#### Encyclopedia of Microtonal Music Theory

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# Analysis of frequencies in Stockhausen's "Studie II", actual and implied

[Joe Monzo]

In 2012 I did an intensive study of Karlheinz Stockhausen's landmark piece Studie II, composed in 1954. This was one of the earliest compositions to be made entirely from electronic sound sources, RC oscillators. Ordinary complex harmonic timbres (i.e., those of most orchestral instruments) can be analyzed using Fourier analysis as being built up from sine-wave partials whose frequencies follow the harmonic series, as integer multiples of the fundamental frequency. Stockhausen's goal in this piece was to use an analogy of this, to create new timbres by using amalgamations of sine waves tuned not to the harmonic series, but rather to an unusual tuning of 25 equal divisions of the 5:1 ratio. This tuning is commonly described in standard musical academia as "25th root of 5", but would be abbreviated by 21st-century tuning theorists as "25ed5" ("ed" = "equal divisions of"). For at least 2 centuries, the standard tuning in Western musical culture has been 12ed2, which uses 12 equal divisions of the 2:1 ratio ("12th root of 2") and is commonly described as "12-tone equal temperament".

Except for the flute, which is fairly close to a pure sine wave, ordinary instrument timbres have a very complex pattern of harmonic sine-wave frequencies and also of the amplitude envelopes for those frequencies. Stockhausen based this entire piece on the number 5, and used the serial method for its manipulation. So every timbre he creates employs 5 sine-waves in the 25th-root-of-5 tuning, the most closely-spaced using each of the 4 other degrees above his analogy of the fundamental, the next using every 2 degrees above, the third using every 3 degrees above, etc. He calls each set of 5 timbres on the same "fundamental" a Group.

Studie II was also the first electronic music score to be published. Stockhausen prefaced the score with detailed information about how he composed the piece and rendered it onto magnetic tape. Included in this preface is a listing of the frequencies to which he tuned the oscillators. With my interest in microtonal tuning systems, I noticed that the actual frequencies implied by 25th-root-of-5 tuning are not quite the same as the ones Stockenhausen used and published in the score. Stockhausen used a base frequency of 100 Hz, which is ~35 cents above G2 (see our MIDI note number, frequency table -- the G on the bottom line of the bass staff), calculating all the others from that. The actual frequencies would be calculated thus: 5^0/25 = 100 Hz, 5^1/25 * 100 = ~106.64949422 Hz, etc. But Stockhausen's published frequency for the next degree of the tuning is 107 Hz, about 5.65 cents too high -- a difference large enough to be noticeable to someone who is listening carefully.

So with all of the careful attention to detail that Stockhausen put into creating this piece, this got me wondering why his frequencies are so far off. I wrote a little python program to query these discrepancies, the output of which I publish below. By comparing the actual and ideal frequencies, which are printed in the first two columns, it becomes evident that there is a pattern to the rounding error of Hz: accuracy to the nearest unit digit up to 999 Hz, accuracy to the nearest 10 digit between 1000 and 9999 Hz, and accuracy to the nearest 100 digit above 10000 Hz. I guessed that the rounding was due to limitations of the oscillators used by Stockhausen, but had also thought that perhaps it was due to calculators or computers that he may have used. But Richard Toop, who later worked with Stockhausen personally, specifies that Stockhausen employed printed logarithm tables in calculating every single frequency by hand [Toop 1981, p 170]. So my conclusion is that the oscillators available to Stockhausen in 1954 had a resolution to only the 3 most significant digits, and that he simply got as close as he could.

One significant reason I am publishing this is that there have been re-creations of this piece by others. One such example is a real-time Csound file which creates the piece using algorithms. However, if a process like that simply tunes the oscillators to the 25th-root-of-5 tuning, the resulting audio file will not exactly match the sound acheived by Stockhausen in his 1954 rendition.

I will also note that 8 years after my analysis, and also shortly after creating this webpage, I discovered that these discrepancies were also written about in 2014 by Glenn P Llorente. Llorente states that Stockhausen rounded the frequencies to the nearest integer, and also takes note that the rounding changes to 10s at one point and then to 100s at a higher point. [Llorente 2014, p 5]. Llorente references Stockhausen's own preface to the score [Stockhausen 2000] and also in a footnote, that the Cologne WDR Studio for Electronic Music, where Stockhausen created this piece, had sine wave generators which could only produce "whole numbered cycles per second" [Llorente 2014, p 6]. But while Llorente points out the change in rounding from 1s to 10s to 100s, he seems not to notice the pattern in the rounding error which I noticed, leading to my conclusion of 3-digit accuracy (the three most significant digits) for the oscillators.

