Integrated Frequency
Brief History Bound by Mindsets A Solution History of Integration Playing with the System
Breaking the Series into Functional Segments Experiments
My Story
As a graduate of Juilliard and Eastman, a professional musician and composer for over 40 years, I have always had an appreciation for intonation, tuning systems, and the overtone series. My father, a symphony conductor for several decades, discussed with me various tuning systems at work in symphony orchestras, allowing them to blend well in certain passages, particularly in the woodwind section.
In my own experience as a violinist, playing the Unaccompanied Sonatas and Partitas by J.S. Bach, I would notice certain chords would sound better by slightly lowering the 3rd scale degree, causing them to resonate better. At other times, raising pitches in a melody to ‘lead’ upwards to the next note, or lowering them in a descending passage, helped shape the musical phrase.
I always admired Bach’s development of equal-temperament in his compositions. I believe he established compositional examples for countless other composers to follow in Western music history. Bach established new standards in motivic development through intervallic patterns. At times he would even break rules of counterpoint for the sake of the integrity of the intervallic lines. After hundreds of years of using equal-temperament, composers elevated the use of intervallic patterns above harmony altogether in atonal composition.
My father was my first composition teacher. He had compositional training from teachers in Vienna who had known Schoenberg. Some of my first lessons were based around the understanding of serialism techniques.
Something, however, always bothered me about serialism’s ethos. It seemed disconnected from what was deeply meaningful to me about music: a sense of tonality or ‘home-base.’ In my own composing, even though I employ a high level of chromaticism, I always maintain a tonal center.
As I would write for string groups or choral ensembles, in dense chromatic textures and strong lines, I noticed something unusual happen. The players would instinctively bend the pitches to sound more homophonic. Instead of leaving the pitches in equal-temperament, they would (as an entire group of players or singers) intuitively morph into resonation what sometimes were very strident passages, causing them to resonate better harmonically.
In my father’s office, he had a ‘frequency’ chart posted on his wall that fascinated me as a young man. It wasn’t just about musical frequency, but showed how sound, light, and other scientific frequencies were all part of a larger system.
This idea stayed in my imagination for years. I had an idea of using the overtone series compositionally, since the 1980s, but I didn’t have the technical tools to do so until recently.
As a violinist, I have studied and performed Classical works by composers across the centuries, from the Baroque, Classical, Romantic, and Modern eras. It concerned me that composers in the modern era could not seem to reconcile harmony, serialism and microtonality successfully. Modern composers had broken away, generally, from any harmonic moorings found in earlier Classical music. I noticed since the mid-20th century, composers tried to find a way to reconnect with harmony, but it seemed like they were trying to recreate the past- a compromise to the forward motion of history. With all of the experimentation going on, I couldn’t feel musically comfortable with the schools of thought I was seeing and hearing.
What I have set forth in this document are early experiments in a new way to compose, based on tools we have available for the first time in history. These experiments are my attempt to reconcile the disparities between compositional divides that have existed now for decades. In my view, I see a new possibility of congruity between all the components of music: rhythm, harmony, diatonicism, chromaticism, and microtonality, all functioning seamlessly in one system. I have given it the name Integrated Frequency.
Brief History
Throughout the history of Western Art Music, following the progression of its development, there are several ‘defining moments’ in terms of the tools that have been available. I am not referring to the musical instruments, per se, but rather the methodologies.
Pre-Bach, the key of D Major and the key of Ab Major could not successfully exist together in the same composition. When equal-temperament emerged, it was as if Bach had a new toy to play with, seeing how far he could push its boundaries, experimenting.
Broadly speaking, pre-Bach, we had tonality, but limited chromaticism. Starting with Bach, we had a new direction in the unification of tonalities. A composer could travel to multiple harmonic centers within one composition rather easily, and still sound harmonically congruent.
The subsequent Classical period explored this possibility further, and the Romantic Period began pushing the boundaries of equal-temperament even more, fluidly transitioning from key to key and even losing a clear tonal center.
The early music of Schoenberg started here, but he pushed into what he called pan-tonality, which became known as atonal, 12-tone, or serial music. This system relied heavily on intervallic relationships rather than any harmonic foundations, albeit composers in this era could still manage to use the techniques of serialism and maintain a sense of tonality (i.e. Berg).
