A Unified Theory of Tempo Relativity

(Updated: November 2020) This article was originally written in 1992 by Cory Hall while a graduate student at the University of Kansas where he was pursuing a D.M.A. in piano performance and M.M. in historical musicology. At the time, he was on a quest to determine perfect tempi, and in addition, as part of a larger project was in the midst of analysing the dimensional and temporal characteristics Bach’s complete works. “A Unified Theory of Tempo Relativity” was first published in 2008 on the newly established BachScholar® website. When the BachScholar® website changed its format and purpose in 2012, the article was deleted. In 2022, Dr. Hall re-published the article on the current website. “A Unified Theory of Tempo Relativity” proposes that “mathematically perfect” or “theoretically ideal” tempi exist, they are all proportionally related, and they can be expressed in a matrix with rows and columns.

The Subdivision-Consolidation Threshold

The eminent eighteenth-century theorist and Bach disciple, Johann Philip Kirnberger (1721-83), aptly explains how humans perceive tempo and metrical hierarchies:

If one hears a succession of equal pulses that are repeated at the same time interval, as in the first example, experience teaches us that we immediately divide them metrically in our minds by arranging them in groups containing an equal number of pulses; and we do this in such a way that we put an accent on the first pulse of each group or imagine hearing it stronger than the others. This division can occur in three ways. That is, we divide the pulses into groups of two, three, or four. We do not arrive at any other division in a natural way. No one can repeat groups of five and even less of seven equal pulses in succession without wearisome strain. It can be done more easily with six, especially when the pulses go rather quickly; however, one will notice that groups of six or more pulses are not easily comprehended without thinking of a subdivision, in which case they once again resemble the above-mentioned groups of two, three, and four. (Italics added.) (See Endnote 1.)

As Kirnberger explains, and, as simple logic tells us, metrical and temporal perception result from a hierarchy of two phenomena––(1) subdivision of one slower pulse into more than one faster pulses, and; (2) consolidation of two or more faster pulses into one slower pulse.

For example, beginning with a moderate pulse, say, 72 beats per minute, and continually decelerating, a speed “x” will inevitably be reached that can no longer be perceived as one pulse, but rather will be perceived as consisting of two or three subdivisions of the very slow pulse of “x”. In the case of a duple meter like 2/4 such a tempo “x” applies to the half note, meaning that upon subdivision quarter notes would have the tempo of “2x”. In the case of a triple meter like 3/4 such a tempo “x” applies to the dotted half note, meaning that upon subdivision the quarter notes would have the tempo “3x”.

Conversely, beginning with a moderate pulse (72 beats per minute) and continually accelerating, a speed “y” will inevitably be reached that can no longer be perceived as one pulse, but rather will be perceived as a consolidation of two or three very fast pulses. In the case of a duple meter like 2/4 such a tempo “y” applies to the quarter note, meaning that upon consolidation the half note would have the tempo of “y/2”. In the case of a triple meter like 3/4 such a tempo “y” also applies to the quarter note, meaning that upon consolidation the dotted quarter note would have the tempo “y/3”.

The problem now becomes to determine these hypothetical tempos, which is possible by employing logic and process of elimination, and thus, determining what the “subdivision-consolidation threshold” at all levels may or may not be.

Let us refer to a familiar piece of music in order to make this concept a little less abstract: Bach's C-major Prelude from book 1 of The Well-Tempered Clavier. When played at a moderate tempo, say, q = 72, the quarter note is clearly perceived as the main pulse unit. (See Endnote 2.) But how slow must the quarter note be to cause the eighth note rather than the quarter note to be perceived as the main pulse unit: q = 40? q = 35? q = 30? Similarly, how slow must the quarter note be to cause the sixteenth note rather than the eighth note to be perceived as the main pulse unit: q = 20? q = 15? q = 10? Conversely, how fast must the quarter note be to cause the half note rather than the quarter note to be perceived as the main pulse unit: q = 80? q = 85? q = 100? Similarly, how fast must the quarter note be to cause the whole note rather than the half note to be perceived as the main pulse unit: q = 160? q = 185? q = 200?