Well one day later, now I have found a photo of exactly the sine-wave generator used by Stockhausen to compose this piece: the MG-60 Messgenerator 10 Hz .... 11 kHz, produced by Wandel u. Goltermann - Reutlingen.

Sine-wave generator MG-60, by Wandel and Goltermann, used by Stockenhausen to create Studie II

In the middle section of the front faceplate, the three knobs on the top row are measured in units of kHz (i.e., 1000 Hz), 100 Hz, and 10 Hz, respectively. The left knob on the bottom row is similar to those three and measured in units of single-digit Hz. The right knob on the bottom row appears to be some sort of calibration, and it is unclear whether the knobs are discrete selector switches or continuous potentiometers -- I welcome more information from anyone who knows more about this device. [Many thanks to Scott Thompson for pointing me to the photo and for his description of the device.] In any case, it is clear the Stockhausen selected his frequencies according to the measurement marked on the four frequency knobs, which confirms my hunch.

```2012.0620 - studie-ii.py output, sorted by cents error

studie-ii.py (c)2012.0619.2344 by Joe Monzo
re: frequencies used in Stockhausen _Studie II_
prints the 25th-root-of-5 values of exact frequencies used
and idealhz, actualcents, idealcents, and centserror

monz's attempt to determine the cause of the rounding error
mentioned in the score by Stockhausen ... my finding:

apparently his oscillators could step in hz as follows:

10000 to ...  step 100
1000 to 9999 step  10
100 to  999 step   1

in other words, stockhausen had 3-digit accuracy available
and had to choose the frequencies within that limitation

in the tables below i present data comparing the frequencies he
actually used with the theoretical "ideal" frequencies resulting
from the calculation 5^(x/25), with 5^(0/25) = 100 Hz

note that stockhausen did not use the frequencies corresponding
to the degrees 74, 76, 77, 78, 79 of this tuning

=============== new run =====================

data presented in columns thus:

col data
1 . the integer actual frequency in Hz used by stockhausen
2 . the real true frequency of 5^x/25 tuning
3 . the real value of x in 5^x/25 of stockhausen's actual frequencies
4 . the error in Hz of stockhausen's frequencies from the true
5 . the real cents value of stockhausen's frequencies
6 . the real cents value of the true frequencies
7 . the error in cents of stockhausen's frequencies from the true

in descending order of frequency:

actlhz idealhz    25root5  hzerror actlcents  idealcents  centerror

17200  17246.621   79.958     -46.6   8911.518  8916.204      -4.686
--   16171.310  [79]         --        --        --           --
--   15163.045  [78]         --        --        --           --
--   14217.643  [77]         --        --        --           --
--   13331.187  [76]         --        --        --           --
12500  12500.000   75.000       0.0   8358.941  8358.941       0.000
--   11720.637  [74]         --        --        --           --
11000  10989.866   73.014      10.1   8137.632  8136.036       1.596
10300  10304.659   71.993      -4.7   8023.801  8024.583      -0.783
9660   9662.173   70.997      -2.2   7912.742  7913.131      -0.389
9060   9059.746   70.000       0.3   7801.727  7801.678       0.049
8500   8494.879   69.009       5.1   7691.269  7690.226       1.043
7970   7965.232   68.009       4.8   7579.809  7578.773       1.036
7470   7468.607   67.003       1.4   7467.644  7467.321       0.323
7000   7002.947   65.993      -2.9   7355.140  7355.868      -0.729
6570   6566.320   65.009       3.7   7245.386  7244.416       0.970
6160   6156.916   64.008       3.1   7133.830  7132.963       0.867
5770   5773.038   62.992      -3.0   7020.599  7021.511      -0.911
5410   5413.095   61.991      -3.1   6909.068  6910.058      -0.990
5080   5075.593   61.013       4.4   6800.108  6798.605       1.502
4760   4759.135   60.003       0.9   6687.468  6687.153       0.315
4460   4462.407   58.992      -2.4   6574.766  6575.700      -0.934
4180   4184.180   57.984      -4.2   6462.517  6464.248      -1.731
3920   3923.301   56.987      -3.3   6351.338  6352.795      -1.457
3680   3678.687   56.006       1.3   6241.961  6241.343       0.618
3450   3449.324   55.003       0.7   6130.229  6129.890       0.339
3230   3234.262   53.980      -4.3   6016.155  6018.438      -2.283
3030   3032.609   52.987      -2.6   5905.495  5906.985      -1.490
