The Theory of Harmonic Rhythm

by Stephen Jay

In an effort to understand the hidden design of the physical components of music, I have focused on the basic elements of harmony and rhythm, and their relationship to complex musical events. In doing this I have observed that harmony and rhythm are really the same "thing", happening at two radically different speeds. They are aspects of each other. Harmony can be converted into rhythm and vice versa, and the special features that make each of them work as music, are translated analogously between their respective domains. The two seemingly diverse elements are really an occurrence of the same physical phenomenon, follow identical mathematical rules of consonance, and coincide in the effect of specific characteristics, across the range of their musical activity.

By looking at harmony "under a microscope", that is by slowing it down to the point where pitches become pulses, I have observed that only the more consonant harmonic intervals become regularly repeating rhythms. And the more consonant the interval, the more simple its rhythm. Looking at rhythm in the opposite way, by speeding it up, reveals identical physical processes involved in the creation of both. Harmony is very fast rhythm.

Any harmonic interval, or group of intervals such as a triad, lowered five to ten octaves, becomes a rhythmic pattern, or group of patterns in a sequence. The more "consonant" the interval or chord is, the more regular and mathematically simple its rhythmic pattern. The opposite is also true. Rhythm is converted to harmony by raising it five to ten octaves, and again its inherent relative consonance translates perfectly between the two scales. Adding notes to a harmony increases its degree of relative dissonance by raising the level of complexity in the rhythmic pattern producing it. The actual number of octaves that a harmony must be lowered in order to hear its rhythm, or vice versa, depends on its specific frequency range or tempo.

This presents a new way of connecting harmony and rhythm, listening to music can be compared to seeing normally and through a microscope at the same time, or a fabric with a greatly enlarged image of its own microscopic texture, printed upon it. The basic characteristics of harmony and rhythm appreciated as positive aspects, translate into each other perfectly, through the shared structure of their physical generation. By changing "beats per minute" to "cycles per second", pulse is converted to pitch. Inversely, interpreting "cycles per second" as "beats per minute" converts frequency to tempo. The ratio of difference between harmony and rhythm is roughly equal to the difference between minutes and seconds. What happens in one second at the harmonic level contains a full minute of specific rhythmic events. The exact nature of those rhythmic events, and all of the details of their geometric symmetry, comprise the physical information which we interpret as specific harmonic intervals and chords. On a theoretical musical instrument with a range that extended a few octaves below audible pitch, which sounded the rhythmic patterns of the wave peaks and beat frequencies in that range, the interval of a perfect fifth would be heard as illustrated in Example 1-1.

Example 1-1

If this pattern were played as a rhythm, but about 60 times faster, the interval of a perfect 5th would be heard. Keeping the same tempo, and changing the rhythmic pattern as found in Example 1-2, would change the 5th to a minor 3rd.

Example 1-2

The basic mathematical characteristics of the harmonic rhythms are analogously linked to the musical qualities evoked by each interval. The dynamic characteristics of what can be created with each of the two basic elements of music, harmony and rhythm, are unified by a common mathematical framework, based on the natural harmonic series.

Diatonic scales produce sonic events with a high percentage of mathematical "sympathy", or "consonance". The range of rhythmic styles contained within the harmonic intervals of the diatonic scale is enormous. Certain intervals and chords when converted to rhythm produce very stylized patterns which are associated with specific cultures. Major, minor, diminished, augmented, and perfect intervals all have characteristic harmo-rhythmic stylistic traits. The interval of a perfect fifth, C and G, contains a rhythmic pattern predominant in West African folk music. Dotted, triplet, polyrhythm, swing, African, and Latin rhythmic elements are among those found in harmo-rhythmic patterns. Individual rhythmic styles and patterns which have been associated with specific cultural and geographic areas around the world, have always been contained inside the harmonic intervals of the natural harmonic series, and the diatonic scale.

What society has regarded as being musically "consonant", has been steadily increasing in complexity throughout history. In Western music the acceptance of smaller harmonic intervals has been accompanied by the allowance of shorter and more mathematically irregular subdivisions of time in rhythm, from Gregorian Chants to Karlheinz Stockhausen. This evolution has proceeded sequentially up the natural harmonic series, with a consensus of opinion accepting the higher and higher intervals which occur naturally between the partials as being relatively "consonant", or more musically "useful" than previously thought. The same applies to all of the overtones of a given timbre and all of the complex rhythms produced by their "harmo-rhythmic" alliance. The collective ability to perceive increasingly complex harmonic rhythm patterns in instrumental and vocal timbre appears to be on a similar evolutionary path.

