Mari Kimura SUBHARMONICS String Theory: A RevolutionarY technique for the violin

In April 1994, at a solo recital given by Mari Kimura in New York City, subharmonics were introduced to the musical community as a revolutionary bowing technique to extend the violin's range by a full octave below on the open G string without changing the tuning. Subharmonics require precise control of the bow pressure and speed, in order to freely utilize this extended bowing technique. Subharmonics can be used as a musical element for composers to explore additional possibilities for the violin and other string instruments. Kimura is further developing the technique and continues to write works for the violin utilizing subharmonics.

Subharmonics were introduced to the public as a musical element in the third movement of ALT (1992) for solo violin written by Kimura. This extended bowing technique allows one to play notes below the fundamental by applying precise amount of bow pressure and speed. As a result, the range of the violin was expanded one octave below the open G, the lowest string of the instrument. The musical effects of subharmonics were noted as a "revolutionary bowing technique [with] astonishing effect" [1]. During the past year, subharmonics have been developed further. It is now possible to control the various chromatic intervals below the fundamental pitch, highly dependably for performance purposes. Each interval such as second, third, fifth, and octave below the fundamental pitch, requires different speed, pressure, and location of the bow on the string. One can isolate different subharmonics by normalizing the speed and pressure of the bow accordingly; precise control achieves regular, repeatable, and dependable results.
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Review in 'New York Times'; A Violinist Tests Limits In Music Of Her Time

By EDWARD ROTHSTEIN ........... The history of the violin is partly a history of liberation: liberation from technical constrictions, styles, attitudes. Composers and players have always been challenged to push at the limits of this compact collection of wire and wood, to make it do something it has never done before. It was seen as the instrument of angels in the 18th century, of the Devil in the 19th, of the irrational and inexpressible in the 20th.
In a remarkable debut recital on Tuesday night at Merkin Hall, presented by the League of Composers/ ISCM, the nation's oldest organization devoted to contemporary music, Mari Kimura set about pushing virtuosically at any boundaries that might still exist. Ms. Kimura's teachers have included Joseph Fuchs, Roman Totenberg, Toshiya Eto and Armand Weisbord. She has studied composition with Mario Davidovsky at Columbia University and has played widely at international festivals. Her interest, though, is not in the traditional repertory, but in the most recent attempts at liberation from it. With a relaxed agility and a mastery of 20th-century bowing and plucking techniques, Ms. Kimura created a series of musical worlds at the border of the technically possible.
The major promise of the recital was to reveal Ms. Kimura's "revolutionary bowing technique" that allowed her to sound notes a full octave below the instrument's G-string, reaching into the realm of the viola and the cello. This was an astonishing effect, used in the third movement of the violinist's own composition "ALT" (1992). Ms. Kimura bore down steadily on the G string in a way she cannot, even now, successfully describe, sounding particular pitches, including a low D sharp, a low A and a gravelly cellist's G on the bottom line of the bass-clef stave.

Researchers in acoustics, physics and electrical engineering at both Stanford University and the Massachusetts Institute of Technology are trying to establish just how Ms. Kimura succeeds in producing these subharmonics (bow-hair tension, amount of rosin and the quality of the string affect the sound). She discovered the pitches while practicing Russian-school bowing exercises. Ms. Kimura did not use them for novelty's sake, but as elements in a sweet, ghostly composition meant to expand musical territory as well. She wrote in the program notes: "I wanted to free myself from the boundaries of Western musical idioms, associated with traditional violin literature, to reflect my own Japanese heritage in my compositions."

MARI KIMURA BIOGRAPHY

Mari Kimura is at the forefront of violinists who are extending the technical and expressive capabilities of the instrument.  As a performer, composer, and researcher, she has opened up new sonic worlds for the violin. Notably, she has mastered the production of pitches that sound up to an octave below the violin’s lowest string without retuning.  This technique, which she calls Subharmonics, has earned Mari considerable renown in the concert music world and beyond.  She is also a pioneer in the field of interactive computer music.  At the same time, she has earned international acclaim as a soloist and recitalist in both standard and contemporary repertoire.