2840   2843.529   51.981      -3.5   5793.383  5795.533      -2.150
2670   2666.237   51.022       3.8   5686.521  5684.080       2.441
2500   2500.000   50.000       0.0   5572.627  5572.627       0.000
2340   2344.127   48.973      -4.1   5458.124  5461.175      -3.051
2200   2197.973   48.014       2.0   5351.318  5349.722       1.596
2060   2060.932   46.993      -0.9   5237.487  5238.270      -0.783
1930   1932.435   45.980      -2.4   5124.635  5126.817      -2.183
1810   1811.949   44.983      -1.9   5013.501  5015.365      -1.863
1700   1698.976   44.009       1.0   4904.955  4903.912       1.043
1590   1593.046   42.970      -3.0   4789.146  4792.460      -3.314
1490   1493.721   41.961      -3.7   4676.689  4681.007      -4.319
1400   1400.589   40.993      -0.6   4568.826  4569.554      -0.729
1310   1313.264   39.961      -3.3   4453.794  4458.102      -4.308
1230   1231.383   38.983      -1.4   4344.704  4346.649      -1.946
1150   1154.608   37.938      -4.6   4228.274  4235.197      -6.922
1080   1082.619   36.962      -2.6   4119.551  4123.744      -4.193
1010   1015.119   35.921      -5.1   4003.540  4012.292      -8.752
952    951.827   35.003       0.2   3901.154  3900.839       0.315
893    892.481   34.009       0.5   3790.392  3789.387       1.006
837    836.836   33.003       0.2   3678.273  3677.934       0.339
785    784.660   32.007       0.3   3567.231  3566.482       0.750
736    735.737   31.006       0.3   3455.647  3455.029       0.618
690    689.865   30.003       0.1   3343.916  3343.576       0.339
647    646.852   29.004       0.1   3232.519  3232.124       0.395
607    606.522   28.012       0.5   3122.036  3120.671       1.364
569    568.706   27.008       0.3   3010.114  3009.219       0.896
533    533.247   25.993      -0.2   2896.963  2897.766      -0.804
500    500.000   25.000       0.0   2786.314  2786.314       0.000
469    468.825   24.006       0.2   2675.506  2674.861       0.644
440    439.595   23.014       0.4   2565.004  2563.409       1.596
412    412.186   21.993      -0.2   2451.173  2451.956      -0.783
386    386.487   20.980      -0.5   2338.321  2340.504      -2.183
362    362.390   19.983      -0.4   2227.188  2229.051      -1.863
340    339.795   19.009       0.2   2118.642  2117.598       1.043
319    318.609   18.019       0.4   2008.268  2006.146       2.122
299    298.744   17.013       0.3   1896.175  1894.693       1.481
280    280.118   15.993      -0.1   1782.512  1783.241      -0.729
263    262.653   15.021       0.3   1674.075  1671.788       2.287
246    246.277   13.983      -0.3   1558.390  1560.336      -1.946
231    230.922   13.005       0.1   1449.471  1448.883       0.588
217    216.524   12.034       0.5   1341.234  1337.431       3.803
203    203.024   10.998      -0.0   1225.776  1225.978      -0.202
190    190.365    9.970      -0.4   1111.199  1114.525      -3.326
178    178.496    8.957      -0.5    998.253  1003.073      -4.820
167    167.367    7.966      -0.4    887.818   891.620      -3.803
157    156.932    7.007       0.1    780.917   780.168       0.750
147    147.147    5.984      -0.1    666.979   668.715      -1.736
138    137.973    5.003       0.0    557.602   557.263       0.339
129    129.370    3.955      -0.4    440.845   445.810      -4.965
121    121.304    2.961      -0.3    330.008   334.358      -4.349
114    113.741    2.035       0.3    226.841   222.905       3.935
107    106.649    1.051       0.4    117.133   111.453       5.680
100    100.000    0.000       0.0      0.000     0.000       0.000

---------

in descending order of cents error:

actlhz idealhz    25root5  hzerror actlcents  idealcents  centerror

1010   1015.119   35.921      -5.1   4003.540  4012.292      -8.752
1150   1154.608   37.938      -4.6   4228.274  4235.197      -6.922
107    106.649    1.051       0.4    117.133   111.453       5.680
129    129.370    3.955      -0.4    440.845   445.810      -4.965
178    178.496    8.957      -0.5    998.253  1003.073      -4.820
17200  17246.621   79.958     -46.6   8911.518  8916.204      -4.686
121    121.304    2.961      -0.3    330.008   334.358      -4.349
1490   1493.721   41.961      -3.7   4676.689  4681.007      -4.319
1310   1313.264   39.961      -3.3   4453.794  4458.102      -4.308
1080   1082.619   36.962      -2.6   4119.551  4123.744      -4.193
114    113.741    2.035       0.3    226.841   222.905       3.935
217    216.524   12.034       0.5   1341.234  1337.431       3.803
