Research project on the development of new tools for musical expression at the University College Ghent

School of Arts

<Rodo>

an automated and interactive set of bronze tines

Godfried-Willem RAES

2013 - 2017


<Rodo>

Rods clamped on one side and free to vibrate at the other side are the acoustical base of quite many musical instruments and sound installations: reed organs, mouth organs, bandoneon, music boxes with a comb, nail violin, toy piano, Fender-Rhodes piano, Waterphones, Harry Bertoia's installations, African lamellophones, clock gongs to name just a few. Two classes of instruments using this sound source should be distinguished: instruments that only use the fundamental resonant tone (reed organs, Fender-Rhodes, music boxes) and that are always considered pitched instruments and at the other hand those that use the broad spectrum of overtones these rods can generate, when tuned to very low fundamental resonant frequencies (clocks, Waterphone, toy piano...). The fundamental resonant frequency of these rods is inversely proportional to the square of the length of the rod: With: f= frequency in Hz, L= length of the rod, k=diameter of the rod, Q= modulus of elasticity, r= density of the material. (Olson, p.76)

The <Rodo> robot was designed to be either an extension or a generalization of the toy piano robot <Toypi>. Just like in the toy piano, the sounds all stem from massive rods clamped at one end in a solid cast iron bar. We started the project, as the automation of the small instrument was very successful and appeared to have many more sonic possibilities than we grasped at the start. So we thought of rescaling the design such that the range would extend much lower and the maximum sound level quite a bit higher. At the same time, we aimed at making the instrument a lot more sturdy than the toy instrument, that needed all too many repairs because the tines broke very easily on very fast note-repetition rates. Also quite some new features were added in this design: individual dampers, an e-drive mechanism and a set of sensors to allow gesture interactive activation and playing modes.

We started off by doing experiments on different metals and alloys for the tines: martensic stainless steel, hardened spring steel, brass, aluminum, phosfor-bronze, aluminum-bronze. We even experimented with some non metals such as bamboo, glass, carbon-fibre. Those experiments made us drop the nonmetals very fast as the sound was too weak or the rods too fragile. Obviously the evaluation of sonic quality has to remain a quite subjective issue, since there is no standard to compare to as we are designing a new instrument. After all these experiments were performed we decided to go for the aluminum-bronze alloy. These rods produce a very rich tone, though not as brilliant and loud as spring steel rods. As we couldn't get beryllium copper (after the properties, it ought to be the perfect material here) nor phosphor bronze (CuSn8) ,we cannot judge on these promising alloys. In the design of <Rodo> we took into account the possibility for tuning and adaptation to different tuning systems. To allow this, set screws are used to fix the vibrating length of the tone rods. This arrangement makes it possible to exchange the rods for other sets, as long as the diameter is 8 mm. By default the tuning is chromatic, equal temperament. The electromagnetic driver feature however, can work in different tuning systems, including just intonation. Due to the high inharmonicity of vibrating bars clamped at one end, it is perfectly possible to consider the instrument as 'non-pitched' in the context of orchestra compositions conceived for our robots. In this respect, the instrument would sound like a set of gongs.

The acoustics of rods clamped at one end, after the theory books (Rayleigh,V1,p.278; Talukdar) allow us to predict the frequency of the overtones.

f0 (fundamental) 1
f1 6.276
f2 17.547
f3 34.155
f4 56.84
f5 84.91
The literature on the subject doesn't give any values for higher overtones than f5. Also, different authors do not seem to agree on the exact values of the factors. The factors derived from theoretical calculus, generally simplify the problem by neglecting the torsional forces occurring in vibrating rods, only considering lateral vibration. So, these factors can only be used as guidelines and experiments were necessary to obtain more exact results.