Born in Tokyo, Japan to two professors (father, architecture; mother, law), Mari began violin lessons at the age of five with Armand Weisbord, a student of Eugène Ysaÿe and former concertmaster of the CBC Orchestra in Ottawa.  After earning a Bachelors’ degree in violin performance from the Toho School, Japan’s top conservatory where she studied with Toshiya Eto, she moved to the US to study with Roman Totenberg at Boston University.  One semester away from a Masters’ degree, she needed an extra credit to maintain her student visa.  Out of curiosity, she chose an electronic music course, setting her on a new artistic path – in her words, “carrying on the old traditions of the violin while using the tools of our age.”

Mari is also active as an improvising musician; three recordings feature her in that role.  Her first CD, Acoustics, released in 1993 on the Victo label, is a collaboration with guitarist/world music producer Henry Kaiser, together with guitarist Jim O’Rourke (formerly of Sonic Youth) and saxophonist John Oswald.  Irrefragable Dreams, an album of improvisations with avant-garde flutist Robert Dick, followed in 1996; Allmusic called it “poetic…highly recommended.” Mari teamed up with improvising multi-instrumentalist Roberto Morales Manzanares for Leyendas (1999), described by Strings magazine as “simply stunning… Kimura brings a rare level of excitement and grandeur to improvised music.”

Technique

Some subharmonic intervals are obtained by exerting the same amount of bow pressure. The only notable element that separates these intervals is the slight shift in the emplacement of the bow on the string. [Fig. 1] shows the relative bow location on the string for playing different subharmonics on open G string. ("open G" is called here as a musical note "G2", and other notes are musically labeled along with each interval). For example, subharmonic minor second (Sm2), major second (SM2), minor third (Sm3) and major third (SM3) are obtained by using almost identical bow pressure and speed, which is called {P1}(see [Fig. 1] No. 1) . Similarly, the bow pressure {P2} is identical for the subharmonic perfect fifth (SP5), minor sixth (Sm6), diminished fifth (Sd5) (see [Fig. 1] No. 2), and so is the bow pressure {P3} for subharmonic octave (S8), minor seventh (Sm7) and major seventh (SM7) (see [Fig. 1] No. 3). Larger intervals such as ninth, eleventh (octave & third), and thirteenth (octave & fifth), can be obtained in a similar manner.

Quality

The sound quality of subharmonics can be described as more nasal than the violin played normally. The different proportions of harmonic components shows a clear difference in sound quality. [Fig. 2] shows the FFT analysis of open G "G2" ([Fig. 2] No. 1) and subharmonic octave "G1" ([Fig. 2] No. 1). The fundamental frequency of G2 is approximately 196Hz, followed by the harmonics at 392Hz, 588Hz and 784Hz. The frequency that acts as the fundamental frequency for the subharmonic octave "G1" is 98Hz, as is shown in [Fig. 2] No. 2. The amplitude of the first harmonic of G1 at 196Hz, which is also the fundamental frequency of G2, is suppressed almost as half as much as the first harmonic of G2. The loudness of subharmonics does not vary much and rather uniform; it is usually rather hard to control.
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Variables

There are several variable elements that are included in order to obtain subharmonics: the amount of rosin, the age and thickness of the string, and the composition of the string. These variables are often combined and create technical problems, mainly affecting the location of the bow on the string. The amount of bow pressure is the most crucial element, however, which one must imagine precisely before playing subharmonics.

Musical Creations

Subharmonics usually sound somewhat harsher than notes played normally, and they can be used as an effective element in musical compositions. Some composers have written works for Kimura, utilizing subharmonics. For example, Variants by Jean-Claude Risset (1994), and Effective by Robert Rowe (1996), along with works by Kimura herself: ALT (1992, revised 1994), Gemini (1993) and Caprices (1996-).
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Discovery

One day nearly 20 years ago, violinist Mari Kimura was practicing a son filé exercise she’d long used to improve her sound on the E string, and on a lark decided to try it down on her G string. She drew her bow very slowly, and applied a bit more pressure than usual. Suddenly she heard a crunch and a scrape—and a G note a whole octave below what the violin is supposed to be able to play without changing the tuning.

