Statistical Analysis of Tunings and
Acoustical Beating Rates in Balinese Gamelans

CHRISTINE N. SOUTHWORTH
June, 2001



1. Introduction Gamelan music is the traditional music of Indonesia. The word gamelan refers to this type of music, the instruments that the music is played on (generally an orchestra of metallophones, gongs, and hand-drums, or a smaller group containing only a few metallophones), and the group of people who play it, much like the western "orchestra". Gamelan music is played on a set of instruments and each particular group of instruments has its own tuning system so that all of the different gamelans, of which there are hundreds, have their own distinct sound. The particular tuning system has a starting frequency, or tonic note, and then subsequent intervals between frequencies to make up an octave scale of between five and seven notes. This set of intervals is a scale, and each instrument has one to several scales on it. The tunings of the various Indonesian gamelans generally differ in both starting frequencies and subsequent intervals.

In Bali, an island in Indonesia, tunings of gamelans are generally either pelog or slendro. Pelog is a five tone (pentatonic) scale based on an ancient seven tone scale. The scale has three small intervals and two large intervals, the third and fifth. Slendro is also pentatonic, but the intervals are fairly equal. This makes pelog tunings have a natural tonic built into the system, whereas slendro tunings do not. So on slendro instruments, melodies can be transposed up and down the instrument with ease; in pelog, transposing a melody will completely change the melody. Other tunings besides these two do exist in Bali as well. Recently, a seven tone scale based on the ancient pelog system has been brought back into use, and is called Semar Peguligan. Other, less widely used tunings are also found around the island.

There is an interesting phenomenon in Balinese music that does not occur anywhere else in Indonesia. The instruments are paired so that they produce acoustical beats. So within a gamelan, half of the instruments are tuned to one tuning, and the other half are tuned slightly off of the first half. The beating rates vary, both within each gamelan and between different gamelans.

This paper presents a comparison of the variations in beating rates in five different gamelans. Three of the gamelans were recorded in Bali last summer, and two belong to MIT. I recorded many more gamelans last summer, but the data was corrupted so I cannot use it in this study. The gamelans studied here are a Gamelan Angklung from Bali Utara (22 instruments) and a Gender Wayang pair, recorded in the STSI Museum, Denpasar, Bali, a Gender Wayang pair belonging to Pak Wayan Loceng, Sukawati, Bali, Gamelan Gong Kebyar from MIT's Galak Tika (20 instruments), and Gamelan Gender Wayang from Galak Tika (4 instruments). All of these instruments were made in Bali. The Galak Tika gamelan has been tuned since then by members of the group.


2. Method The frequencies of the instruments were collected in two ways: the instruments in Bali were recorded on DAT (digital audio tape) and then brought into a computer later to obtain frequencies, while for the instruments here I used a chromatic tuner to obtain frequencies. The latter proved to be a much easier and just as accurate method. In Bali, I recorded seventeen sets of instruments at the music museum at STSI, a university in Denpasar. Unfortunately, most of the instruments were recorded poorly and I could not obtain frequencies from the tapes once I returned home. I measured four sets of instruments here once I returned, however. These instruments were measured using a standard chromatic tuner which gives the note name plus or minus cent value (eg. F# + 30 cents). I converted these values to frequencies by first normalizing each note (F# etc.) by taking the base 2 logarithm of its frequency given by Sheibler's Stuttgart table [2, pp. 437], an index of the standard frequency values for the Western equal tempered scale. By taking the value of one note minus the note a half step down from it (F# - F) and dividing that number by 100 (there are 100 cents between two consecutive notes), I found that each normalized cent has a value of .0008 times the actual cent value. So to find the normalized frequency of each sampled note I added the normalized note value plus .0008*number of cents off. Then by taking two to this power, I found the frequency of the note.

Balinese gamelans are organized into different sections of instruments, ranging in pitch from the very low gongs to very high gangsa (ten-keyed metallophones). The gangsa is a set of up to ten instruments divided into low pemade and high kantilan. Each individual instrument has two octaves - the lower octave of the instruments in the kantilan is the same as the upper octave of the instruments in the pemade. Within the pemade and the kantilan, the instruments are grouped into two groups of instruments tuned slightly off each other, called polos and sangsih. The polos are tuned lower, the sangsih higher. They are only tuned off by a matter of a few hertz, and the amount by which they differ causes beating at different rates.