One would expect that the perceived pitch of a struck and clamped rod would correspond to at least one of these overtones, however -at least for the rods we selected- this is not the case. In the table below we give the measured frequencies of a single bronze rod, 8 mm in diameter and cut to a length of 961 mm. (An extra length of 20 mm serves for clamping and thus can be considered acoustically dead). The perceived pitch of this bar was C#3 raised 20 cents (139.45 Hz), yet this pitch was not found back under forced resonance conditions::

  measured calculated remarks
f0 5 Hz 4.937 Hz =MM296, checked by comparison to a metronome set to MM150
f1 30.86 Hz 30.987 Hz almost unmeasurable
f2 86.19 Hz 86,638 Hz  
f3 168.64 Hz 168.64 Hz reference tone used for best match with calculated spectrum
f4 280.47 Hz 280.647 Hz this corresponds to the octave above the perceived pitch (139.45 Hz)
f5 416.00 Hz 419.24 Hz  
f6 581.66 Hz -  
f7 773.50 Hz -  
f8 986.80 Hz -  
f9 1235.57 Hz -  
f10 1807.5 Hz -  
f11 2128.27 Hz -  

Measurements were performed under forced excitation using a transducer driven by a Thurlby Thandar TG1006 DDS function generator with a 6 digit frequency readout. Resonance's were determined by very slowly sweeping the sine wave frequency and fine tuning to find the peeks in amplitude. When we tried to make the rod resonate to the perceived pitch of 139.45 Hz, it clearly responded with the pitch found in the table for f4. The conclusion is that our ear adds a 'missing fundamental' here. This is not something novel, as similar phenomena have been described with regard to the perceived pitches of tubular bells.

To conclude our measurements on the test rod, we tried to excite it with a bow and could find all overtones from f2 to f9 back, be it sometimes after quite some attempts as it seems difficult to predict the pitch one will get when bowing.

Starting from the measurements and experimental data obtained, we designed a small computer program to calculate all 31 rods based on a perceived pitch range from C3 (midi note 48) to F#5 (midi note 78) for equal temperament and a basic tuning to A=440 Hz..

