# Mean Harmonics

Sit back, get a cup of coffee or tea, and we will explain what we have learned… some of our education has come the hard way.

Musical instruments are complex. Middle C (C4) sounds different on a piano, flute, guitar, trumpet, and marimba because each has a unique timbre.  Each may play the same note (the fundamental), but the tone is accompanied by dozens of harmonics, usually interesting multiples of the basic note. A common harmonic to be carried along with the fundamental tone is an octave. (twice the fundamental).  For example, the “A” above middle C is usually 440 Hertz, or 440 cycles per second.  Or, if you can imagine an old fashioned speaker vibrating in and out, it would have to do that 440 times a second to play a A4.  The A5 an octave above that is double in frequency, or 880 Hertz.  A6 is double that, or 1760 Hz. So integer multiples of the base frequency (2f, 3f, 4f, etc) are the common harmonics that are heard along with the fundamental (the pitch being played)

That sounds so simple.  Such beautiful math.  Of course, doubling (2X) the frequency (Hertz) for every change in ONE octave means the pitch and frequency are related through a logarithm… Moving a note up THREE octaves (from A4 to A7 for example) means multiplying the frequency by EIGHT.  Moving a note down TWO octaves (from A4 to A2) means dividing the frequency by FOUR. Again, beauty.  We will skip the discussion of the equal temperament scale (12th roots are required) for now.  Just marvel at the beauty of how frequencies with wonderful ratios are perceived as different pitches by the human ear.

A simple penny whistle or flute has a very clean sound.  The note that is played (the fundamental) is what you hear loudest. That’s because the flute is a very pure instrument.  The column of air inside the flute is what vibrates to make the sound, not the metal flute. So even though Harold Hill may try and sell you on something better, a silver, gold, platinum, and nickel flute will all sound mostly the same — it is the column of air that is vibrating that makes the sound, and air is free.

However for a marimba, there are two parts needed to make music, the wooden bar, and the resonator (the column of air in a tube below it).  Unlike the resonator, with it’s simple column of air, the wooden bar vibrates in VERY complex ways and therefore produces several harmonics simultaneously (its timbre). The choice of wood affects how the wood will “sustain” after being struck, how fast certain harmonics decay, and the strength of various vibrational modes. A marimba bar does not just vibrate in a simple up and down wave motion (transverse mode), like a guitar string, but also twisting back and forth (torsional mode) and wiggling side to side (lateral mode).  However for Paul and I, we are only going to try and tune the transverse modes of the “first” and maybe the “second” harmonic after the fundamental.

Whew.

Ok, that was the background :-)  Now the problem with tuning….

A “C” on our Baldwin baby grand. The leftmost peak with the arrow above it is the fundamental. The other peaks are the 6 harmonics

Using a combination of the spectrum analyzer and how we strike the bar, we can measure the pitch of the fundamental (the primary note that you should hear) and the first harmonic. It turns out that for lower bars (C4 up to C5), tuning is pretty simple.  At least we figured it out after studying several documents on the topic.  Under each bar we cut an arch.  In the middle, where the arch is thin, the bar vibrates up most.  Where the bar is thick (on the ends), the bar is sluggish, with more mass.

The fundamental tone (the pitch we are trying to shape each bar for) is dominated by the thinness of the arch.  The thinner the arch, the slower the bar ends bounce up and down (vibrate).  A really thick arch is stiff, so the energy of the mallet is carried efficiently through the bar, with the ends vibrating quickly.  So, cut the arch deep (so the bar is thin in the middle) and the pitch drops.  However, the LENGTH of the arch is what determines the harmonic.  Stretch the arch LONG across the bottom of the bar and the harmonic becomes lower.

What frequency is the harmonic we want?  That my friend, is the question.

For bars in the first octave we shaped, C4 to C5, the first harmonic was TWO OCTAVES above the fundamental.  So you can imagine that the sound the marimba bar makes when struck is like middle C combined with the C two octaves above that.  Pleasant.  We will skip discussion of the third harmonic for now.

So, a quick FFT spectrum analysis of the bars from middle C (C4) up to C5, and you see a strong fundamental (the basic pitch) combined with a very pleasing C6 to C7 harmonic.

However… that ratio of fundamental and harmonic cannot last.

We noticed as we tuned our way up the scale from C5, to C#, D, D#, E, F, F#, G, etc, that the arch we were cutting in the bottom was getting smaller and smaller.  See the picture below. The bar at the top of the picture is C5.  As the arch gets smaller and smaller, the margin for error also becomes bigger.  With such a very small area to sand, removing just a little bit of wood changes the pitch quite dramatically.  We began to worry about how well the bars would stay in tune after final tuning.

C5 to A5

However, more even more critical was the ever-smaller arch — it could not continue.  Remember that the depth of the arch (resulting thinness of the bar) is what sets the fundamental frequency (pitch). So the problem here is really about the first harmonic, which was two octaves higher.

