<Plus - Minus>

2019

 

'Plus-Minus', an interactive and demonstrative acoustic audio art project

After having build some 76 musical robots, all acoustic in their sound generation but extensively controlled through both digital and analog electronics, the latest project is very different in nature. The robots, no matter how successful they are musically, if used in a non-conventional interactive way -for instance, gesture controlled- with audiences, the results were rarely acceptable from a musical point of view. Provided with traditional interfaces (such as keyboards) at the other end, we felt ourselves merely reinventing the piano or for that matter any other existing instrument, thus making the automation of instruments a completely unnecessary mediated step. 'Plus-Minus', in contrast, became an audio art installation allowing participants to explore the world of sum- and difference tones -the base of musical harmony- in a truly interactive way. The musical aesthetics are confined to a very well defined range of possibilities. However, some intelligence is still required to make the artistic result into a seducing and surprising experience.

The 'Plus-Minus' project consists of 18 analog ring modulators (multiplier circuits) combined with 38 compressors. Purely analog as well. The output is through three large megaphone speakers. The input -and that's were the project really gets to become interactive- is a very large set of precisely tuned thick aluminum tubes, to be freely selected in pairs by the participants.

The musical range of the tubes is:

The tubes can be suspended in three rows on a triangular welded structure in stainless steel mounted on three poles. The power amplifiers found a place inside the bases of the poles. Each horizontal carrier accommodates 12 tubes, arranged in pairs. For each pair of tubes, there is an electronic circuit board holding two tiny microphones in immediate proximity to the tubes. The microphone signal is preamplified, filtered and compressed (using a 1:10 bulb compressor) before it is presented to a multiplier, causing the sum and difference tones that will be sounded on striking the tubes. Here is the complete circuit diagram:

A second, more frequency selective, compressor, using legacy AA119 Germanium diodes, follows the multiplier circuit, thus avoiding the exponential amplitude curve (and the harsh attacks...) that would otherwise be characteristic for multiplier outputs. The single sided and hand made boards -the project has 18 of them, all in the Eurocard format and using through hole components only- look like this:

The microphones are protected against shocks, impact and physical dammage with10 mm thick felt rings glued to the PCB, The 18 boards are mounted on the steel structure with shock absorbing rubber mounts. This way not only feedback is avoided but also awkward noises caused by accidentally hitting the carying structure instead of the tubular bells.

The transfer function of the multiplier, in the time-domain is

The scaling factor k, for the AD633 chips used here is k = 10. The absolute value of Ux and Uy must be <= 10 V.

Suppose we now feed the circuit with two periodic functions, for instance and to keep math simple, sinewaves:

fx and fy are the frequencies and phase angles of the input and Uxp and Uyp the peak amplitudes of both signals.

If we now develop the product we get:

thus leading to the conclusion that the output signal contains the sum and difference frequencies of the input frequencies. Generalising for arbitrary periodic waveforms, the circuit will produce all sum and difference tones to be found in the two input signals.