167    167.367    7.966      -0.4    887.818   891.620      -3.803
190    190.365    9.970      -0.4   1111.199  1114.525      -3.326
1590   1593.046   42.970      -3.0   4789.146  4792.460      -3.314
2340   2344.127   48.973      -4.1   5458.124  5461.175      -3.051
2670   2666.237   51.022       3.8   5686.521  5684.080       2.441
263    262.653   15.021       0.3   1674.075  1671.788       2.287
3230   3234.262   53.980      -4.3   6016.155  6018.438      -2.283
1930   1932.435   45.980      -2.4   5124.635  5126.817      -2.183
386    386.487   20.980      -0.5   2338.321  2340.504      -2.183
2840   2843.529   51.981      -3.5   5793.383  5795.533      -2.150
319    318.609   18.019       0.4   2008.268  2006.146       2.122
1230   1231.383   38.983      -1.4   4344.704  4346.649      -1.946
246    246.277   13.983      -0.3   1558.390  1560.336      -1.946
1810   1811.949   44.983      -1.9   5013.501  5015.365      -1.863
362    362.390   19.983      -0.4   2227.188  2229.051      -1.863
147    147.147    5.984      -0.1    666.979   668.715      -1.736
4180   4184.180   57.984      -4.2   6462.517  6464.248      -1.731
11000  10989.866   73.014      10.1   8137.632  8136.036       1.596
2200   2197.973   48.014       2.0   5351.318  5349.722       1.596
440    439.595   23.014       0.4   2565.004  2563.409       1.596
5080   5075.593   61.013       4.4   6800.108  6798.605       1.502
3030   3032.609   52.987      -2.6   5905.495  5906.985      -1.490
299    298.744   17.013       0.3   1896.175  1894.693       1.481
3920   3923.301   56.987      -3.3   6351.338  6352.795      -1.457
607    606.522   28.012       0.5   3122.036  3120.671       1.364
8500   8494.879   69.009       5.1   7691.269  7690.226       1.043
1700   1698.976   44.009       1.0   4904.955  4903.912       1.043
340    339.795   19.009       0.2   2118.642  2117.598       1.043
7970   7965.232   68.009       4.8   7579.809  7578.773       1.036
893    892.481   34.009       0.5   3790.392  3789.387       1.006
4460   4462.407   58.992      -2.4   6574.766  6575.700      -0.934
5770   5773.038   62.992      -3.0   7020.599  7021.511      -0.911
5410   5413.095   61.991      -3.1   6909.068  6910.058      -0.990
6570   6566.320   65.009       3.7   7245.386  7244.416       0.970
4460   4462.407   58.992      -2.4   6574.766  6575.700      -0.934
4460   4462.407   58.992      -2.4   6574.766  6575.700      -0.934
569    568.706   27.008       0.3   3010.114  3009.219       0.896
6160   6156.916   64.008       3.1   7133.830  7132.963       0.867
533    533.247   25.993      -0.2   2896.963  2897.766      -0.804
10300  10304.659   71.993      -4.7   8023.801  8024.583      -0.783
2060   2060.932   46.993      -0.9   5237.487  5238.270      -0.783
412    412.186   21.993      -0.2   2451.173  2451.956      -0.783
785    784.660   32.007       0.3   3567.231  3566.482       0.750
157    156.932    7.007       0.1    780.917   780.168       0.750
7000   7002.947   65.993      -2.9   7355.140  7355.868      -0.729
1400   1400.589   40.993      -0.6   4568.826  4569.554      -0.729
280    280.118   15.993      -0.1   1782.512  1783.241      -0.729
469    468.825   24.006       0.2   2675.506  2674.861       0.644
3680   3678.687   56.006       1.3   6241.961  6241.343       0.618
736    735.737   31.006       0.3   3455.647  3455.029       0.618
231    230.922   13.005       0.1   1449.471  1448.883       0.588
647    646.852   29.004       0.1   3232.519  3232.124       0.395
9660   9662.173   70.997      -2.2   7912.742  7913.131      -0.389
3450   3449.324   55.003       0.7   6130.229  6129.890       0.339
837    836.836   33.003       0.2   3678.273  3677.934       0.339
690    689.865   30.003       0.1   3343.916  3343.576       0.339
138    137.973    5.003       0.0    557.602   557.263       0.339
7470   7468.607   67.003       1.4   7467.644  7467.321       0.323
4760   4759.135   60.003       0.9   6687.468  6687.153       0.315
952    951.827   35.003       0.2   3901.154  3900.839       0.315
203    203.024   10.998      -0.0   1225.776  1225.978      -0.202
9060   9059.746   70.000       0.3   7801.727  7801.678       0.049
12500  12500.000   75.000       0.0   8358.941  8358.941       0.000
2500   2500.000   50.000       0.0   5572.627  5572.627       0.000
500    500.000   25.000       0.0   2786.314  2786.314       0.000
100    100.000    0.000       0.0      0.000     0.000       0.000