This extended bowing technique, producing what Kimura calls “subharmonics,” wasn’t exactly new; Paganini is thought to have toyed with it during his practice sessions. But with a few exceptions, such as George Crumb’s Black Angels, it wasn’t commonly found in compositions. Kimura, who first described the technique in a 2001 Strings article, resolved to master the technique, stabilize the sound, and use subharmonics in her own music—and encourage other composers to use them, too.

Through practice, Kimura managed to eliminate most of the scratchy transient noise associated with the technique, and she incorporated subharmonics into a recital she gave in 1994. Since then, she’s been refining the technique. By carefully controlling the location, speed, and pressure of the bow on the string, she can play almost all the chromatic intervals below the string’s fundamental notes. When she plays the open G, she can produce the G an octave below; when she plays middle C on the G string—by pressing the string down to the fingerboard normally, not barely touching it as with “regular” harmonics—she can produce the viola’s open C. On the open G string, she can also get minor seconds, minor thirds and, she says, “on a good day,” a perfect fifth.

Still, not all the notes are stable, even on the best string for the purpose, the G. (Subharmonics are possible on the higher strings, but the sound isn’t so good.)

“The hard part is to play reliably on cue,” she says. “Anybody can do this by accident, but not necessarily on demand. It takes a lot of control of my right arm, as well as mental preparation.

“When I first started I thought it was all in my head: when I believed I could play it, I could play it, but when I had doubts, I couldn’t. It’s like ‘the force is with you.’ But that’s not good enough for a performance. I have to keep a kinetic memory of my right arm where that vibration locks in—one way to play the octave, another way to play the third. The fifth comes only on a very, very good day.”

In performance, Kimura says, “That’s not good enough for me to use.”

Even with the notes solidly in her arsenal, Kimura can’t really play a full line using subharmonics. She can play a few notes in succession, but only so long as the single stroke continues. And she can’t play them reliably with the upper bow because, like most string players, she has better control at the frog.

“I can sustain about three seconds worth of stroke for this,” she says, “and that’s enough to play a trill.”

Scientists aren’t quite sure how subharmonics work, aside from the octave effect. To help understand the phenomenon, Kimura recently visited Stanford University’s Max Mathews, a pioneer in the world of computer music who is devising a special microphone to measure subharmonic vibrations and help analyze the effect. Kimura looks forward to learning the results of Mathews’ study.

Meanwhile, she has mastered sufficient “good enough” notes to incorporate them into an entire CD’s worth of original compositions using subharmonics. The self-produced disc, The World Below G, includes Kimura’s Six Caprices for Subharmonics, a three-movement solo suite called Alt, and two versions (one with interactive electronics) of a piece titled Gemini. Subharmonics also abound on her most recent CD, Polytopia (Bridge 9236), a collection of violin-and-electronics pieces by Kimura and six other composers.

In 2007, Kimura premiered a violin concerto written for her by Jean-Claude Risset, who allowed Kimura to write her own cadenza that doubles as a subharmonic showcase (see music on pages 36 and 37).

It’s not an easy technique to master. Kimura admits that she’s encountered several students who try repeatedly to produce subharmonics but fail until she demonstrates it, and then something suddenly seems to click. “The other day at Juilliard an undergraduate asked me to show him how to do it,” she says. “He just couldn’t make it come out. So I took his violin, loosened his bow hair a little, and I found his octave below G in about five seconds. Then in about ten seconds he could do it, too.”

The first secret, she says, is maintaining loose bow hair. “You don’t want a lot of tension,” Kimura says. “You need enough elasticity on the bow hair that you can really grab the string. That way, you don’t have to use a tremendous amount of pressure. Focus more on how you draw the bow: slowly, maintaining a consistent speed with just a little more pressure than usual.”