For the sets of tubes we tuned to equal temperament and provide with the project, the sum- and difference tones, expressed as fractional midi-notes, can be found in this table:
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
72
0.
23.07
35.62
43.18
48.67
53.04
56.68
59.89
62.77
65.38
67.74
69.92
71.99
73.93
75.78
77.55
79.24
80.86
82.43
83.96
85.45
86.9
88.29
89.67
91.02
92.34
93.64
94.92
96.17
97.41
98.63
99.84
101.03
102.22
103.38
104.54
105.69
73
84.5
0.
24.16
36.67
44.19
49.66
54.
57.69
60.93
63.81
66.37
68.72
70.93
72.99
74.94
76.79
78.55
80.24
81.86
83.44
84.97
86.46
87.89
89.3
90.68
92.02
93.34
94.64
95.92
97.17
98.41
99.63
100.84
102.04
103.21
104.38
105.54
74
85.02
85.5
0.
25.18
37.66
45.16
50.6
55.
58.72
61.96
64.79
67.35
69.73
71.93
73.99
75.94
77.79
79.54
81.23
82.86
84.45
85.98
87.45
88.89
90.3
91.67
93.02
94.34
95.64
96.91
98.17
99.41
100.63
101.84
103.03
104.22
105.38
75
85.56
86.02
86.5
0.
26.14
38.6
46.07
51.59
56.03
59.76
62.94
65.76
68.36
70.72
72.93
74.99
76.94
78.78
80.54
82.23
83.88
85.45
86.97
88.45
89.89
91.3
92.67
94.02
95.34
96.63
97.91
99.17
100.41
101.63
102.84
104.03
105.22
76
86.11
86.56
87.02
87.5
0.
27.05
39.49
47.07
52.64
57.08
60.74
63.91
66.78
69.35
71.72
73.93
75.99
77.93
79.78
81.54
83.25
84.88
86.44
87.97
89.45
90.89
92.3
93.68
95.02
96.34
97.63
98.91
100.17
101.41
102.63
103.84
105.03
77
86.67
87.1
87.55
88.02
88.5
0.
27.91
40.54
48.16
53.72
58.07
61.72
64.94
67.78
70.36
72.74
74.94
76.99
78.93
80.78
82.56
84.25
85.87
87.45
88.98
90.45
91.89
93.3
94.68
96.02
97.34
98.63
99.91
101.17
102.41
103.63
104.84
78
87.24
87.66
88.1
88.55
89.01
89.49
0.
29.14
41.72
49.3
54.73
59.07
62.77
65.95
68.8
71.39
73.75
75.94
78.
79.95
81.81
83.57
85.25
86.88
88.46
89.98
91.46
92.9
94.31
95.67
97.02
98.34
99.64
100.92
102.17
103.41
104.64
79
87.84
88.24
88.66
89.1
89.55
90.01
90.48
0.
30.28
42.82
50.25
55.69
60.09
63.76
66.96
69.81
72.38
74.74
76.94
79.
80.96
82.81
84.56
86.25
87.89
89.45
90.98
92.46
93.9
95.3
96.67
98.02
99.34
100.64
101.92
103.18
104.41
80
88.45
88.84
89.25
89.67
90.11
90.55
91.01
91.49
0.
31.35
43.69
51.16
56.68
61.05
64.74
67.95
70.79
73.36
75.73
77.93
80.01
81.96
83.79
85.56
87.26
88.88
90.45
91.98
93.46
94.89
96.3
97.67
99.02
100.34
101.64
102.92
104.17
81
89.08
89.45
89.85
90.26
90.68
91.11
91.55
92.02
92.5
0.
32.03
44.52
52.12
57.62
62.02
65.72
68.92
71.76
74.34
76.71
78.93
81.
82.93
84.78
86.56
88.24
89.87
91.45
92.97
94.44
95.88
97.29
98.67
100.02
101.34
102.64
103.91
82
89.71
90.07
90.45
90.84
91.25
91.67
92.1
92.55
93.02
93.5
0.
33.
45.62
53.14
58.65
63.05
66.73
69.92
72.77
75.35
77.74
79.95
81.99
83.94
85.8
87.55
89.25
90.88
92.45
93.97
95.45
96.89
98.3
99.68
101.02
102.34
103.64
83
90.35
90.7
91.07
91.45
91.84
92.25
92.66
93.1
93.55
94.02
94.5
0.
34.22
46.65
54.19
59.69
64.07
67.74
70.93
73.78
76.38
78.76
80.94
83.
84.96
86.8
88.56
90.26
91.88
93.45
94.97
96.45
97.89
99.3
100.68
102.02
103.34
84
91.01
91.35
91.7
92.07
92.45
92.84
