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CALCULATING HAND PAN DIMENSIONS: DING, DIMPLES, NOTES. Do you want to build your own Hand Pan? Struggling to get the right dimensions of Ding, Notes and Dimples? These notes can help you.

By Andrea Rubano ([email protected])

Calculating Hand Pan dimensions: ding, dimples, notes. by Andrea Rubano (pesante



@gmail.com)

INTRO Starting from the very precious measurements by Marco Selvaggio (thanks a lot!! I added the notes at the end of this document) freely downloaded from the internet, I started a quantitative analysis for answering a simple question: “When I build an Hand Pan, how do I build my own templates for any scale I choose?” The point is to get free from the semi-empirical templates which are circulating in the internet. Marco Selvaggio measured several quantities, for a number of different Hand Pans he played with:

He observed that all dings were semi-spherical (so he measured only the radius), while all notes and the dimples were elliptical (and thus he measured the longest and shortest diameter, and also how deep is the dimple). I used those numbers to extract a general rule about how to calculate the note dimensions.

Octave → Note ↓

1

2

3

4

5

6

C

16.3520

32.7030

65.4060

130.8100

261.6300

523.2500

C#/Db

17.3240

34.6480

69.2960

138.5900

277.1800

554.3700

D

18.3540

36.7080

73.4160

146.8300

293.6600

587.3300

D#/Eb

19.4450

38.8910

77.7820

155.5600

311.1300

622.2500

E

20.6020

41.2030

82.4070

164.8100

329.6300

659.2600

F

21.8270

43.6540

87.3070

174.6100

349.2300

698.4600

F#/Gb

23.1250

46.2490

92.4990

185.0000

369.9900

739.9900

G

24.5000

48.9990

97.9990

196.0000

392.0000

783.9900

G#/Ab

25.957

51.9130

103.8300

207.6500

415.3000

830.6100

A

27.5000

55.0000

110.0000

220.0000

440.0000

880.0000

A#/Bb

29.1350

58.2700

116.5400

233.0800

466.1600

932.3300

B

30.8680

61.7350

123.4700

246.9400

493.8800

987.7700

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Calculating Hand Pan dimensions: ding, dimples, notes. by Andrea Rubano (pesante



@gmail.com)

I started from the table above, where each note (first column on the left) corresponds to a specific frequency in Hz, for each single octave (columns). So for instance an A8 is a sound with about 55 Hz frequency. Please note that the human ear works in a logarithmic scale (the frequency difference of one note to the following changes from one octave to another) therefore all my calculations are referred to the natural logarithm (base e, not the base10!!) Log(frequency(Hz)) rather than the frequency itself. As Marco Selvaggio did not write explicitly the octave of the notes he measured, I assumed (reasonably) that all notes were between the 4th and the 5th octave, except from the Akebono Hand Pan, which was not fitting this hypothesis, and thus I put it between the 5th and the 6th octave. All dings I assumed are one octave down, so in the 3rd octave. In case my assumption is wrong, nothing bad because you can always shift my results by one or two octaves.

DING Regarding the Ding, the data are unfortunately not so many, as some notes are equal and some values too. Therefore there are only 4 significant points to be used to get the formula out. In the following plot you see the different diameters of the Dings for the different frequencies: DING 7.8 7.6

Ding Diameter(cm)

7.4 7.2 7 6.8 6.6 6.4 6.2 6 3.5

4

4.5

5

Log(f(Hz))

The green line represents the “ideal” best fit line that goes through the points. From calculation it is seen that if you want to have a Ding Diameter (D) of a given note (N) you must apply the formula: D(N )= -0.9712 * Log(N(Hz)) + 11.0235 (result in cm) Which means that you look at the Table, choose one note on the first column, choose one octave on the first row and you pick up the frequency of that note, then you put it into the formula and you calculate the “Ideal Diameter” of a Ding that sounds with that note.

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Calculating Hand Pan dimensions: ding, dimples, notes. by Andrea Rubano (pesante



@gmail.com)

Example: We want to build an Hand Pan with a Gb3 Ding. The table says that Gb (7th row) octave 3 (3rd column) corresponds to 92.4990 Hz. Therefore the natural logarithm is 4.5272, and so the diameter of the Ding will be: D(Gb3) = -0.9712 * 4.5272 + 11.0235 = 6.6267 cm Let us note that unfortunately we have no information about the length and width of the tone-field around the Ding. We’ll come back on this point later.

