Rhythmiconic Sections

RHYTHMICONIC SECTIONS by American Composer David Mooney. Computer Music . . . Truly Unique, Dynamic and Engaging!! AUR CD 3087 $16.95

Built in 1931 by Leon Theremin for American composer Henry Cowell, the rhythmicon generated rhythmic beats according to the intervals of the harmonic series. For every one beat of the fundamental, the second harmonic beat twice, the third three times, the fourth four times, and so on, through the 16th harmonic. The performer could sound any combination of the harmonics by means of a 17 key keyboard. (The 17th key sounded a syncopating beat.) As long as the key was depressed, the harmonic sounded. The performer could change the tempo and the pitch of the fundamental, and therefore of all the harmonics, with levers and a rheostat. On a second improved model, the performer changed pitch and tempo with foot pedals.

The rhythmicon used an "electric eye" to create the rhythmic patterns, an engineering solution that was proposed by Cowell. Each key had a corresponding cog wheel that rotated while the key was depressed. As the wheel rotated it interrupted a beam of light focused on photoreceptors (the "electric eyes"), which divided the beam into impulses corresponding to the beat pattern of the harmonic, thus triggering the sound. Theremin built two machines. After losing interest in the device, Cowell lent the original one to the Psychology Department at Stanford University. It was discarded in 1938, by which time it had ceased to function. The second machine was commissioned by Charles Ives for Nicolas Slonimsky. After using it for a while, and even writing a composition for it, Slonimsky eventually sold it to Joseph Schillinger, who had known Theremin since the early 1920s and had a lifelong interest in technology and music. Schillinger's widow donated the device to the Smithsonian Institution in 1966. A third machine was built in Russia after Theremin returned the Soviet Union in the late 1930s.

As early as 1915, Cowell had discussed with Russell Varian the concept of a mechanical device that could play the complicated rhythms he was exploring at the time. In particular, two string quartets, the Quartet Romantic and the Quartet Euphometric, employed his method of converting chords into rhythms. The rhythms that resulted from this process -- 6 beats against 5, 7 against 3, 10 against 4, etc. -- were so complex that Cowell devised differently shaped note heads to represent the rhythms: triangles for "third" notes, square for "fifth" notes, diamond for "seventh" notes, etc. In notes for the quartets written in 1964, Cowell states, "...the meters for the most part were necessarily so complex that they were obviously unperformable by any known human agency, and I thought of them as purely fanciful." Both quartets, however, have been recorded since the composer's death.

Unfortunately, the rhythmicon did not have the melodic flexibility Cowell desired. In his notes for the quartets he wrote, "But since there was no way of giving melodic freedom by varying the note lengths in a single part, and no method of accenting, these early quartets still could not be played on it." Instead, Cowell composed works for the machine: Rhythmicana, later renamed Concerto for Rhythmicon and Orchestra, and Music for Violin and Rhythmicon. He also wrote several solo works which he played when demonstrating the instrument. After demonstration concerts in New York on January 19, 1932, and San Francisco on May 15, 1932, (A Paris concert scheduled for February was canceled), Cowell lost interest in the Rhythmicon and composed no further works for it.

Rhythmicana remained unperformed at Cowell's death in 1965. Leland Smith, an American composer then at Stanford University, became interested in the piece and in the rhythmicon in 1970. He was able to locate the score and proceeded to realize the rhythmicon part using his SCORE notation program on a DEC PDP10 computer. Each harmonic was assigned to a separate instrument. Tempo was assigned as a relative value based on the TEMPO feature of SCORE, which stored the absolute tempo. The pitch of each of the harmonic instruments was similarly a multiple of an otherwise silent instrument that stored the value of the fundamental pitch. This approach -- a separate instrument for each harmonic -- was chosen, Smith writes, "...since most of the time only a few of the rhythms are used at once, it was more economical, from the point of view of computer time, to set up an 'orchestra' wherein each instrument played a single rhythm." Rhythmicana was premiered by the Stanford Symphony Orchestra, conducted by Sandor Salgo, on December 3, 1971.

Sources

Cowell, Henry. New Musical Resources. New York: Cambridge University Press, 1996

"Preface". Quartet Romantic; Quartet Euphometric. New York: C. F. Peters, 1974

Davies, Hugh. "Rhythmicon (Polyrhythmophone)." New Grove Dictionary of Musical Instruments. Ed. Stanley Sadie. 3 vols. London: MacMillan Press, 1984

Glinsky, Albert. Theremin: Ether Music and Espionage. Urbana: University of Illinois Press, 2000

Mead, Rita. Henry Cowell's New Music, 1926-1936: The Society, the Music Editions, and the Recordings. Ann Arbor, UMI Research Press, 1981

Schonberg, Harold C. "Music Leon Theremin; Inventor of Instrument Bearing His Name Is Interviewed in the Soviet Union." New York Times, (April 26, 1967), 40

