Chapter 1: Groundbreaking Discoveries
“A great discovery is a fact whose appearance in science gives rise to shining ideas, whose light dispels many obscurities and shows us new paths.” – Claude Bernard
The Epiphany
In June 1992 I had just finished my first year at the University of Kansas working towards a D.M.A. in piano performance and M.M. in historical musicology. One of the courses I really enjoyed that year was a “Performance Practice Seminar” taught by Professor J. Bunker Clark. For this course I wrote a term paper on “Tempo in the Baroque,” which was my first exposure to the scholarly world of tempo and historical performance. This was just the tip of the iceberg, which led to a lifelong fascination and obsession with tempo. I was determined to figure out Baroque tempi if it was the last thing I ever did. Around this time (the spring semester of 1992) I developed my “Unified Theory of Tempo Relativity,” which uses logic and limits to determine mathematically or theoretically correct tempos which can be expressed in a matrix of whole numbers. This complete article may be read on the BachScholar® website. (See Table 1.)
This simple matrix of numbers has special and unique properties unlike any other matrix of numbers:
A “fastest common note value” (FCNV) is assigned for each row or family of tempos – This is the only way a tempo matrix can be designed, that is, in which each row is defined by a common note value. Then, each column presents a different grouping of this common note value.
Consists exclusively of whole numbers or integers – Except for avant garde composers like Karlheinz Stockhausen, composers and performers prefer whole numbers when measuring tempos. For example, no composer ever gives a fractional value like 62.7 as a tempo suggestion, but rather, a whole number like 63. This is because human beings are naturally attracted to whole numbers. If Bach used a tempo system in which his tempos were expressed in beats per minute, it can be assumed as an axiom that such numbers were integers and not numbers with decimals.
Does not include any numbers divisible by 10 – In the article “A Unified Theory of Tempo Relativity,” it is shown mathematically that if the hypothetical “tempo x” that demarcates the subdivision and consolidation thresholds is assumed to be a number divisible by 10, like 40, then this excludes all tempos divisible by 10 like 40, 60, 80, 100, or 120. In other words, base-ten tempos, namely, 40, 60, 80, 100, 120, cannot belong to the matrix if they function as threshold tempos.
Includes all the main note groupings – Musical beats for most styles of music can be subdivided into twos, threes, fours, in which the threes and fours can further be subdivided into sixes and eights. Therefore, a practical matrix that applies to common practice music should exclude columns in the matrix that include fives and sevens.
Each row going downwards accelerates at equal proportions – Any system of tempo expressed in a matrix that strives to be “mathematically ideal” should consist of rows that speed up at the same rate as all other rows. This creates a distinct “tempo family” for each row that is equidistant from the slower row above and faster row below.
The only mathematically perfect matrix of numbers in existence – Due to all the unique properties listed above, this is the only “mathematically ideal” or “theoretically ideal” tempo matrix that can possibly be derived.
When I had developed this tempo matrix in the spring of 1992, I began applying the tempos to a wide selection of styles and composers. I discovered that the “ideal” tempo for any piece of music must be found somewhere in this matrix, which makes process of elimination the best way to locate such a tempo. For example, the only possible theoretically correct tempos for a work like Prelude in C Major are the tempos in the “Fours” column, which can be narrowed down to perhaps three or four possibilities: 54, 63, 72, 84. Trying out these four tempos indicates that 63 or 72 are probably the best choices. For many years I was convinced that the best tempo for this prelude was 72 bpm, but now several years later, I have gravitated towards 63 bpm, which, as we will see throughout this study, was one of Bach’s most common tempos. (See Example 1.)
