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.

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?

Harmonic Analysis of Tibetan Singing BowlsFatima 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 malletProcedure: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: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 JonathanI took the analysis a little further, out of personal interest. Here is what I noticed.

Table 1 Percent Difference in Frequency: Striking vs Rubbing

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?