"A Short History of the Rhythmicon." [WWW document]. URL http://www.mutelibtech.com/mute/came/camemore.htm (1997, Dec 1)

Smith, Leland. "Henry Cowell's Rhythmicana." Yearbook for Inter-American Musical Research, 9 (1973), 134-147

Szigeti, Joseph. With Strings Attached: Reminiscences and Reflections. 2nd ed. New York: Knopf, 1967

Theremin: An Electronic Odyssey. Videocassette. Dir Steven M. Martin, 1995

THE VIRTUAL RHYTHMICON ENVIRONMENT

Far more than Cowell’s theories, it was the device, Theremin’s machine, and the conceptual framework for composition that the rhythmicon suggests which inspired Rhythmiconic Sections. After a few "proof of concept" studies, the initial virtual rhythmicon expanded into a complex “virtual rhythmicon environment.” Cowell’s concept was extended up through the 24th harmonic, and control was enabled over far more parameters than pitch and tempo. As the work progressed this environment grew into a set of 25 omni-related rhythmicons. This environment and almost all work on the pieces completed using the Symbolic Sound’s Kyma system.

Analysis of the rhythmic structure and internal harmonic relationships of the first experimental rhythmicon, the “basic rhythmicon” that recurs throughout Rhythmiconic Sections, provided the tools for my compositional approaches to the rhythmicon environment. For Rhythmiconic Sections the fundamental pitch (first harmonic) of the basic rhythmicon was set at 65.406 hz, two octaves below middle C This was the lowest pitch of Theremin's rhythmicon. The pitches of the remaining 23 harmonics are whole number multiples of the fundamental. This can be heard in its most unadulterated form in the first section, “Albert’s Bicycle.”

Cowell's objections to the rigidity of Theremin's rhythmicon were easily overcome in Kyma’s graphical programming environment. Similarly to Leland Smith's use of SCORE to realize the rhythmicon part of Cowell's Rhythmicana, I used the scripting capability in Kyma to set tempo, fundamental frequency and other variables. Changing the value of variables in the script changed the relative values of any scripted parameter I needed to adjust for all 24 harmonics.

After creating the basic rhythmicon, I developed additional timbres and means of control during the course of composing Rhythmiconic Sections. These other timbres varied considerably from the original clear and clean approach used in the basic rhythmicon. They included synthesized gong/bell sounds, waveform sounds in which MIDI controllers let me shift among a variety of waveforms to vary timbre, various sustained tones for low harmonics, amplitude modulation sounds with real time control of the pitch of the modulators for timbre shifts, layered waveforms with real time detuning control, and samples. Short samples play through during each beat of the harmonic(s) to which they were assigned with appropriate time and pitch shifting. Section 16, “Pork with Ham,” contains an example of this approach. Long samples which sustained perhaps over many beats were enveloped, allowing them to be heard when triggered

In addition, I developed the ability to sustain the harmonics as constant tones. The sustained tones could vary in volume relative to the intensity of beats to the extent that the beats could be completely submerged. The use of sustained tones is especially evident in Section 19, “Screws in Their Shoes.” Another development was the capability of enveloping the beats of a rhythmicon of quick tempo to form beating-beats of a slow meta-rhythmicon. This can be heard in Section 23, “Violation: Time Expired.” The most extreme timbral variation was the use of spectral analysis on spoken text and the subsequent splitting of the spoken words into beating harmonic layers. This can be heard in Section 4, “Malfunction 54.” I have continued to refine these techniques and means of control in works composed after Rhythmiconic Sections.

Thus in addition to the frequency and tempo settings of Theremin's electromechanical rhythmicon, the following parameters and aspects of the rhythmicon were controllable in the virtual rhythmicon environment developed in Kyma for Rhythmiconic Sections. Control was accomplished via MIDI controllers sent to Kyma from sequencing software, MIDI faders, or by changing variables in Kyma scripts.

Attack-Decay-Sustain-Release (ADSR): ADSR envelopes controlled the beating of the harmonics. The ability to vary these parameters in real time controlled the expressive quality of the rhythmicon and could also affect the timbre of the sound of the rhythmicon as a whole. Generally I confined envelope parameters to the length of the beat of the harmonic, but when needed, the release parameter could extend beyond the beat for any desired length of time.

Volume: All harmonics of the basic rhythmicon are "on" all the time. A separate MIDI controller determined the volume level of each harmonic and, thus, which sounded at what level at any given moment.

Pan (stereo position): Another set of controllers determined the stereo position of each harmonic. I developed a set of default pan settings that was used throughout Rhythmiconic Sections, though these could move as in the spread from center to default positions heard in Section 1 and the collapse back to center at the end of Section 24.