NOTES AND DIMPLES Regarding the Notes and the Dimples, we have more details from Marco Selvaggio’s work. For example, the Length of the Notes (L) for all Hand Pans he measure is represented here: NOTE LENGTH 20 Halo Bell Integral Akebono Halo2

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Note Length (cm)

16

14

12

10

8

6 4.5

5

5.5

6

6.5

7

Log(f(Hz))

It is nice to see that so many different scales and notes, tuned on different instruments and with different manufacturers are more or less on the same line (blue line, linear best fit). The formula for the “ideal” length (L) of a given note N is: L(N )= -5.0767 * Log(N(Hz)) + 42.0239 (result in cm) A similar figure is seen in the case of the Width (W) of the notes as well, and the formula in that case is: W(N )= -6.2447* Log(N(Hz)) + 47.9443 (result in cm) And for the length of the inner dent “l” l(N )= -3.3803* Log(N(Hz)) + 24.3689 (result in cm) 3

Calculating Hand Pan dimensions: ding, dimples, notes. by Andrea Rubano (pesante



@gmail.com)

And for the width of the inner dent “w” w(N )= -3.5468* Log(N(Hz)) + 25.0046 (result in cm) On the contrary, the height (depth) of the dent is not linear with the frequency logarithm, as seen in this figure: Height of the dent 2.5 measured h cubic fit linear fit

Dent height h (cm)

2

1.5

1

0.5

0 4.5

5

5.5

6

6.5

7

Log(f(Hz))

The points are not on a straight line (green line) as before, but rather on a curve (magenta line). In this case the formula to calculate the height “h” is a bit more complicated: h(N) = -0.65943*x3 + 12.566*x2 - 79.788*x + 168.99 (result in cm) here we indicated x = Log(N(Hz)). What is quite surprising is that for both Notes and Dimples, the ratio between the length and the width is very similar to 1, as shown here: Length/Width Ratio 1.5

Ratio L/W

1

Note Dimple

0.5

0 4.5

5

5.5

6 Log(f(Hz))

This means that the ellipses are (basically) circles. 4

6.5

7

Calculating Hand Pan dimensions: ding, dimples, notes. by Andrea Rubano (pesante



@gmail.com)

It has been said in different blogs and forums that the shape of the notes must be elliptical, but apparently it is not so important for the pitch, and possibly it could affect the timbre and color of the sound. The mean value of the ratio L/W for the notes is 1.0736, while the mean value for the dimples is 1.0827: as these two values are very similar and very close to one, we can approximate the Notes an Dimples with circles with radius rnote and rdimple given by this formula: rnote (N ) = -5.6607 * Log(N(Hz)) + 44.9841 (result in cm) rdimple(N ) = -3.4635 * Log(N(Hz)) + 24.6868 (result in cm) I can tell more about this when I make my own Pan. In case you’re not happy with the circular approximation, you can anyway get back to the ellipse by imposing one length and one width L and W so that the area of the Ellipse is the same as the area of the circle: L = 1.0736 * W; L * W = r2note → 1.0736 * W 2 = r2 → W = rnote/1.0361 l = 1.0827 * w; l * w = r2note → 1.0827 * w2 = r2 → w = rdimple/1.0405 Last remark on the large note on the Ding: we do not have a measurement of its diameter, so all we can do is to “guess” it by looking at the other notes: Dimple/Note diameter ratio 0.5 0.45

Dimple/Note diameter ratio (cm)

0.4 0.35 0.3 0.25 Halo Bell Integral Akebono Halo2

0.2 0.15 0.1 0.05 0 4.5

5

5.5

6

6.5

7

Log(f(Hz))

The ratio between each dimple and note diameters (calculated as the average between length and width) is not constant with the frequency. Anyway, as the Ding always corresponds to the lowest tone, it is reasonable to assume that the ratio must be similar to the lower part of the frequency range, which is about 0.425. This means that we should calculate the Ding diameter, the diameter of the surrounding note must be 2.35 times larger.

Using these formulas, everyone can build his own templates and scales as he/she wishes.

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Calculating Hand Pan dimensions: ding, dimples, notes. by Andrea Rubano (pesante



@gmail.com)

I hope that this work can be beneficial for you, whoever you are. In the case that you have questions/criticisms/problems of any kind, please write me, and I’ll be happy to help you (or to change these notes if there are errors). In case you want to have my files in order to perform your own calculations, please write and I’ll send them. In case you will use and/or cite this work, please just inform me with an email ([email protected]).

Andrea Rubano, July 2014, Napoli (Italy).

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