Bach lived in the Enlightenment Era, also known as the “Age of Reason,” in which the main esthetics of reason, order, and balance reflected the pure ideals of the ancient Greeks. Imagine living in the Enlightenment and your good friend asked you to specify your preferred tempo in beats per minute for your favorite composition. Since metronomes had not been invented yet, you used a clock or watch to help you determine this. So, you sang, conducted, or played the composition counting the number of beats or musical pulses that occur in a minute’s time. What are the chances of you telling your friend that your preferred tempo is, say, 65.7 or 82.3 beats per minute? Or perhaps 74.1 or 95.6 beats per minute? The chances of choosing fractional values like these are virtually zero, because human beings by their very nature (regardless of the time period) are attracted to whole numbers or integers rather than fractions and decimals. Let us now make an initial assumption or “if-then” hypothesis based on this logic:

IF Bach had a special system of tempos for his music, THEN it can be assumed that this system consisted exclusively of whole numbers or integers.

It now becomes desirable to devise such a system. (Please understand that at the moment this is merely an assumption and I am not claiming these to be Bach’s tempos.) In order to include all possibilities of integer tempos, it is necessary to set up a tempo table or matrix that consists of rows and columns in which each row represents related tempos and each column represents all possible metrical groupings, and, which consists exclusively of integers.

More specifically, each row consists of tempos unified by a fastest moving note value, what I call the “fastest common denominator” or FCD, and each column groups this FCD into all its possible combinations as either duplets (for example, a group of two eighth notes), triplets (for example, eighth-note triplets), quadruplets (for example, a group of four sixteenth notes), or sextuplets (for example, two groups of triplet sixteenths). Quintuplets are excluded because they do not occur in most music. An “octuplet” column is not included because these values are simply the “quadruplet” values halved. In essence, the tempos in each row can be understood as belonging to a distinct tempo “class” or “family” unified by a “fastest common denominator,” or FCD, that is grouped into all possible combinations (except by fives).

Devising such a complete and comprehensive system of tempo that consists exclusively of integers and includes a wide range of tempo levels or classes (i.e., rows) initially seems like a problem for someone with a Ph.D. in mathematics using a super-computer. Surprisingly, this is not the case at all. In fact, devising such a system or matrix of tempos is easy to do and requires nothing more than setting up a grid and applying elementary arithmetic. What may be even more surprising is that there exist not several possibilities for such a system, but only one. Setting up a grid and applying some easy division will eventually lead to nothing other than this special and unique matrix:

Mathematically ideal tempo matrix (FCD = “fastest common denominator”)

FCD	Duplet	Triplet	Quadruplet	Sextuplet
144	72	48	36	24
168	84	56	42	28
192	96	64	48	32
216	108	72	54	36
252	126	84	63	42
288	144	96	72	48
336	168	112	84	56
384	192	128	96	64
432	216	144	108	72

Adding some more rows between some of the present rows may initially seem possible, but can be excluded for one or more reasons. For example, although the FCD of 240 produces exclusively integer tempos—duplet of 120, triplet of 80, quadruplet of 60, and sextuplet of 40—this tempo class can be excluded, since its speed is not different enough from the two rows on either side, FCD 216 and 252. Similarly, although the FCD of 324 does produce exclusively integer durations—duplet of 162, triplet of 108, quadruplet of 81, and sextuplet of 54—this tempo class can also be excluded, since its speed is not different enough from the next fastest class, FCD 336. It is significant that in this matrix each row is faster than the next at a consistent percentage (see below), in which case adding the FCD 240 or 324 classes would obstruct this natural order. And finally, although the FCD of 320 produces a quadruplet tempo that is an integer, 80, this tempo class can automatically be eliminated, since it results in fractional values for the triplet and sextuplet tempos, 106.66 and 53.33.

This matrix, which could be continued in either direction by the addition of more rows that continue the logical sequence, possesses very unique and special qualities. First, it is comprised exclusively of integers. Second, it includes all possible tempo gradations from very slow to very fast. Third, the difference between these tempo gradations represented in each row remains consistent at virtually the same ratios. That is, the ratio of each FCD value to the next faster one remains constant at virtually the same value ranging from .857 to .888 (144/168 = .857, 168/192 = .875, 192/216 = .888, etc.). This third quality is very important because it shows that each tempo class (i.e., row) is slow or fast enough to distinguish itself from the classes on either side. In sum, this tempo matrix is ideal and complete within itself. It is perfect, in that no tempos can be added and no tempos can be taken away.

This matrix also possesses the unique quality of existing as a complete system of tempo in its own right and is not necessarily specific to Bach's music. In fact, I had developed this matrix and discovered its unique qualities even before I had conceived the present theory of tempo and duration in Bach’s music. After I had discovered this tempo matrix and all its unique qualities, I began to apply these tempos to a variety of styles of music by a variety of composers (even to pop and rock music) and found that the ideal tempo for any work exists somewhere in this matrix. This happens to be the case, since this is quite simply the best and most mathematically ideal matrix of tempos that can possibly be derived, which is not composer and style specific but universal in its application. (For more on the topic of ideal or perfect tempos, please refer to my study, “Determining Ideal Tempos: A Unified Theory of Tempo Relativity.”)