The consonance of a harmony or a rhythmic pattern is also related to the duration of what is perceived by the listener to be one complete cycle. Interestingly, the natural complications which result from adding notes to a chord multiply harmo-rhythmically in a dynamic range similar to the effect of harmony on the listeners emotions. This also occurs in increasing complexity as you move up the natural harmonic series by intervals. Just as certain rhythmic patterns appear to have natural downbeats, harmonic intervals and chord combinations have root, or tonic notes. Harmony and rhythm sympathize, sustain and mutually vitalize each other.

Examining the unified field between pitch, harmony, and rhythm, leads to some interesting new theoretical connections. Since melody is also horizontal harmony, timbre can be seen as an equivalent of vertical rhythm, with the same mathematical rules of consonance and dissonance in effect. Synchronization in the physical relationships between the two basic elements, harmony and rhythm, and the way in which they mesh, creates a "global" system of sympathetic harmonic dynamics. Harmo-rhythmic relationships exist horizontally in time as linear melodies, vertically in harmonic intervals, internally between the overtones of each individual timbre, and in the combination of all three. This myriad of mathematical and sonic coincidence is further expanded by the harmo-rhythmic relationships of all of the "resulting tones" (sum and difference tones).

All of this comes together in creating music. Harmo-rhythmic sympathy increases its internal symmetry. When we listen to music, and it sounds good to us, there is something special happening in the relationships between harmony and rhythm, a symmetry to which we are emotionally sympathetic. That "something special" has the same character in both harmony and rhythm. The ear-catching "golden" results of harmonic mathematics in synchronized motion during the simultaneous occurrence of two or more different frequencies of oscillation.

The harmonic rhythms of chords are so fast and complex that their total conscious perception is impossible. They appear different to each listener not only because of varying perspective, but also because only a portion of the actual sonic events can be consciously perceived during the limited time of their occurrence. This connects the listener with the functioning subconscious auditory facilities in order to take in all the complexity, and produces a personal, unique feeling. It is like throwing a handful of rocks into a pond, where the pond is the music and the rocks are the total attention of the listener. Everyone may hit the pond with all of their rocks, but they always land in different places, and never cover the entire surface.

In the five octave range that separates our perception of the active ranges of harmony and rhythm, there is no audible middle ground where the two aspects mix and are indistinguishable. Only inaudibly low pitches played at extremely fast tempos would qualify. In this zone of transition, the physical activity of harmony and rhythm are outside our realm of perception. This causes the original illusion of separation between the two. The existence of this natural blind spot, caused by the limited response of our auditory sense is what makes music happen the way it does for us. Otherwise we might perceive the scale on which harmony and rhythm connect, as continuous, like it really is, and that would change everything.

Table 1-1: Intervallic Harmonic Rhythms

Seventh: E/D# 82/156
Minor Seventh: E/D 82/147
Major Sixth: E/C# 82/139
Minor Sixth: E/C 82/131
Perfect Fifth: E/B 82/123
Augmented Fourth: E/A# 82/117
Perfect Fourth: E/A 82/110
Major Third: E/G# 82/104
Minor Third: E/G 82/98
Major Second: E/F# 82/92

(Note: The rhythms of the minor second are not easily perceived as repeating or phasing patterns.)

The relativity between harmony and rhythm can be represented mathematically by the equation:
H = Any interval or group of intervals produced by two or more simultaneous pitches,
R = A rhythmic repeating pattern or group of patterns, and
N = The number of octaves necessary to transpose a given example. (N > 4, N < 11).
And the inverse:
R2N =H

Given that pitch and tempo are perceived as separate aspects solely because of their frequency, like color and heat, the differentiation of two parts of the same numeric scale is appropriate. The relationship between pitch and tempo can be represented by the equation:

P/2N = T
and inversely,
T2N = P

The perception of two or more simultaneous low frequencies of oscillation as a rhythmic pattern, is the mathematical equivalent of being able to "feel" a ratio. This process of musical perception turns a numeric "interval", or space, into something strong and tangible, a pattern in the physical activity producing the harmony which evokes our response. For illustrating specific mathematical patterns arising from the simultaneous occurrence of different pitches, rhythmic notation is a convenient way of representing their ratio and its specific mathematical properties as an analog combination in a linear sequence.