Staccato/Sostenuto: All harmonics could be set individually at any point along a continuum between clearly separated beats and non-beating sustained tones. This control was separate from the ADSR controls, thus, for example, sharply articulated beats could occur partly or largely submerged in sustained tones, or conversely, beats could rise like smooth bubbles on the surface of sustained tones.

Timbre shifting: The timbre of a number of the sounds could be shifted in real time using MIDI controllers.

The Kitchen Sink: I tossed in non-rhythmicon sounds as needed. This is most evident in Section 21, “Maxine’s Buzz.”

The combination of timbre shifting, ADSR shifts, and shifts in the balance of sustained tones to beats, not to mention panning and relative volume, provided a complex and fascinating compositional environment.

RHYTHMICONIC SECTIONS

The image along the bottom of this page represents half of a “rhythmicon measure” employing 24 harmonic tones. The second half is a mirror image of the first half. Harmonic one (H1), at the bottom, is the fundamental. The beats of H24 are numbered in the top row. The tempo of the fundamental is the base beat of the rhythmicon. The tempos of the harmonics are whole number multiples of the base. Thus the measure can be thought of as consisting of a single beat, or 1/1 time if you will, that is further subdivided into 23 distinct rhythmic patterns. The image reveals many graceful interlocking curves that describe visually the swaying or rocking pattern of sound heard when all harmonics of the rhythmicon are sounding. The accordion-like squeezing and spreading of beats are clearly evident in many of the sections.

There are many possible points of entry, numerous curves and patterns that could be used for compositions based on the structure of the beats. For Rhythmiconic Sections I chose to work with the curves that begin (descending) or end (ascending) on each of the 24 beats of the highest harmonic. I call these curves "cascades," a kind of special case arpeggio.

On the downbeat all harmonics sound simultaneously. They are vertically aligned. The first descending cascade occurs, then, on the second beat of the 24th harmonic, the second descending cascade on the third beat, etc., with the 24th descending cascade beginning on the downbeat of the next measure. Each harmonic of the cascade begins to sound along a hyperbolic curve. The curve is longer for each descending cascade, and the spread between the initial sounding of each harmonic increases from higher to lower harmonic.

In the first descending cascade, each harmonic begins within the first measure. The fundamental sounds on the downbeat of the second measure. In the second descending cascade harmonics H24 down through H3 sound in the first measure. H2 sounds on the downbeat of the second measure and H1 on the downbeat of the third measure. In the third descending cascade H3 sounds on the downbeat of the second measure, H2 at the midpoint of measure two, and H1 on the downbeat of the fourth measure. The gap between H2 and H1 spreads in succeeding cascades. This makes sense algebraically, but is nonsense to the ear. Thus I adopted the compositional (and programming) rule that in terms of the cascades H1 would never sound more than one measure by itself This artificially shortens the curves. Internally, however, in the longer cascades, measures may be entirely skipped over before the next harmonic begins to sound. In this scheme, then, the shortest cascade is one measure long, and the longest spreads over 13 measures.

Each beat of the 24th harmonic is also the end of an ascending cascade. Ascending cascades are viewed from right to left. Thus the first ascending cascade begins on the downbeat of the measure and ends on the 24th beat of the 24th harmonic. The second ascending harmonic ends in the second measure on the 23rd beat of the 24th harmonic, and so on in reverse of the descending cascades. The spread decreases as higher harmonics sound. Because H1 always begins on the downbeat of the first measure in ascending cascades and always begins on the downbeat of the next measure in descending cascades, descending cascades are always one measure longer than the corresponding ascending cascade.

The cascades represent starting and stopping points for the sounding of the beats of the harmonics. Once started, a harmonic might continue to sound, sound only for a short while, or sound only once. Any subset of any cascade can start or stop at any point along any of the 48 curves. In addition, using Cowell's concept of tone clusters, any subset of harmonics can start or stop at once rather than along a cascade curve. That is, the volumes can be turned up from 0 at any time. Depending on the sound used and parameter settings, each harmonic will be heard instantly in mid-beat at some point of its ADSR curve or as a sustained tone, or not heard until its next beat. All of these methods in many combinations can be heard throughout Rhythmiconic Sections.

In addition to the sounds used for the harmonics, the tempo of the rhythmicon is highly relevant. Since the sections of Rhythmiconic Sections are short I elected to use the rhythmicon for most part at a fairly fast tempo. The basic rhythmicon sounding at four seconds to the measure is the maximum speed at which I can hear all 24 harmonics as discrete sounds, and thus the setting of the four second measure as the basic tempo in many of sections. At faster tempos the harmonics begin to blend. At extreme speeds, the rhythmicon becomes an exercise in additive synthesis -- a single complex sound that can be varied by changing the volume levels of selected harmonics. Sections 3 and 20, “Wave Motion Machine,” make use of this effect. At slower tempos melodic attributes of the rhythmicon begin to reveal themselves. As the tempo continues to slow there comes a point -- about 20 seconds per measure to my ears -- when the harmonics become separate notes, and the rhythmicon begins to lose coherence as a unified entity. Listen to section 23 to hear one minute measures.