When advancing from the open G, the higher the finger position, the closer the bow should be to the bridge to keep a correct subharmonic octave position. This is something you just have to tune by ear. Beyond this, by moving your bow toward or away from the bridge, you can produce subharmonic notes other than the octave. Move the bow toward the fingerboard, and you can get a half-step above your core note (a subharmonic major seventh); move the bow toward the bridge, and the result is a half-step lower (a subharmonic minor ninth).

Old strings are better than new for subharmonics, Kimura adds, because old strings allow you to bow farther from the bridge, where the sound is superior. If you must use a new string, try taking it at the tailpiece end and giving it one counterclockwise twist. This will increase the distance of the correct bow location from the bridge.

“Anyone who can do a son filé can do subharmonics,” Kimura encourages. “But it takes practice. It goes from no sound to ‘My god!’”

Research

In June 2006, I was invited by Prof. Alfred Hanssen at the Physics Dept. of University of Tromsø, Norway to record my Subharmonics.  Prof. Hanssen, a non-linear physicist, and his team are now analyzing the data.  As the technique has grown significantly since the initial discovery more than 15 years ago, it was apparent to me that Subharmonics needed a closer look in the most controlled environment.  I recorded inside an anechoic chamber located in the basement of the University hospital, recording pretty much everything technically possible for me at this time

Strange musical sounds draw scientific scrutiny
Mari Kimura, a New York-based solo violinist who lectures at the renowned Juilliard School of Music, is one of a handful of people who can produce controlled violin tones known as “subharmonics.”

This means she plays sounds that are of lower pitch than is normally possible for a violin, or written into musical scores for the instrument. The tones she plays are more typical of the lower-pitched cello.

The violinist, who plays modernist and contemporary music, has turned her ability to a signature trait in her own compositions and improvisations. But she is as stumped as anyone as to how she does it.

“I don’t really know what it is I do,” she said, because she learned it by “trial and error.”

A team of scientists at the University of Tromsø, Norway, is the latest to take a crack at the puzzle. They claim their unique combination of physics knowledge and musical interest will help them succeed where other researchers have stumbled.

“We have worked with strange and exotic sound systems,” said physics professor Alfred Hanssen.

Kimura said she has made the unique sounds for more than a decade. “I showed it to top researchers in U.S. and Japan. They have all been very enthusiastic.” Some of them tried to study the effect but gave up, she added. “Hanssen’s team is the one who is making the commitment.”

Violins and all stringed instruments follow the same principles. A string vibrates and produces a note whose pitch depends on the frequency—that is, speed—of vibration. Shortening the string leads to faster, smaller vibrations and a higher pitch. Violinists accomplish this by using a finger to clamp the string down onto the wood, reducing the length available to vibrate.

Halving a string’s length raises the pitch one octave, or one trip up the scale. So in theory, a violinist can produce a note as high as he or she wants. It just requires moving the clamping finger further and further along the string, in effect halving the length repeatedly.

But at the other end of the pitch spectrum, the low notes, the instrument has a natural limit: the lowest-pitched string, played fingerless. A player can’t normally play a deeper pitch because he or she can’t lengthen the string.

There are some tricks that enliven the palette of notes available. For instance, holding a finger at the middle of a string only gently—without clamping fully—produces a “harmonic,” or a distinctively bright-sounding note. The brilliant timbre comes from the fact that the whole string vibrates. But, because the finger places constraints on this oscillation, the pitch is the same as if the finger had been clamping.

The deep tones emanating from Kimura’s instrument are thought to be an unusual form of harmonics, which is why they are called subharmonics.
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Hanssen and colleagues analyzed the sounds in an echoless chamber at the university hospital. By putting even pressure on the string through fine, steady bow movements, Kimura can conjure many different tones from one place on the string, the scientists noted.

