Dancing the Helmholtz on a String (2,104 Hours)

Every cellists desires a sweet and resonant tone, but what is the secret that differentiates a rough scraping bow from a beautiful and pure buzz?   In an effort to answer this question, a curious German physicists named Hermann von Helmholtz, peered into a “vibration microscope” in 1863 and saw something that changed our understanding of sound. They didn’t have slow motion video technology and wouldn’t for another 70 years, but what Helmholtz discovered was as clear to him as the video below…

You may have already heard of the “stick-slip” action of the string and the bow hair, but the way the string slips and returns to the hair is fundamental to a pure tone quality.   In the first fraction of a second, all is chaos, but the pattern that emerges within the dancing string is a magical side effect of the nature of bowed instruments, like pistons on a train wheel transforming that straight line motion of the bow into a rolling rotation on the string itself.  This circular wave of a cello string is called Helmholtz Motion after our intrepid German physicist mentioned above.

When viewed from above the wave appears to be tracing out a familiar elliptical “eye” shape (shown in blue below) and this standing wave pattern is what our eye actually sees when we bow the string.   This very unique vibratory pattern is specific to bowed instruments such as the violin or cello, and is what gives these marvelous instruments their characteristic rich bumblebee sound.

Figure 1

Well this is all very cool display of physics, but I imagine at this point you are wondering “how does this impact my cello playing”?   Well for starters, the motion described above is the idealized version and is only the result of very good bowing technique.  When we use improper technique, the pattern gets distorted which results in poor tone and unwanted noises.  For example, using too much or too little bow speed, or letting contact point drift up and down the string can cause the hair to stick to the string too long or in wrong part of the wave which results in a diminished irregular wave, a more gritty tone, and plenty of unnecessary string noise…

Basic Elements of Good Bow Technique:

There are TWO basic ingredients to maintaining a pure Helmholtz wave:

  1. Steady contact point:   This means no sudden changes of the distance from the bow hair to the bridge.  This is easier if your bow is parallel to the bridge.  Changing the contact point changes where the wave re-sticks to the bowhair, and should only be done with great intention during longer notes.  For shorter notes, alter the contact point by lifting and resetting the bow entirely at opportune points between notes.
  2. Matching your bow speed with your bow pressure:  the more weight you apply to the bow, the faster the bow speed must be.   However, to keep things interesting, when you are closer to the bridge, you need more pressure but less speed!  Confused?  Welcome to the cello.

The Schelling Diagram (below) clears things up a little bit by showing the range of viable pressures at various contact distances from the bridge.  The range of clean sounding Helmholtz motion is shown in yellow.   By wandering into the areas outside of the yellow region you will discover the source of sound effects for horror movies.   The term β simply describes the contact point as a fraction of the total string length, so smaller values of  β are closer to the bridge, and larger values are closer to the finger board.  Notice how the range narrows  as you get closer to the bridge, requiring more force and greater precision, and widens as you get closer to the finger board, requiring less force and less precision.

Schelleng Diagram

The range of viable forces near the bridge vary by around 30%, but near the fingerboard they can vary by more than 5000%.   Since the different strings require such different forces to activate them, this makes string crossings and chords quite a bit more challenging when playing near the bridge because the pressures required are far more precise and differentiated.  Whereas playing nearer to the fingerboard, the required forces are much more similar for adjacent strings and can even overlap.


The Importance of Equipment, Strings, Intonation, and Bowing Tricks:

There are many other variables that impact Helmholtz motion, such as the diameter and elasticity of the string, the stickiness of rosin, resonance and responsiveness of the cello itself, the difference in mass and tension of strings during string crossings, etc.   All of these elements must be taken into account, and adjusted each time you change rosin or try out a new brand of strings.  Playing in tune will also make Helmholtz motion a bit easier to begin or maintain because the cello won’t resist new vibrations that match the overtones of an open string or those of a note that is already resonating in the wood.   Granted this resonance doesn’t work as well for dissonant intervals (ie major and minor 2nds), but it does still work if your strings are particularly resonant (Evah Pirazzi Golds are excellent at this).