Let us assume the hypothetical speed “x” beats per minute for the initial subdivision-consolidation threshold, which represents the border or threshold that demarcates the quarter-note and eighth-note hierarchies. Thus, when the Prelude is played slightly slower than q = x, like q = x - 1, then the eighth note becomes the main pulse unit: e = 2(x - 1). Similarly, when played slower than q = x/2, like q = (x - 1)/2, then the sixteenth note becomes the main pulse unit: s = 2(x - 1). Conversely, when played slightly faster than q = 2x, like q = 2(x + 1), then the half note becomes the main pulse unit: h = x + 1. Similarly, when played faster than q = 4x, like q = 4(x + 1), then the whole note becomes the main pulse unit: w = x + 1.

Now let us assume the Prelude were played at q = x, precisely. In this case, which note value would be perceived as the main pulse unit? The eighth note or the quarter note? Logically, it can be concluded that neither value would be perceived as the main pulse unit, but the pulse unit in this case must be ambiguous — that is, q = x would be neither quite fast enough for quarter-note pulse perception nor quite slow enough for eighth-note pulse perception.

Since an “ideal” tempo, by definition, should be a speed at which temporal and metrical ambiguities are avoided, and a speed that conveys at the very least a main pulse unit — in the case of this Prelude, the quarter note — then the tempo of the Prelude should be faster than but not equal to “x” beats per minute: q > x.

Even a tempo slower than q = x (q < x) would be better than q = x precisely, since this would at least avoid ambiguity by causing the eighth note to be perceived as the main pulse unit. Similarly, even a tempo slower than q = x/2 (q < x/2) would be better than q = x/2 precisely, since this would at least avoid ambiguity by causing the sixteenth note to be perceived as the main pulse unit. Conversely, a tempo faster than q = 2x (q > 2x) would be better than q = 2x precisely, since this would at least avoid ambiguity by causing the half note to be perceived as the main pulse unit.

Therefore, it follows logically that in order for the sixteenth note to be the main pulse unit the tempo must be x/8 < q < x/4, in order for the eighth note to be the main pulse unit the tempo must be x/4 < q < x/2, in order for the quarter note to be the main pulse unit the tempo must be x < q < 2x, in order for the half note to be the main pulse unit the tempo must be 2x < q < 4x, and in order for the whole note to be the main pulse unit the tempo must be 4x < q < 8x.

Since these hierarchies can, theoretically, continue infinitely to both slow and fast extremes, then the ideal tempo of this Prelude or any other work consisting of binary subdivisions of the metrical unit shall be slower or faster but not equal to the tempo “ax/b” where “a” and “b” represent 1, 2, 4, 8, 16 . . . and “x” is constant. In other words, although we have not yet determined a tempo for the C-major Prelude, we at least know at this point that an ideal tempo cannot be q = x, x/2, x/4, x/8 . . . etc. nor can it be q = 2x, 4x, 8x, 16x . . .etc.

The tempo exclusions for the C-major Prelude are limited not only to multiples and factors of two as shown above (x/8, x/4, x/2, 2x, 4x, 8x, 16x, etc.). Ultimately, since “x” is the subdivision-consolidation threshold, then “x” must also be excluded as an ideal tempo for all possible combinations of the fastest common denominator, the sixteenth note. This results in a deductive reasoning set in which the “since” half expresses “x” for each possible grouping of the sixteenth note and the “then” half expresses the equivalent tempo in terms of the quarter note – that is, as a fraction in terms of “x”. Logically, since “x” (in the “since” half) must be excluded, then its quarter-note equivalent (in the “then” half) must also be excluded. In sum, the infinite succession of quarter-note tempos belong to the algebraic sequence “ax/4” where “a” is any integer, “x” is constant, and the denominator, 4, is the number of notes per beat of equal value. (See Deductive Reasoning Set 1.)

The tempo exclusions are limited not only to fractions with denominator 4. More specifically, the existence of 32nd-note subdivisions excludes tempos belonging to the fraction “ax/8” while the existence of 64th-note subdivisions excludes tempos belonging to the fraction “ax/16”. Such subdivisions can, theoretically, multiply infinitely, leading to the rule that the ideal tempo of any work whose beat consists of 2, 4, 8, 16 . . . etc. subdivisions cannot be “ax/2b” where “a” represents any integer and “b” represents any integer divisible by two. In other words, not only must the tempos q = x and q = ax/4 be excluded, but also q = 17x/16, 18x/16 (9x/8), 19x/16, 20x/16 (5x/4) 21x/16, 22x/16 (11x/8) . . . etc. to the fast side of “x”, q = 15x/16, 14x/16 (7x/8), 13x/16, 12x/16 (3x/4), 11x/16, 10x/16 (5x/8) . . . etc. to the slow side of “x”, as well as tempos belonging to the sequences ax/32, ax/64, ax/128 . . . etc.