```

The tuning of the frequencies actually used by Stockhausen thus constitute a rational intonation (RI), which in this case is an approximation to his ideal 25ed5 tuning. This is very similar to the implementation of Hammond Organ tuning, also an RI but in that case intended to emulate the usual 12ed2, and generally with much closer approximations.

Below I present a table in descending pitch order in three columns. The first column shows the number of the note in the tuning, which equates to the ideal exponent x in 5^x/25, Stockhausen's intended 25ed5 tuning. The second column shows the actual frequency used by Stockhausen with the oscillators he had available. The third column shows the factoring of the ratios of Stockhausen's frequencies compared to his reference frequency of 100 Hz, notated using a combination of the prime subgroup in parentheses on the left, followed by the monzo of those primes in ket brackets, . Note that the entire tuning using 39 different prime-factors, which is why the (subgroup)|monzo> format was employed, to simplify the notation; most of the ratios only employ from 1 to 4 factors, so if this simplified notation had not been employed, most of the monzos would contain long strings of zeros.

```calculate the monzo of a list of frequencies
using the lowest frequency as the 1:1

5^~x/25   Hz    (prime.subgroup) | monzo >

80   17200   ( 2 . 43 ) | 2 1 >

75   12500   ( 5 ) | 3 >

73   11000   ( 2 . 5 . 11 ) | 1 1 1 >

72   10300   ( 103 ) | 1 >

71    9660   ( 3 . 5 . 7 . 23 ) | 1 -1 1 1 >

70    9060   ( 3 . 5 . 151 ) | 1 -1 1 >

69    8500   ( 5 . 17 ) | 1 1 >

68    7970   ( 2 . 5 . 797 ) | -1 -1 1 >

67    7470   ( 2 . 3 . 5 . 83 ) | -1 2 -1 1 >

66    7000   ( 2 . 5 . 7 ) | 1 1 1 >

65    6570   ( 2 . 3 . 5 . 73 ) | -1 2 -1 1 >

64    6160   ( 2 . 5 . 7 . 11 ) | 2 -1 1 1 >

63    5770   ( 2 . 5 . 577 ) | -1 -1 1 >

62    5410   ( 2 . 5 . 541 ) | -1 -1 1 >

61    5080   ( 2 . 5 . 127 ) | 1 -1 1 >

60    4760   ( 2 . 5 . 7 . 17 ) | 1 -1 1 1 >

59    4460   ( 5 . 223 ) | -1 1 >

58    4180   ( 5 . 11 . 19 ) | -1 1 1 >

57    3920   ( 2 . 5 . 7 ) | 2 -1 2 >

56    3680   ( 2 . 5 . 23 ) | 3 -1 1 >

55    3450   ( 2 . 3 . 23 ) | -1 1 1 >

54    3230   ( 2 . 5 . 17 . 19 ) | -1 -1 1 1 >

53    3030   ( 2 . 3 . 5 . 101 ) | -1 1 -1 1 >

52    2840   ( 2 . 5 . 71 ) | 1 -1 1 >

51    2670   ( 2 . 3 . 5 . 89 ) | -1 1 -1 1 >

50    2500   ( 5 ) | 2 >

49    2340   ( 3 . 5 . 13 ) | 2 -1 1 >

48    2200   ( 2 . 11 ) | 1 1 >

47    2060   ( 5 . 103 ) | -1 1 >