In the 20th Century, composers in the Western tradition began seeking to use intervals smaller than the half-step on the piano with micro-tonality as well as varied tunings outside of the traditional Western scale.
Another development in the mid-20th Century, electronic music began to be explored (i.e. Stockhausen, Babbitt). With computer technology, specific tunings have now become readily available and accurately achieved through the use of software programs such as Scala, in which user-defined tunings can be created rather easily.
To summarize, music in the pre-Bach era was harmonically key-centric. Post-Bach equal-temperament became increasingly less tonal and finally non-harmonic. Since that point, experimentation outside the traditional Western scale has been underway.
In terms of the ‘tools’ available to composers, history starts with harmony, moves to chromatic harmony into linear chromaticism, serialism, and finally micro-tonality. Along with this transition, rhythm begins simplistically in the early eras of harmony, progressing to a high level of complexity using macro-polyrhythms seen in works by Conlon Nancarrow.
What if all of these systems could somehow co-exist?
Serialism and tonality are compositionally polarized. Micro-tonality clearly departs from the 12-note scale, and the incongruities are plentiful among the many directions of our day.
Bound by Mind-sets
For some reason, we have relegated the subjects of tempo and pitch to two different ends of the musical thought-process, when in fact they both have a common thread: frequency.
We have come to measure tempo by BPM (Beats Per Minute) and pitch to BPS (Beats Per Second), and we have never challenged ourselves to see them in one system. I once asked an acoustician if there is any sound below 1Hz and he answered me, ‘No.’
But what about .5Hz?
Human hearing doesn’t go lower than 20Hz, but that doesn’t mean that we don’t experience frequency lower than this. We simply call it rhythm.
But rhythm and pitch are unequivocally in the same system.
Here’s another box we have put ourselves in. The traditional Western scale of 12 notes (in all of its various permutations, be it the majors, minors or the church modes) has been etched into our consciousness as always existing within one octave. Whatever octave you’re in, the intervallic relationships stay the same. That was handy for organizational purposes for a while, but even with our recent experimentations of new tuning systems, we still stay within this one-octave boundary.
Currently, only a few software instruments allow pan-octave tunings to be scripted. And major DAW platforms are either currently locked into one-octave tuning systems or limited altogether in the subject of tuning, requiring 3rd party plug-ins to help their lack of development.
A Solution
Our understanding of harmony comes from the Overtone Series. In fact, all frequency is similar, be it electrical, light, etc. Frequency functions with a fundamental and upper partials (or nodes).
Our traditional understanding in music in the use of the Overtone Series has always been within the boundaries of human hearing (20Hz to 20 kHz), but what if we extended the fundamental below human hearing, say to 4, or 2, or .5? What would the overtones of this system look like?
The higher the overtones go from the fundamental, the smaller the interval. The closer the intervals are to the fundamental, the wider they are. (i.e. the first node is an octave from the fundamental, the 29th node is a much closer microtone interval to the 30th node.)
Normally, if the fundamental is in the sing-able range of human hearing, the overtones easily extend beyond the scope of human hearing when reaching into the areas of node 30. But if the fundamental is dropped below human hearing, then the 30th node, for example, becomes a usable musical frequency.
Here is the main point: I believe that the subject of rhythm, harmony, melody, chromaticism, serialism, and micro-tonality can live together in one system. This system starts from the fundamental below human hearing (rhythm); the lower overtones are spacious (bass); the overtones above this are usable as chords (harmony); higher than this are closer intervals resembling the Western scale (melody); and finally higher intervals are smaller, chromatic and microtonal.
History of Integration
As the equal-temperament system was at its furthest distance from any harmonic foundations, a new approach to music-creation was in its embryonic stages.
In the early 20th Century, experiments with musique concrete, ambient sounds, spectralism, and organically derived sounds from the overtone series were being conducted by Pierre Schaeffer, John Cage, Gerard Grisey, and Harry Partch, respectively. The influential book New Musical Resources, by Henry Cowell, exposed areas of rhythmic expansion within the Western traditon never before explored.