93.24
93.67
94.1
94.56
95.02
95.5
0.
35.07
47.62
55.18
60.67
65.04
68.72
71.93
74.8
77.38
79.74
81.94
84.01
85.95
87.8
89.56
91.25
92.87
94.44
95.97
97.45
98.9
100.3
101.68
103.02
85
91.68
92.01
92.35
92.7
93.07
93.45
93.84
94.24
94.67
95.11
95.56
96.02
96.5
0.
36.16
48.67
56.19
61.66
66.05
69.73
72.96
75.81
78.37
80.74
82.95
85.
86.95
88.8
90.56
92.25
93.87
95.45
96.97
98.46
99.89
101.3
102.68
86
92.37
92.68
93.01
93.35
93.7
94.07
94.44
94.84
95.25
95.67
96.11
96.55
97.02
97.5
0.
37.18
49.66
57.16
62.66
67.04
70.76
73.96
76.79
79.37
81.75
83.94
86.
87.95
89.8
91.55
93.24
94.87
96.45
97.98
99.45
100.9
102.3
87
93.07
93.37
93.68
94.01
94.35
94.7
95.07
95.45
95.84
96.25
96.67
97.11
97.56
98.02
98.5
0.
38.14
50.6
58.14
63.65
68.07
71.76
74.94
77.79
80.38
82.74
84.94
87.01
88.95
90.79
92.55
94.24
95.88
97.45
98.97
100.45
101.89
88
93.78
94.07
94.37
94.69
95.01
95.35
95.7
96.07
96.45
96.85
97.25
97.67
98.11
98.56
99.02
99.5
0.
39.05
51.59
59.14
64.69
69.08
72.74
75.94
78.8
81.37
83.74
85.95
88.01
89.94
91.79
93.55
95.25
96.88
98.45
99.97
101.45
89
94.5
94.77
95.07
95.37
95.68
96.01
96.34
96.7
97.07
97.45
97.84
98.25
98.67
99.1
99.55
100.02
100.5
0.
40.12
52.64
60.22
65.72
70.07
73.75
76.96
79.8
82.38
84.75
86.95
89.
90.94
92.79
94.56
96.25
97.88
99.45
100.98
90
95.23
95.5
95.78
96.07
96.37
96.68
97.
97.34
97.7
98.07
98.45
98.84
99.25
99.67
100.1
100.56
101.02
101.5
0.
41.14
53.72
61.24
66.69
71.07
74.77
77.95
80.8
83.39
85.75
87.94
90.
91.95
93.8
95.56
97.25
98.88
100.45
91
95.98
96.23
96.5
96.78
97.07
97.37
97.68
98.
98.35
98.71
99.07
99.45
99.84
100.25
100.67
101.11
101.56
102.02
102.5
0.
42.28
54.73
62.2
67.69
72.09
75.76
78.96
81.81
84.38
86.74
88.94
91.
92.95
94.8
96.56
98.25
99.88
92
96.74
96.98
97.24
97.5
97.78
98.07
98.37
98.68
99.01
99.36
99.71
100.07
100.45
100.85
101.25
101.68
102.11
102.56
103.02
103.5
0.
43.18
55.61
63.16
68.68
73.05
76.74
79.95
82.79
85.36
87.73
89.93
92.
93.95
95.79
97.56
99.25
93
97.5
97.74
97.98
98.24
98.51
98.78
99.07
99.37
99.69
100.02
100.36
100.71
101.08
101.45
101.85
102.26
102.68
103.11
103.56
104.02
104.51
0.
44.03
56.6
64.18
69.66
74.05
77.74
80.94
83.78
86.36
88.73
90.93
93.
94.94
96.79
98.56
94
98.28
98.5
98.73
98.98
99.23
99.5
99.78
100.07
100.37
100.69
101.01
101.35
101.71
102.07
102.45
102.84
103.25
103.67
104.11
104.56
105.02
105.5
0.
45.16
57.69
65.19
70.69
75.08
78.76
81.94
84.78
87.36
89.74
91.95
94.
95.95
97.8
95
99.06
99.28
99.5
99.74
99.98
100.23
100.5
100.78
101.07
101.38
101.69
102.01
102.35
102.71
103.07
103.45
103.85
104.25
104.67
105.11
105.56
106.03
106.5
0.
46.22
58.65
66.19
71.69
76.07
79.74
82.93
85.78
88.37
90.74
92.94
95.
96.95
96
99.86
100.07
100.28
100.5
100.74
100.98
101.23
101.5
101.78
102.08
102.38
102.69
103.02
103.36
103.71
104.08
104.46
104.85
105.25
105.68
106.11
106.56
107.03
107.5
0.
47.07
59.62
67.18
72.67
77.04
80.72
83.93
86.78
89.37
91.74
93.94
96.
97
100.66
100.86
101.06
101.28
101.5
101.74
101.98
102.23