The algebraic sequence in Deductive Reasoning Set 1 applies to all simple meters with binary divisions of the pulse—works in 2/4, 4/4, 3/4, 2/2, 4/2—the only difference being that the note values are doubled when in alla breve. In the case of ternary divisions of the pulse—works in 3/8, 6/8, 9/8, 12/8, 6/16, 9/16, 12/16, 24/16—the algebraic sequence of tempo exclusions do not have a base-two but a base-three number as a denominator. For example, if “x” must be excluded as an ideal tempo for each possible grouping of the eighth note in a work in 6/8 with prevailing eighth-note motion, then all dotted-quarter-note tempos belonging to the sequence “ax/3” must also be excluded. (See Deductive Reasoning Set 2.)

Likewise, if “x” must be excluded for each possible grouping of the sixteenth note in a piece in 3/8, 6/8, 9/8, or 12/8 with prevailing sixteenth-note motion, then all dotted-quarter-note tempos belonging to the sequence “ax/6” must also be excluded. (See Deductive Reasoning Set 3.)

Similarly, if “x” must be excluded for each possible grouping of the sixteenth note in a piece in 9/16 (or each possible grouping of the eighth note in 9/8), then all dotted-sixteenth-note tempos belonging to the sequence “ax/9” must also be excluded. (See Deductive Reasoning Set 4.)

Since all denominators divisible by 2 and 3 for the fraction “ax/b” must be excluded as ideal tempos, then the only possibilities remaining are denominators divisible by 5 or 7. Thus, we have arrived at the preliminary conclusion that the denominator of the fraction “ax/b” may consist of integers divisible by 5 or 7 only—for example, ax/10, ax/15, ax/20, ax/25 . . . etc. or ax/14, ax/21, ax/28, ax/35 . . . etc. The next step is to narrow down the possibilitiy of 5 or 7 still further until the answer becomes unequivocal.

Temporal Unification of Duple and Triple Meters

Putting the conclusions reached so far temporarily aside, it is now necessary to determine the “base” or “original” duple- and triple-meter tempo from which all other tempos will be derived and to establish the “lowest common temporal denominator” between a duple and triple meter. We can determine the lowest common temporal denominator by setting up 2/4 with two quarter notes and 3/4 with three quarter notes, assigning the lowest common denominator of the number of pulses in each measure as the tempo of each measure — the LCD of 2 and 3 is 6, so each measure gets six beats per measure — which results in two quarter-note tempos: 12 and 18 beats per minute, respectively. In other words, two beats at twelve beats per minute and three beats at eighteen beats per minute can both be reduced to one large beat of six beats per minute.

The two tempos 12 and 18 can be represented on a plane, each with boundaries symbolizing “thresholds” to the slow and fast sides; that is, the thresholds are boundaries to speed categories which may be termed as slow, moderate, fast, moderately-fast, etc. The only way to fixate these tempos directly between the thresholds is to equalize the ratio of the difference between the tempo and its threshold to the value of the threshold itself. Thus, for 12 we assign the left threshold the value 10 and the right threshold 15, resulting in the ratios 2/10 and 3/15 which both equal 1/5. Likewise, the thresholds to 18 become 15 and 22.5, resulting in the ratios 3/15 and 4.5/22.5 which also equal 1/5. Furthermore, the ratio of the difference between each tempo and the thresholds and the ratio of the thresholds themselves are equal at 2/3 (2/3 = 10/15 = 2/3 and 3/4.5 = 15/22.5 = 2/3).

Since the ratio of the difference between the tempos and thresholds to the thresholds themselves is 1/5, then the preliminary conclusion that the denominator of “ax/b” must be 5 or 7 can be finally be narrowed down to one possibility only: 5. This being the case, it can be concluded that the threshold tempo “x” must be an integer divisible by 5 — that is, something like 25, 30, 35, 40, 45, etc. Upon further analysis, the actual value of this threshold will become apparent.