Composers were just beginning to experiment with electronic music, as the tools were in their infancy.
In May 1948 Pierre Schaeffer created the first example of what became known as ‘musique concrete’: Etude aux Chemins de fer, a three-minute piece made by manipulating recording of railway trains.
Experiments with discs had been conducted before the war, notably (and independently) by Milhaud, Hindemith, and Varese, but it remained to Schaeffer to discover and use the basic techniques of sound transformation: reversing a sound by playing its recording backward, altering it in pitch, speed, and timbre by changing the velocity of play-back, isolating elements from it, and superimposing one sound on another. Just as important as these possibilities was the change to the art of composition. Every example of musique concrete was an improvisation created by the composer working directly with the sounds available. [1]
Amazingly, this marked the beginnings of techniques DJs and EDM artists would use decades later. These early 20th Century composers would have loved the technology available today, several decades away from their creative exploits.
Schaeffer even describes sampling techniques that would come about. He had hoped
to employ an array of gramophone turn-tables as ‘the most general musical instrument possible’, providing facilities for altering any sound derived from the real world. [2]
John Cage, in 1937, described today’s modern computer technology, and
had expressed his optimistic view of the potential electronic evolution of music, and in 1942- after he had made his first electronic experiments, beginning with the 1939 turntables with frequency recordings- he had been more specific: ‘Many musicians,’ he had written, ‘the writer included, have dreamed of compact technological boxes, inside which all audible sounds, including noise, would be ready to come forth at the command of the composer. [3]
This is an extremely provocative statement envisioning what would be state-of-the-art technology. Technology, such as Ableton Live, running on any laptop, producing time and pitch altering loops was possible in the early 2000’s, sixty years later.
Cage’s concept of rhythm, which again ties into looping technology, was
for music to be structured on the basis of duration (possessed by all kinds of sound, and silence) rather than harmony (possessed only by pitched tones in combination). [4]
Cage applauded other composers who saw a correlation between tempo and pitch, crediting
Satie and Webern for correctly using duration as the measure: ‘There can be no right making of music that does not structure itself from the very roots of sound and silence- lengths of time…to bring into co-being elements paradoxical by nature, to bring into one situation elements that can be and ought to be agreed upon- this is, Law elements- together with elements that cannot and ought not to be agreed upon- that is, Freedom elements- these two ornamented by other elements, which may lend support to one or the other of the two fundamental and opposed elements, the whole forming thereby an organic entity’. [5]
Duration, amplitude, frequency, and timbre were the four characteristics of sound that composers could manipulate using serialism. Seeking integration of pitch and rhythm, Messiaen experimented with serialism’s high degree of organization in Mode de valeurs, in which
each of the thirty-six pitches is permanently associated with one of the thirty-six durations. [6]
Elliot Carter was also experimenting with time and duration.
Carter’s polyphony, in the First Quartet as in his later music, is one proof against subjectivity: because the music is happening in several speeds simultaneously, it has no speed of its own, and therefore allows no presumption that it speeds or sings (or, given the abundant pulsed rhythms, dances) the thinking of one person at one time…liberation from psychological time…has an effect on the notion of what constitutes a musical movement. [7]
Stockhausen, also seeking integration of pitch and rhythm, focused his attention on experimentation with such things as
the conversion of a complex event into a single sound, or the treatment of duration as a variable with the same capacity for complex relationships as pitch. [8]
Stockhausen was working on Studien, to prove
a practical demonstration of how pitch and duration- the two parameters whose parallel ordering had been such a problem to total serialism- are aspects of a single phenomenon, that of vibration. A vibration of, say, 32 Hz will be perceived as a pitched note, whereas one of 4 Hz will be heard as a regular rhythm, and somewhere in between the one will merge into the other. So for different reasons- to do with acoustics rather than mathematics- Stockhausen came to the same conclusion that Babbitt had reached a little earlier, that some deep coherence had to be sought between the principles applied to pitch and to rhythm in forming a work. The scale of chromatic durations was inadequate, in Stockhausen’s view, because it contradicted acoustical reality, being an additive series and not a logarithmic one, such as lay behind the chromatic scale of pitches; moreover, it led to absurdities and inconsistencies, such as the tendency towards regular pulsation when many lines are superimposed, or the undue weight of long durations. [9]
(He) wanted to base his organization on the nature of sound. To create a true confluence with the phenomenon of pitch, he introduced a logarithmic scale of twelve tempos- a scale that could be ‘transposed’ by altering the rhythmic unit: for example, a change from crotchets to semibreves, and therefore a deceleration by a factor of four, would be the equivalent of a downward shift of two octaves. Within this system, the obverse of the one proposed earlier by Boulez, a rise of a perfect twelfth would have its analogue in a change of tempo in the ratio 3:2 (the frequency ratio of a perfect fifth, if one discounts the small discrepancies of temperament) coupled with a halving of the rhythmic unit. So any pitch line could be turned into a duration-tempo succession, a melody of rhythm, and one could also change the timbre of the rhythm, as it were, by adding ‘partials’ in the form of other duration-tempo successions going on at the same time, their number limited only by the practicalities of performance. [10]
Stockhausen tried to accomplish this in his composition Gruppen with traditional orchestral instruments and players, since he didn’t have access to successful computer technology. This was in 1957.