midi note frequency rod length id. with clamp f0 note0 f1 note1 f2 note2 f3 note3 f4 note4 f5 note5
48 130.8 989.16 1009.2 4.603 -9.946 28.89 21.85 80.77 39.65 157.2 51.18 261.6 60 390.8 66.95
49 138.6 961 981 4.876 -8.946 30.6 22.85 85.57 40.65 166.6 52.18 277.2 61 414.1 67.95
50 146.8 933.64 953.64 5.166 -7.946 32.42 23.85 90.66 41.65 176.5 53.18 293.7 62 438.7 68.95
51 155.6 907.06 927.06 5.474 -6.946 34.35 24.85 96.05 42.65 187 54.18 311.1 63 464.8 69.95
52 164.8 881.24 901.24 5.799 -5.946 36.4 25.85 101.8 43.65 198.1 55.18 329.6 64 492.4 70.95
53 174.6 856.15 876.15 6.144 -4.946 38.56 26.85 107.8 44.65 209.8 56.18 349.2 65 521.7 71.95
54 185 831.78 851.78 6.509 -3.946 40.85 27.85 114.2 45.65 222.3 57.18 370 66 552.7 72.95
55 196 808.1 828.1 6.896 -2.946 43.28 28.85 121 46.65 235.5 58.18 392 67 585.6 73.95
56 207.6 785.1 805.1 7.306 -1.946 45.86 29.85 128.2 47.65 249.6 59.18 415.3 68 620.4 74.95
57 220 762.75 782.75 7.741 -0.946 48.58 30.85 135.8 48.65 264.4 60.18 440 69 657.3 75.95
58 233.1 741.03 761.03 8.201 0.054 51.47 31.85 143.9 49.65 280.1 61.18 466.2 70 696.4 76.95
59 246.9 719.94 739.94 8.689 1.054 54.53 32.85 152.5 50.65 296.8 62.18 493.9 71 737.8 77.95
60 261.6 699.44 719.44 9.206 2.054 57.77 33.85 161.5 51.65 314.4 63.18 523.2 72 781.6 78.95
61 277.2 679.53 699.53 9.753 3.054 61.21 34.85 171.1 52.65 333.1 64.18 554.4 73 828.1 79.95
62 293.7 660.18 680.18 10.33 4.054 64.85 35.85 181.3 53.65 352.9 65.18 587.3 74 877.4 80.95
63 311.1 641.39 661.39 10.95 5.054 68.71 36.85 192.1 54.65 373.9 66.18 622.2 75 929.5 81.95
64 329.6 623.13 643.13 11.6 6.054 72.79 37.85 203.5 55.65 396.1 67.18 659.2 76 984.8 82.95
65 349.2 605.39 625.39 12.29 7.054 77.12 38.85 215.6 56.65 419.7 68.18 698.4 77 1043 83.95
66 370 588.16 608.16 13.02 8.054 81.71 39.85 228.4 57.65 444.6 69.18 740 78 1105 84.95
67 392 571.41 591.41 13.79 9.054 86.56 40.85 242 58.65 471.1 70.18 784 79 1171 85.95
68 415.3 555.15 575.15 14.61 10.05 91.71 41.85 256.4 59.65 499.1 71.18 830.6 80 1241 86.95
69 440 539.34 559.34 15.48 11.05 97.16 42.85 271.7 60.65 528.8 72.18 880 81 1314 87.95
70 466.2 523.99 543.99 16.4 12.05 102.9 43.85 287.8 61.65 560.2 73.18 932.3 82 1393 88.95
71 493.9 509.07 529.07 17.38 13.05 109.1 44.85 304.9 62.65 593.5 74.18 987.8 83 1476 89.95
72 523.2 494.58 514.58 18.41 14.05 115.5 45.85 323.1 63.65 628.8 75.18 1046 84 1563 90.95
73 554.4 480.5 500.5 19.51 15.05 122.4 46.85 342.3 64.65 666.2 76.18 1109 85 1656 91.95
74 587.3 466.82 486.82 20.67 16.05 129.7 47.85 362.6 65.65 705.8 77.18 1175 86 1755 92.95
75 622.2 453.53 473.53 21.89 17.05 137.4 48.85 384.2 66.65 747.8 78.18 1244 87 1859 93.95
76 659.2 440.62 460.62 23.2 18.05 145.6 49.85 407 67.65 792.3 79.18 1318 88 1970 94.95
77 698.4 428.08 448.08 24.58 19.05 154.2 50.85 431.2 68.65 839.4 80.18 1397 89 2087 95.95
78 740 415.89 435.89 26.04 20.05 163.4 51.85 456.9 69.65 889.3 81.18 1480 90 2211 96.95

The negative fractional midi note numbers in the note0 column (giving the fundamental frequency of the rods) are colored brown, as these cannot be properly expressed in MIDI. Beyond that, it will be clear that they are absolutely inaudible. To measure their frequency, a calibrated stroboscope can be used. The notes given as overtones are notated as fractional midi, the integer part being the midi note and the fractional part the fraction in cents. Note that the (nearly) fifth intervals between the notes 4 and 5 are the cause of our perception of a missing fundamental, heard as f4 / 2.


To strike the tines, we decided to use Kuhnke thrust type solenoids with conical face armatures. Type HM157 has an armature weighting only 8 g. The nominal voltage for 100% duty cycle is 24 V, but at that voltage the attack time is 34 ms for a traject of 5 mm. The data sheet specifies:

duty cycle power attack time force voltage
100% 2.8 W 34 ms 1 N 24 V
70% 4.3 W   1.5 N 30 V
45% 6.5 W   2 N 36 V
25% 10 W   2.5 N 45 V
15% 18 W   3 N 60 V
5% 5 2W 8 ms 5 N 103 V