The Slow, Moderate, and Fast Tempo Hierarchies

In order to use the two base tempos 12 and 18 to generate more possible tempos, it is necessary to assume a simple axiom from the outset. Being that the general speed level of tempos can be described in objective terms such as “slow”, “moderately-slow”, “moderate”, etc., the axiom becomes:

“Fast” is precisely two times “slow”, and “moderate” is precisely midway between these two extremes.

The actual tempo assumed is irrelevant to this hierarchy as is also the difference in two listeners' subjective opinions. For example, if one considers the tempo q = x “slow” for any given piece, then “fast” for the same piece to the same person must be q = 2x and “moderate” must be q = 3/2(x). Seen from another perspective, we can draw a parallel in the physics of colors. For example, since pure yellow is created from mixing “x” amount of pure red with “x” amount of pure green, then pure yellow can be considered as being precisely midwaybetween or a precise hybridof red and green:

red + green = yellow

Were the amount of pure red and pure green not mixed in a precise 1:1 ratio, then the result would not be “pure yellow”, but a slightly different hybrid shade of yellow.

Yellow is precisely midway between red and green

This same principle applies to the slow, moderate, and fast tempo levels as well as their hybrids. More specifically, “moderate” is precisely midway between or a precise hybrid of “slow” and “fast”. Likewise, “moderately-slow” is precisely midway between or a precise hybrid of “slow” and “moderate”, while “moderately-fast” is precisely midway between or a precise hybrid of “moderate” and “fast”.

Moderate is precisely midway between slow and fast

Moderately-slow is precisely midway between slow and moderate

Moderately-fast is precisely midway between moderate and fast

It is now possible to arrange the tempos 12 and 18 into a logical hierarchical system. More specifically, if 12 is slow, then 18 is moderate and 24 is fast, whereby the thresholds to these categories become 10–15, 15–22.5, and 20–30, respectively. It is worth noting that each tempo is two-thirds of the way between its two thresholds. The fast threshold to “moderate” and the slow threshold to “fast” are the only thresholds to intersect. Thus, setting up these boundaries as a new set of thresholds, 20–22.5, and assuming a tempo two-thirds of the way between, 21, another tempo level is discovered, “moderately-fast”. The tempo 21 not only fits the requirement of lying two-thirds of the way between its thresholds, but also lies midway between the “moderate” and “fast” tempos—hence, “moderately-fast”. We now have four tempos special to four distinct, non-intersecting threshold limits.

Determining Theoretically Correct Tempi and the Threshold Tempo “x”

In order to determine the comprehensive gamut of theoretically correct tempos, it is necessary to begin with the smallest integer and perhaps the slowest tempo possible: 1. Assuming this tempo to be “slow”, its related tempo categories are most readily determined by multiplying 1 by 3/2 to attain “moderate”, multiplying “moderate” by 7/6 to attain “moderately-fast”, and multiplying “moderately-fast” by 8/7 to attain “fast”. The next fastest tempo, 1.5, shall now be assumed as “slow”, whereby its “moderate”, “moderately-fast”, and “fast” equivalents are calculated using the same ratios. Likewise, the next fastest tempo, 1.75, can be assumed as “slow”, whereby its three related tempos are calculated. In sum, this is an infinite self-generating progression, which is the most complete and efficient way of producing a complete comprehensive gamut of possible theoretically correct tempos. (See Table 1.)

The most noticeable idiosyncracy of the self-generating progression is that certain integers are absent. Most notably, the integer 5 does not appear, making it inevitable that all integers divisible by 5 also do not subsequently appear. Furthermore, prime numbers greater than 7 and all multiples thereof also do not appear — more specifically, 11, 13, 17, 19, 22 (11 x 2), 23, 26 (13 x 2), 29, 31, 33, 34 (17 x 2), 37, 38 (19 x 2), 41, 43, 46 (23 x 2), 47 . . . etc. That 5 and integers divisible by 5 do not appear corroborates the preliminary conclusion that the threshold tempo “x” must be an integer divisible by 5. In other words, since Table 1 lists the entire gamut of possible theoretically correct tempos, then the absence of a tempo (i.e., 5 or an integer divisible by 5) automatically makes such a tempo theoretically incorrect.