102.5
102.78
103.07
103.37
103.69
104.01
104.35
104.71
105.07
105.45
105.84
106.25
106.68
107.11
107.56
108.02
108.51
0.
48.16
60.67
68.19
73.66
78.05
81.73
84.94
87.8
90.37
92.74
94.94
98
101.48
101.66
101.86
102.06
102.28
102.5
102.73
102.98
103.24
103.51
103.78
104.07
104.37
104.69
105.01
105.36
105.71
106.07
106.45
106.85
107.26
107.68
108.11
108.56
109.03
109.5
0.
49.18
61.66
69.16
74.66
79.04
82.74
85.95
88.79
91.37
93.74
99
102.3
102.48
102.67
102.86
103.07
103.28
103.5
103.74
103.98
104.24
104.51
104.78
105.07
105.37
105.69
106.02
106.36
106.71
107.07
107.45
107.85
108.26
108.68
109.11
109.56
110.03
110.51
0.
50.14
62.6
70.14
75.65
80.05
83.74
86.94
89.79
92.37
100
103.13
103.3
103.48
103.67
103.86
104.07
104.28
104.5
104.74
104.99
105.24
105.5
105.78
106.07
106.38
106.69
107.02
107.36
107.71
108.07
108.46
108.85
109.26
109.68
110.12
110.56
111.03
111.51
0.
51.05
63.59
71.14
76.66
81.06
84.74
87.94
90.79
101
103.97
104.13
104.3
104.48
104.66
104.86
105.06
105.27
105.5
105.74
105.98
106.23
106.5
106.78
107.07
107.37
107.69
108.01
108.35
108.7
109.07
109.45
109.85
110.25
110.68
111.11
111.56
112.03
112.5
0.
52.12
64.64
72.19
77.69
82.07
85.75
88.95
102
104.81
104.97
105.13
105.3
105.48
105.66
105.85
106.06
106.27
106.5
106.73
106.98
107.23
107.5
107.78
108.07
108.37
108.69
109.01
109.35
109.71
110.07
110.45
110.85
111.26
111.68
112.11
112.56
113.03
113.5
0.
53.14
65.67
73.21
78.69
83.07
86.75
103
105.67
105.81
105.97
106.13
106.3
106.47
106.66
106.85
107.06
107.28
107.5
107.73
107.98
108.23
108.5
108.78
109.07
109.37
109.69
110.01
110.36
110.71
111.07
111.45
111.85
112.25
112.68
113.11
113.56
114.02
114.5
0.
54.19
66.69
74.2
79.69
84.07
104
106.53
106.67
106.81
106.97
107.13
107.3
107.48
107.66
107.86
108.07
108.28
108.5
108.74
108.98
109.24
109.5
109.78
110.07
110.37
110.69
111.02
111.36
111.71
112.08
112.46
112.85
113.26
113.68
114.11
114.56
115.02
115.5
0.
55.18
67.65
75.19
80.68
105
107.4
107.53
107.67
107.82
107.97
108.13
108.3
108.48
108.67
108.86
109.07
109.28
109.5
109.74
109.98
110.24
110.51
110.78
111.07
111.37
111.69
112.02
112.36
112.71
113.08
113.46
113.85
114.26
114.68
115.11
115.56
116.03
116.51
0.
56.11
68.64
76.18
106
108.27
108.4
108.53
108.67
108.82
108.97
109.13
109.3
109.48
109.67
109.86
110.06
110.28
110.5
110.74
110.98
111.24
111.5
111.78
112.07
112.38
112.69
113.02
113.36
113.71
114.08
114.46
114.85
115.26
115.68
116.11
116.56
117.03
117.51
0.
57.16
69.66
107
109.15
109.27
109.4
109.53
109.67
109.82
109.97
110.13
110.3
110.48
110.67
110.86
111.07
111.28
111.5
111.74
111.98
112.24
112.5
112.78
113.08
113.38
113.69
114.02
114.36
114.71
115.08
115.46
115.85
116.26
116.68
117.11
117.56
118.03
118.51
0.
58.14
108
110.04
110.15
110.27
110.4
110.53
110.67
110.81
110.97
111.13
111.3
111.48
111.67
111.86
112.07
112.28
112.5
112.74
112.98
113.24
113.5
113.79
114.08
114.38
114.69
115.02
115.36
115.71
116.08
116.46
116.85
117.26
117.68
118.11
118.56
119.03
119.51
0.