Example 1. Prelude in C Major (BWV 846)
On the other hand, the ideal tempo for any piece with triplets or three notes per beat is found in the “Threes” column. For example, the ideal tempo for Invention 10 in G Major, in 9/8, can also be narrowed down to perhaps three or four possibilities: 64, 72, 84, 96. Considering that this invention has extended trills in both hands (at different times), the ideal speed should be that which allows the proper playing of two trill notes for every eighth note. How fast one can play trills may vary greatly from pianist to pianist, but a “safe” tempo that seems fast enough and allows for the playing of the trills by the average pianist is either 72 or 84 bpm. Thus, one of these tempos is most likely the one Bach intended. Currently, I believe it is 72 bpm:
Example 2. Invention 10 (BWV 781)
Most likely as an aftermath of my semester of being preoccupied with tempo, during that summer – specifically on July 2, 1992 – a powerful force hit me. I could barely sleep for two days due to an epiphany that hit me like a bolt of lightning. I will remember this epiphany forever. My wife thought I was crazy, not sleeping for two days and rattling on and on about “tempo” which made no sense to her being that she was not a musician or a scholar. I told some of my closest graduate student friends that I had figured out Bach’s tempo system and they also all thought I was crazy. I admit at the time I may have been a bit overzealous about my new theory, which explains why my theory was often misinterpreted to imply that there are “correct” tempos and “incorrect” tempos. After all, musicians hate being told how fast or slow to perform Bach’s music, since this takes away from their personal freedom. Just to set this straight 30 years after the fact, my intention at the time was not to dictate to performers how fast or slow Bach’s music should be played, but rather, to use logic and analysis to recreate Bach’s standard operating procedures or modus operandi. I am willing to bet that even if Bach specified his intended tempos precisely in beats per minute with authentic, handwritten instructions most performers would disregard these instructions and simply play the tempos they feel like playing. And this is certainly permissible, especially considering the conclusion reached in the Preface that there is no such thing as a “correct” tempo.
On July 2, 1992, I was certain that some unknown “spirit” revealed to me the system of tempo Bach used in his works, which had to do with tempo (in beats per minute), the lengths of movements (numbers of measures), and resulting proportionally related durations (in minutes and seconds). I stopped by Professor Clark’s office the next time he was available and described to him the epiphany I experienced. I exclaimed to him that I had figured out Bach’s system of tempo and implied that this was the “discovery of the century.” I felt like Moses after talking to God through the burning bush. I also showed Professor Clark my newly written “A Unified Theory of Tempo Relativity” which made me feel like the “Einstein of Musical Tempo.” I could tell Professor Clark was skeptical about my theories and claims, yet at the same time, he was fascinated with my theories, and he eventually became my biggest supporter and advocate. I kept in contact with Professor Clark for several years after my graduation, after his retirement, and even after I had moved out-of-state until he unfortunately died of cancer in 2003. This study is dedicated to my late, great mentor.
Back to my epiphany. Immediately after the interactions I had with the “spirit” I immediately went to the university music library and checked out about a dozen cantata scores. I also owned editions to most of Bach’s keyboard works. That weekend I embarked on serious analysis of many keyboard and vocal works. My “analysis” consisted of counting all the numbers of measures in Bach’s works (at least at this point, a few preludes and fugues and cantatas) and recognizing “red flags” or patterns regarding Bach’s choice of number of measures. I did this because I was informed by this spirit that Bach used tempo and number of measures to achieve certain desired proportional relationships. For example, a cantata aria at its natural tempo may last two times longer than the following chorus at its natural tempo, showing a 2:1 duration ratio, or perhaps a chorus-aria-chorus sequence may have 2:3:2 duration ratio.
I began applying the tempos in my “mathematically correct tempo matrix” to many of Bach’s preludes and fugues from The Well-Tempered Clavier as well as tracking the number of measures for each movement. I did the same with the Inventions & Sinfonias. I also investigated many cantata movements with my library scores, mainly arias and choruses. With virtually every prelude and fugue, most of the inventions and sinfonias, and every cantata I made an intriguing and jaw dropping discovery. I discovered that Bach primarily sought the duration ratios 1:1, 1:2 (or 2:1), 2:3 (or 3:2) in his music assuming the “natural” tempo for the movement, known in Bach’s time as tempo giusto. (This term will be discussed further in a later chapter.)