46    1930   ( 2 . 5 . 193 ) | -1 -1 1 >

45    1810   ( 2 . 5 . 181 ) | -1 -1 1 >

44    1700   ( 17 ) | 1 >

43    1590   ( 2 . 3 . 5 . 53 ) | -1 1 -1 1 >

42    1490   ( 2 . 5 . 149 ) | -1 -1 1 >

41    1400   ( 2 . 7 ) | 1 1 >

40    1310   ( 2 . 5 . 131 ) | -1 -1 1 >

39    1230   ( 2 . 3 . 5 . 41 ) | -1 1 -1 1 >

38    1150   ( 2 . 23 ) | -1 1 >

37    1080   ( 2 . 3 . 5 ) | 1 3 -1 >

36    1010   ( 2 . 5 . 101 ) | -1 -1 1 >

35     952   ( 2 . 5 . 7 . 17 ) | 1 -2 1 1 >

34     893   ( 2 . 5 . 19 . 47 ) | -2 -2 1 1 >

33     837   ( 2 . 3 . 5 . 31 ) | -2 3 -2 1 >

32     785   ( 2 . 5 . 157 ) | -2 -1 1 >

31     736   ( 2 . 5 . 23 ) | 3 -2 1 >

30     690   ( 2 . 3 . 5 . 23 ) | -1 1 -1 1 >

29     647   ( 2 . 5 . 647 ) | -2 -2 1 >

28     607   ( 2 . 5 . 607 ) | -2 -2 1 >

27     569   ( 2 . 5 . 569 ) | -2 -2 1 >

26     533   ( 2 . 5 . 13 . 41 ) | -2 -2 1 1 >

25     500   ( 5 ) | 1 >

24     469   ( 2 . 5 . 7 . 67 ) | -2 -2 1 1 >

23     440   ( 2 . 5 . 11 ) | 1 -1 1 >

22     412   ( 5 . 103 ) | -2 1 >

21     386   ( 2 . 5 . 193 ) | -1 -2 1 >

20     362   ( 2 . 5 . 181 ) | -1 -2 1 >

19     340   ( 5 . 17 ) | -1 1 >

18     319   ( 2 . 5 . 11 . 29 ) | -2 -2 1 1 >

17     299   ( 2 . 5 . 13 . 23 ) | -2 -2 1 1 >

16     280   ( 2 . 5 . 7 ) | 1 -1 1 >

15     263   ( 2 . 5 . 263 ) | -2 -2 1 >

14     246   ( 2 . 3 . 5 . 41 ) | -1 1 -2 1 >

13     231   ( 2 . 3 . 5 . 7 . 11 ) | -2 1 -2 1 1 >

12     217   ( 2 . 5 . 7 . 31 ) | -2 -2 1 1 >

11     203   ( 2 . 5 . 7 . 29 ) | -2 -2 1 1 >

10     190   ( 2 . 5 . 19 ) | -1 -1 1 >

9     178   ( 2 . 5 . 89 ) | -1 -2 1 >

8     167   ( 2 . 5 . 167 ) | -2 -2 1 >

7     157   ( 2 . 5 . 157 ) | -2 -2 1 >

6     147   ( 2 . 3 . 5 . 7 ) | -2 1 -2 2 >

5     138   ( 2 . 3 . 5 . 23 ) | -1 1 -2 1 >

4     129   ( 2 . 3 . 5 . 43 ) | -2 1 -2 1 >

3     121   ( 2 . 5 . 11 ) | -2 -2 2 >

2     114   ( 2 . 3 . 5 . 19 ) | -1 1 -2 1 >

1     107   ( 2 . 5 . 107 ) | -2 -2 1 >

0     100   ( n ) | 0  >

```

### REFERENCES

Stockhausen, Karlheinz. 2000. Studie II: Elektronische Musik. Stockhausen-Verlag. K¨rten, Germany.

Toop, Richard. 1981. "Stockhausen's Electronic Works: Sketches and Work-Sheets from 1951-1967". Interface - Journal of New Music Research, 10. Rotterdam, Netherlands.

Llorente, Glenn P. 2014. Stockhausen's Studie II: Elektronische Musik (1954): Exploring the Extent of Multiple-Serialism in Electronic Music. Musicology 245: Early Music Models in Post-War European Musical Modernism. Herb Alpert School of Music, UCLA.

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