Studying Tonal Evolution of Western Choral Music: A Corpus-Based Strategy

Abstract

The availability of large digital music archives combined with signi昀椀cant advances in computational analysis methods have enabled novel strategies for musicological corpus studies. This includes approaches based on audio recordings, which are available in large quantities for di昀erent musical works and styles. In this paper, we take up such an audio-based approach for studying the tonal complexity of music and its evolution over centuries. In particular, we examine the tonal evolution of Western choral and sacred music exploiting a novel audio corpus (5773 tracks) with a rich set of annotations. The data stems from one of the world's leading music publisher for choral music, the Carus-Verlag, which is specialized on scholarly-critical sheet music editions of this repertoire and also runs an own record label. Based on this corpus, we revisit a heuristic strategy that exploits composer life dates to approximate work count curves over the years, validate this approximation strategy, and optimize its parameters using the reference composition years annotated in the Carus dataset. We then apply this strategy to derive evolution curves from the full Carus dataset. We compare the results to a study based on a purely instrumental dataset and test three hypotheses on tonal evolution, namely that (1) global complexity increases faster than local complexity, that (2) major keys are tonally more complex than minor keys, and that (3) instrumental music is more complex than vocal music. The results provide interesting insights into the choral music repertoire and suggest that well-curated publisher data constitutes a valuable resource for the computational humanities.

Keywords

• eol>Computational Musicology
• Corpus Analysis
• Musical Style Evolution
• Tonal Analysis

1. Introduction

As digitization progresses, more and more comprehensive archives of cultural data become available. In combination with the further development of analysis algorithms, such archives provide promising opportunities for quantitative analyses and large-scale corpus studies in computational humanities. This also applies to music data, which exists in a variety of styles and digital data types, including graphical sheet music, symbolic (i. e., machine-readable) scores, and audio recordings. While symbolic scores, which explicitly encode musical symbols, usually allow for the most detailed analyses (as 2in0, [ 2, 10, 7, 26, 12 ]), such data is hard to 0.7

Beethoven, Piano Sonata Schoenberg, Orchestral Piece Evolution Curve 1700 1750 1800 1850 1900

1950 acquire. Manual creation of symbolic data is tedious, and automated conversion of graphical sheet music to symbolic scores known as optical music recognition (OMR4)] [or automatic music transcription (AMT) for converting audio recordings to symbolic sco3r]eso昀琀e[n lead to unsatisfactory results, thus requiring labour-intensive post-processing.

For e昀케ciently scaling up computational music analyses, corpus-based studies have also been approached directly based on raw data such as sheet music image2s1[, 14] or audio recordings [ 16, 1, 9, 22, 23 ]. This requires advanced computational techniques that convert the data into semantically meaningful representations that can be directly interpreted by music experts. An example for such a representation is the measurement of tonal complexi2t4y][, which has been applied for corpus analyses of jazz22[] and Western classical music2[ 3 ] based on pitch-class representations (chroma) of audio recordings.

Beyond the computational tools, comprehensive and carefully curated datasets are essential for conducting corpus analyses1[ 7 ]. While some well-annotated public datasets of limited size and scope are available (see e. g.,1[ 3, 6, 15, 18 ]), a good coverage of a larger repertoire is required to draw more general conclusions. However, annotations and historical metadata is o昀琀en hard to acquire for large corpora. In previous wor2k3][, we made an attempt to compile a diverse, medium-sized datasetC(ross-Era) of 2000 classical music recordings (piano and orchestral music) spanning roughly 350 years of Western music history. Since this dataset did not contain any 昀椀ne-grained annotations of composition yearsw(ork dates), we proposed a workaround to map tonal analysis results onto a historical time axis (“evolution curves”, compare Figure1) based on composers’ lifetimec(omposer dates). Until now, this simplifying approach has not been systematically tested on any dataset with composition year annotations.