Acoustical beating is a phenomenon that occurs when two simple harmonic vibrations of equal amplitude but slightly different frequencies are sounded at the same time. They add up to form a sound with the frequency being the average of the two frequencies but the amplitude varies over time to form a larger wave envelope. [see drawing below from 1, p.607]

Figure 1 Two pitches of same amplitude but slightly different frequencies (A and B) sounded together produce C, a sound with frequency equal to the average of A and B but varying amplitude produced by interference formed by A+ B. This effect is called beating. This variation in amplitude is called beating. The number of beats per second is measured by the difference in the frequencies of the initial pitches. Therefore if one key were tuned to 441 Hz and its counterpart were tuned at 453 hertz, the resulting beating rate would be 12 beats/second.

Within a gangsa section of five instruments, there are either three low, polos, instruments and two high, sangsih, or three sangsih and two polos. Although all of the polos and all of the sangsih are supposed to be tuned the same, that is not the case. So in order to examine beating rates effectively I have looked at the six combinations of polos and sangsih in each section. So for a polos-dominated gangsa section with instruments tuned to the frequencies (p1, p2, p3, s1, s2), the beating rates used in this paper are for the frequency differences (s1-p1, s2-p1, s1-p2, s2-p2, s1-p3, s2-p3). The beating rates themselves vary widely, but when we look at the averages of all six beating rates within a section, we find some interesting trends.

3. Observations Plotting beating rates on their own gives no interesting correlations or trends. The beating rates vary between 5 and 30 beats per second and do not follow any sort of trend from one end of the instrument to the other. From talking to Balinese instrument makers, I would expect that the beating rates were roughly constant along the instruments or, in the very least, would vary linearly. But no such trends are found, meaning that the instruments are tuned somewhat randomly. However, if we look at the average beating rates of the pemade and kantilan of the three different large (more than 2 instruments) gamelans, we find that the pemade and kantilan beating rates seem to be roughly correlated. [see Figure 2 below]. The lines plotted are the averages of the six beating rates (((s1-p1) + (s2-p1) + (s1-p2) + (s2-p2) + (s1-p3) + (s2-p3)) / 6) for each key of the instruments. So, for instance, in the plot of Gamelan Angklung from Bali Utara, for each of the six keys in the scale the average beating rates are plotted in blue for the pemade, lower instruments, and in green for the kantilan, higher instruments.


Figure 2 Average Beating Rates of Gangsa Pemade (blue) and Kantilan (green), plotted beating rates (y) versus key number (x). Pemade and kantilan beating rates appear to roughly correlate.

The beating rates of the pemade and kantilan appear to be correlated. In order to test this, we first look at the averages of beating rates in each of the three gamelans plotted pemade on the x-axis versus kantilan on the y-axis, so that we can see how closely they match [see Figure 3 below]. If the two were completely the same, the points would all lie along the central axis. The Bali Utara Angklung (from now on referred to as angklung) is plotted in red circles, Galak Tika Gong Kebyar (gong kebyar) is plotted in blue triangles, and the Galak Tika Gender Wayang (gender) is plotted in green plus signs. The angklung and gong kebyar lie quite close to the axis, while the gender points are further away. However, for all three examined gamelans over 70% of the points lie very close to the axis. Notice also that most points lie above the axis, implying that generally the kantilan have faster beating rates than the pemade.

Figure 3 Averages of Beating Rates for Angklung (red), Gong Kebyar (blue), Gender Wayang (green)

To further test how well these beating rates are correlated, we look at their correlation coefficients. A correlation coefficient is the probability that two variables are linearly correlated. It is the covariance of the two variables divided by the product of the individual variances, i.e. Sigma(xy)/(sigma(x)*sigma(y)) where sigma(xy) is the covariance. This should give a value between +1 (perfect positive correlation) and -1 (perfect negative correlation), with 0 meaning uncorrelated.