Relative consonance is inversely proportional to the value of the lowest common denominator of the two frequencies in a given interval. The higher the common denominator, the more frequent the occurrence of mathematical regularity. For example, the interval of an octave, A 440 and A 220, with its high denominator, is a simple rhythmic pattern of eighth notes over quarter notes. An interval higher up in the series, such as a perfect fourth, is the more complicated rhythm, as found in.

Example 1-3

This goes on increasing with movement higher up the natural harmonic series and as notes are added to the harmony. Each additional note complicates the harmonic rhythm in a proportional dynamic to the color it adds to the harmony of the chord. Smaller intervals with lower common denominators form repeating patterns of longer duration, and vice-versa. When intonation is imperfect, the rhythm of a harmonic interval becomes a sequence of patterns through which the activity moves in regular cycles each time the phase shift crosses the center axis of the frequencies involved. This rhythmic phasing occurs at a rate relative to the percentage of "out of tuneness". These sequences of changing multi-meter rhythmic patterns occur constantly in the case of some harmonic intervals, as a result of the particular mathematics of the tempered scale.

Table 1-2: Harmonic Rhythm Phasing Sequences


Major Seventh

Major Sixth

Minor Sixth

Perfect Fifth

Perfect Fourth

Major Third

Minor Third

New rhythmic cycles also form in a regular sequence as the frequencies become offset during the slight tuning variations in vibrato. Some intervals produce a sequence of rhythmic patterns in shifting meters which are similar in dynamic character to the multiple meter sequences performed by traditional drummers in West Africa.

Table 1-3: Djerma Rhythmic Cycle, Dundun/Talking Drum (Niger)

Musical performance at a tempo which is an even subdivision of the "key", or pitch that predominates the harmonic organization, will contain physical patterns of symmetry that display a greater frequency of mathematically regular events. "Sympathetic tempo" is referred to here as the key to tempo relationship that produces the simplest harmonic rhythms. The connections between each individual pitch and its relative tempo are subject to the perception of rhythmic pulse when interpreted by the listener. The pitch "A" at 110 cycles per second is equal to 6600 beats per minute. The specific numeric sequence for lowering this "A" by octaves (dividing by 2), is given in Table 1-4.

Table 1-4: Pitch/Pulse Conversion for "A" 110

Original Pitch6600.00110.00
1 octave down-3300.0055.00
2 octaves down-1650.0027.50
3 ocataves down-825.0013.75
4 ocataves down-412.506.87
Presto5 ocataves down-206.253.43
{Rhythmic range}Allegro6 ocataves down-103.121.71 1
Largo7 ocataves down-51.56.85

An understanding of the rhythmic basis for harmonic sonorities can aid in the overall integration of the components of music. For example a tempo, when viewed as tonic, can be chosen accordingly. The key of "C" for example would have the greatest degree of harmo-rhythmic "sympathy" at 61,123, or 246 beats per minute. Performance techniques can synchronize, offset or fragment harmo-rhythmicity. Arppegiations can be played in time with the harmo-rhythmics to create desirable combinatory tones. Certain scale temperings, timbre voicings, interval bending, and off-sets produce unusually perfect harmo-rhythms. Bending notes and glissandos equal ritards and accelerandos on the harmo-rhythmic level. When relative synchronization is performed, and the frequency of the dominant pitches are multiples of the tempo and rhythm at which they are being played, a wide natural resonance occurs between the most basic aspects of music.

Table 1-5, Sympathetic Tempo



The study of harmonic rhythm and its many wide ranging theoretical connections may lead to a better understanding of certain unusual musical occurrences, i.e. : why certain tempos seem to flow more naturally for a given key; why some rhythm/harmony textural combinations produce strikingly beautiful musical symmetry in themselves without regard to specific compositional elements; why slightly "out of tune" notes actually sound better in certain harmonic contexts; what makes harmonic elements "ring", or sustain in varying degrees; why certain harmonic sonorities seem to imply specific rhythms, and sympathize with particular rhythmic styles. Awareness of harmonic rhythm may also lead to the realization of a more cohesive musical symmetry through subtle adjustments and nuance in performance and compositional techniques.