In this paper, we approach this problem by studying the distribution of work dates over the lifetime of a composer. To this end, we consider the data repository of tChaerus Verlag,1 a German music publisher specializing in choral and sacred music. Carus produces high-quality editions conforming to a historical-critical standard, also employing leading musicologists with comprehensive expertise on their repertoire. Since Carus is also active as a record label releasing reference recordings of their own editions, their repository comprises a large number of audio recordings (more than 7000) with a rich set of detailed and well-curated metadata, including information about work dates, composer dates, instrumentation, singing language, key, and other annotations2.

Based on the Carus audio corpus (CAC), we make the following contributions in this paper. First, we revisit the heuristic strategy for approximating work count curves and evolution curves based on Tukey windows (Figure1) proposed in [ 23 ]. We systematically validate this strategy and optimize the Tukey window parameters by comparing the approximation curves with reference curves derived from the work dates annotated in the CAC. As an exemplary application, we then consider the measurement of tonal complexity as proposed2i4n],[visualized over composer dates and work dates, respectively. Second, using this strategy, we perform multiple analyses regarding the tonal evolution. In contrast2t3o],[the CAC allows us to go beyond instrumental music and focus on vocal/choir and sacred music instead. Moreover, we consider a substantially extended time span of 450 years in CAC (as opposed to roughly 300 in [ 23 ]). Finally, the detailed annotations in the CAC allow for testing di昀erent hypotheses about the tonal complexity of Western (vocal) music, i. e.: (1) Global complexity increases earlier than local complexity. (2) Major keys are tonally more complex than minor keys. (3) Instrumental music is more complex than vocal music. The computed evolution curves provide interesting insights regarding these questions and indicate that well-curated publisher data can be of high value for the computational humanities.

The remainder of this paper is organized as follows: Secti2onpresents information and statistics of the CAC. Section3 deals with the approximation of work count curves and evolution curves, tests the validity of this strategy, and determine optimal parameters based on the reference annotations in CAC. In Section4, we use this strategy to compute evolution curves on tonal complexity and to test three hyptheses on tonal evolution. Secti5onconcludes the paper. Further related work is discussed in the respective sections.

2. The Carus Audio Corpus

The Carus-Verlag, founded near Stuttgart, Germany, in 1972 is a family business focusing on vocal and sacred music. Their sheet music editions include around 45,000 works (most of them vocal compositions) and re昀氀ect the development of 昀椀ve centuries of choral music, ranging from Gregorian chant, madrigals, and motets of the Renaissance, to contemporary choral music, and works for jazz and pop choi3r.Carus o昀ers scholarly-critical music editions of the most important oratorios, masses, and cantatas in music history, oriented towards historically informed performance practice. Being also active as a record label, Carus releases reference recordings based on their own editions. A core mission of the company is to help amateur and semiprofessional choirs to improve their skills. To this end, digital tools such aCsathrues music app have been created.

The CAC4 comprises the majority of the Carus CD releases (as of 2019), totalling 7115 tracks corresponding to individual works (for one-movement works) or movements (for multi2Since the audio recordings are commercial releases, we cannot publish the dataset. However, detailed information about individual recordings is provided at the publisher’s webshitetp(s://www.carus-verlag.com/en)/. 3https://www.carus-verlag.com/en/ueber-carus/ 4This corpus has been made available to us for research puposes based on a collaborative project. movement works and work cycles). Since we want to focus on original art music compositions, we perform a 昀椀rst cleaning step where we remove works without composer, works without composer life dates, arrangements, pop music, children songs, and christmas songs. A昀琀er this, 5773 tracks (movements) remain belonging to 2409 di昀erent works with a total duration of 389:52:20 (hh:mm:ss). On average, a work has 2.4 movements and a duration of 9:43 (mm:ss). However, we note that the number of movements per work is highly unbalanced, with many one-movement works on the one hand and many large-scale works (oratorios, passions, etc.) with more than 30 movements on the other hand. In the following, we present all statistics and analysis results at thework level, where information such as key or instrumentation always refer to the overarching work (note that e. g., a mass in C minor for choir and orchestra may also contain individual movements in other keys and instrumentations).

