University of South Carolina Scholar Commons Theses and Dissertations Spring 2020 The Mathematics of Rubato: Analyzing Expressivetiming in Sergei Rachmaninoff’s Performances of Hisown Music Meilun An Follow this and additional works at: https://scholarcommons.sc.edu/etd Part of the Music Performance Commons Recommended Citation An, M.(2020). The Mathematics of Rubato: Analyzing Expressivetiming in Sergei Rachmaninoff’s Performances of Hisown Music. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/ etd/5905 This Open Access Dissertation is brought to you by Scholar Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. THE MATHEMATICS OF RUBATO: ANALYZING EXPRESSIVE TIMING IN SERGEI RACHMANINOFF’S PERFORMANCES OF HIS OWN MUSIC by Meilun An Bachelor of Music Eastman School of Music, 2015 Master of Music Texas State University, 2017 Submitted in Partial Fulfillment of the Requirements For the Degree of Doctor of Musical Arts in Piano Performance School of Music University of South Carolina 2020 Accepted by: Joseph Rackers, Major Professor Charles Fugo, Committee Member Phillip Bush, Committee Member David Garner, Committee Member Cheryl L. Addy, Vice Provost and Dean of the Graduate School © Copyright by Meilun An, 2020 All Rights Reserved. ii ACKNOWLEDGEMENTS Thank you to Joseph Rackers, Charles Fugo, Phillip Bush and David Garner. Each of you has supported me at different moments during this degree. It is an honor to have you all guide me through this final step in my education. A special thank you to my husband, Cameron Dennis, for his help in guiding the mathematical portion of this document. Thank you to my professor, Joseph Rackers, for the motivation to finish this document on time. You have been a mentor to me in my Doctor of Musical Arts Degree by supporting me through yearly MTNA competitions and all my degree recitals. Thank you to Charles Fugo who has served as my advisor for the duration of my studies. You have helped me with instrumental accompanying and taught me a lot about ensemble playing. Thank you to Phillip Bush who has coached me on chamber music and made my Schnittke Piano Quintet group sound great! Finally, thank you to David Garner whose coursework opened my eyes to this topic of research. iii ABSTRACT The purpose of this paper is to quantitatively investigate the nature of Rachmaninoff’s playing, with a specific focus on his own compositions. The primary source of data for this study is inter-onset intervals, which will be captured through the use of Sonic Visualiser. Inter-onset intervals are calculated by taking the difference between the onset of two adjacent notes, and can be used to calculate an instantaneous tempo. Various statistical methods will be used, including the variance, moving average, Pearson correlation coefficient, and a custom defined metric. The calculation of variance is especially useful in detecting major deviations from the average tempo in certain sections. These tempo fluctuations are either accelerandos or ritardandos whose data can be fit to a curve using a program called MATLAB. The resulting equations can then be compared with other sections of the piece, or with other pianists’ recordings of the same piece. This study will also include a comparison between Rachmaninoff and several other pianists; his playing will be compared to the group average as well as to each individual pianist. The individual comparisons will make use of the Pearson correlation coefficient, which provides a measure of how similar two datasets are to one another. iv TABLE OF CONTENTS Acknowledgements ............................................................................................................ iii Abstract .............................................................................................................................. iv List of Tables ..................................................................................................................... vi List of Figures ................................................................................................................... vii List of Excerpts ....................................................................................................................x List of Equations ................................................................................................................ xi Chapter 1: Introduction ........................................................................................................1 Chapter 2: Prelude in G Minor, Op. 23 No. 5 ....................................................................10 Chapter 3: Prelude in C-Sharp Minor, Op. 3 No. 2 ...........................................................47 Chapter 4: Rachmaninoff vs. Other Pianists ......................................................................60 Chapter 5: Summary and Conclusions ...............................................................................77 References ..........................................................................................................................86 Appendix A: Annotated Scores .........................................................................................88 Appendix B: Degree Recital Programs ..............................................................................99 v LIST OF TABLES Table 2.1 Formal Analysis of A Section ............................................................................13 Table 2.2 Variances for A Section .....................................................................................14 Table 2.3 Formal Analysis of B Section ............................................................................21 Table 2.4 Formal Analysis of A’ Section ..........................................................................32 Table 2.5 Variances for A’ Section....................................................................................33 Table 3.1 Formal Analysis of A Section ............................................................................48 Table 3.2 Formal Analysis of B Section ............................................................................50 Table 3.3 Formal Analysis of A’ Section ..........................................................................57 Table 4.1 Rachmaninoff vs. Group Tempo Comparison ...................................................62 Table 4.2 Rachmaninoff vs. Group Correlation Coefficients ............................................69 Table 4.3 Rachmaninoff vs. Group γ Values .....................................................................69 Table 4.4 Rachmaninoff vs. Group Average Tempo Per Beat Correlation Coefficients ..76 Table 4.5 Rachmaninoff vs. Group γ Values .....................................................................76 Table 5.1 Rachmaninoff vs. Group Correlation Coefficients ............................................85 Table 5.2 Rachmaninoff vs. Group γ Values .....................................................................85 vi LIST OF FIGURES Figure 2.1 Inter-onset Intervals for S2t (mm. 23 – 24) .......................................................15 Figure 2.2 Inter-onset Intervals for S3t (mm. 30 – 34) .......................................................16 Figure 2.3 Inter-onset Intervals for S3m and S3t (mm. 25 – 34)..........................................17 Figure 2.4 Inter-onset Intervals for S3t R2 (mm. 31 – 32) ..................................................18 Figure 2.5 Inter-onset Intervals for S3t R3 (mm. 33 – 34) ..................................................19 Figure 2.6 Inter-onset Intervals for S3t R3 (mm. 33 – 34) with Curve Fit ..........................20 Figure 2.7 Tempo for S4a (mm. 35 – 36) ...........................................................................21 Figure 2.8 Tempo for S4b (mm. 37 – 38) ...........................................................................22 Figure 2.9 Comparison between S4a and S4b with Curve Fits ............................................23 Figure 2.10 Rate of Change of Tempo for S4a and S4b ......................................................24 Figure 2.11 Tempo for S4c (mm. 39 – 41) .........................................................................25 Figure 2.12 Comparison between S4c RE1 and S4c RE2 .....................................................26 Figure 2.13 Rate of Change of Tempo (RE1 and RE2) ......................................................28 Figure 2.14 Comparison between S5a and S5b with Moving Average ................................29 Figure 2.15 Tempo for S5c with Moving Average .............................................................30 Figure 2.16 Tempo for B Section with Moving Average ..................................................31 Figure 2.17 Tempo for A’ Section with Moving Average .................................................32 Figure 2.18 Tempo for S6a by Rhythmic Content ..............................................................34 Figure 2.19 Tempo for S6b by Rhythmic Content ..............................................................35 Figure 2.20 Tempo for S6c by Rhythmic Content ..............................................................36 vii Figure 2.21 Variances for S6a by Rhythmic Content .........................................................37 Figure 2.22 Variances for S6b by Rhythmic Content .........................................................38