It is for this reason that the tempos I assign to Bach’s works throughout this web site, and throughout my entire corpus of analyses, should not be considered “my tempos” at all because I have simply discovered what has always existed. Therefore, I consider the tempos I use to be something closer to “God’s tempos.” To use an analogy, the tempos in this system are just as much “my” tempos as a² + b² = c² is Pythagorus’ “own” theorem. Pythagorus merely “discovered” this theorem to be true as applied to the area of triangles, a law of geometry that was always in existence even before his discovery. Similarly, I have merely “discovered” the unique and special qualities of this tempo matrix, the tempos and their ratios having always been in existence even before this discovery. (Perhaps someone before me devised this matrix and discovered its unique qualities, of whom I am unaware. I would not be surprised if this were the case, since deriving this tempo matrix and realizing its unique qualities is nowhere near as specialized and complex as, say, Einstein's theory of relativity.)

The best or most appropriate tempo for any one work can usually be found by process of elimination. For example, trying out all the possible tempos in the quadruplet column for a piece in 4/4 with sixteenth-note motion, say, the C-major Prelude from book 1 of The Well-Tempered Clavier, will most likely lead to the tempo of quarter = 72. This tempo will have been arrived at through a process of elimination, in that the tempo of quarter = 84 seems a little too fast while the tempo of quarter = 63 seems a little too slow.

I find something inevitably “right” or “perfect” about the tempo of quarter = 72 for this prelude. But even if this is just my own personal opinion and you disagree, then you will undoubtedly be able to find your own preferred tempo, say, quarter = 63, or whatever you have determined using your own judgment based on process of elimination. Please understand that I am not trying to dictate to performers what tempos they should choose. Rather, the whole thesis of this essay is that the present matrix represents the most complete and mathematically ideal system of tempos that can possibly be derived, which is neither composer nor style specific.

This matrix consists of all the tempos one could possibly want for any music, since it encompasses all possible gradations in speed. This can be seen by assigning generic descriptions of tempos to each tempo family (i.e., each row). One could debate on which descriptions go with which classes, but nevertheless, one would still have to put descriptions somewhere. But whatever the case may be, this matrix encompasses all gradations of tempos from very slow to very fast, making it the most complete and theoretically ideal gamut of integer tempos that can possibly be derived.

Mathematically ideal tempo matrix with generic descriptive speeds

FCD	Duplet	Triplet	Quadruplet	Sextuplet
144	72	48	36 (very slow)	24
168	84	56	42 (slower)	28
192	96	64	48 (slower than 48)	32
216	108	72	54 (moderately slow)	36
252	126	84	63 (moderate)	42
288	144	96	72 (moderately fast)	48
336	168	112	84 (fast)	56
384	192	128	96 (faster than 84)	64
432	216	144	108 (very fast)	72

Let us now translate these generic tempo descriptions to the usual Italian terminology, which are all terms that Bach used frequently in his music (except for “Moderato,” which Bach only used once). The tempo word hierarchy here is based on process of elimination and common sense and should not be taken literally. For example, Adagio and Largo are both “slow” tempos that I assign in this order from slow to fast, which does not necessarily mean that Adagios are always slower than Largos. To determine this, a variety of factors need to be considered, which include such defining characteristics as meter and prevailing note values. Similarly, Allegro and Vivace are often used interchangeably, and although I find Vivaces usually to be a little faster than Allegros in Bach’s music, the hierarchy here does not imply that this is the case all the time. Please understand that these are just words and should not be taken so literally; they are merely used here to provide descriptive titles to the general hierarchy of the possible tempo classes or families.

Mathematically ideal tempo matrix with Italian descriptions

FCD	Duplet	Triplet	Quadruplet	Sextuplet
144	72	48	36 (molto Adagio)	24
168	84	56	42 (Adagio)	28
192	96	64	48 (Adagio, Largo)	32
216	108	72	54 (Largo, Andante)	36
252	126	84	63 (Andante)	42
288	144	96	72 (Moderato)	48
336	168	112	84 (Allegro)	56
384	192	128	96 (Allegro, Vivace)	64
432	216	144	108 (Presto)	72

Let us now return to our original inquiry, although modified a little. Suppose you were the most methodical and systematic of all composers in the history of music and you wished to devise a mathematically ideal system of tempos. You would undoubtedly arrive at the matrix derived here, since there are no other possibilities—Pythagorus or Plato could have asked for nothing better.

Please understand that at the moment I am not claiming this matrix to represent Bach’s tempos or even presupposing that Bach would have reduced his tempos to a special kind of system. Rather, I am just using simple logic that can be expressed in a simple hypothesis using “if-then” reasoning, which is the initial hypothesis posed at the beginning of this essay:

IF Bach had a special system of tempos for his music, THEN it can be assumed that this system consisted exclusively of whole numbers or integers.

This initial hypothesis can now be made even more specific:

IF Bach had a special system of tempos for his music, THEN this system is most likely the present tempo matrix.

We now really have no other choice but to begin applying these tempos to Bach’s music. The results are bound to be intriguing and maybe even a little shocking.

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