Resonance, Tuning, Tone, and Intonation (Quick Update)

I recently “misplaced” my cell phone for a few days and was forced to realize how much I depend on this little device to play the cello.  This is because my favorite tuner is an app on my cell phone called PitchLab, which I use tune the instrument to a very high degree of accuracy.  Although when I lost my cell phone, I wasn’t too worried because I usually just use it tune the cello, and not during actual practice.  Besides, I do have other tuners and an old phone with ClearTune on it.  However, the extra degree of accuracy on PitchLab was far more important than I realized.  It can tune my cello to such a high degree of accuracy that string resonances are easily apparent when I play in tune which allows me to actually practice without a need for a tuner.   Yes  …my tuner makes my tuner unnecessary.  By contrast, when I used my old ClearTune app, I was able to get the strings to within 2 cents of the actual note, but this wasn’t good enough for the sympathetic string resonances to really kick in, so my pitch accuracy during practice was only in the 10 cent range.  2 cents off in tuning leading to 10cents off in the ear?   That is what is referred in math as a multiplier effect, and this is caused by missing an audible and tactile cues of string resonance that is in my case apparently more accurate than pitch hearing itself.  To be able to benefit from string resonance, at least on my student cello, I need an accuracy tighter than 0.1hz or about 0.5 cents on the A string.

So what causes string resonances?  Well they are related to overtones.  When you play a note, it vibrates at multiple frequencies other than the strongest one (ie the one that shows up on your tuner).  These other frequencies (called overtones) will align with the main frequencies and overtones of the open strings causing them to vibrate in sympathy.  This causes the whole cello to resonate in subtle yet quite lovely and complex ways which makes the tone much more enjoyable.

String resonance is a gentle feedback mechanism that cello players use to tell when they are in tune.  The other side of string resonance coin is that when you play slightly out of tune, the overtones won’t quite line up, and the resonances can even work against you causing the cello to vibrate in ways that make the note unstable in pitch or somewhat muffled, particularly when you play the notes in quick succession.  This can even happen on a cello that is in tune but being played with bad intonation.   On a cello where all the strings are a little bit off pitch, it is impossible not to have negative or misaligned resonances with at least two strings, and if you are resonating with one of them, then you are still playing out of tune and likely messing up your ear!  In terms of tone production, misaligned resonances are essentially what creates a wolf note, and a cello that is out of tune is basically playing lots of little wolf notes, but instead of being loud and obnoxious sounding, they will instead be dull, muted, or less stable in pitch or volume.

Playing out of tune can make legato notes sound choppy, double stops feel like one string just won’t activate quickly enough, produce dull muted tone, and alter the general response of your cello.  Needless to say, playing on a cello that is even slightly out of tune can result in a wasted or at least frustrating practice session, because the response of the cello is so radically different that you might as well be playing with a rubber mute.

You could also make a case that playing on a cello that is tuned to equal temperament will cause these kinds of bad resonances on the lower two strings when compared with Pythagorean or Just intonation.  This is because, unlike the latter two tuning methods, many notes on an Equal Tempered cello won’t have overtones that are pure frequency multiples of the lower strings, unless you’re playing a basic key like C or D or G where the 5ths will at least be somewhat accurate.  I haven’t tested Pythagorean tuning thoroughly enough on my cello to make any solid claims about this, and will have to do a little more experimentation and research.  It should be noted that string resonance matters mostly for playing solo music.  If you are playing with a piano or with an ensemble, then Equal Temperament is pretty much required, and the resonances will be mostly harmonic ones between instruments and this is happening directly in the wood, the air, and the room itself, and not just between the strings of a single instrument.

…After rereading this post, I realize that if my teacher reads it, she will most certainly redouble her efforts to get me to use my tuning fork more often, and I am sure she would be right!


Good Vibrations (1,126 Hours)

As much as I dream of owning a better instrument some day, I know from experience that the wide realm of possibilities that define a cellist’s expressiveness is more the result of mastering of the principles of technique rather than the pedigree of their equipment.  This has been made evident to me on the many occasions upon which my teacher has borrowed my humble student cello in order to demonstrate a new technique or to inspect my setup, and the quality of her tone retains the same “personality” and “sweetness” despite the instrument’s limitations.  While a higher quality and more resonant instrument (or bow) can be a great benefit in molding your talent and understanding of the cello, there are also some basic principles of physics that when applied to the cello playing (or purchasing) can aid in your personal search for that elusive rich & dulcet timbre.

Likely by now you have heard of such things as overtones, or upper partials when describing a certain quality of sound of string, bow, or instrument, and harmonics when describing the left hand technique of producing more ephemeral pure tones.  What you  may not have been aware of is that all of these terms are all synonyms describing the same types of vibrations on a string.    Hidden within the broad waving motion of each note are shorter and faster vibrations at specific fractional wavelengths (eg 1/2, 1/3, 1/4, 1/5, etc).  It can seem somewhat bizarre that a string can be vibrating simultaneously at so many frequencies at the same time (much like Schrodingers mysteriously bi-modal Cat).  Yet we know from experience that this is the case, because we can actively select each of these frequencies when we play harmonics or “false” harmonics  by isolating the individual overtones (and their octaves) by damping all the other vibrations with our finger tips:

The quality of a cello’s timbre is often described by a “color” as being warm, deep, mellow, clear, sweet, bright, rich, dull, round, metallic, rumbling or even harsh.   All of these terms are actually describing the strength and distribution of the overtones.   Warm, deep, mellow, pure or often what is generally referred to as “core” sound is actually describing weakness of overtones and stronger fundamental vibrations (eg all of the A’s ringing on the A string, 220hz, 440hz, 880hz, etc).  Rich, bright, metallic, rumbling, clear, or harsh is describing an increase in the strength of the overtones (eg all of the E’s, C’s, and G’s etc, which are also ringing on the open A string) compared to the strength of the fundamental vibrations.  Perhaps you’ve already noticed that occasionally when you play the open A string with a slightly harsh sounding technique, your tuner might register an E instead of an A.   This is because brighter sounds have stronger overtones, and the E is normally the strongest non-A overtone of the open A string!   As you will see later, it is no mere coincidence that the E and A sound so pleasing when played in harmony.

When a cello has a more core sound, it is easier to discern the intonation for that singular note.  However, when a cello is rich in overtones, it is easier to discern the relationship and intonation for the distance between notes.   Having more core sound vs rich overtones is something that you can alter by equipment selection and setup and also by bowing technique and fingering choice for enharmonics.  For example, playing with your bow closer to the bridge will produce stronger overtones, where as playing closer to the fingerboard will produce a more core or mellow sound.   In the case of enharmonics (ie equivalent pitches played on different strings) playing the same note closer to the bridge will reduce the strength of the overtones and produce a warmer sound, as I am sure you have noticed by now, playing the Open A string produces a much brighter (overtone rich) sound than playing the same note on the D string.   You can combine these techniques & setups to create various combinations of brightness and warmness.   Other factors that effect the richness of the overtones are the quality of the bow,  the gauge and tension of the strings, the suppleness of the bow hand/arm/shoulder, the shape of the bridge, the thickness/density/age of the wood, dryness/humidity, the quality and application of the varnish, and even the tension on the bow hair.   Generally speaking, the louder you play, and the more resonant your set up, the greater the component of overtones will be.  Some mutes will selectively dampen high or low frequencies, and depending on the note being played this will make the cello produce a dim rumble or a hollow squawk.

Though we do not hear them consciously as individual notes, overtones are what define our concept of harmony and dissonance.    This is because the part of the temporal lobe (auditory cortex) that is connected to our inner ear is designed to detect patterns and predict their significance.  For example. the simplest harmonic ratios between note frequencies are  2:3 (major 3rd), 3:4 (perfect 4th), or 4:5 (perfect 5th).  Because of the simplicity of their ratios, these notes share some of their strongest overtones, which means our brain has less work to do to detect the patterns involved, and this results in the general positive and uplifting association with these Major and Perfect intervals and this is what we perceive as “harmony”.  Not surprisingly, much of the earliest written music is based on the structure of these primal harmonies.   Other intervals have frequency ratios that share only weaker overtones (or upper partials) which means more work for the brain to decode that patterns, such as 5:6 (minor 3rd) 8:9 (major 2nd) or 8:15 (major 7th).   This extra load on our auditory cortex is more troubling to our psyche and leaves us with a feeling of discontent or unresolved tension (evidenced by the Riot of the Rite of Spring in 1913) and this what we call “dissonance”.   For better or worse, our brains are remarkably adaptable when it comes to learning new patterns and our sense of harmony is somewhat malleable & plastic both individually and as a society, which is why we enjoy music now (like Jazz and Heavy Metal) that likely would have wilted Mozart’s ears. 

The same is true when detecting and manipulating harmonic resonances on the cello.  Simple intervals create stronger sympathetic vibrations which is useful when trying to get better intonation by paying attention the resonance (ie “ringing”).   This is because the most resonant overtones are also the simplest ratios of frequencies.  As a general rule, the lower numbered overtones are stronger than the higher numbered ones even on setups with strong overtones. The strength of these basic overtones is why you can actually see your strings vibrating sympathetically when you play certain notes on the cello.  For weaker resonances, you will simply hear an increase in volume and texture of the sound and perhaps even feel your other strings vibrating by touching them with your finger tips.  Below is a map of the first 7 overtones on the cello (not counting the fundamental resonant frequency of the cello body itself which is usually F3 or F#3).  A complete map goes off the page to an infinite number of overtones, but these ones should be the most useful for checking intonation… remember, that even though the overtones are several octaves higher than the other open strings, every note is a strong overtone of the octave below it (eg G3 and G4 are strong overtones of the Open G2 string as well as the open C2 string).

Root 8va 5th 8va 3rd 5th m7 8va
1:1 1:2 1:3 1:4 1:5 1:6 1:7 1:8
C2 C3 G3 C4 E4 G4 Bb4 C5
G2 G3 D4 G4 B4 D5 F5 G5
D3 D4 A4 D5 F5 A5 C5 D5
A3 A4 E5 A5 C6 E6 G6 A6

What is truly amazing about this information is not only that it is useful for creating beautiful music, but that these mathematical relationships actually define what music is!!   This is why music is so fundamental to the human experience, and might even be universal to any life form with a sense of sound.   Or, as math-musician Vihart  puts it:  all sound is essentially music…