Table 1 provides statistics over the CAC’s annotations at the work level. Roughly half of the works (1151 out of 2409) has annotations regarding the year of composition (work date). The majority (1964 out of 2409) is annotated regarding instrumentation. As expected, there is a strong focus on vocal music (1764) in general and on choral music speci昀椀cally (1400 out of 1764).5 From the perspective of tonal analysis, the availability of key annotations for roughly half of the works (1166 out of 2409) is of particular relevance. As one might expect for this repertoire, there is a bias towards major keys as well as a considerable number of other keys (church modes such as dorian in early works).

As mentioned above, CAC spans roughly 450 years, covering the period from about 1570– 2020. In total, the works stem from 234 di昀erent composers. Figur2e shows a historical view on the composer dates for composers with at least 昀椀ve works. Well-known composers like Felix Mendelssohn Bartholdy, Johann Sebastian Bach, or Wolfgang Amadeus Mozart make up a signi昀椀cant part. However, CAC also comprises less known composers such as Heinrich Schütz (featuring the complete edition) or Max Reger. Carus even makes great e昀orts to bring almost 5Please note that, due to the work-related annotations, individual solo vocal movements (e. g., an aria) within a choir work (e. g., an oratorio) are counted towards choral works. Lechner, Leonhard [ 19 ] Eccard, Johannes [ 12 ] Calvisius, Sethus [ 5 ] Gesualdo di Venosa, Carlo [ 6 ] Hassler, Hans Leo [ 23 ] Monteverdi, Claudio [ 7 ] Praetorius, Michael [ 18 ] Sacco, Salvatore [ 5 ] Grandi, Alessandro [14]

Schütz, Heinrich [321] Schein, Johann Hermann [31] Scheidt, Samuel [ 18 ]

Bertali, Antonio [ 12 ]

Hammerschmidt, Andreas [ 19 ]

Pohle, David [11]

Buxtehude, Dieterich [31]

Charpentier, Marc-Antoine [ 16 ] Biber, Heinrich Ignaz Franz [9] Krieger, Johann Philipp [ 7 ] Förtsch, Johann Philipp [11]

Purcell, Henry [56] Fux, Johann Joseph [ 5 ]

Bourgeois, Thomas-Louis [ 5 ] Vivaldi, Antonio [ 12 ] Schieferdecker, Johann Christian [ 6 ] Zelenka, Jan Dismas [ 5 ] Telemann, Georg Philipp [55] Graupner, Christoph [ 12 ] Heinichen, Johann David [ 7 ] Händel, Georg Friedrich [28] Bach, Johann Sebastian [112] Scarlatti, Domenico [9] Galliard, Johann Ernst [ 5 ]

Hasse, Johann Adolf [ 7 ]

Bach, Wilhelm Friedemann [32] Kayser, Isfrid [ 7 ] Bach, Carl Philipp Emanuel [11] Homilius, Gottfried August [63] Altnickol, Johann Christoph [ 5 ]

Bach, Johann Christian [11] Haydn, Johann Michael [13]

Knecht, Justin Heinrich [ 5 ] Kraus, Joseph Martin [ 5 ] Mozart, Wolfgang Amadeus [ 24 ] Rosengart, Æmilian [ 15 ]

Spohr, Louis [ 5 ] Silcher, Friedrich [ 22 ] Rossini, Gioachino [ 18 ] Schubert, Franz [43]

Mendelssohn Bartholdy, Felix [138] Schumann, Robert [32] Nicolai, Otto [14] Liszt, Franz [31]

Gounod, Charles [ 7 ]

Bruckner, Anton [ 15 ] Cornelius, Peter [ 15 ]

Brahms, Johannes [67] Becker, Albert [ 22 ] Rheinberger, Josef Gabriel [259] Tschaikowsky, Peter I. [11] von Herzogenberg, Heinrich [60]

Wolf, Hugo [9]

