Harmonic Analysis of Tibetan Singing Bowls
Fatima and Alex

Purpose: Compare overtones present with running mallet around Tibetan singing bowls and hitting bowls with mallet. This may help to figure out how the bowls produce sound.

Equipment: Logger Pro, Vernier microphone, 2 Tibetan singing bowls, and mallet

Procedure: We set up Logger Pro and the microphone with the computer. We hit the bowl 1 and measured the frequencies on the FFT graph. Then we used the mallet to go around the bowl 3 times and measured the frequencies present. (Instructions for playing can be found at http://www.bodhisattva.com/playing.htm). On the FFT graph we recorded the peaks as the main frequencies. These peaks were the frequencies that were playing the loudest and so we viewed them as the “main” ones in comparison to the much smaller peaks around them. We took the lowest Hz measure as the tonic and measured the others as its overtones. For each of the three trials we took the average of the peak frequencies that were present and then recorded those. We repeated all of these steps for bowl 2 to get the frequencies for hitting and going around.

Data Analysis:
Bowl 1 around (measured in Hz)
Ratios

Bowl 1 Hitting
Ratios
205
1

200
1
586
2.858537

580
2.9
1101
5.370732

1100
5.5
1738
8.478049

1750
8.75
2481
12.10244

2500
12.5



3345
16.725



4280
21.4





Bowl 2 around
Ratios

Bowl 2 hitting
Ratios
180
1

200
1
500
2.777778

600
3
960
5.333333

1100
5.5
1530
8.5

1720
8.6
2170
12.05556

2480
12.4
2930
16.27778

2850
14.25
3700
20.55556

3340
16.7



4260
21.3
To get the ratios we divided each of the peak frequencies by the tonic/fundamental which was the lowest peak frequency. This division gave us the overtones.

When we are going around the singing bowl with the mallet we are predicting that the sound is caused by the mallet chasing a node of a certain wavelength around the bowl and thereby screening out some of the overtones. The leftover overtones make up the sound that comes from the bowl. This may be why hitting the bowls creates more overtones than going around—when you are just hitting it you are not dealing with nodes that will block out some of the overtones. The ratios represent the values divided by the lowest frequencies in order to get the overtones. It is clear from the data that there is a distinct correlation between the ratios of going around the bowl, and hitting it. If we take out some of the values of the hitting then it is an even clearer matchup. We ruled the overtones produced by hitting the bowl as insignificant because the amplitude of these extra overtones was much lower than all of all the other frequencies.

Conclusion: There are more overtones present in hitting the bowls than when simply going around but both scenarios have similar ratios meaning similar overtones.


Addendum by Jonathan
I took the analysis a little further, out of personal interest. Here is what I noticed.
Table 1 Percent Difference in Frequency: Striking vs Rubbing

Bowl 1 rubbed (Hz)
Bowl 1 struck (Hz)
% Difference from striking
Bowl 1 struck (Hz)
Bowl 1 struck (Hz)
% Difference from striking
Fundamental
205
200
2.44
180
200
-11.1
1st overtone
586
580
1.02
500
600
-20.0
2nd overtone
1101
1100
0.09083
960
1100
-14.6
3rd overtone
1738
1750
-0.6904
1530
1720
-12.42
4th overtone
2481
2500
-0.7658
2170
2480
-14.29
5th overtone
---
3345
---
2930
2850
2.730
6th overtone
---
4280
---
3700
3340
9.730
7th overtone
---
---
---
---
4260
---
What I read from this is
1) Striking the bowl produces more overtones that rubbing it.
2) The frequencies of the overtones differ between rubbing and striking.
3) Sometimes rubbing gives an overtone of a lower frequency and sometimes of a higher frequency. Is there any pattern to which it does?
On Bowl 1, the frequency differences are all within 20Hz, so that could just be the limit of the mike sensitivity. On the other hand, the overtone ratios in Bowl one were always higher for striking that rubbing…
On Bowl 2, the frequencies are higher in the first five trials. Is there some mechanism you can see that would explain the higher frequencies in a struck bowl (or the lower ones in a rubbed one)? Does the mechanism also lead to expect the reverse effect with the higher overtones?