As you saw in the first video above, Helmholtz  Motion is not necessarily instantaneous.  This is because the force required to start the string moving is greater than that required to simply maintain that motion.  Any delay in responsiveness no matter how short can dramatically impact sound quality. As a general rule a faster responding setup usually means a brighter sound.   Brighter is not always better, and many cellists who can’t afford cellos that have both speed and depth will sacrifice depth of sound for speed by fitting their cello with brighter and more responsive strings because a higher pitched sounds can still be fairly sweet if you can achieve more pure Helmholtz motion.   Hence the popularity of tungsten wound C and G strings…  extremely responsive compared to their chromium wound counterparts, but also quite bright sounding.  There has also been a move towards altering the mass, tension, and elasticity of strings so that adjacent strings aren’t quite so different.  This can be seen in the evolution from standard Jargar strings to the more modern Larsens, EPs, and Thomastiks.

Some cellists will compensate for a reluctant C string by plucking it right before bowing it.  This  little trick gives the string an initial momentum and reduces the force required to start Helmholtz motion.  This is also why starting an accented bow stroke is relatively easy: the initial force is always sufficient.  To compensate for the initial resistance when playing an unaccented note, you can start off with low pressure, and gradually sink more weight into the string in the middle of the bow stroke to get a fuller sound, then ease off the pressure at the end of the stroke.  This is like the gentle sinking motion a boat makes when lolling in and out of the water.


Consequences for Continuous Bowing

One important consequence of the stick-slip wave motion is that the direction the wave travels (clockwise vs counter clockwise) depends on the direction of the bow.  If the bow is going in the opposite direction, then the string would release in a clockwise direction (instead of counter-clockwise).   This has significant implications for bow direction changes.  Some controversy has arisen over whether or not it is possible to create an inaudible bow change by continuing the momentum of the original string vibration: the so-called “endless bow”.  However a Helmholtz wave moves very much like a train wheels, propelled in a circular motion by straight line cranks and pistons (ie the bowing itself).    Much like a rolling iron train wheel, the circular energy of the wave has it own momentum once pushed into motion.  Changing bow directions reverses the rotational momentum of the wave and inverts the wave pattern, which requires both time and energy.   It is impossible to invert the wave without cancelling or at least disrupting the original wave.  This means that there will always be a small disruption in the generation of sound from the strings no matter how good your technique is.  All you can do is minimize the pause by making it as short as possible and maximizing residual resonance of the sound via room acoustics and if you are very lucky: an extremely resonant cello.




9 thoughts on “Dancing the Helmholtz on a String (2,104 Hours)

  1. I’ve been reading you for a while now, and I’m excited to read more about your journey! (you actually inspired me to write my own cello-learning blog =) ) Thank you.

  2. Pascal says:

    Hey Mindful Cellist, it was great to read you again,

  3. Thanks for the positive feedback. I truly hope this information is useful. Despite the long breaks, I am usually working on a several posts at a time, and deciding between what seems most relevant and developed enough to share. I have a “practice” post that has been sitting in the queue for over a year that is long overdue. Just made a few breakthrough discoveries worth sharing that I hope to post soon. For now, I just updated the booklist with a new title that is blowing my mind about how to practice effectively.

    • Pascal says:

      The information is really useful. I often go back to earlier posts (for instance “Targeting Tone” which contains a lot of useful information).

      There is a topic I’m specially interested in, but have some difficulty finding exactly what I’m looking for, maybe you came across something.

      My teacher has me practising the Prelude 1 of Bach’s suites in particular because it is a great piece to practice playing in tune (in part due to the multiple string crossings and arppegios with notes that keep vibrating). And so for instance, when playing the B in the opening measure, it will be played slightly sharper than the E in the next sequence (G-E-C-B…) which might be 10 cents flatter or so (at least, compared to what would be played on the piano).

