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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 >
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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