The DC resistance of the coil is 206 Ohms. We decided to design the robot for operation on 60 V and thus the duty cycle should be restricted to 15%. Thus the maximum note repetition speed at full power should be brought down from the theoretical maximum of 50 strokes a second to ca. 15 strokes a second. Power supply requirements can now be derived from these data: we have 31 solenoids and thus the required power becomes: 18 W * 31 * 0.15 = 84 W. In order to stay on the safe side, a 48 V ac transformer rated at 100 VA should do the job. After rectification and smoothing, this will give us a dc voltage of ca. 65 V without load, going down to 48 V al full load. By raising the voltage, we can improve the attack time considerably as shown, but one should keep in mind that the fall back time is limited by the force of gravity. When we calculate this for a fall trajectory of 5 mm ( t = SQR(0.005/4.9) we get 32 ms. Thus the fastest full up-down cycle would take 40 ms, leading to a fastest possible repetition speed of 25 Hz. However, in practice when driven much faster than these repetition speeds, it still works fine but the anchors will not fully fall back between strokes. As a consequence, the amplitude of the produced sound will decrease.

The mass of the longest tine being ca. 400 g entails that the striking force for a good excitation ought to be between 0.39 N and 0.78 N. These values will be reached with midi velocity values around 80.

An extra and new feature of the <Rodo> design, as compared to <Toypi> is the electromagnetic feedback driver mechanism. To this end we mounted a powerful (100 W) electromagnet very close (leaving just an air gap less than 0.1 mm) to and underneath the cast iron bar. This electromagnet is driven by a high voltage digital driver whose input comes from a Microchip 24EP128MC202 microprocessor. The input for the driver can be either a signal picked up with a piezo transducer from the soundboard filtered and processed by the microcontroller, or an injected drive signal under midi-controll. This mechanism enables bowed and sustained sounds to be produced from this instrument. Rodo can sound very much like a bowed string instrument in this mode, although sound build-up is rather slow due to the inertia of the mass of the rod assembly.

An improvement over the <Toypi> robot certainly consists in the damping possibilities offered here. Each rod has an individual damper mounted exactly above the point of excitation. The duration of the contact between the felt covered damper and the tone rod can be controlled with the release value of the note-off command. A sustain controller (#64) is implemented, to disable the damping functionality completely. A controller is used to set the default damper-contact-time for note-off commands with release parameter 0. This was done for user friendliness, as most commercial sequencers do not offer the user the possibility to send note-off with release values, although these are part of the standard midi specification.

For the damper mechanism, we went for tubular push type solenoids (Lucas-Ledex type...). This entailed the use of return springs, calculated such that at rest their force just keeps the damper and the plunger above the tone rods without making contact. As the plungers on these tubular solenoids have an open cylindrical head, we attached 10 mm brass disks to the ends and fitted the return spring over the plunger. We used the same springs as we had custom made for our player pianos. There is a slight problem with the damper mechanism: as the anchors are free to move above the rods, mechanical resonances can occur at high resonance peaks of the rods. This causes a rattling noise.

A special feature of the <Rodo> robot is that it can be used fully stand-alone as a gesture controlled robot. To achieve this <Rodo> has two microwave radar systems on board. These radars, operating in the X-band at 10.587GHz, can measure and detect body movement up to a distance of 10 meters. The amount of movement as well as the velocity and accelleration of the gestures can be detected. The radar sensors as well as the associated microprocessors responsible for the data analysis are mounted on the left and right sides of the robot. If no external midi signal is connected, <Rodo> will allways operate as a stand alone interactive robot. If an external midi signal is connected, the operational parameters can be changed.

The <Rodo> robot is available as a stand-alone interactive audio art installation.


Midi implementation and mapping:

Note off: steers the dampers on the rods, if sustain is switched off. Value 0, makes use the the default damper time set by controller #14. Value 127 keeps the damper on as long as a note-on is not received. Values between 1 and 126, control the time the damper felt stays in contact with the rod. Note that it is possible to play a nice pp by using the dampers as very soft beaters. This is done by sending only note-off commands with the release value set for the required velocity in the range 1 to 126.

Note on: The velocity byte steers the force of the strokes. For very fast repetition rates, low values for velocity should be used.

