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<\/a>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<\/p><\/div>\n
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.\u00a0 At least we figured it out after studying several documents on the topic.\u00a0 Under each bar we cut an arch.\u00a0 In the middle, where the arch is thin, the bar vibrates up most.\u00a0 Where the bar is thick (on the ends), the bar is sluggish, with more mass.<\/p>\n
The fundamental tone (the pitch we are trying to shape each bar for) is dominated by the thinness of the arch.\u00a0 The thinner the arch, the slower the bar ends bounce up and down (vibrate).\u00a0 A really thick arch is stiff, so the energy of the mallet is carried efficiently through the bar, with the ends vibrating quickly.\u00a0 So, cut the arch deep (so the bar is thin in the middle) and the pitch drops.\u00a0 However, the LENGTH of the arch is what determines the harmonic.\u00a0 Stretch the arch LONG across the bottom of the bar and the harmonic becomes lower.<\/p>\n
What frequency is the harmonic we want?\u00a0 That my friend, is the question.<\/p>\n
For bars in the first octave we shaped, C4 to C5, the first harmonic was TWO OCTAVES above the fundamental.\u00a0 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.\u00a0 Pleasant.\u00a0 We will skip discussion of the third harmonic for now.<\/p>\n
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.<\/p>\n
However… that ratio of fundamental and harmonic cannot last.<\/p>\n
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.\u00a0 See the picture below. The bar at the top of the picture is C5.\u00a0 As the arch gets smaller and smaller, the margin for error also becomes bigger.\u00a0 With such a very small area to sand, removing just a little bit of wood changes the pitch quite dramatically.\u00a0 We began to worry about how well the bars would stay in tune after final tuning.<\/p>\n
<\/a>C5 to A5<\/p><\/div>\n
However, more even more critical was the ever-smaller arch — it could not continue.\u00a0 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.<\/p>\n
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.\u00a0 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).\u00a0 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.\u00a0 But what?<\/p>\n
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.\u00a0 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.<\/p>\n
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.<\/p>\n
However, this discussion seems so matter of fact, so rational, so clear.\u00a0 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.\u00a0 Paul drummed his fingers to strange rhythms.\u00a0 Shaving, scratching, and drumming.\u00a0 Finally, we decided to go for it, and with a lot of angst we shaped G#5 to the new harmonic.<\/p>\n
Viola!<\/p>\n
It sounded great.<\/p>\n
We then shaped A4. Like with the previous keys, the arch cut in the bottom was a little shorter and little shallower.\u00a0 We predict this pattern will proceed until we need to once again switch to a new harmonic.\u00a0 We will then have to choose between using a major 3rd or just move to a single octave between fundamental and harmonic.<\/p>\n
What do other marimbas do? \u00a0Well, 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. \u00a0Until we bring our recording gear to someone’s 5 octave marimba and look at the FFT, we won’t know for sure. \u00a0However, 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).<\/p>\n
I’m exhausted.\u00a0 We have finally moved back into a standard machining process, and can probably crank out another 6 bars this weekend.\u00a0 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.\u00a0 However aluminum tubing is both expensive and difficult to work with…<\/p>\n
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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.\u00a0Middle C (C4) sounds different on a piano, flute, guitar, trumpet, … Continue reading