The same table, but here expressed as frequencies:
523
554
587
622
659
698
739
783
830
880
932
987
1046
1108
1174
1244
1318
1396
1479
1567
1661
1760
1864
1975
2093
2217
2349
2489
2637
2793
2959
3135
3322
3520
3729
3951
4186
523
0
31
64
99
136
175
216
260
307
357
409
464
523
585
651
721
795
873
956
1044
1138
1237
1341
1452
1570
1694
1826
1966
2114
2270
2436
2612
2799
2997
3206
3428
3663
554
1077
0
33
68
105
144
185
229
276
326
378
433
492
554
620
690
764
842
925
1013
1107
1206
1310
1421
1539
1663
1795
1935
2083
2239
2405
2581
2768
2966
3175
3397
3632
587
1110
1141
0
35
72
111
152
196
243
293
345
400
459
521
587
657
731
809
892
980
1074
1173
1277
1388
1506
1630
1762
1902
2050
2206
2372
2548
2735
2933
3142
3364
3599
622
1145
1176
1209
0
37
76
117
161
208
258
310
365
424
486
552
622
696
774
857
945
1039
1138
1242
1353
1471
1595
1727
1867
2015
2171
2337
2513
2700
2898
3107
3329
3564
659
1182
1213
1246
1281
0
39
80
124
171
221
273
328
387
449
515
585
659
737
820
908
1002
1101
1205
1316
1434
1558
1690
1830
1978
2134
2300
2476
2663
2861
3070
3292
3527
698
1221
1252
1285
1320
1357
0
41
85
132
182
234
289
348
410
476
546
620
698
781
869
963
1062
1166
1277
1395
1519
1651
1791
1939
2095
2261
2437
2624
2822
3031
3253
3488
739
1262
1293
1326
1361
1398
1437
0
44
91
141
193
248
307
369
435
505
579
657
740
828
922
1021
1125
1236
1354
1478
1610
1750
1898
2054
2220
2396
2583
2781
2990
3212
3447
783
1306
1337
1370
1405
1442
1481
1522
0
47
97
149
204
263
325
391
461
535
613
696
784
878
977
1081
1192
1310
1434
1566
1706
1854
2010
2176
2352
2539
2737
2946
3168
3403
830
1353
1384
1417
1452
1489
1528
1569
1613
0
50
102
157
216
278
344
414
488
566
649
737
831
930
1034
1145
1263
1387
1519
1659
1807
1963
2129
2305
2492
2690
2899
3121
3356
880
1403
1434
1467
1502
1539
1578
1619
1663
1710
0
52
107
166
228
294
364
438
516
599
687
781
880
984
1095
1213
1337
1469
1609
1757
1913
2079
2255
2442
2640
2849
3071
3306
932
1455
1486
1519
1554
1591
1630
1671
1715
1762
1812
0
55
114
176
242
312
386
464
547
635
729
828
932
1043
1161
1285
1417
1557
1705
1861
2027
2203
2390
2588
2797
3019
3254
987
1510
1541
1574
1609
1646
1685
1726
1770
1817
1867
1919
0
59
121
187
257
331
409
492
580
674
773
877
988
1106
1230
1362
1502
1650
1806
1972
2148
2335
2533
2742
2964
3199
1046
1569
1600
1633
1668
1705
1744
1785
1829
1876
1926
1978
2033
0
62
128
198
272
350
433
521
615
714
818
929
1047
1171
1303
1443
1591
1747
1913
2089
2276
2474
2683
2905
3140
1108
1631
1662
1695
1730
1767
1806
1847
1891
1938
1988
2040
2095
2154
0
66
136
210
288
371
459
553
652
756
867
985
1109
1241
1381
1529
1685
1851
2027
2214
2412
2621
2843
3078
1174
1697
1728
1761
1796
1833
1872
1913
1957
2004
2054
2106
2161
2220
2282
0
70
144
222
305
393
487
586
690
801
919
1043
1175
1315
1463
1619
1785
1961
2148
2346
2555
2777
3012
1244
1767
1798
1831
1866
1903
1942
1983
2027
2074
2124
2176
2231
2290
2352
2418
0
74
152
235
323
417
516
620
731
849
973
1105
1245
1393
1549
1715
1891
2078
2276
2485
2707
2942
1318
1841
1872
1905
1940
1977
2016
2057
2101
2148
2198
2250
2305
2364
2426
2492
2562
0
78
161
249
343
442
546
657
775
899
1031
1171
1319
1475
1641
1817
2004
2202
2411
2633
2868
1396
1919
1950
1983
2018
2055
2094
2135
2179
2226
2276
2328
2383
2442
2504
2570
2640
2714
0
83
171
265
364
468
579
697
821
953
1093