      Another example when playing the Swan accompanied by a piano, I must adjust to play in tune (we both play the melody part which makes things worst), but since we practice this only once a week and I don’t really know when to play sharper/flatter, some notes are more out of tune than when I play alone or along a YouTube player.

      Other that using the ear, I was curious to see specifically on the cello when (for instance: in this key signature) this note should be flatter/sharper, but the only information I come across is more generally related to “just tuning” vs “equal temperament”.

      • To get a clearer picture of how the notes actually harmonize, I practice the intonation for the 1st Prelude by playing actual double stops for the Arpeggio sections (for the whole piece really). I tune the B in the first sequence first by tuning the G3 (196hz)on the D string with open G octave (98hz), and then by tuning the G3 with the B3 on the A string. To my ear this sounds best if the B is a bit flatter than piano tuning. Just for reference, standard piano tuning (Equal Temperament) puts the B at 146.9 hz. If you listen closely to Yo Yo Ma recordings, he plays the B between 144.5 and 145.5 with a A=440 tuning. To my ear, he seems to play it consistently “flat” throughout the entire piece. His is also the most relaxed and pleasing sounding to me, but it could just be because his was also the first recording I heard. In other the recordings by other artists, the B is closer to piano tuning and has a bit more tension and anticipation surrounding it. There isn’t really a wrong answer here. It’s all about the beats between thirds and sixths that sound most pleasing to YOU. Whether you decide to constantly re-tune the B to match the notes immediately surrounding it or decide to have it consistently match the root of the key (G), it is entirely up to your artistic sense of what sounds best. To me the G is the pedal tone of the first section, so it makes sense to keep the B steady in relation to the root of the Key, at least in first 4 bars.

        The Swan is a whole different story however because it tends to be performed with piano accompaniment. This means that you have to conform as much as possible to piano tuning. Piano tuning is always the same distance between notes, and the only thing that changes is that in the US we tune to 440 and in Europe they can tune roughly a quarter tone higher. In order to practice with a piano, I recommend grabbing an human-performed accompaniment track and then slow it down with software (or apps) so that you can check your tuning and your rhythm. It’s also great practice to play in more a structured setting so you get more used to the demands of ensemble playing. I’ll post a link on my blog later for expressive human recordings of piano accompaniments – if I don’t have one already. Midi tracks can be OK in a pinch, but they are somewhat mechanical especially for such expressive music.

        • Pascal says:

          You know what, you are perfectly right about B being best played flat. I don’t know how I came with this idea that it should be sharp. Maybe I was confused but the fact that E sounds best (at least to me) when played even flatter. I tune using perfect fifths (that is, I tune A at 220 then tune the other strings using the best sounding fifths, which from memory gave when I tested it the open D at -2 cents, G at -4 cents and C at -6 cents when referring to a “equal temperament” chart. I thought it was the standard way to tune, but I must confess that it is not a topic we discussed much after my initial lessons.

          When I read too much about this topic of tuning and temperament, it is as though my world falls apart and I no longer have any references. I sometimes wish there were fewer “it depends” 🙂 Yet it is a fascinating subject.

          • The standard is almost always ET in a modern ensemble setting to avoid confusion or dissonance between bass and treble clef instruments. If you are playing solo, then use whatever tuning method you think sounds best. Pianos all use ET. Fortepianos and harpsichords are a different story.

  4. I just noticed that you were posting really about equal temperament vs just intonation (basically “well temperament”). That is a complex physics and math topic, and I would recommend that you google the term “pythagorean comma” if you want to understand the heart of the issue. I can’t recall if I’ve posted on this subject, but I have researched it thoroughly. In general, I think it might be best to start training your ear to hear ET since that is the standard for ensemble performance. Once you have that in your ear, then the nuances of playing more harmonious 3rds and 6ths will just jump out at you and seem more obvious which sounds best in a given situation. Be aware that some cellists tune their instruments in perfect 5ths and most others will tune all the strings to ET. This will impact where the notes line up on the fingerboard, so the Bs and Es and the As and Ds won’t always line up perfectly depending on the key.

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