“Kimura makes a violin string vibrate in a totally new way. In physics we call this a driven and damped non-linear system, which we are particularly preoccupied with in our research,” Hanssen said. Driven and damped systems are, respectively, ones in which an outside force either stimulates or quashes a vibration. Nonlinear systems are ones in which there is no simple relation between a disturbance and the response to it.

Kimura said that if Hanssen’s team comes up with any answers, these may be useful to her, by suggesting yet new avenues to manipulate tones. “As an artist you are always searching for ways to expand the sound,” she said.

In 2008, Prof. Max Mathews, prof. Emeritus at Stanford University, visited me at Juilliard then in February 2009, we took some measurements and video recordings of Subharmonics at Stanford

Music Technology Pioneer Visits Juilliard
How many of us have not listened to a digitally recorded and mastered CD, or to synthesized sounds or MP3 files? Today, virtually all music is recorded and mastered digitally—in many cases also being digitally enhanced, such as with reverberation and equalizing. Today, we do not get our music into the world without having it go through some kind of digital process, unless it is only heard live, with no amplification or modification.

All this would not have been possible if it weren’t for Max Mathews, known as “the father of computer music,” who changed our lives for good. Max was one of the first people in the world to think, “What if we used computers to make music?” back in the 1950s at AT&T Bell Labs, where he directed research in acoustics and speech synthesis, among other pioneering fields.

On October 10, Max—together with his son, Boyd Mathews, who also works at Bell Labs—visited me at Juilliard. Born in Columbus, Neb., in 1926, Max is vigorously continuing research as professor emeritus at Stanford University in California. He has been interested in a bowing technique I have been working on called subharmonics, which allows notes to be played below the open G string on the violin without changing the tuning, and he wanted to record my sounds during his trip. When faculty member Michael Czajkowski learned of Max’s visit, he suggested that we conduct our recording session in front of his Introduction to Music Technology class. The fortunate students got to see this historic figure up close at work, and were treated to an impromptu lecture on string acoustics and a fascinating high-speed video of string oscillation.

As Max explained, “I think that, by analyzing Mari’s subharmonic string vibrations, we may learn some interesting new things about how violins work. Herman Helmholtz, in the mid-19th century, was able to show that the normal vibration of a ‘good’ violin sound is made by a simple triangular waveform of string vibration. He also showed the physics of how this waveform is generated by the ‘stick-slip’ motion of the bow on the violin string, and how the pitch is properly controlled by the string length, weight, and tension. In order to get subharmonics, Mari’s bow technique must produce a very different string vibration waveform.”

In 1994, I first introduced subharmonics as a musical element during a recital at Merkin Hall. Since then, I have been continuing to improve the technique, producing several intervals other than the “pedal tone” (one octave below), which has long been known to musicians. I have also met with numerous physics and acoustics professors who wanted to study and extract the theory behind this phenomenon, which has taken me around the world from Tokyo to Massachusetts to Tromso, Norway.

Musically, I have been working extensively with composers using this technique. One, a French composer named Jean-Claude Risset, is a former director of research at IRCAM and—as another pioneer of musical acoustics in the ’60s at Bell Labs—a good friend of Max. Last year, Risset wrote a violin concerto titled Schemes, which received its premiere at Suntory Hall in Japan with the Tokyo Symphony Orchestra. I performed my own cadenza, also incorporating subharmonics. Schemes is the world’s first violin concerto using subharmonics, aside from my own concerto, which I premiered in 1999.

It is for this musical purpose—not for the sake of novelty, but to expand the possibilities for future violin repertoire—that I have been most keen on collaborating with scientists in attempts to uncover the theory of subharmonics. In the past, I have found new intervals and new techniques only empirically. If I could learn the theory behind subharmonics, it might suggest to me what else can be achieved and how.

Max Mathews is the latest scientist, and one of the world’s foremost authorities in the field, to tackle this scientific “riddle.” After all, as an enthusiastic amateur violinist himself, he has all the incentive he needs to find out.

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