The lights are mapped on midi notes as follows: (left and right hereunder are defined from the viewpoint of the listener)

Key pressure: this command should be used to send precise pitches to the e-drive mechanism independently from the operation of beaters and dampers. Polyphony is not implemented here. The note range here is 48 to 97. The key pressure value sets the attack level of the note. To switch the note off, the pressure value should be set to zero. This will only work if controllers 3, 5 and 6 are switched ON.

Controllers:

Pitch bend: used for changing a frequency for the e-drive mechanism. This will only work if controller #3 is set to a value > 0. The range is a quartertone up or down. The command should be given before the note-on command.

Program change: used to steer the tuning of the notes injected in the ebow.

Key pressure: if controllers #3 , #5 and #6 are switched on, keypressure command can be used to steer note playing with the e-bow independently from the beater mechanism. The range implemented here is larger: 48 to 97. However, the implementation is monophonic. A new note will always replace the playing note. Sending the key pressure command with a zero value for pressure, switches the note off.

<Rodo> uses midi channel 7 (counting 0-15)


Technical specifications:



Design and construction: dr.Godfried-Willem Raes (2013-2017)

Collaborators on the construction of this robot:


Music composed for <Rodo>:

 

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Last update: 2017-05-03

by Godfried-Willem Raes

Further reading by the author on the topic of musical robot design:

Bibliographical references:

MECHEL, F.P., "Formulas of Acoustics"

OLSON, Harry F. "Music, Physics and engineering"

PHILIPS Semiconductors, "Power Semiconductor Applications"

RAES, Godfried-Willem, "Gesture controlled virtual musical instruments" (1999)

RAES, Godfried-Willem, "Distance sensing" (2007)

RAES, Godfried-Willem, "BumbleBee" (2009)

RAES, Godfried-Willem, "Microwave Gesture Sensing" (2009)

RAES, Godfried-Willem, "A reminder on comparator design" (2017)

RAYLEIGH, John William Strutt "The Theory of Sound" (2 volumes)

TALUKDAR, S, "Vibration of Continuous Systems",


Technical data sheet, design calculations and maintenance instructions:

Technical data and design elements. Maintenance instructions and replacement parts.
Technische gegevens, ontwerpberekeningen en instrukties voor onderhoud en demontage:

Solenoids:

Tone rods: Material: Aluminum-bronze, Cu Al10 Ni5 Fe4, diameter 8 mm. Ordered from Demar-Lux bvba.

Cast iron bar: 40 x 40 x 1020, GG25. Ordered from Demar-Lux bvba. http://www.demar-lux.be

 

Polystyreen Eigenschappen:

Notatie EPS 60 EPS 100 EPS 150 EPS 200 EPS 250
Druksterkte bij 10% vervorming (korte duur) se=10% of CS (10) 60kPa 100kPa 150kPa 200kPa 250kPa
Lange-duur druksterkte se=2% of CS (2) 18kPa 30kPa 45kPa 60kPa 75kPa
Buigsterkte sb of BS 100kPa 150kPa 200kPa 250kPa 350kPa
E-modulus Et 4000kPa 6000kPa 8000kPa 10000kPa 12000kPa
Afschuifsterkte t 50kPa 75kPa 100kPa 125kPa 170kPa
Treksterkte st 100kPa 150kPa 200kPa 250kPa 350kPa
WrijvingscoŽfficiŽnt c 0.5 0.5 0.5 0.5 0.5
WarmtegeleidingscoŽfficiŽnt lD = lR W/m.K 0.038 K 0.036 K 0.034 K 0.033 K 0.033 K
Diffusieweerstandsgetal m 20 30 40 60 90
Vochtopname bij onderdompeling % v/v 5.0 4.0 3.5 3.0 2.0
Lineaire uitzettingscoŽfficiŽnt a m/m 7.10-5 7.10-5 7.10-5 7.10-5 7.10-5
Warmtecapaciteit C J/kg K 1450 1450 1450 1450 1450
Temperatuurbestendigheid (min/max) T -180 +80 -180 + 80 -180 +80 -180 +80 -180 + 80

Robody Pictures with <Rodo>