Boulez had difficulty embracing Stockhausen’s path into frequency-based methods.
Boulez was mistrustful of one of Stockhausen’s new departures, finding in it ‘a new sort of automatism, one which, for all its apparent opening the gates to freedom, has only really let in an element of risk that seems to me absolutely inimical to the integrity of the work’…In a crucial article on aleatory composition, Boulez proposes that chance can be ‘absorbed’ in musical structures dependent on a degree of flexibility, perhaps in tempo: there were already examples of such structures in his own music. [11]
Stockhausen also worked towards what ultimately would become known as the keyboard synthesizer, common to performers today.
A new musical architecture demanded new material, not refashionings of the old. The work of Helmholtz and Fourier had suggested that any sound could be analyzed as a collection of pure frequencies, of sine tones, and this was something that Stockhausen thought he had confirmed, in analyzing instrumental sounds in Paris. So it seemed reasonable to suppose that the process could be reversed, that timbres could be synthesized by playing together a chosen group of sine tones at chosen relative dynamic levels. One could thereby form a repertory of artificial timbres that were related in defined ways, and therefore suited to serial composition. This Stockhausen tried, working with a sine-wave generator at the postal headquarters in Paris, but the practical problems were insuperable. Instead, in December 1952, he turned to using initial moments from prepared piano sounds in his first electronic composition, the Etude. [12]
Also in this era, exploration of Spectralism or Spectral Music was being developed,
music whose composition is informed by the overtone spectra of sounds, especially instrumental sounds. [13]
Gerard Grisey became the predominant composer here, focusing on the ‘texture’ of sounds. He saw the inherent integration of pitch and rhythm in the overtone series.
From an understanding of sound, therefore, came an understanding of rhythm. For Grisey it was important to keep in view the rudimentary periodic states of both: the harmonic series and pulsation. Periodes owes its title to the regular rhythm to which it returns from time to time, as at the end, a rhythm that is not metronomic but organic. [14]
This work is similar to Stockhausen’s exploration of the synchronicity of pitch and rhythm, mentioned above.
Cowell’s New Musical Resources sparked yet another branch of experimentation and exploration by Conlon Nancarrow, Georgy Ligeti, and others.
Cowell extends the concepts of ratios to dynamics, meter and time combinations, tempo, scales of rhythm, and time.
In the scale of time-values a quarter-note is taken as the equivalent of C. The reason for this is evident- a low C in tone is produced by sixteen vibrations per second. If this C be carried down two octaves, the result is a sub-audible C of four vibrations per second. If the time-value of a whole note is taken to be one second, four quarter-notes will be produced in one second, just as four vibrations will produce a sub-audible C. The corresponding time-values of the other tones of the scale are such that the number of times a given note is contained in a whole noted will equal the number of vibrations per second of the corresponding tone, on a base of C equal to four vibrations. [15]
Conlon Nancarrow, inspired by Cowell’s writings, found a way to produce on a player-piano what was impossible to achieve otherwise, and foreshadowed our modern Digital Audio Workstation MIDI sequencer.
