Musical Instrument Research Report
Hadi Sumoro
I. Statement of the Research Topic
The main focus on the musical instrument research I did is vibration and fundamental frequency of a bar/pipe. The characteristic of pipe’s vibration is interesting and has a lot of inharmonicity phenomenon. Unlike calculating the fundamental frequency of a string, the fundamental frequency of a bar/pipe depends on a lot of thing, especially the density of the material, which correlates directly with the speed of the sound in the medium.
II. Summary
The basic idea of the pipe experiment I did came from a wind chime I calculated few weeks before I started this project. One of my friends asked me to calculate a 6m pipe to be cut into 5 pieces and to sound a major Chinese pentatonic scale for a huge wind chime (not to be in tune, but maximizing the length of the pipe). From the calculation of this wind chime, I compared the basic principle of it with common percussion instruments like xylophone/vibraphone/glockenspiel and came down to a decision to make a similar marching band glockenspiel type of instrument.
Started with the
basic principle of the wind chimes and I found out that the calculation of the
fundamental frequency is mostly based on
. From this point, I started to find information about the
various densities of materials and the way to get pipes and the cutter around
the
The tricky part
of this project is starting the calculation for the constant as shown in this
equation:
I started the project by finding a reference (from www.wikipedia.com) of the copper density to calculate the speed of the sound in this material. It turns out that they have the speed of the sound in their website. After clicking for some links provided in the wikipedia.com about copper material, I found out that the speed of the sound is 3750m/s – 3810m/s in an ideal copper. I use the average (3780m/s) of those 2 numbers for my calculation. By doing this we can the constant of the pipe as I’m changing the length to find a certain frequency.
I started with by
hitting a 60.64cm pipe and found out that the strike tone is closer to the Db4
on the piano. According to my set up above (to be explained in 3rd
chapter), 
and this frequency is very close with the Db4 frequency which
is about 277.18Hz. From this calculation I can start to calculate the other
length of the pipes.
After doing the calculation for the fundamental frequency (discussed in the chapter 3 of this research report), I cut the pipe and also calculated the nodal points of each pipe. Since I don’t have a drill to put a hole on the pipe, and had difficulties to put those pipes to be aligned, so I was thinking to hang them with rope/string. Having a problem with the exceeded initial budget I’ve prepared, I can only provide broken cables to hang them, and they sound just like what I’ve expected.
The tuning of my
instrument is a Japanese scale, (Ab) C Db F (F#)
G Ab C (Db). The 3 notes in the parentheses are
additional notes for performance purposes. The F# will alter the scale into a
Javanese sounding scale with the G natural is being avoided (a traditional
Indonesian scale), (Ab) C Db F Gb (G) Ab C (Db).
III. Acoustical Principles and Calculation
A. Acoustical Principles and Calculation
The only formula
I used to calculate the fundamental frequency of the copper pipe I bought is where I consider the 
as a constant. I didn’t really worry about the longitudinal
frequency calculation since the nodal point of the transverse frequency of the
pipe will affect the longitudinal frequency, and the pipes are meant to be
stroked in the middle to give more amplitude to the transverse vibration.
As stated in the chapter 2, the speed of sound inside the copper I used is 3780 m/s. Since the frequency that I’m dealing is only the fundamental, I take m equals 3.0112. This gives us:
Diameter of the pipe: outer=2.2cm and inner =2.1cm
K = 
=
m = 3.0112,5,7,…
Substituting the number of frequency to get the length as in:
and fundamental frequencies are:
|
Note name (Equal Tempered) |
Ab3 |
C4 Mid-C |
Db4 |
F4 |
F#4 |
G4 |
Ab4 |
C5 |
Db5 |
|
Fundamental Frequency (Hz) |
207.65 |
261.63 |
277.18 |
349.23 |
369.99 |
392 |
415.3 |
523.25 |
554.37 |
|
Length (cm) |
70.2 |
62.54 |
60.76 |
54.13 |
52.59 |
51.09 |
49.64 |
44.22 |
42.96 |
To prove and to make the calculation more accurate, I used length ratio as in:
This will give ratio of (considering that C is the base)
Ab3 : C4 : Db4 : F4 : G4 : Ab4 : C5 : Db5 (equal temperament) =
1.12246L : L :
0.971532L : 0.865537L : 0.816958L : 0.793701L : 0.707107L : 0.686977L.
F#4 isn’t included in the length ratio since it’s going to be the average of F4 and G4.
The result I have
is very close with the table above, about 
correction.
The node of the pipe can be calculated by 22.42% L
This gives us the nodal points from the end of each bar:
|
Pipe’s tone |
Ab3 |
C4 |
Db4 |
F4 |
F#4 |
G4 |
Ab4 |
C5 |
Db5 |
|
Nodal point from the end (cm) |
15.73 |
14.02 |
13.62 |
12.13 |
11.79 |
11.45 |
11.12 |
9.91 |
9.63 |
B. Calculation Error
After I build the instrument it seems that all of the fundamental frequencies are shifted major second down from what I’ve calculated above. My instrument sounds closely to F#3 Bb3 Cb3 Eb4 E4 F4 Ab4 Bb4 B4.
I’m not surprised with the result. I believe that the rubber band I put on the nodal points of each pipe doesn’t have anything to do with the frequency alteration. The cutting of the pipe will have some kind of contribution for this frequency alteration because I couldn’t really cut the pipes neatly.
On the other
hand, the speed of sound depends on the density and the elasticity of the
material. Since the calculation I did above is applied for an ideal copper,
I’ve expected if there’s a difference between the result and the calculated
numbers. Putting back the fundamental frequency of each pipes
to the formula: and using the length
I’ve calculated above, I can find out that the real speed of sound in the
copper pipe I’m using for this project is about 3370 m/s. I believe this copper
pipe is mixed with some additional materials. At the bottom line, the shifting of
the major second down is acceptable because the ratio of the frequencies among
pipes are considered the same and still sounds like the scale I wanted it to
be.
To proof that the speed of the sound inside this copper is about 3370m/s, I check it with 2 frequencies:
F#3 is about
185Hz. 
Bb4 is about 466.57Hz.
C. Acoustics Principles Summary
The acoustics
principles of the pipes that are supported on the nodal points follow the basic
principles of open bars/pipes at both ends. Pipes and bars have inharmonic
overtones, the fundamental frequency depends on the length
, density of the material, elasticity of the material, the
shape of the bar (radius of gyration), and the mode of the frequencies are
almost proportion to the squares of the odd integers.
IV. Conclusion
The inharmonicity of bars/pipes has a lot of contributions to
the sound of the instruments built with them. By taking the simple rule of 
someone can make a wind chime to sound in a certain scale
without being really in tune. The inharmonicity of the
pipes hung for a wind chime can make a rich sounding scale. As far as my
instrument is being constructed and hung, it can be played by placing them on a
flat surface, with the rubber bands supporting each pipe on the nodal points
and stuck them with our right hand (with a mallet or something like that), and
damp (the decay) them by touching the pipe with our left hand after being
struck. The other possible way is to make my instrument as a wind chime, it’s a
little big, but it will work just fine. This is one big reason I’m doing pipe
instead of rectangular bars, I can put the pipes in many way to be constructed
as an instrument.
V. Documentation
The way it’s being played
The instrument itself
MP3 files : Japanese Scale - Javanese Scale