For the three gamelans above, the correlation coefficient for the averages of beating rates of the pemade versus kantilan are 0.6209 for the angklung, 0.2224 for the gong kebyar, and 0.6367 for the gender wayang. Below is a chart of explaining the significance of a correlation according to these values. How "significant", ie. how likely that the observed variables are linearly related, depends on the number of individual observations, which in this case is the number of keys on the instrument.

NΚΚΚΚΚ 0ΚΚΚ 0.1ΚΚΚ 0.2ΚΚΚ 0.3ΚΚΚ 0.4ΚΚΚ 0.5ΚΚΚ 0.6ΚΚΚ 0.7ΚΚΚ 0.8ΚΚΚ 0.9

Κ6ΚΚΚΚ 100Κ 85ΚΚΚΚ 70ΚΚΚΚ 56ΚΚΚΚ 43ΚΚΚΚ 31ΚΚΚΚ 21ΚΚΚΚ 12ΚΚΚΚ 5.6ΚΚΚ 1.4

10ΚΚΚ 100Κ 78ΚΚΚΚ 58ΚΚΚΚ 40ΚΚΚΚ 25ΚΚΚΚ 14ΚΚΚΚ 6.7ΚΚΚ 2.4ΚΚΚ 0.5ΚΚΚ ~0

The 0.1 to 0.9 on the top are values of correlation coefficient. The values below "N" (6,10) are the number of observations. The values are the percent probabilities that two uncorrelated variables would give a correlation coefficient this high. So if a scale has notes and the correlation coefficient is 0.6, there is a 21% chance that the variables are not correlated.

Since there are 6 observations which we are comparing for the angklung, there is an 89% probability that the beating rates of the pemade and kantilan are correlated. For the gong kebyar, with 10 keys, there is a 44% probability that they are correlated. And for the gender, also with 10 keys, there is a 94% probability that the pemade and kantilan beating rates are correlated.

4. Conclusion Although the gong kebyar shows a weak correlation, it is still significant. Oddly, this trend was not intentional by the instrument makers in Bali. They intend to make all of the beating rates constant. Yet this strange trend between the beating rates along the pemade and kantilan occurs. More data is needed to prove whether this trend is real or if it is coincidental in this small data set, however the strong correlation coefficients lead us to believe that it is indeed a real trend. More investigation must be done, however, to discover how well gamelans all over Bali do tend to follow this trend, and also to look for reasons why this strange phenomenon occurs.



REFERENCES / BIBLIOGRAPHY
[1] George Bekefi and Alan H. Barrett, "Electromagnetic Vibrations, Waves, and Radiation," Sixth Printing, MIT Press, 1990.
[2] Hermann Helmholtz, "On the Sensations of Tone," Second English Edition, Dover Publications, 1954.
[3] Robert V. Hogg and Elliot A. Tanis, "Probability and Statistical Inference," Fifth Edition, Prentice Hall, 1997.
[4] Mark Lindley and Ronald Turner-Smith, "Mathematical Models of Musical Scales," Verlag fur systematische Musikwissenschaft, 1993.
[5] Colin McPhee, "Music in Bali," Yale University Press, 1966.
[6] Robert S. Pindyck and Daniel L. Rubinfeld, "Econometric Models and Economic Forecasts," Fourth Edition, McGraw Hill, 1998.
[7] William A. Sethares, "Tuning, Timbre, Spectrum, and Scale," Rudolph Steiner Press, 1992.
[8] Michael Tenzer, "Balinese Music," Periplus Editions, 1998.

APPENDIX 1: Frequencies of Gamelans in Hertz


Gamelan Angklung (Bali Utara)
Gangsa Pemade
P1 309.4350 358.8270 404.8380 464.3730 544.6350 629.9685
P2 308.0385 358.9740 404.6175 464.5935 546.3990 630.1890
P3 310.0230 358.9740 404.3970 465.4755 547.2810 630.6300
S1 313.9920 365.1480 412.1145 470.3265 552.5730 638.1270
S2 314.4330 365.5890 411.8940 470.5470 552.7935 637.0245