Ligeti, who greatly appreciated Nancarrow’s work, sought to find a way to include the kinds of tempo ratios used by Nancarrow in live performance, also with a desire for convergence between frequency and pitch.
Microtonality has always been an important feature of Ligeti’s work. Rather than employing it in an arbitrary manner in Hora lunga, he utilizes the natural microtonal discrepancies offered by the harmonic series, and in so doing parallels certain developments in the spectral music of Tristan Murail, Horatiu Radulescu and Gerald Grisey. [16]
Could Nancarrow’s combination of breath-taking speed, rhythmic intricacy and unrelated metres be replicated by a human pianist? Ligeti pondered the challenge. [17]
He answered the question in his Piano Concerto. In this work, Ligeti finally emerged with a style he forged that was uniquely his own. He began using a combination of independent meters and multi-tonal harmonies.
The key to the breakthrough was not only polymetric. Polymetre, after all, had been an occasional component of Ligeti’s earlier music...True, the actual first movement is more ingeniously polymetric than anything Ligeti had composed before. But crucially it is supported by something else: the layers are also differentiated harmonically.
Above all, it was the partnership between polymetre and Ligeti’s unique brand of multi-tonality that emphatically secured the ground for the achievements of his final years. [18]
Even though Pierre Boulez did not share Stockhausen’s excitement in some of the experimentation he championed, he was yet influential in music’s technological advancement through his leadership of IRCAM (Institute for Research and Coordination in Acoustics/Music) in Paris, from which a team of computer savvy musicians later developed flexible software designed for performance: Max/MSP.
As synthesizers became usable, connected through MIDI, and easily manipulated with computers, the desire to merge computer technology with synthesis came together in the DAW (Digital Audio Workstation). The next goal was to bring this technology to the live stage. The DJ’s LP turntables were soon replaced by computers with human interfaces, and particularly with the program Ableton Live. IRCAM’s Max/MSP was a perfect fit for Ableton, and Ableton’s version 10 absorbed it with Max for Live. Ableton is currently the most widely used platform for EDM (Electronic Dance Music).
Whether the creators of Ableton knew it or not, their platform became a convergence point for decades-old avant guarde Western Classical theoretical exploration and Pop music’s DJs and ultimately the EDM artist’s computer-reliance.
Yet another example of Modern Classical composition being ahead of pop-culture, Phillip Glass, with his minimalism, was remarkably creating ‘looping’ with orchestral instruments decades before EDM became vogue, or DAW’s were technologically available. Glass marked the convergence of classical music, pop music, ethnic rhythms, foreshadowing EDM decades prior to its arrival.
Throughout the 20th Century, composers can be seen searching for ways to find unity in the separate subjects of rhythm and pitch, harmony and serialism, diatonic, chromatic, and microtonal scales.
Early thoughts of the integration of these disparate systems could be seen in Hindemith’s desire for harmonic integration and tonality in a world of high chromaticism. From a solidly European Western tradition of equal-temperament, he would never leave that world. Neither could he, since the technology to do so was not yet available. His desire for complete chromatic/harmonic unity was simply unachievable in the construct of equal-temperament.
Conlon Nancarrow could have brought his rhythmic canons into harmonic unity with the overtone series, if the technology had been available. His compositional approach of using ratios to derive his rhythmic counterpoint would have been a perfect fit for harmonic series tunings that were congruent to those tempi. His player-piano, of course, was equal-tempered. His music has successfully been transferred into MIDI files, so it may yet be possible to add harmonic series tuning scales congruent with the tempos he was using.
Playing with the System
What if we started with a fundamental quite low, below human hearing, and imagined it as the trunk of a tree. From this trunk, branches stemmed off at different points, each having their own fundamental and corresponding overtone branches (complete with tempo, harmony, melody, microtones). All of these systems would ultimately be related in one congruent system (the same ‘tree’).