There is no reason to limit the music of the rhythmicon to a fixed fundamental sounding at a fixed tempo. Initially I thought to vary the fundamental and tempo of a single rhythmicon, as was possible with Theremin's instrument, but a more elegant and musically satisfying solution to me is to posit multiple rhythmicons. Taking Cowell's idea of "polyharmony" as a cue -- that each harmonic of a given harmonic series is itself the fundamental of a new series -- I devised a set of 24 theoretical rhythmicons. Each harmonic of the basic rhythmicon (R1) is the fundamental of a new rhythmicon. Thus the second harmonic of R1 becomes the fundamental of the second rhythmicon: R1/H2 = R2/H1; the third harmonic of R1 becomes the fundamental of the third rhythmicon: R1/H3 = R3/H1; and so on through R1/H24 = R24/H1.

Using 65.406 hz as the fundamental of R1 in this ever rising scenario, the limits of human hearing are reached (about 20k hz) long before the higher rhythmicons reach their 24th harmonic. For example, R21/H14 is 19229.364 hz and R24/H12 is 18836.928 hz. I added R0, the "bass rhythmicon", with a fundamental one octave below R1 at 32.703 hz. R0 adds weight to the system and helps close the rather large frequency gaps at the low end of R1. This completed the set of 25 rhythmicons. It was not my intention to create a scale or a system of tonalities as such so I accepted the still comparatively wide gaps among the lower tones as an inherent quality of the rhythmicon and composed Rhythmiconic Sections accordingly.

Rhythmically, the rhythmiconic idea of whole number multiples of basic units applies nicely to multiple rhythmicons, in this case inversely so with each R lasting a fraction of the length of R1. If an R1 measure lasts four seconds, an R2 measure lasts two seconds: R1 * 1/2; an R3 measure lasts 1.333... seconds: R1 * 1/3; and so on through R24 at 0.1666… seconds per measure: R1 * 1/24. R0 is twice as slow at eight seconds: R1 * 2. The length of the measure of each R is thus equivalent to the length of the beat of the harmonic of R1 that is the fundamental of that R. For example, an entire measure of R8 would complete between successive beats of R1/H8. R1/H8 = R8/H1 both in frequency and time. Section 17, “The Circassian Chicken Dispute,” uses multiple rhythmicons in a particularly clear manner.

Within the context described above I used two approaches in varying degrees of admixture to compose and realize Rhythmiconic Sections: a deterministic approach in which I made decisions about elements based on creative intent, and an algorithmic approach in which elements were determined by processes based on rules.

For the algorithmic compositional approach I used several data sets, principally the set of 24 numbers. These I dipped into with dice, slips of numbered paper and various homegrown random number programs to choose subsets of harmonics or beats, to determine the rate the volumes of different harmonics would fade over time and the like. A larger data set was derived from the cascades. The cascades extend over different numbers of measures, and each has a unique spread of the harmonics. If each cascade is broken down to subsets consisting of events that occur along its curve in single measures, keeping in mind my shortening of the curves by bringing H1 in closer, there are 360 unique measures. "Unique" in an abstract sense; an empty measure in which no event occurs is counted as unique to that curve. Algorithmic processes might, for instance, determine which of the 360 measures occurred next. The full data set might be grouped into subsets: for example, a subset consisting of all measures in which H6, H7 and H8 occur. Occurrences of measures might overlap to varying degrees. One is fading out as another starts; several are at different points of fading out when a new measure starts, etc. Panning, timbre changes, and the relative level of sustained tones were also determined algorithmically in some sections. In addition Kyma provides the capability of building randomness into the structure of a sound, which was used on occasion as well. Since all harmonics of the virtual rhythmicon are always "on", the degree to which any one is heard or not is a matter of relative volume levels. When using multiple layers of algorithmically determined events, this approach is not unlike dipping my hand into a stream of sound and moving my fingers to create a constantly changing pattern of ripples on the surface.

Larger structural concerns overlay the specifics of the individual sections. For example the lengths of the sections bear the same whole number relationship to one another as that used in tempo and frequency of the rhythmicon. Other multiples exist. The four sections that make use of text or vocal sounds are 4, 8, 12, 16 and 24. Certain timbres recur in similar ways. I will leave it up to the listener to discover more subtle interrelationships.

Titles for most of the sections were applied more or less randomly from a list derived from things read, overheard, seen, or otherwise discovered floating unanchored in my thoughts during the time the sections were composed (August, 1998 – November, 2000). Programmatic implications are therefore inferred by the listener. However, a few notes on some sections are in order.

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