A 1:1 duration ration occurs when two movements have equal durations, or at least, equal durations within a small margin of error of less than about 5%. A 1:2 or 2:1 duration ratio occurs when one movement lasts two times longer than the other, while a 2:3 or 3:2 duration ration occurs when one movement lasts one-half longer than the other. Analysis shows that these were the only three duration ratios Bach sought in his music. For some reason, 3:4 or 4:3 ratios can never be found in Bach’s works at least when using the present analytical system and the tempi applied within. Also, 1:3 (or 3:1) and 1:4 (or 4:1) duration ratios seem to occur infrequently if at all. As to why Bach would do such a thing as plan his movements to have proportionally related durations, this will be answered in later chapters.
At this time, in 1992 throughout 1994, I was working as an accompanist for ballet classes at the School of Dance at the University of Kansas. I also began a musical collaboration with a professor of dance and dance history who specialized in Renaissance and Baroque dance, Professor Joan Stone. I reasoned that if my Bach tempo theory had any validity, the tempos must at least be tested out on and approved by an experienced Baroque dancer and historian. To my amazement, virtually all my dance tempos worked perfectly for dancing according to Professor Stone. That is, I had no knowledge of how to dance the dances, yet I had good knowledge of which dances were slower or faster and how all the dance tempos relate hierarchically. For example, I have no idea how to dance a courante but when I put my metronome on 54 bpm and played my “courante tempo” Professor Stone exclaimed this was the first time she ever heard a pianist play a correct courante tempo. I was ecstatic! My tempo theory had been tested, validated, and confirmed.
Discovery of a New Bach Motif
A fascinating and important byproduct of my intense study of Bach’s works was my discovery of Bach’s use of the “SDG” musical motif, which is arguably the most ubiquitous motif in Bach’s music. It is well known among scholars that at the end of his church compositions Bach always wrote the letters “S. D. G.”, which means “Glory to God Alone.” In German, E-flat is called “Es” which is also the spelling for the letter “S”. Hence, combining the pitches E-flat – D – G shows that this musical motif consists of a descending half step followed by an ascending perfect fourth, which may be presented in three permutations: 1) inversion 2) retrograde 3) retrograde inversion. (See Diagram 1.)
Diagram 1. The SDG Motif and its Permutations
Not all combinations of musical letters translate into effective musical motifs; however, the SDG motif is one of the few exceptions. It is one of the most diverse and effective motifs in music due to its unique contour and intervallic make-up and has been used by composers for time immemorial. Bach apparently was aware of the unique properties of the SDG motif as well as its theological significance, since he composed an entire fugue that uses two permutations of the motif in the opening six notes of the subject, Contrapunctus 10 from The Art of Fugue. Out of all the countless books and articles analyzing and discussing The Art of Fugue, still to the date of this writing (2022), there is no mention of the SDG motif in Contrapunctus 10 or anywhere else. It seems to have eluded all scholars until my discovery of it in 1992. (See Diagram 2.)
Diagram 2. The SDG Motif in Contrapunctus 10 (bars 1-5)
Another effective musical motif Bach used that did not elude scholars is the “BACH” motif, which has been known about since 1750. In German, B-flat is called “B” while “B natural” is called “H,” which contributes to a four-note motif consisting of a descending half step, ascending minor third, and descending half step. The most famous occurrence of this motif is in the final subject of the incomplete Contrapunctus 14 where Bach states it in its original non-transposed and non-permutated form. Like the SDG motif, the BACH motif can also be presented in three permutations. (See Diagrams 3 and 4.)
Diagram 3. The BACH Motif and its Permutations
Diagram 4. Contrapunctus 14 (bars 193-204)
It is well known among scholars that Bach also used the BACH motif in the second and third subjects of Contrapuncti 8 and 11, both triple fugues. However, what apparently no scholar knew until my discovery in 1992 was that Bach accompanied his personal signature in these two monumental triple fugues with various permutations of the SDG motif. Each of the three subjects in Contrapuncti 8 and 11 contain SDG or BACH so that when the subjects appear together these two motifs occur simultaneously. This was Bach exclaiming symbolically “I am Bach and I believe in the Glory of God Alone.” (See Diagram 5.)