Rachmaninow, Sergei [34] Reger, Max [84] Schreker, Franz [ 7 ]

Boulanger, Lili [ 15 ]

Distler, Hugo [11]

Tormis, Veljo [41]

Miskinis, Vytautas [ 19 ]

Schanderl, Hans [ 15 ] Johannsen, Kay [11] Schwemmer, Frank [ 23 ]

Mocnik, Damijan [ 18 ] 1600 1700

1800

Year

1900 2000 works by each composer is indicated in square brackets and encoded by the darkness of the bars. forgotten works by Gottfried August Homilius or Josef Gabriel Rheinberger back into the focus of the German choir scene and beyond. A particular interesting fact is the good coverage of the late 15th and 16th century (which is not covered in23[]). In the 20th century, however, we 椀昀nd a lower number of works, almost observing a gap around 1950.

3. Approximation Strategy for Work Count Curves

We now outline the approximation of work count curves and the strategy for computing evolution curves as done in2[ 3 ]. We then validate this strategy and optimize the involved parameters by comparing approximation curves based on composer dates to the reference curves based on true work dates using the annotations in CAC. In the following, we simplify all temporal information by only considering the respective year.

3.1. Work count curves

To analyze musical styles in their historical context, one ideally has information about the true work dates, which we assume to be the year work ∈ ℕ, where a composition was completed. Musical styles may evolve rapidly, and composing is subject to trends and in昀氀uenced by other composers, the taste of audiences, or extra-musical stimuli such as political events. One might think of composers with several “creative periods,” such as Ludwig van Beethoven or Arnold Schönberg. However, collecting reliable work date annotations for larger datasets requires a substantial amount of manual research, and this information is unknown or in doubt for quite a number of works. Even if one knows all composition dates, it becomes di昀케cult to create a dataset with a balanced coverage of all years.

Because of such problems, we adopted in previous wor2k3[] a pragmatic approach by projecting works onto the historical time axis based ocnomposer dates, i. e., the information on birth year birth, death year death, and overall ag e death = death − birth, which is considerably faster to acquire. We proposed an approximation of work counts over the course of a composer’s life. For this distribution, we assumed that a typical composer starts composing not before a certain (昀椀xed) age given by start ∈ ℕ years (with start = 10 in [ 23 ]). For the remaining years (ages)[ start ∶ death] ∶= { start, start + 1, … , death}, we computed a roughly 氀昀at distribution with smooth edges. To this end, we used a so-calleTdukey window (ortapered cosine window) ∶ ℕ → ℝ with parameter ∈ ℝ :

1, ⎨ ⎩ ( − ), () = ⎧0.5 (1 − cos ( 2 )) , 0 ≤ <

2 ≤ ≤

2 2 2 < ≤ (1) with = [0 ∶ ] and = death − start being the window length. In 2[ 3 ], the parameters were heuristically chosen to a start age ofstart = 10 and a Tukey parameter of = 0.35 . Figure1 shows the resulting distribution for Beethoven and Schönberg. The total distribution is then amplitude-normalized to∑ () = 1 and weighted with the total number of works by a composer in the dataset, resulting in a so-callewdork count curve (WCC). That way, each work contributes to the part of the time axis that corresponds to the composer’s lifetime, as indicated in the distribution. This means that a composer with more works in the dataset will have a greater in昀氀uence on the WCC.

3.2. Validating and optimizing the approximation strategy

In [ 23 ], the Tukey window and its parameters were chosen heuristically without any further validation since work date annotations were not available for the dataset used. The CAC contains such annotations for roughly half of the works (compare Ta1b)l.e Using these an notations, we now validate the approximation strategy and search for optimal values of the parameters and start. We do this in a stepwise fashion: First, we determine the start age start, i. e., the age at which we expect an average composer to start composing. To this end, we calculate the percentage of all works that were composed at a speci昀椀c absolute age in years (blue curve in Figure3a). To counteract the e昀ect of imbalanced composition ages, we slightly smooth this curve by convolution with a 5-year kernkel= (0.1, 0.2, 0.4, 0.2, 0.1)T. Since composers have died at di昀erent ages, the red curve slowly decreases a昀琀er an age of approximately 60. We then de昀椀ne a half Tukey window for the range[ ∶ 60] preceded by zeros (red curve start in Figure3a). For each value of start

