Complex rhythms, I have always argued, demand complex harmonies. And really, there is no reason to choose one over the other, as both have their basis in periodicity,-- in pitch, periodicity is consonance and aperiodicity is dissonance, and in rhythm, periodicity is the ictus and aperiodicity is syncopation. As such, rhythm and harmony are immanent in each other; rhythm is simply harmony on a larger scale. It must be stressed at the beginning that the author believes consonance and dissonance to be a continuous spectrum of simple to complex relationships between fundamentals based on physical phenomena, and follows in the beliefs of Arnold Schoenberg, who thought dissonance to be intrinsically more "complex" than consonance; there is, however, a camp of people who believe that harmony and dissonance are a "societal construct,"-- exactly how else a "societal construct" could be identified is an issue left unattended, but this camp gains the upper hand in explaining anomalies like the fourth and the third, the confusion entering into the scene when we consider the classical definition of a dissonance as something requiring "resolution." What exactly resolution entails is complex and varies from interval to interval, but its effects can be intuitively felt when you consider the tension-filled penultimate chord of any Classical-era piece, and the sense of resolution that comes from the final chord. The tension-filled chord is called the dominant, and can often be very dissonant, and the resolving chord called the tonic, which is often repeated a few times for good measure, to show us that we have, in fact, reached the end of the line. This repetition of the tonic is an important concept we'll be visiting later when we get back to issues of rhythm. Basically, since the Classical era, dissonance is often associated with something requiring resolution, but the Second Viennese school and most jazz music illustrates that dissonance can be left unresolved and still be musically interpretable. Basically, what we're all in agreement about is that there's a hierarchy of consonance to dissonance that has remained basically unchanged throughout the history of Western music, the precise threshold of "dissonance," what exactly is categorized in the discrete groups of "consonant" and "dissonant," what exactly requires resolution, all being issues subject to the whim of composers, and all being the source of much consternation to theorists. Metric modulation is a concept invented by one Elliott Carter, and involves modulations between meters, much as one would modulate between keys. This technique puts new demands on performers and listeners alike. What I will be dealing with in this article is kind of a dumbed-down version of metric modulation, involving only changes in tempo, and presumes a 4/4 metrical grid. Sure, there are more interesting time signatures, but there is still much great music to be written in 4/4. Generally, the idea put forward in this article is that drum 'n' bass could benefit from the addition of tempo changes, either subtle changes of the form (n+1)/n, or drastic changes of the form 2n/(n+1). One's first thought is that this is some revolutionary tract urging the overthrow of the hegemony of the DJ, and it's not, but rather a polite suggestion that we, producers and DJs alike, can be working a little harder for our craft. The simplest relation two notes can share is of course, the unison, or the octave. Rhythmically, this would translate to either no change in the groove of a piece, or a doubling of the tempo. A doubling of the tempo often falls somewhat flat, and I have two theories as to why this is the case. Firstly, a doubling of the tempo, if the original tempo is in the 70-100 BPM range, simply doesn't obey the human dynamic, it is only natural that a human drummer would play music notated at double the tempo at slightly less than double the speed. Secondly, imagine a person is tapping their foot and snapping their fingers to music at 83 BPM, they're tapping their foot to beats 1 and 3 and snapping their fingers to beats 2 and 4, now imagine the music doubles in tempo to 166 BPM,-- the person, neither expecting nor being immediately aware of this shift in tempo, continues tapping their foot and snapping their fingers as before, only now they're tapping their foot to the ones and snapping their fingers to the threes, so the rhythm their body moves to now goes kick-snare-snare-snare, kind of a breakbeat oom-pah... spectacularly unfunky! Wouldn't it be cooler if the rhythm their body carries over from the rhythmic tonic could be playing dotted notes or triplets against the new metrical grid,-- indeed, this is what we'll be exploring as we go into fifths and fourths. Part of drum 'n' bass's heritage comes from American hip-hop; however, drum 'n' bass is, for the most part, almost consistently double the tempo of hip-hop, occupying a narrow band ranging from around 165 to around 185 BPM. In some artists' attempts to bridge the gap, there is often some ambiguity in tempo, one is unsure if the music is moving to a hip-hop or drum 'n' bass pulse,-- this is largely effected by highly syncopated use of the kicks and snares, and can be used to wonderful effect in skilled hands. However, this is an entirely different animal than a blatant doubling in tempo, harmonically equivalent to more of a barber-pole scale than anything else. A superimposition of two rhythms differing by a factor of two is even worse precisely because the aforementioned oom-pah effect is made explicit, this is why I think Keyboard magazine's "breakbeat breakthrough" sounds so lame. In the end, a doubling and halving in tempo fails for the same reason an ascent and a descent of an octave fails as a melody, or why a Classical piece that stays in the same key fails as a piece,-- the relationships are too simple to be musically pleasing. Give me a major seventh or a minor ninth any day over an octave. More complicated than the unison