Here is a chart, showing the division of the THH (Threshold of Human Hearing) and the convergence of BPM (Beats per Minute) and Hz (Beats per Second). Setting A = 440 as the tuning note of the entire system, extending backwards, down the series, 20Hz = 64th note, 10Hz = 32nd note, and on down to 0.3125Hz = a whole note.
This frequency can be called the ‘root’ or base of the tree. Then a branch could come off of this root, multiplying it by, say 3, or 7. Then, off of these branches, other branches could come from any of the available nodes, creating their own harmonic series.
What this makes available is a variety of tempos and micro-tonalities between the ‘frequency scales,’ yet all of them related, much like a family tree.
In the compositional process, rhythm relates to the overtone series metaphorically. That is, we hear pitch, but we feel rhythm. Functioning in the above-mentioned frequency scale, the Quarter-note = 75. Any rhythmic duration that would be desired will fall in line with the overtone system at that tempo. A half-note, in the above-mentioned system is the value of: 18.75 (whole note) x 2. An eighth-note is 18.75 x 8. Any rhythm we could desire: triplet, septuplet, dotted-eighth/sixteenth, etc. falls exactly in line with the overtone series at that tempo. After all, this is how we derive the overtone series itself (fundamental x1, x2, x3, etc.). Any expression of rhythm is exactly congruent with the overtone system, as long as it is in tempo with it.
For the first time in human history, we have the tools available to us to have exact integration of the entire frequency spectrum, perfectly timed, perfectly in-tune. What composers for the last century have been searching for is now before us.
To me, this is where music history has brought us. It represents a wide-open new system in which to experiment, and brings the history of Western Art Music into ‘harmony’ with itself.
Breaking the Series into Functional Segments
Below, I have endeavored to create short compositions utilizing the various sections of an overtone series: Rhythm, Rhythm/Bass, Harmony, Diatonic Melody, Chromaticism, and Microtonality.
I used a variety of expressions from various fundamental roots and their corresponding harmonic series ‘scales.’
Using Max for Live inside of Ableton, I created a Polyphonic Additive synthesizer and inserted my own 30-node overtone series using the coll object in Max. For the various scales, I gave the name R60 (Root 60) or R40 (Root 40), etc.
I realized I needed to have a convergence table to represent the overtone series (starting from any root), depicted simply in the treble clef, starting on Middle C, MIDI note #60, going up chromatically. Each written half-step represents the next node of the overtone scale. I set a standard scale of 30 nodes, including the fundamental.
I used Ableton Live as a host, importing MIDI files of compositions created in Finale.
Once I had tweaked the Max instrument’s tone settings for a track, I exported the audio file from Ableton, with the exact tempo I wanted for that particular track. For example, R40 is in the frequency tree of quarter-note = 75BPM. So, I exported the MIDI notes of the R40 track at 75BPM. Then I did the same process for the R60 track at half-note = 112BPM, since these are congruent with their own frequency trees.
The MIDI compositions I created, I freely used rhythms as if I were writing music traditionally. As mentioned above, the rhythms heard in the compositions represent the rhythms that exist below the threshold of human hearing. (In the 18.75BPM frequency tree, 18.75 (or 0.3125Hz) = a whole note; 37.5BPM (or 0.625) = a half note; 75BPM (or 1.25Hz) = a quarter-note, etc.) So, representationally, the rhythms I used in the composition are the same values.
Once the audio files were exported from Ableton, I imported them into Logic, setting Logic’s tempo at 75, so the measures would line up with my audio files. Then the composition process continued as I structured the placement of the audio files, creating polyrhythms between the various frequency trees, similar to Conlon Nancarrow’s work. Since the frequency trees were related, but distinct, this created microtonal scales in counterpoint against each other.
Experiments
Rhythm
Beginning with the lowest fundamentals, here is an experiment in polyrhythms. Using the root frequencies of R5, R10, R20, R40, R80, and R120 at 75BPM as one rhythmic block, R30 at 112.5BPM and 225BPM as another rhythmic block, and R35 and R70 at 131.25BPM and 525BPM as the last rhythmic block, many polyrhythms are created. At one point there is a ratio of 12:28:11.