∈ [0 ∶ 24], we 昀椀t the Tukey parameter (see Eq. (1)) as well [ start

∶

Using ages from [ as a magnitude scaling factor using non-linear least squares. We obtain a minimal squared dis(compare Figure3a), which is slightly higher than the valu estart = 10 used in [ 23 ]. tance (Euclidean distance) between the curve and the half-Tukey approximation asttart = 13 start = 13, we now 昀椀t the window parameters for the remaining years, i. e., the interval death . To counteract the e昀ects of di昀erent overall ages, we normalize the overall ∶ death] to [ start

∶ 60] by interpolating work dates accordingly followed by smoothing with the kernelk (blue curve in Figure3b). Since the curve ends steeper than it begins, we allow the 昀椀tted Tukey window to cover a range[ start ∶ 60 + add] (the additional years will be set to zero later). With the same 昀椀tting strategy as above (non-linear least squares), we then 昀椀nd an optimal value of add

= 6. For the Tukey paramete r , we determine the optimal value to = 0.72 , which is considerably larger than the value o=f 0.35 used in [ 23 ]. The 椀昀tted curve is shown in red in Figure3b.

We 昀椀nally set the curve to zero for all age>s 60 and normalize the window weights such that the total weight amounts to 1. The resulting curve is shown in Figu3rce. For a given composer with 昀椀nal age

] by suitable interpolation.

death, we then re-normalize this window length back from the range

With these optimized window parameters, we now validate the approximation strategy for the work count curve. To this end, we 昀椀rst compute the reference curve using the work date annotations for 1151 works that have these annotations. We post-process the curve with an average 昀椀lter of length 15 years (red curve in Figur4e). We then compare this reference curve with our approximation curve based on composer dates and our optimized Tukey window (blue curve in Figure4). Overall, the approximation seems to be suitable. In some periods (e. g., around 1680), the approximation curve is ahead, for others (e. g., at 1770), it lags behind the add reference curve. In a quantitative comparison, we measure an Euclidean distance of 0.046 (averaged per year). In contrast, when using the parameters o2f3[], i. e., , = 0.35 , start = 10, and = 0, we measure an average distance of 0.068. We conclude that the approximation based on Tukey windows is a suitable strategy to compensate for missing work date annotations. (a) ) (3 # s k r o f2 W o n i1 o t c a r F 0 (b) ) # (3 s k r o fW2 o n i1 o t c a r F 0 (c) ) # (3 s k r o fW2 o n i1 o t c a r F 0 0 20 40 60 80

100

Age 10 20

30

Average counts relative

Tukey model fit 40

Age (normalized) 50 60

Average counts

Half Tukey model fit

Average counts

Tukey model fit 0 10 20 40 50 60 30

Age (normalized)

4. Studying the Evolution of Tonal Complexity

With the validated strategy, we now investigate the tonal evolution of choral music in the CAC. First, we summarize the computational approach for measuring tonal complexity from audio recordings. Then, we compare the results to the study i2n3[] and then use our evolution curves to test three common hypotheses about the repertoire.