Diagram 5. Contrapunctus 8, first subject (bars 1-4)
Diagram 6. Contrapunctus 8, first and second subjects (bars 39-46)
Diagram 7. Contrapunctus 11, second and third subjects (bars 89-92)
After discovering Bach’s use of the SDG motif and its combination with the BACH motif throughout The Art of Fugue, I made the startling discovery that the SDG motif was perhaps the most prevalent and ubiquitous musical motif in all of Bach’s music. Any time Bach uses three consecutive pitches that consist of the intervals of a half step followed by a perfect fourth as shown in Diagram ?, this symbolizes SDG:
INSERT EXAMPLES OF SDG PERMUTATIONS IN MELODIES
Around 1995 I wrote a long and detailed paper titled “Discovery of a New Bach Motif” that showed with dozens of musical examples that Bach combined the previously undiscovered SDG motif with the well-known BACH motif. I showed all the significant examples in The Art of Fugue as well as the occurrence of SDG (and its permutations) in Bach’s fugue subjects and melodies. I submitted the paper to be considered for publication in the prestigious Riemenschneider BACH Journal and it was rejected. Apparently, according to one of the reviewers, this had “all been done before” and my article was “lacking anything original.” I also had the same paper continually rejected from its presentation at national theory and musicology conferences. So here I was, an unknown and unpublished struggling graduate student who made one of the significant musicological discoveries of the century – Bach’s use of the SDG motif – yet I was basically blacklisted and ignored by the academic community if I ever attempted to have it published. I realized at this time the ugliness of professional jealousy. I reasoned that if Bach’s very easy to recognize and clear-as-day use of the SDG motif had been undiscovered for all these years by even the most prestigious and published of scholars and reviewers, then literally anything was possible. And that literal thing is the determination and explanation of Bach’s “secret tempo code.”
Post-Epiphany
Shortly after my epiphany on July 2, 1992, I acquired a microfiche machine and purchased a complete collection of Bach’s works for microfiche. I then spent the next 10 years counting measures, applying tempi, calculating durations, and compiling tempo and duration charts (or spreadsheets) for Bach’s complete works. Then, around in the early 2000s, life and the real world got in the way, and I put all my research and data aside, where it lay dormant for about 20 years in a large folder called “Bach Tempo.” In 2022 I decided to resurrect my theory and re-evaluate the conclusions made 20-30 years previously.
Back in the 1990s some educated colleagues and skeptics predicted that I most likely would relegate my theory all to coincidence and abolish it after re-evaluating it in a few years after taking a rest. This, however, has never happened. I admit that many of my assigned tempos are different now than 20-30 years ago, as I have gravitated towards slightly slower tempos for many movements. But this is not a problem and does not contradict my theory in any way since it simply reflects more current research.
The present theory of tempo in Bach’s music is not necessarily contingent on the individual tempo (i.e., metronomic speed) for any given movement, but rather, the crux of the theory lies in the objective analysis of the numbers of measures in each movement and the relationships between tempos and their resulting in specific duration ratios when these tempos are applied. And finally, the tempo matrix assumed for this study is cast into stone, and has been for thirty years, as it contains the only theoretically or mathematically correct tempos in existence. (Please refer to “A Unified Theory of Tempo Relativity.”)
I concluded in 1992 that if Bach planned duration ratios by applying certain tempos with specific numbers of measures, then he must have used a proportionally related system of tempo like the tempo matrix shown here. It is necessary that the performer or listener makes tempo decisions with an open mind and open ears and comes to his/her own conclusions rather than relying on preconceived biases. Ideally, one should approach a Bach work as if it had been newly discovered yesterday and never recorded yet by any artist. To erase biases and form a more objective mind, it is necessary to take a short excursion into music history, which is the topic of the next chapter.