Kantilan
P1 629.7480 725.8860 814.0860 940.2120 1100.7360 1265.4495
P2 629.9685 726.5475 813.4245 940.4325 1100.5155 1265.2290
P3 628.8660 725.5920 813.6450 939.9915 1104.0435 1267.6545
S1 636.3630 733.3830 820.4805 945.5040 1109.1150 1274.4900
S2 635.7750 731.3985 821.1420 947.2680 1111.0995 1274.4900

Jublag
J1 316.8585 366.0300 411.8940 471.4290 553.0140 637.2450

Jegogan
P1 151.9245 183.8970 207.0495 235.9350 277.1685 315.7560
S1 158.5395 176.8410 200.8755 228.8790 270.7740 309.1410

Tawa-Tawa
P 99.2250
S 104.2965

reyong 278.7120 319.725 389.1825 431.298 488.8485 549.7065 631.7325 731.3985 834.5925 983.4300

gongs 57.3300 60.6375
58.8000 63.7000

Gong Kebyar (galak tika)
Gangsa Pemade
P1 303.4258 326.5137 417.8380 451.6307 571.3377 612.2597 660.3101 829.2136 893.7948 1149.0294
P2 304.4370 327.2387 418.0697 451.1301 571.3377 611.9203 661.7763 828.2945 894.7866 1152.2196
P3(u) 305.9601 328.1473 418.0697 451.1301 571.3377 611.9203 662.8782 827.3764 885.8999 1150.3044
S1 310.2312 336.9988 426.9726 456.1612 576.1098 619.7746 669.8990 837.5317 904.7654 1159.9122
S2 309.3722 334.5782 425.7905 457.4277 575.4712 622.5301 670.2706 839.8570 905.7694 1160.5556

Kantilan
P1 611.5810 661.0428 827.8353 893.7948 1152.2196 1232.0114 1333.1283 1663.0317 1792.5529 2295.5115
P2 611.9203 661.0428 827.3764 893.7948 1153.4981 1234.0627 1333.1283 1663.0317 1791.5591 2298.0588
S1(u) 621.84016 667.6739 837.5317 902.2603 1160.5556 1242.9908 1340.5413 1674.1348 1802.5205 2313.4015
S2 621.4954 669.89909 837.0674 904.7654 1160.5556 1240.9248 1341.2849 1672.2791 1799.5244 2299.3334
S3 619.7746 669.5277 836.1395 904.7654 1152.2196 1242.9908 1340.5413 1672.2791 1797.5298 2309.5563

Jegogan
P1 138.9286 148.7968 162.0842 197.5398 222.4596
S1 144.4273 154.8577 166.8259 266.5015 228.0806

Jublag
P1 283.3025 303.0894 326.8760 464.0061 452.3827
S1 291.2673 310.2312 334.9495 533.0030 456.1612

Calung
P1 212.3058 226.1913 285.6688 307.6614 331.6229 417.6063 449.8811
S1 214.0790 229.7307 288.0549 309.2007 335.8795 420.3945 453.6387


Gender Wayang (galak tika)
P1 356.0411 395.5181 457.2375 533.4465 639.6770 710.1106 822.3450 945.9368 1078.7912 1258.2474
S1 362.6163 415.5274 469.3108 541.7937 633.3240 718.8266 835.2127 957.0165 1087.8017 1268.7569
P2 710.8986 808.7780 943.8409 1078.1932 1257.5499 1428.9106 1691.8664 917.2197 2190.1287 2597.3054
S2 720.8223 835.2127 954.3667 1086.5960 1272.2795 1442.4444 1707.8907 1927.8806 2209.6466 2618.9993




Appendix 2: Beating Rates in beats/second for all polos/sangsih pairs S1: Bali Utara Angklung


l11 = Gangsa Pemade
4.5570 4.9980 5.9535 6.3945 3.9690 4.4100
6.3210 6.7620 6.1740 6.6150 6.1740 6.6150
7.0560 7.2765 7.2765 7.4970 7.4970 7.7175
5.9535 6.1740 5.7330 5.9535 4.8510 5.0715
7.9380 8.1585 6.1740 6.3945 5.2920 5.5125
8.1585 7.0560 7.9380 6.8355 7.4970 6.3945

t11 = stats for s11 1-6 (mean, standard deviation, variance)
5.0470 6.4435 7.3868 5.6228 6.5783 7.3133
0.9432 0.2532 0.2313 0.5356 1.2111 0.6748
0.8896 0.0641 0.0535 0.2869 1.4667 0.4554