4.1. Measuring tonal complexity

We now revisit the measurement of tonal complexity from audio recordings as performed in [ 23 ]. First, we discuss related work regarding complexity. Then, we present the method applied here, closely following22[]. 14 ) te12 u lso10 b a (s 8 k r o 6 W fo 4 . o N 2 0

Musical complexity is a highly relevant (yet vague and multi-faceted) notion for analysis, which has been approached by various researchers. In19[], several aspects of complexity regarding acoustic, timbral, or rhythmic properties were investigated. Concerning tonality, several authors1[ 9, 5, 8 ] have focused on sequential complexity including chord sequence5s].[In contrast, we introduced in2[ 4 ] tonal complexity measures that locally describe distributions of energy across the twelve chromatic pitch classes used in the Western tonal system. As one principle, these measures quantify the variety of pitch classes used such that 昀氀at distributions (e. g., chromatic clusters) result in high complexity values while sharp distributions (e. g., single notes) result in low ones (see Figur5e), thus indicating an average degree of dissonance. Such features have shown good correspondence to an intuitive understanding of tonal complexity over the course of an individual work, which we have veri昀椀ed on a set of chords as well as for segments of Beethoven’s piano sonatas [ 24 ]. Averaging such complexity features over many works provides meaningful and stable results, which has been demonstrated by a large-scale study of musical evolution in classical musi2c3[] and jazz [ 22 ].

Following 2[3, Fig. 6], we select a geometric complexity measure that accounts for the harmonic relationship between pitch classes and is capable of describing the pitch-class content on various temporal levels (昀椀琀h-width complexity, see [ 24 ]). We now summarize the de昀椀nition of this measure encoded by the functionΓ ∶ ℝ12 → [ 0, 1 ]. First, we extract a chroma representation from the audio data using the 昀椀lter-bank method presented in11[], with a resolution of 10 Hz (ten chroma vectors per second). As a result, we obtain chroma vectors c = ( 0, 1, … , 11)T ∈ ℝ12

with positive entries ( ≥ 0) normalized with respect to thℓe1-norm (∑1=10 = 1). The entries with ∈ [0 ∶ 11] indicate the salience or energy of the twelve pitch classes C, C♯, …, B, respectively. Because of octave invariance, the features are of a cyclic nature (a transposition results in a cyclic shi昀琀).

For computing the complexityΓ(c) ∈ [ 0, 1 ], we map the chroma features onto the circle (having a size of 7 semitones) resulting in the vectorcfifth: of 昀椀琀h. To this end, we 昀椀rst re-order the chroma values according to perfect 昀椀琀h intervals r(c): Based on the reordered vectocrfifth, we compute circular statistics using the resultant vector Then, the complexityΓ(c) relates to the inverse length orf(c) and is de昀椀ned as:

fifth = (⋅7 ) mod12. r(c) = 1 ∑=−01

fifth exp (2 1i2 ). Γ(c) = √1 − |r(c)|. (2) (3) (4) This measure corresponds to the angular deviation (the circular equivalent to the standard vectors yield intermediate complexity value0s<( Γ(c) < 1). deviation) and describes the spread of the pitch classes around the circle of 昀椀琀hs. Figure5 illustrates the de昀椀nition of the complexity feature and the resultant vectro(rc)(in red) showing examples for three input chroma vectorcs. For a sparse vector (le昀琀), the complexity is minimal (Γ(c) = 0). For a 昀氀at vector (middle), we obtain maximal complexity(Γ(c) = 1). Other chroma

Finally, we note that there are di昀erent strategies of aggregation to track-wise (i. e., movement-wise) values. First, we de昀椀ne a local measureΓlocalby calculatingΓ(c) for all 10 Hz chroma vectorsc (i. e., ten chroma vectors per second) and then averaging over these features. Second, we 昀椀rst compute a global chroma statistics by averaging andℓ1-normalizing the features and then calculating a single complexity valΓugelobalfor each movement. Aggregation to works is then done by averaging over the complexity values for all movements.

4.2. Evolution curves

In Section 3, we have studied the total number of works in CAC over the course of the years (work count curves) using the work dates or our approximation strategy based on composer dates. As an example for a quantitative analysis, we now apply these strategies to our measurements of tonal complexity as de昀椀ned in Section4.1. For the approximation curves, we again use the window parameters as determined above. For the reference curves, we use a 15-year average 昀椀lter for smoothing.