Clearly, we needed to start tuning for a DIFFERENT harmonic. So, what tone would we mix with the fundamental to make a pleasing sound? For example, mixing a fundamental of C4 with a harmonic of C#5 would be painful.  Some cymbals or gongs have an almost dissonant irritating sound because they mix harmonics that do not play well together. Having the first harmonic be TWO octaves away was too far (and therefore the arch too short).  Tuning to have the first harmonic only ONE octave away would not be far enough, the arch would not fit on the bar. We would have to choose something in between.  But what?

After a bit of piano playing, we decided that either the major 3rd or the major 5th sounds very nice together with the fundamental. The major 3rd would be like playing the C and the E together.  The 5th (perfect 5th) would be like playing a C and a G together. The 5th is the most consonant tone to play along with a note after the octave, and would therefore be our target.

So, with the arch diminishing to nothing to tune the first harmonic to two octaves higher, we decided to switch to one octave and a fifth higher for the harmonic.

However, this discussion seems so matter of fact, so rational, so clear.  In fact, we spent hours tinkering with bars, testing pieces of wood, shaving off bits from different edges, and then spent hours more scratching our heads.  Paul drummed his fingers to strange rhythms.  Shaving, scratching, and drumming.  Finally, we decided to go for it, and with a lot of angst we shaped G#5 to the new harmonic.

Viola!

It sounded great.

We then shaped A4. Like with the previous keys, the arch cut in the bottom was a little shorter and little shallower.  We predict this pattern will proceed until we need to once again switch to a new harmonic.  We will then have to choose between using a major 3rd or just move to a single octave between fundamental and harmonic.

What do other marimbas do?  Well, it appears they ignore the harmonics in the higher register. Do they really just shape the bottom of the bar with a convenient arch? We don’t know.  Until we bring our recording gear to someone’s 5 octave marimba and look at the FFT, we won’t know for sure.  However, we have read at least one description that indicated the harmonics were not tunes above C5. We are much too geeky to just let the harmonics fall where they may for high frequencies. We want a harmonic nice, not a harmonic mean (mathematicians, forgive my pun).

I’m exhausted.  We have finally moved back into a standard machining process, and can probably crank out another 6 bars this weekend.  We continue to make slow progress, but I’m getting anxious about the resonators…. After all the work we are doing with the gorgeous African Padauk, I can’t bear the thought of using PVC pipe for the resonators.  However aluminum tubing is both expensive and difficult to work with…

## One thought on “Mean Harmonics”

1. Pete & Paul,

You are going through a very meticulous and painstaking process. Great for the mind and “father/son” relationship. You will be very gratified and proud of your work when you finish. You both have certainly done the research and chosen the best materials for your project. I was able to follow your deliberations on the “science of sound” and the tuning of each pitch. Courses in the psychology of music includes tuning, temperament, pitch and tuning. You are actually learning this formulae as you construct the marimba.

A couple of comments. When referring to the overtone series and intervals between notes in a scale, the common language for temperament includes
naming of intervals as follows. Octaves, 4ths, 5ths, and unisons are “perfect.”
2nds, 3rds, 6ths, & 7ths are Major or minor. Therefore the reference to intervals in the major scale are PU, M2, M3, P4, P5, M6, M7 ,PO. Glad you did not get into tuning from an historical basis. It gets more complicated. While all tuning and music
is based on sounds generating from the overtone series, intonation as, such, began with “just” intonation, then, “mean tone” temperament, and now “equal” temperament…thanks to JS Bach and his 24 classics on equal temperament including all major and minor scales.

The evolution from just intonation (based purely on the overtones series) created a system in which the black notes on the keyboard were actually split. That is the upper split black note was i.e.for C#, while the lower was for Db. Very difficult for keyboard players to negotiate. When you run this to the extreme upper and lower range on the keyboard, the lower notes “sound” flat to our ears and the upper notes sound sharp. Therefore, the new equal temperament system theoretically divided the 1/2 step (distance between each white/black note) on the keyboard was divided into 100 cents. Herein lies a mathematical formula to accommodate the harmonics that you are working with.

Today, the best piano tuners who tune by ear still tune the extreme upper register on the piano slightly lower and the extreme upper range a little flat to accommodate the overtone series and our adjustments that are necessary to play in an ensemble.
Symphony orchestra string players, historically, like to play on the “high” side of the pitch. Some orchestras tune to A444…until they have a guest artist who plays on the piano, and then the entire orchestra must “tune down” to the piano that was tuned to A440. I hear this every time a pianist is a guest with any orchestra. It is easier for a violinist to move their fingers “up” on the string to play a note than “down.” String players hate to play in flat keys. Their favorite key is D major.

Now to one last thought…I see that both of you are very careful to cover your nose and mouth to protect from breathing in the dust from wood cutting. May I also request that you “clean up my garage floor” when you finish…just a practical matter.