l12 = Kantilan
6.6150 6.0270 6.3945 5.8065 7.4970 6.9090
7.4970 5.5125 6.8355 4.8510 7.7910 5.8065
6.3945 7.0560 7.0560 7.7175 6.8355 7.4970
5.2920 7.0560 5.0715 6.8355 5.5125 7.2765
8.3790 10.3635 8.5995 10.5840 5.0715 7.0560
9.0405 9.0405 9.2610 9.2610 6.8355 6.8355

t12 = stats for s12 1-6 (mean, standard deviation, variance)
6.5415 6.3823 7.0928 6.1740 8.3422 8.3790
0.6132 1.1718 0.4712 0.9861 2.0751 1.1996
0.3760 1.3731 0.2220 0.9724 4.3061 1.4392

t14 = stats for s14 1-6 Calung
6.6150 7.0560 6.1740 7.0560 6.3945 6.6150

S2:Gamelan Galak Tika Gong Kebyar

l21 = Gangsa Pemade
6.8054 5.9464 5.7942 4.9352 4.2710 3.4121
10.4851 8.0645 9.7601 7.3395 8.8515 6.4309
9.1347 7.9525 8.9029 7.7207 8.9029 7.7207
4.5304 5.7969 5.0310 6.2975 5.0310 6.2975
4.7721 4.1335 4.7721 4.1335 4.7721 4.1335
7.5149 10.2705 7.8544 10.6099 7.8544 10.6099
9.5890 9.9605 8.1227 8.4943 7.0209 7.3925
8.3181 10.6434 9.2372 11.5626 10.1553 12.4807
10.9705 11.9745 9.9787 10.9827 18.8655 19.8694
10.8829 11.5263 7.6927 8.3361 9.6079 10.2512

t21 = stats for b21 1-10 (mean, standard deviation, variance)
5.1941 8.4886 8.3891 5.4974 4.4528 9.1190 8.4300 10.3995 13.7736 9.7162
1.2343 1.5145 0.6585 0.7403 0.3498 1.5194 1.1702 1.5157 4.3902 1.4788
1.5234 2.2938 0.4336 0.5480 0.1223 2.3086 1.3693 2.2973 19.2741 2.1870

l22 = Kantilan
10.2591 9.9198 9.9143 9.5751 8.1936 7.8544
6.6311 6.6311 8.8562 8.8562 8.4849 8.4849
9.6964 10.1553 9.2321 9.6910 8.3043 8.7632
8.4655 8.4655 10.9705 10.9705 10.9705 10.9705
8.3361 7.0575 8.3361 7.0575 0 -1.2786
10.9794 8.9282 8.9133 6.8621 10.9794 8.9282
7.4130 7.4130 8.1565 8.1565 7.4130 7.4130
11.1031 11.1031 9.2474 9.2474 9.2474 9.2474
9.9676 10.9614 6.9715 7.9653 4.9769 5.9706
17.8900 15.3428 3.8219 1.2747 14.0447 11.4975

t22 = stats for b22 1-10 (mean, standard deviation, variance)
9.2861 7.9907 9.3070 10.1355 4.9181 9.2651 7.6608 9.8660 7.8022 10.6452
1.0070 1.0662 0.6820 1.2936 4.3613 1.5494 0.3840 0.9583 2.3118 6.6513
1.0140 1.1367 0.4651 1.6734 19.0210 2.4006 0.1474 0.9182 5.3446 44.2403

l23 = Jegogan
2.7494 3.0305 2.3709 3.1402 2.8105

l24 = Jublag
3.9824 3.5709 4.0367 2.9310 1.8893

l25 = Calung
1.7733 3.5394 2.3860 1.5393 4.2566 2.7881 3.7576


S3: Galak Tika Gender Wayang Quartet
l31 = low 6.5752 20.0093 12.0733 8.3472 -6.3531 8.7160 12.8678 11.0797 9.0106 10.5095
l32 = high 9.9238 26.4348 10.5258 8.4029 14.7296 13.5337 16.0243 10.6609 19.5179 21.6939