While the windows for each work were weighted with the value of 1 to account for the total number of works, we now use the complexity valuΓeof the respective work for weighting. We sum up all weighted windows and divide by the respective work count curve for normalization. We obtain a so-calledevolution curve (EC) that indicates the average complexity of the works

Individual works

Approximation curve (composer dates)

Reference curve (work dates), smoothed 1800

Year

1800 Year 1600 1650 1700 1750 1850 1900 1950 2000

Approximation curve (composer dates)

Reference curve (work dates), smoothed

Combined 1600 1650 1700 1750 1850 1900 1950 2000 along the historical time axis. That way, each work contributes to the part of the time axis that corresponds to its work date (for the reference curve) or its composer’s life dates (for the approximation curve).

Denoting our full dataset a s , we 昀椀rst consider the subset work ⊂ comprising all works with available work date annotations (1151 works in total). Figu6rae shows the resulting EC for the global complexity both as approximation curve (blue) and reference curve (red), together with the individual works’ complexity values (gray crosses). Compared to the work count curves (Figure4), the approximation is still good but the deviations are slightly higher. However, we observe such deviations only in regions where only few works contribute, e. g., around the years 1600, 1750, 1800, or 1920–1950. As long as there is su昀케cient coverage of works/composers, the approximation curve closely resembles the reference curve.

Based on this 昀椀nding, we now analyze the full dataset applying a combined strategy: For the subset work ⊂ (1151 works), we make use of the work date annotations and map them directly to the time axis (smoothed as above) as done for the reference curves (red curve in Figure6b. For the subset comp ⊂ (1258 works), which contains the works without work date annotations, we use the mapping based on our optimized Tukey windows as done for the approximation curves (blue curve in Figur6eb). The resulting combined EC is shown as the black curve in Figure6b. We observe a stabilized curve where minor outliers are removed (e. g., around the years 1700, 1760, or 1920) while not loosing the interesting trends.

4.3. Three hypotheses on tonal evolution

We now apply this mixed strategy for investigating the evolution of the tonal complexity in CAC, for comparing the results to those in23[], and for testing three musicological hypotheses. To this end, we use our mixed approach for computing various variants of the combined EC, always using the full datase t .

Comparison to related work. We start with two of the combined ECs, one based on the local complexityΓlocaland the other based on the global complexitΓyglobal, respectively (Figure 7). Looking at the global EC (black), we observe an increase in complexity over the course of the 17th and 18th century. Interestingly, we do not observe any drop around 1750, in contrast to [ 23 ] where the demand for more “simplicity” a昀琀er the Baroque era was clearly visible (however, this trend is supported by a small number of works available for the period around 1800). On the other hand, the increase during the 19th century observed 2in3][is not visible for CAC. Even more remarkably, CAC does not show any major increase in complexity during the 20th century. The modernism in tonality, pushed by expressionist and dodecaphonic composers such as Arnold Schoenberg or Igor Stravinsky, does not seem to be re昀氀ected in choral music to the same degree. This could be based on di昀erent stylistics trends in choral music, but also be a property of the CAC, where complex atonal works might not be in the focus since they are hard to be performed by amateur choirs.

Global versus local complexity. We now test di昀erent hypotheses starting with the assumption that the global complexity evolves independently from the local one. This behavior was observed in [ 23 ] especially within the 19th century, where the local complexity (refering to the complexity of e. g., chords) was fairly stable while the global complex (refering to the complexity of modulations across the whole piece) was clearly increasing. For CAC, we do not observe such a behavior. Comparing ECs for global and local complexity, we mostly observe a parallel evolution. The distance beween the curves only marginally increases a昀琀er 1820. A possible reason might be the typical movement length, which can be considerably higher in instrumental works such as string quartets or symphonies, as opposed to the shorter movements of oratorios or masses. This shorter length might restrict the number and tonal distance of modulations occuring within a movement. However, this hypothesis needs further investigation.

