 2 surpasses his Aria 1 by 30%, commencing after only a very slight ritardando at the end of variation 30 and thereby providing a particularly fluid transition. Hence, one might consider Kempff’s Aria da capo a recall of his Aria, and Takahashi’s Aria da capo a logical result of his Quodlibet.

Figure 9: Tempo diagram for variation 30 and Aria 2

Audio Example 18a: Bach, “Goldberg Variations,” Takahashi 2004, variation 30 (mm. 1–2, 15–16), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Yūji Takahashi, CD Avex Classics AVCL-84069, ℗&© 2014, Tracks 31, 32)

At the other end of the spectrum, the two recordings creating the highest tempo contrast between a fast variation 30 and a slow Aria 2 are Gould 1981 (62% difference; Audio Ex. 18b: 73.3–27.9 bpm) and Ernst 2020 (61% difference; Audio Ex. 18c: 73.1–28.5 bpm), their tempo choices matching almost exactly.[58] Moreover, these two interpretations are simultaneously among those displaying the highest difference between Aria 2 and Aria 1 (Ernst 2020: 27%; Lang 2020a: 24%; Gould 1981: 17%). This disparity combined with the tempo-related detachment of the Aria da capo from the Quodlibet could suggest a concept which designs variation 30 as the conclusion of the variation cycle and Aria 2 as a mellow, reminiscing echo of Aria 1, orchestrating a farewell.

Audio Example 18b: Bach, “Goldberg Variations,” Gould 1981, variation 30 (mm. 1–2, 15–16); Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Glenn Gould, CD Sony Classical 52619-10, ℗ 1982 © 1993, Tracks 31, 32)

Audio Example 18c: Bach, “Goldberg Variations,” Ernst 2020, variation 30 (mm. 1–2, 15–16); Aria 2 (mm. 1–4)
(Moritz Ernst, unpublished live recording, Stadtkirche Bayreuth, 23/07/2020, used with kind permission)

Remarkably enough, Chen 1985 and Verlet 1992 apply virtually the same tempo to Aria 2 as to variation 30 (Chen: 6% lower than variation 30; Audio Ex. 18d: 61.5–57.7 bpm. Verlet: 2% higher than variation 30; Audio Ex. 18e: 64.9–66.4 bpm). As both interpretations only feature very slight ritardando at the end of the Quodlibet, it could be argued that the performers view these two pieces as strongly belonging together, Aria 2 being a sort of continuation of variation 30. This performance concept could be interpreted as conceiving the “end” of the cycle as both pieces together. On the other hand, both performers choose to play the Aria da capo slightly slower than the Aria, also conveying the impression of an echo of the opening piece.

Audio Example 18d: Bach, “Goldberg Variations,” Chen 1985, variation 30 (mm. 1–2, 15–16), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Pi-hsien Chen, CD Naxos 8.550078, ℗&© 1987, Tracks 31, 32)

Audio Example 18e: Bach, “Goldberg Variations,” Verlet 1992, variation 30 (mm. 1–2, 15–16), Aria 2 (mm. 1–4)
(Bach, “Goldberg Variations,” Blandine Verlet, CD Astrée Auvidis E 87459, ℗&© 1993, Tracks 31, 32)

The slowest interpretation of variation 30 is found in Nikolayeva’s 1992 recording. As with Verlet and Chen, there is only a small tempo difference between Quodlibet and Aria da capo (11%), missing the bracket for the green 1:1 proportion between Aria 2: variation 30 by a tiny margin (1%). Taking this connection into account, all four tempo relations are present in the pattern for this recording (green/orange/orange/[green], tempo proportions 1:2:1:1), as is made obvious by the similarity in tempi (56.3–104.7–58.1–51.8 bpm). In this case, it seems possible that the calm tempo and melancholic atmosphere of the Arias has permeated variation 30. As with Verlet and Chen, the “end” could again be seen as a combination of both Quodlibet and Aria da capo; here, however, both Aria 2 and variation 30 project an arc back towards Aria 1.

Audio Example 18f: Bach, “Goldberg Variations,” Nikolayeva 1992, variation 30 (mm. 1–4, 13–16), Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” Tatiana Nikolayeva, CD Hyperion Records Limited A66589, ℗&© 1992, Tracks 31, 32)

Conversely, Peter Serkin’s 1994 tempo (111.2 bpm) for variation 30 is by far the fastest among all examined recordings, essentially making it alla breve (the tempo also corresponding to his choices for the virtuoso variations). Ipso facto, Serkin achieves a very high contrast between variation 30 and Aria 2 (59% slower than the Quodlibet). These interpretative decisions are very strongly reminiscent of Gould 1981 and Ernst 2020, matching their amount of contrast almost exactly. Consequently, he appears to apply a very similar dramaturgical concept, his festively performed Quodlibet achieving a finale atmosphere before letting the Aria da capo echo the beginning of the cycle (green 1:1 proportion between arias).