or octave is the fifth, an interval of seven semitones. The corresponding rhythmic modulation would be 3/2, so, if we were starting from 120 BPM, we would "modulate" to 180 BPM. One issue desperately needs attending to here, and that is that, in 12-tone equal temperament, a fifth is a hair shy of a perfect 3/2 proportion to the root tone. That is, if we were to pitch our 120 BPM piece up so that the drum hits formed a saw wave at 440 Hz, the part that modulates to 180 BPM, which would be, in turn, 660 Hz, would be slightly "sharper" than a fifth. In 12-tone equal temperament, a fifth differs from its root by a factor of about 1.4983 in pitch. However, this is a negligible difference when you consider that this differs from a perfect 3/2 relationship by about two cents, and that Just Notisable Difference- the interval just large enough to be perceived as a different tone by the human ear,- is about ten cents over the greater range of human hearing. What is important to remember is that, in practice, a fifth stands in 3/2 relationship to its root, and that, were a solo violinist to play a fifth, they would naturally play it slightly sharper than a 12-tone equal temperament tuned instrument. Before we explore the rhythmic properties of the fifth, we must go into greater detail about what I call "carrying-over",-- the phenomenon I illustrated with the man tapping his foot and snapping his fingers to music. I already made mention of the dominant-tonic formula as it is used in the Classical tradition; it is important here because the root of the chord of the dominant is a fifth higher than the root of the chord of the tonic. The final chord of a Classical piece is the tonic, and falls on the downbeat; the dominant chord that precedes it may very well fall on the weaker beats. As well, a piece either begins on the tonic or begins on the dominant and resolves to the tonic on the downbeat. This association between the tonic and the downbeat is no accident; the sense of resolution that comes from the tonic is due in large part by the accentuation of strong beats,-- an effect that has ramifications for the large-scale rhythm attendant to large-scale harmony. That is to say, the tonic, either harmonic or rhythmic, small-scale or large-scale, is the place syncopation is least likely to occur. To illustrate the phenomenon of carrying-over as it applies to harmony, let us say we are working in the key of C major. The root chord is comprised of the notes C E and G. The dominant seventh chord contains the notes G B D and F. Together, these comprise six of the seven notes of the diatonic, white-key, scale. The dominant and tonic chords may appear on any beat and with any note as the bass note throughout strings of chord sequences, but a progression from the dominant to the tonic in root position, that is, with the dominant and tonic notes as the respective bass notes, has special significance for tonal music, and almost exclusively occurs with the tonic falling on the downbeat, usually either starting a new phrase or ending the piece. This is what is known as small-scale harmony, and it is my theory that this chord sequence's dynamic properties is due to the "carrying-over" of the tonic;-- that is, the notes of the tonic are mentally superimposed over the notes of the dominant, resulting in a dissonant chord containing most of the notes of the white key scale, a sense of closure effected only by a return to the tonic,-- the chord we started with. So what then is large-scale harmony, large scale rhythm, large-scale carrying-over? Well, in Classical practice, it is customary for a composer to modulate to another key, that is, to have another note temporarily displace the beginning key as the tonic. Any other key can be modulated to, but, for a major key, the first choice is, you guessed it, the dominant. So instead of our dominant-tonic formulas going GMaj7-CMaj as before, they are now going DMaj7-G. So what happened to the seventh, to the F that originally accompanied the G major chord when we were in C major? Well, the chord GMaj7 may very well occur in chord sequences in GMajor (usually going to C major, which is now called the "subdominant",) but, generally, doesn't occur in dominant-tonic formulas. So, technically, the dominant-tonic formulas, with the chords in root position and the tonic falling on the downbeat, are still occurring, so the piece could very well end right there in G major, right? If it were not for the *key* of C major carrying-over this time. This is what is known as a resolution in large-scale harmony, the key of G resolving back to the key of C, just as the *chord* of GMaj7 resolves to the chord of C; the sense of displacement accompanied by the chord of GMaj7 has a large-scale analogue in the sense of displacement accompanied by a modulation to the key of G major, and this is the source of all the drama in Classical-era music. So what about rhythm in large-scale harmony? Well, if the small-scale harmonic association of the tonic is with the downbeat, and the respective association of the dominant is the weak beat, we can expect that, in large-scale harmony, the dominant would be more syncopated than the tonic;-- indeed, even as the chord progressions circle around a displaced tonic (in our example, G major,) the resolutions are never quite as heavy-handed as they are in the final measures of a piece, when the original tonic is emphasized. This equating of the tonic, rhythmic or harmonic, with rhythmic simplicity is an important aesthetic tool in the arsenal for makers of modern music, a real-world example of which we'll soon be witness. However, I'd just like to say that this association of the harmonic tonic with the downbeat has always been my axe to grind with tonality as it is used in drum 'n' bass music,- which is so non-downbeat oriented,- and which is, in turn, why I urge beginners to explore atonality. I've already expressed a dislike for octaves, either melodic or rhythmic, and I