Major versus minor keys. Our second hypothesis is based on the observation that minor keys usually exhibit more chromatic in昀氀ections as compared to major keys. To this end, we consider the data subset with key annotations (major and minor) and compute an EC for each of them (Figure8). For the global complexity (solid lines), both curves follow a similar trend. However, we see a small but consistent o昀set of the minor curve (red) over the major curve (green). This con昀椀rms our hypothesis that minor keys use a larger pitch-class range and, thus, are tonally more complex. For the local complexities (dashed curves), we do not observe this o昀set. Moreover, for the 20th century, we see some 昀氀uctuating behavior, which is due to the 1.0 2000

Vocal: Global

Vocal: Local

Instrumental: Global

Instrumental: Local 1600 1650 1700 1750 1800 1850 1900 1950 2000 Year

fact that there is little data with key annotations for that period (for atonal and free tonal music, key is o昀琀en not a relevant concept).

Vocal versus instrumental music. Next, we investigate the hypothesis that instrumental music is more complex than vocal music. We expect such behavior since vocal compositions need to account for the higher di昀케culty in producing pitches when singing, especially for large and complex intervals. Moreover, musicologists o昀琀en claim that compositional “revolutions” were o昀琀en happening in compact instrumental settings such as the string quartet. To test our hypothesis, we use the instrumentation annotations and compute a vocal as well as an instrumental EC (Figure9). As a downside of CAC, we 昀椀nd an unbalanced situation (compare Table1), resulting in a small number of works available for the instrumental EC. Nevertheless, we observe a clear tendency that contradicts our hypothesis: Vocal music seems to be more complex than instrumental music for most time periods. In particular, the o昀set is large for the local complexity (dashed lines). However, we suspect a technical reason for this behavior. Our chroma features are based on a signal processing approach, which maps frequency components extracted from audio recordings to the twelve chroma bands. When dealing with recorded vocal music, this process o昀琀en leads to substantial artifacts since pitch stability is much lower than for instruments and e昀ects such as vibrato, portamento, or typical deviations from the twelve-tone equal temperament (pure tuning) substantially blur the chromagrams. This can lead to quasi-chromatic artifacts that may push the complexity measurements even locally.

5. Conclusions and Future Work

In this paper, we considered an approach for studying the tonal evolution of music based on partially annotated corpora of music audio recordings. As our 昀椀rst contribution, we revisited and validated a strategy for computing work count curves, compensating for missing work composition dates using heuristics based on composer life dates as an approximation. To this end, we exploited the novel Carus audio corpus (CAC), which contains work date annotations for a substantial part of the works. We showed that a good choice of the parameters helps to minimize the deviations of the approximation curve from the reference curve. On this basis, we performed a combined approach for computing evolution curves that map musical features onto the time axis. This strategy allowed us to compare the CAC with previous studies and to test three hypotheses on tonal complexity in this repertoire. In our future work, we plan to substantially extend, improve, and deepen these studies. In particular, we want to investigate potential technical reasons for higher complexity measurements in choral music. To this end, more recent chroma extraction strategies based on deep neural networks are of high potential since they have shown to be successful for deriving tonal information from vocal recordings by reducing typical artifacts2[ 5 ]. Beyond that, a combination of the analysis based on CAC with other datasets such as the one in 2[ 3 ] will provide better insights into the evolution of tonal music and allow for testing further hypotheses. Finally, this paper also aimed for providing high-level insights into the CAC. While the analyses revealed that a very good coverage of the time period under investigation is crucial for obtaining reliable results and that additional data might be bene昀椀cial for some periods, we see a high potential of such well-curated publisher datasets for studies in computational musicology and beyond.

Acknowledgments

We cordially thank the Carus-Verlag Stuttgart (Johannes Graulich and Ester Petri) for enabling the study of the Carus audio corpus. This work was supported by the German Research Foundation within the project “Computational Analysis of Harmonic Structures” (DFG MU 2686/7-2). The International Audio Laboratories Erlangen are a joint institution of the FriedrichAlexander-Universität Erlangen-Nürnberg (FAU) and Fraunhofer Institute for Integrated Circuits IIS.

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