The next idiosyncrasy of the self-generating progression in Table 1 is that only certain tempos appear in all four columns (shown in red font). The slowest tempo that appears in all four columns is 5.25, which multiplies (by 4) to become the smallest integer (non-fractional value) to appear in all four columns, 21. This in turn multiplies to 42, which is the smallest integer to appear in all four columns andto have surrounding tempos (one slower tempo and one faster tempo) to also appear in all four columns. Therefore, 42 belongs to a unique temporal field—the range between 41.3437 and 42.20507 — and is the smallest integer (i.e., slowest tempo) to possess this quality. This means that 42 beats per minute is the slowest tempo that is fast enough to function as one complete non-subdivided beat, which makes the threshold tempo “x” 40 beats per minute.

Since the two original binary and ternary tempos 12 and 18 are integers, then all beats with binary and ternary subdivisions must also be integers. Therefore, the final gamut of theoretically correct tempos for beats with binary and ternary subdivisions includes integers that have 2, 3, 7, or 9 as factors, meaning that tempos with factors of 5 or prime numbers greater than 9 must be excluded.

In sum, theoretically correct tempos may be only those integers in the left column of Table 1 (the “slow” column) — that is, only the tempos shown in bold. We can now organize these tempos into a matrix where the first column consists exclusively of the bold-faced integers in Table 1, and each column lists the corresponding tempo of this integer divided by two to nine. In other words, the single note in each row remains at one constant speed. Since 42 beats per minute is the slowest that a pulse can be before being subdivided, then we will begin the matrix with this speed. The matrix can continue infinitely, but for practical purposes ends with perhaps the fastest speed that is humanly possible. (See Table 2.)

Each row of Table 2 can be referred to as a distinctly different “tempo class”, in that each class is defined by one constant fast-note speed. The previous observation that the tempo of a beat with binary and ternary subdivisions must be an integer — that is, a whole number as opposed to a fraction — makes it necessary to exclude several rows from Table 2 . More specifically, a tempo class must be excluded if any tempo in the duplet or triplet column is not an integer, which includes the following classes: 56, 63, 64, 112, 128, 224, 256, 448, 512. We can now derive a final tempo matrix, which excludes theoretically incorrect tempo classes as well as the extremely slow tempo classes. (See Table 3a.)

Table 3b expresses each tempo in Table 3a as a fraction “ax/b” where “a” represents any integer with a factor of 2, 3, 7, or 9, “b” represents integers with a factor of 5, and “x” represents the speed of the subdivision-consolidation threshold, 40 beats per minute. The purpose of expressing each tempo algebraically is so that were the subdivision-consolidation threshold shown to be is a value other than 40, then this new value would simply be assigned to the constant “x”, whereby a likewise proportionally equivalent gamut of theoretically correct tempos could be computed — hence, the title of this study, “tempo relativity.” In other words, although it is possible to assign the subdivision-consolidation threshold a speed other than 40, it is not possible to alter the proportions in Table 3b. Table 3b is “carved into stone,” in that it represents the one and only gamut of theoretically correct tempos and tempo relationships. (See Table 3b.)

Practical Applications of the Theory

The tempo matrix presented as Table 3acan be used as an aid in determining an ideal tempo for virtually any piece of music, provided that one applies careful process of elimination. As a rule, each column in Table 3a consists of the only possible tempos for the corresponding division of the pulse. For example, given a piece in 2/4, 3/4, 4/4, or 2/2 with prevailing quadruplets, the correct tempo must be one of the twelve tempos in column four. The only exception to this rule is in the case of an extremely slow tempo where the main pulse is subdivided, the correct tempo may be one of the duplet tempos in column two. Likewise, possible tempos for pieces with prevailing quintuplets are found in column five, septuplets in column seven, and nonuplets in column nine.

For example, in searching for the ideal tempo for Bach’s C-major Prelude, one would try out several possibilities from the quadruplet column and apply process of elimination. For example, I find q = 63 a little too slow and q = 84 a little too fast, making the ideal tempo q = 72. As much as one may object to this partially subjective process (for example, another performer may find q = 72 less ideal than, say q = 63), under no circumstances may any tempos other than those in the “quadruplet” column for this particular piece be considered. Applying these rules challenges tradition in that popular base-ten tempos like 60, 80, 100, and 120 beats per minute are, in reality, theoretically incorrect and can be disregarded altogether for any piece of music. The reason for this is that these tempos are directly related to the “non-tempo” or threshold tempo of 40. Since a beat should, ideally, be either slower or faster than 40, but not 40 precisely, then it stands to reason that the tempos slightly slower or faster than 60, 80, 100, and 120 are more ideal than 60, 80, 100, or 120 precisely.