must say that rhythmic modulations of a fifth sound equally unconvincing to my ear. However, the fifth- and its inversion, the fourth,- do have some interesting properties when we take "carrying-over," this time in a rhythmic sense (which is pitch on a larger time-scale,) into consideration. Let us say, as before, that a person is tapping their foot and snapping their fingers to music at 120 BPM, now let us say that the music's tempo increases by a factor of 3/2; the person, again, continues to move their body as before. Only this time, the tempo moves to make their movements move at 2/3 the speed of the new tempo, so they're actually tapping and snapping their foot in dotted rhythms. Here's an illustration: | * * * * * * * * * *... k s k s k s k s k s... t s t s t s t s ... | Where "k" and "s" represent the kicks and snares of the music, and "t" and "s" represent tapping and snapping, and "|" representing the point at which the metric modulation occurs. Note that the person taps their foot at the second snare of the new measure,-- a phase shift not nearly as egregious as the one occurring at a doubling of the tempo. I don't seriously expect you to be programming drums quite as simplistic as the kick and snare I have outlined above, but it's important to consider the pulse of the music as it is affected by metric modulation, and in 4/4 music, the pulse is largely governed by the backbeat. More complicated than the fifth is the fourth, an interval of five semitones. This is referred to as the "subdominant." The fourth is the inversion of the fifth, and together they add up to one octave,-- twelve semitones. That is to say, if you were to count down a fifth from the tonic and then up an octave, the relation in which the tone would stand to the tonic would be the interval of the fourth. Arithmetically, it would be the reciprocal of the proportion of the fifth, 2/3, raised an octave, resulting in the proportion of 4/3 to the tonic. It is easy to see why the fourth is the third most complex relationship between two notes. However, in Classical practice, the fourth is considered dissonant, and must be resolved if it occurs between the bass and an upper voice by one note staying where it is and the other moving towards the other by an interval of a minor or major second, resulting in a major or minor third. Rhythmically, the tonic carries over in a 4/3 proportion to the fourth, that is, in triplets. Interestingly, there is no shift in phase as long as we consider our basic 4/4 beat; the kicks always fall on the kicks, and the snares always fall on the snares. An interesting property of a rhythmic modulation to the fourth becomes apparent when we consider a syncopated lead-in note of a dotted eighth. Rhythms of the type of the following are common in hip-hop music: 1 2 3 4 1 2... k s k st tk s... Where "t" stands for tom. What if we drop the sixteenth-note lead-in to the next measure and modulate to the rhythmic fourth, the rhythm then becomes: 1 2 3 4 1 2 3 4... k s k st k s k s... That is, the syncopation, as it stands in relation to the downbeat, becomes the new pulse of the music. Indeed, the more complex the relationship between the rhythmic tonic and the rhythmic modulation, the more syncopated the lead-in. The harmonic analogue to the lead-in is the dominant preceding the downbeat of a phrase in a new key. Consider the following chord progression: 1 2 3 4 1 2 ... CMaj FMaj Amin7 DMaj7 GMaj CMaj... The modulation is from the key of C major to the key G major. A minor seventh in this case is called the "pivot chord",-- the pivot is a chord common to both keys. D Major seventh is the dominant of G; the dominant needn't be a chord in the key of the tonic, as it essentially "belongs" to the key of the antecedent phrase. However, the dominant of the new key is rhythmically grouped in with the phrase starting with the chord of C Major. The downbeat begins with the tonic of the new key, and it would sound a mess if the downbeat of the new phrase were to begin on the dominant of the new key. Note, however, that the phrase beginning with G Major begins on the dominant of the *tonic* key, and thus that downbeat is "de-emphasized" in the large-scale rhythm. We can also take advantage of the dotted-note feel of the subdominant modulation in other ways. Consider the following rhythm: 1 2 3 4 1 2 3 4 k s k s k s k s tf tf t f t f Where "f" stands for floor tom, and "t" for high tom. Let us say the following transition occurs not in a modulation to the subdominant, but rather from the dominant directly back to the tonic. The reason I say this is because, while the toms seem to be playing dotted-note rhythms when heard with the lead-in, they are actually playing a four to the floor rhythm when heard only in the context of the new tempo. Since the tonic is associated with simple rhythms, this is a jazzy and syncopated way of introducing what would otherwise be unspectacular rhythms. As well, if we are in fact returning to the tonic, there is the shock of recognition of hearing familiar material in a new context. More complex still is the major third, and after that the minor third. Exactly at which point of complexity a change in tempo ceases to be perceived as a ratio of integers and becomes merely an arbitrary change in tempo is subject to debate, but most likely it occurs before respective harmonics cease to approximate any note of the diatonic scale. What I have attempted to codify in this article are merely theoretical tools, and, as with all music theory, can only be hypostatized in accordance to what "sounds good." Indeed, while this is only part of a much larger theory of rhythm, all my theory is experiential, based on my own perceived successes and failures as a musician. If any of the concepts put forward in this article help you to make music that inspires other people, I can claim no credit for that. I do hope that I've inspired others to share my vision of a convergence of complex rhythm and complex harmony.