The pitches were created by making a short note on each fundamental using the additive Max synth that I made, so the tempos and pitches of each rhythmic block are in the same family. There are three main rhythmic families against each other in polyrhythms. All of these families are related to each other, as branches of the same tree, the foundation being 18.75BPM (or .3125Hz).
Rhythm/Bass
This was an experiment in two different overtone scale/rhythms: R5/R10 against R30/R60. The familiar rock style is challenged by the convergence of the two tempos and scales, creating polytonal and polytempo moments.
Only two compositional components were used to create this piece: a bass pattern and a rhythm pattern.
R5/10 is in its corresponding tempo of 150BPM, while R30/60 is in its tempo family of 225BPM.
What interested me, as I brought the bass pattern down to the root of 5Hz, was that it highlighted the overtones of the additive synth I made in Max. In the synth, I use the first five nodes of the overtone scale, detuning the upper nodes slightly for flavor. The lower the synth goes, it becomes possible to hear the individual overtones inside the voice. What surprised me was how accurately this resembled an electric guitar’s distortion, ‘planing’ major-chord harmonies along the bass-line contour.
I created the beat pattern using my additive synth’s lowest nodes at a short attack of the ADSR, so the beats were ‘tuned’ perfectly to their corresponding bass-lines.
Harmony
In this experiment I took a traditional harmonic progression derived from a single overtone scale, in an arpeggiated pattern, and distributed it to other related overtone scales, in their corresponding tempo families.
I used R40, R120, R160, and R200 in their tempo family of 75BPM and 150BMP. I used R60 and R120 in their tempo family of 112.5BPM.
The corresponding relationships created by the harmonic progression in multiple overtone scales and corresponding tempos creates polytonality and polytempo.
I kept R60 in 75BPM to experiment with a Tonic/Dominant relationship between the two instances of R40, then R60. R60 is derived from the 3rd overtone of the root frequency of .3125Hz. The third overtone is the ‘fifth’ scale degree. So, the chord progression was played in R40 (Tonic), then it was played in R60 (Dominant) in the same tempo, then finally back to R40 (Tonic).
Melody
This is a single melodic line, stretching over the span of a several-octave overtone scale. Using the melody canonically in R40, R60, R120, R160, and R200, the lines create counterpoint against each other.
R60 and R40 are related. R40, in this instance represents tonic and R60 represents Dominant.
Chromatic Serial
This experiment plays with the possibility of serialism in the context of the chromatic scale which resides inside of the overtone series, nodes 13-24. I used a 10-note pattern, providing Prime, Retrograde, Inversion, and Retrograde/Inversion of the theme in the overtone scales of R20, R40, and R60.
The R20 and R40 overtone scales are both in the tempo family of quarter-note = 75BPM, while the R60 scale is its tempo family of half-note = 112.5BPM. R60 is related to R20 and R40 as a branch off their family at the 3rd node of .3125Hz (or whole-note = 18.75BPM).
Chords derived from the R40 scale are evenly spaced to provide a harmonic progression, using I, i, V6/4, Vsus4, and ending on I.
Because the chromaticism exists in the same overtone series as the chordal harmonies, they are congruent with one another. Serialism and Harmony are synonymous.
Diatonic, Chromatic, Microtonal Serial
This experiment deals with the same 10-note pattern used in the Chromatic/Serial example above, in three sections of the overtone scale: 1) the lowest ten nodes [diatonic/harmonic], 2) the middle ten nodes [chromatic], and 3) the highest ten nodes [microtonal]. Using the same pattern in each of the sections created the same contour of the theme, but not the same intervals, since the intervallic differences of the lowest ten notes are wide and the intervallic differences of the highest ten are much smaller.
What I found interesting is that the motif could be recognizable, even feeling like an exact reflection, when comparing it against each of the three ranges. The intervallic differences between each of the ranges of the scale didn’t seem to alter the recognizability of the pattern.
From this experiment, perhaps a new way of dealing with motivic development can be obtained: what I call ‘motive ballooning.’ Consider drawing a motif on the outside of a balloon with a marker. Then blow the balloon full of air, and the written motif expands. Let the air out a little, and the motif shrinks, etc. The motif is still recognizable, even though the size of the intervals change.