1241
1397
1563
1739
1926
2124
2333
2555
2790
1479
2002
2033
2066
2101
2138
2177
2218
2262
2309
2359
2411
2466
2525
2587
2653
2723
2797
2875
0
88
182
281
385
496
614
738
870
1010
1158
1314
1480
1656
1843
2041
2250
2472
2707
1567
2090
2121
2154
2189
2226
2265
2306
2350
2397
2447
2499
2554
2613
2675
2741
2811
2885
2963
3046
0
94
193
297
408
526
650
782
922
1070
1226
1392
1568
1755
1953
2162
2384
2619
1661
2184
2215
2248
2283
2320
2359
2400
2444
2491
2541
2593
2648
2707
2769
2835
2905
2979
3057
3140
3228
0
99
203
314
432
556
688
828
976
1132
1298
1474
1661
1859
2068
2290
2525
1760
2283
2314
2347
2382
2419
2458
2499
2543
2590
2640
2692
2747
2806
2868
2934
3004
3078
3156
3239
3327
3421
0
104
215
333
457
589
729
877
1033
1199
1375
1562
1760
1969
2191
2426
1864
2387
2418
2451
2486
2523
2562
2603
2647
2694
2744
2796
2851
2910
2972
3038
3108
3182
3260
3343
3431
3525
3624
0
111
229
353
485
625
773
929
1095
1271
1458
1656
1865
2087
2322
1975
2498
2529
2562
2597
2634
2673
2714
2758
2805
2855
2907
2962
3021
3083
3149
3219
3293
3371
3454
3542
3636
3735
3839
0
118
242
374
514
662
818
984
1160
1347
1545
1754
1976
2211
2093
2616
2647
2680
2715
2752
2791
2832
2876
2923
2973
3025
3080
3139
3201
3267
3337
3411
3489
3572
3660
3754
3853
3957
4068
0
124
256
396
544
700
866
1042
1229
1427
1636
1858
2093
2217
2740
2771
2804
2839
2876
2915
2956
3000
3047
3097
3149
3204
3263
3325
3391
3461
3535
3613
3696
3784
3878
3977
4081
4192
4310
0
132
272
420
576
742
918
1105
1303
1512
1734
1969
2349
2872
2903
2936
2971
3008
3047
3088
3132
3179
3229
3281
3336
3395
3457
3523
3593
3667
3745
3828
3916
4010
4109
4213
4324
4442
4566
0
140
288
444
610
786
973
1171
1380
1602
1837
2489
3012
3043
3076
3111
3148
3187
3228
3272
3319
3369
3421
3476
3535
3597
3663
3733
3807
3885
3968
4056
4150
4249
4353
4464
4582
4706
4838
0
148
304
470
646
833
1031
1240
1462
1697
2637
3160
3191
3224
3259
3296
3335
3376
3420
3467
3517
3569
3624
3683
3745
3811
3881
3955
4033
4116
4204
4298
4397
4501
4612
4730
4854
4986
nh
0
156
322
498
685
883
1092
1314
1549
2793
3316
3347
3380
3415
3452
3491
3532
3576
3623
3673
3725
3780
3839
3901
3967
4037
4111
4189
4272
4360
4454
4553
4657
4768
4886
nh
nh
nh
nh
0
166
342
529
727
936
1158
1393
2959
3482
3513
3546
3581
3618
3657
3698
3742
3789
3839
3891
3946
4005
4067
4133
4203
4277
4355
4438
4526
4620
4719
4823
4934
nh
nh
nh
nh
nh
nh
0
176
363
561
770
992
1227
3135
3658
3689
3722
3757
3794
3833
3874
3918
3965
4015
4067
4122
4181
4243
4309
4379
4453
4531
4614
4702
4796
4895
4999
nh
nh
nh
nh
nh
nh
nh
nh
0
187
385
594
816
1051
3322
3845
3876
3909
3944
3981
4020
4061
4105
4152
4202
4254
4309
4368
4430
4496
4566
4640
4718
4801
4889
4983
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
0
198
407
629
864
3520
4043
4074
4107
4142
4179
4218
4259
4303
4350
4400
4452
4507
4566
4628
4694
4764
4838
4916
4999
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
0
209
431
666
3729
4252
4283
4316
4351
4388
4427
4468
4512
4559
4609
4661
4716
4775
4837
4903
4973
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
0
222
457
3951
4474
4505
4538
4573
4610
4649
4690
4734
4781
4831
4883
4938
4997
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
0
235
4186
4709
4740
4773
4808
4845
4884
4925
4969
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
nh
0