Audio Example 18g: Bach, “Goldberg Variations,” Serkin 1994, variation 30 (mm. 1–4, 13–16), Aria 2 (mm. 1–8)
(Bach, “Goldberg Variations,” Peter Serkin, CD RCA Victor Red Seal – BMG Classics 09026 68188 2, ℗&© 1996, Tracks 34, 35)

Summary and Implications

This study has examined the cyclic potential of the “Goldberg Variations” as implemented by performers through tempo relations between its pieces. Several difficulties inherent to the applied methodology have been observed. First, the method of examining exclusively the initial tempo does not cover possible tempo changes within the individual pieces, and the average values obtained from measuring the opening phrases may be influenced by potential rubato decisions. Second, from the quantitative data collected, it is impossible to reliably determine the extent to which a performer actually intended to implement a certain tempo relation in performance. However, it seems safe to assume at least in a few cases that such relations were indeed a result of conscious planning, but many may equally be a “random” outcome of linear interpretation in real time, of “informed intuition,”[59] or a product of physiological conditions (as implied by Epstein).[60]

Another potential issue concerns the difficulty to relate quantitative data to the compositional properties, musical structure, or shape of a piece. These data cannot always be translated directly into a musical context. Statistical values such as the tempo range certainly depict the difference between the fastest and slowest interpretations but do not illuminate any relational pattern: for example, in spite of the exceptionally high range value for variation 25, both the fastest (Takahashi 2004) and slowest recordings (Nikolayeva 1992) of this variation show an overall pattern of three tempo relations within the cycle.

Still, various conclusions can be made from the results yielded by our quantitative examination. The collected data have shown that performers create tempo relations mostly between adjacent pieces, indicating a progression,[61] or in groups of pieces presented under the aspect of symmetry. Unsurprisingly, the examination of the beginning and end of the cycle established the 1:1 proportion between Aria 1 and Aria 2 as the most frequent relation. However, only a few recordings create a pattern of four relations connecting both Arias with their adjacent variations; this is especially reflected in the relatively low number of relations between variation 30 and Aria 2.

Tempo relations generally seem to occur less in groups selected according to non-linear or asymmetrical criteria. While in the group of minor variations only a single performance (Norton 1942) establishes a tempo relation between all three pieces, about a third of all examined recordings establish (differing) patterns of three relations for variations 24–26. For the variations in a minor key, the most frequently established tempo relation is a 1:1 proportion between variation 21 and variation 25 (as reflected by the mean tempo values 45.1–49.2 bpm). The only exception in this case is Landowska 1945, whose historically informed interpretation is mirrored in Ulrich Siegele’s theoretical tempi. Apart from this congruence, Siegele’s draft of a cyclical temporal structure does not seem to be strongly reflected in the relational data gathered from the recordings (with very few exceptions).

Taking these observations as a vantage point, there is an abundance of research questions to be derived from the examinations presented here. A very rewarding topic might be a detailed analysis of performances of the whole cycle, i.e., of tempo relations established throughout the whole work, or even in a different selection of pieces (following different criteria). The present study has revealed that there are several performers whose interpretations feature a particularly large number of tempo relations, including András Schiff, Eunice Norton, Tatiana Nikolayeva, and Glenn Gould, suggesting that recordings by these performers warrant further investigation.

Another worthwhile research area might be an examination of the interpretational choices made by the same performer at different points in time (several recordings have been made, among others, by Glenn Gould, Angela Hewitt, Ralph Kirkpatrick, Wanda Landowska, Lang Lang, Tatiana Nikolayeva, Karl Richter, András Schiff, Yūji Takahashi, and Rosalyn Tureck), as well as an extensive comparison of “Goldberg” pieces with structurally and musically similar works. Further research aspects could include a systematic comparison of interpretations of the “Goldberg Variations” on the piano versus the harpsichord, the forming of performance traditions potentially handed down from teacher to pupil, and a historical investigation of performance trends.

Appendix

Table 7: Discography (selection of seventy-six recordings)

Table 8: Tempo chart for all seventy-six recordings; this table can be downloaded as a separate PDF-file: https://storage.gmth.de/zgmth/media/1119/Motavasseli_Bach_Tab08.pdf