In this piece, I used two different tempos in the family of R20, 75BPM and 120BPM, so each of the three parts of the R20 scale had eight versions of the motif: Prime, Inversion, Retrograde, Retrograde/Inversion in 75BPM and 150BPM. This, all in all, gave me a total of 24 different versions of the motif, each of them only used once, in the traditional serialist ethos.
Even though serialism is employed, each of the three sections of the overtone scale are harmonically and tonally congruent, as all notes exist inside the same overtone scale. Harmony and Melody or, one could say, Vertical and Horizontal properties are one and the same.
Microtonal 1
This experiment was designed to showcase microtonality as ornamentation. In traditional performance, we use vibrato, portamento/shifting, and even intonation differences to shape a diatonic or chromatic line. These are all uses of microtonality in our traditional Western tradition.
I wanted to take that concept one step further, using the upper partials of the microtonal scale, R20 in ornamentation and melody.
Underneath the melody, using harmonies derived from the central nodes of the R20 scale, I created a progression to accompany the melody.
Aesthetically, I attempted to give some ‘personality’ to the melody by manipulating the attack time of the synth’s ADSR remotely. The melody declares, ‘I’m weird, but I have something important I want to say.’
I also, with a controller, manipulated the first partial of the additive synth I built, to create vibrato (LFO) in the voice, for the accompanying harmony part.
Microtonal 2
In this experiment I used the same ascending and descending scaler pattern in multiple related overtone scales. In each 30-node scale, I used the upper partials to create the microtonality.
In the R5 family, I used R5, R10, R20, R40, R80, and R120 in 75BPM and 150BPM, from their respective tempo family. The root tempo of this family is whole-note = 18.75 (.3125Hz).
In a related branch from that group, I used R30 in 112.5BPM and 225BPM, its respective tempo family. This family is derived by multiplying the above 18.75 x 3.
I also used R35 and R70, another branch from the R5 family in 131.25BPM, its tempo family. This family is derived by multiplying the above 18.75 x 7.
I hoped to create a ‘whirring’ of scales against each other, all related, yet in their own distinct families, like branches off the trunk of a tree.
Polyrhythms and polytonalities are created by the scaler interactions.
Fullrange
This is an experiment using the various parts of the overtone scale in combination in the single R20 scale. Using rhythm, a bass-line, a harmonic ‘comping’ pattern, harmonic counterpoint, chromatic and microtonal patterns, along with a diatonic melody, all ‘sections’ of the overtone scale are utilized.
Using loop-based composition, an extended form emerges from layering of the various elements.
Through-composed
Now that I have conducted several looping experiments with the Integrated Frequency system, this is a short through-composed (non-looping) composition.
The first section is in R20 (essentially the key of E), using the full spectrum of the 30-note scale. The middle section is in R30 (the key of B), again using the full scale-range. The final section returns to R20, however, the ending chord cadences on B Major, while the ending low note is E.
The first and last sections are in 75BPM, congruent with the R20 scale, while the middle section is in 56.25BPM, congruent with the R30 scale.
I endeavored to use interesting aspects of the overtone scale, such as melodic microtonal gestures, contrapuntal chromaticism and harmonic motion.
Integrated Frequency (YouTube) lecture
see, Integrated Frequency - Part 2
Notes
[1] Paul Griffiths, Modern Music and After, 3rd edition (New York: Oxford University Press, 2010), 18
[2] Ibid., 19
[3] Ibid., 22-23
[4] Ibid., 23
[5] Ibid., 23
[6] Ibid., 36
[7] Ibid., 63
[8] Ibid., 51
[9] Ibid., 99
[10] Ibid., 100
[11] Ibid., 110
[12] Ibid., 52
[13] Ibid., 339
[14] Ibid., 343
[15] Henry Cowell, New Musical Resources (New York: Cambridge University Press, 1996), 99-100
[16] Benjamin Dwyer, György Ligeti, of Foreign Lands and Strange Sounds (Rochester: Boydell Press, 2011), 27
[17] Richard Steinitz, György Ligeti, of Foreign Lands and Strange Sounds (Rochester: Boydell Press, 2011), 181
[18] Ibid., 212