Certain tempos in the triplet column do not appear in the duplet or quadruplet columns: 56, 64, 112, 128. This theoretical distinction excludes these tempos for pieces in 2/4, 3/4, 4/4, or 2/2 with prevailing triplets, for the dotted quarter note of pieces in 3/8, 6/8, 9/8, and 12/8, as well as the dotted eighth note of pieces in 3/16, 6/16, 9/16, 12/16, 18/16, and 24/16. Conversely, the tempos in the duplet and quadruplet columns that do not appear in the triplet column — 54, 63, 108, 126 — must be excluded for pieces with prevailing triplets. Thus, a piece in 4/4 with prevailing sixteenth-note motion may have one of the tempos q = 54, 63, 108, 126 but not q = 56, 64, 112, 128, whereas a piece in 6/8 with prevailing eighth- or sixteenth-note motion may have one of the tempos dq = 56, 64, 112, 128 but not dq = 54, 63, 108, 126. Should quadruplets and triplets prevail equally (like in some of Bach’s allemandes), then the faster tempo (triplet) shall overrule the slower tempo (quadruplet). Should quadruplets clearly be dominant and triplets function as occasional embellishments, then the quadruplet tempo shall overrule the triplet tempo; conversely, the opposite also holds true.

 Summary and Conclusion

In order to fully comprehend the implications of this theory of tempo relativity, it is helpful to draw an analogy. Tempo relativity, as exemplified in the fractions of Table 3b, is analogous to the theoretically correct proportional relationships that govern tuning systems. In our well-tempered system, for example, the tuning of each pitch is dependent upon and relative to the tuning of one universal value, A-440. Were one to assume a different value from the outset, say, A-442, then all pitches would sound slightly higher, yet the proportional relationships between the pitches would remain unaltered. Regardless of the type of tuning system one employs, the gamut of pitches will be a direct result of the value of one initial pitch that functions as a value of reference. Such is the case with theoretically correct tempos, in that the tempos must depend upon one initial value, referred to here as the subdivision-consolidation threshold of “x” beats per minute.

Although every tuning system can be codified in a theoretically correct form on paper, it remains a paradox that it is virtually impossible to tune every pitch precisely to its theoretical ideal. For example, as hard as the most competent piano technician tries to tune a piano so that every pitch is perfect, there will, inevitably, be a slight margin of error somewhere. Nevertheless, the theoretically correct prototype of a tuning system, which cannot be precisely duplicated in actual practice, is vital and indispensible in that it functions as a theoretical ideal. Without theoretical ideals, the tuning of our instruments, regardless of tuning system, would be an arbitrary procedure and thus would result in chaos. Similarly, without a theoretically correct system of tempo, the performer who seeks the perfect tempo — like the tuner who strives for the perfect tuning — unfortunately would have no theoretical ideal, thus making the chosen tempos arbitrary.

It becomes desirable then, just as the relationships of the pitches in a tuning system are governed by mathematical ratios, to codify ideal tempos and tempo relationships by means of a mathematically-oriented unified theory. Although one may not be able to play the ideal tempos of 42, 63, or 84 precisely, tempos like these serve the important purpose of functioning as theoretical ideals. To summarize, “perfect” or “ideal” tempos do indeed exist (contrary to what many musicians believe) and are shown in this study to be determinable through recognition of a subdivision-consolidation threshold, referred to here as the tempo “x”.

Endnotes

1 Johann Philipp Kirnberger, Die Kunst des reinen Satzes in der Musik(Berlin, 1774-79), translated by David Beach and Jurgen Thym as The Art of Strict Musical Composition (New Haven: Yale University Press, 1982), 383-4.

2 The following nomenclature is used in this study: w = whole note, dh = dotted half note, h = half note, dq = dotted quarter note, q = quarter note, de = dotted eighth note, e = eighth note, ds = dotted sixteenth note, s = sixteenth note.