Frequencies marked as nh in the tables are inaudible, either because of auditory limitations, or of the filters as designed into our circuitry.

For the calculation of the required tubelengths, we started from the formula given in Harry Olson, 'Music, Physics and engineering' p.77:

Reformulated to units in the SI system this gives:

 

As we use the same material for all tubes (aluminum, AlMgSi 0.5, with density 2.699 kg / liter) and we also keep the thickness the same (30 mm / 20 mm), we can simplify the formula. The part under the square root becomes a constant. The factor 1.133 given by Olson (and blindly taken over in many textbooks on acoustics) , after our measurements for note 84 (f=1046Hz) appears to be 1.203.. The deviation is a lot larger than what can be attributed to the precision of our measurements... Furthermore, the value decreases as the tubes become shorter.

After tuning all 36 tubes and feeding the measurements to a curve-fitting program, we came to the following practical formula to calculate the fundamental frequency in function of the length of the tube:

The 'constants' in the textbook formula are clearly a function of pitch. We do not have a sound explanation for this result. Maybe the force exerted on the tube by its weight when suspended, contributes to the deviation? Very unlikely. Maybe the propagation speed of sound/vibration itself is a function of frequency? Maybe, Olson's formula was never empirically checked? We guess the latter, as nowhere we found empirical data...

Table with exact lengths of the tubes, point of suspension, midi note and frequency for the tubes:

  • absolute error on the length measurements > 310mm : 0.5mm
  • absolute error on the length measurements < 310mm: 0.02mm
  • error on the frequencies measured : 2 cent.
midi note
frequency (Hz)
length (in mm)
suspension point
'constant'
72
523
557.0
124.6
1622602
73
554
541.0
121.0
1621452
74
587
526.0
117.6
1624088
75
622
510.0
114.2
1617822
76
659
496.0
110.8
1621245
77
698

481.0
107.8
1614899
78
739
466.5
104.5
1608228
79
783
453.6
101.7
1611045
80
830
440.0
98.6
1606880
81
880
426.5
95.6
1600739
82
932
414.5
92.9
1601271
83
987
402.0
90.1
1595031
84
1046
391.5
87.7
1603227
85
1108
380.0
85.2
1599952
86
1174
369.0
82.7
1598530
87
1244
357.0
80.0
1585465
88
1318
347.5
77.9
1591567
89
1396
336.0
75.3
1576028
90
1479
326.5
73.2
1576647
91
1576
316.8
71.0
1572676
92
1661
307.73
69.0
1572929
93
1760
298.72
67.0
1570512
94
1864
290.36
65.1
1571518
95
1975
281.18
63.0
1561478
96
2093
272.50
61
1551902
97
2217
264.10
59.2
1546331
98
2349
256.62
57.5
1546906
99
2489
249.14
55.7
1544940
100
2637
238.90
53.6
1505020
101
2793
234.22

52.4
1532212
102
2959
227.28
50.8
1528506
103
3135
219.96
49.3
1516788
104
3322
213.20
47.8
1509989
105
3520
206.70
46.3
1503916
106
3729
199.62
44.8
1485937
107
3951
194.08
43.4
1488225
108
4186
188.34
42.0
1484856

 

The output of the Plus-Minus installation comes from three large megaphone speakers mounted on the verical poles. We used such loudspeakers for they have an intrinsic high-pass filter characteristic (thus avoiding feedback by mechanical coupling) at the one hand, and for their pronounced directional radiation pattern. Thus players can clearly hear what they are doing, the sound source being very close to the origin of the vibrations.

The installation can be played by three persons simultaneously. Vibraphone mallets go with the instrument, as we found them to sound best on our aluminum tubes. Obviously, other mallets and materials can be used as well. A side effect we discovered soon enough, consists in the possibility to sing in close proximity to the tubes. The vocal sounds than get ring-modulated with the struck tube sounds, leading to quite intriguing musical results.

The whole installation is easily transportable and can be set up in less than an hour. It can be rented from the Logos Foundation.

Here is a link to a little sound demo of the Plus-Minus installation (MP3).

dr.Godfried-Willem Raes

Technical data:

Parts list and component data:

 

This paper was presented by the author on the AES conference at De Krook, Ghent, may 14th 2019.