Eduard Heyning



TIERKREIS (German for zodiac) is a musical composition by Karlheinz Stockhausen (1928-2007). The composition consists of twelve melodies, each representing one sign of the zodiac. In bringing music and the stars together Stockhausen stands in a long tradition of West-European cosmology starting from Pythagoras and Plato, inspiring philosophers, poets, designers and composers with the idea of a universe ordered on the proportions of the musical scale. Plato’s Timaeus is the source of a teleological worldview, bringing science, art and nature together in a divine order. Mankind, says Plato, has to look up to the heavens, where the stars and planets produce a harmony, the music of the spheres. Stockhausen says he composed the Tierkreis melodies ‘with all its measures and proportions in keeping with the characteristics of its respective star sign, and one will discover many legitimacies when one hears a melody often, and exactly contemplates its construction ... ’.

TIERKREIS was originally written for music boxes as a component part of a theater piece for percussion sextet titled MUSIK IM BAUCH. When Stockhausen’s youngest daughter was two years old, he used to make her laugh by teasing her about her growling stomach and the “music” she was making there. Later this inspired him to write a performance art piece called MUSIK IM BAUCH. The dreamlike theatrics of this work come to a climax when a performer reaches into the belly of a life-size puppet and pulls out twelve music boxes. Stockhausen’s task of actually writing something these music boxes could play yielded twelve melodies, one named after each constellation of the zodiac. On the Stockhausen CD´s website you can hear them all. As the melodies are composed for music boxes, their range and duration (26-30 seconds) is limited and dynamics are absent. After laboring over the contours of these twelve almost tonal-sounding melodies so that they would symbolically depict the traditional personalities of ancient Babylonian astrology, they were published and performed separately under the title TIERKREIS (work 41), to be played or sung with or without accompaniment.  

"They’re touching in a way that combines awkwardness and grandeur, as if superior beings from outer space were trying to ingratiate themselves with humanity by writing something catchy." (Ivan Hewett)

The twelve melodies of TIERKREIS are character pieces, representing the twelve signs of the Zodiac. A complete performance begins with the melody corresponding to the zodiac sign within which the day of the performance falls, and proceeds through the twelve melodies of the cycle, ending with a return to the starting melody. Each melody is to be played at least three times through, with variations or improvisations. The melodies can also be played individually, or in smaller numbers. In addition to Musik im Bauch, Stockhausen employed the TIERKREIS melodies in the central "wheel" section of SIRIUS (1975–77), an hour-and-a-half-long chamber opera for soprano and bass voices, trumpet, bass clarinet, and eight-channel electronic music. Stockhausen also prepared a number of versions for various specific forces: vocal versions for five different voice ranges (high soprano or high tenor, Nr. 41⅔; soprano or tenor, Nr. 41¾; mezzosoprano, alto, or low tenor, Nr 41⅘; baritone, Nr. 41⅚; bass, Nr. 41 6/7 all 1975), version for octet or chamber orchestra (clarinet, horn, bassoon, strings), Nr. 41 7/8 (1977), a version for clarinet and piano, Nr. 41 8/9 (1981), a "trio version" for clarinet, flute/piccolo, and trumpet/piano, Nr. 41 9/10 (1983), a "version 2003" for soprano or tenor with chording instrument, Nr. 41 10/11 (2003), and, finally, two orchestral versions of five melodies each, titled FÜNF STERNZEICHEN, Nr. 41 11/12, and FÜNF WEITERE STERNZEICHEN, N. 41 12/13. The latter was his last completed composition, finished on 4 December 2007, the night before he died. Stockhausen was planning further work in January 2008, which was probably the orchestration of the remaining two pieces, Cancer and Leo.  

Source: Wikipedia, Jerome Kohl. A lot more background info by Ingvar Loco Nordin you find here.


In 1975 Karlheinz Stockhausen composed 12 melodies for each sign of the zodiac for MUSIK IM BAUCH (Werk 41), which he re-used to create TIERKREIS (Zodiac, 41½), leaving the performer much freedom to make a co-creational version of the piece by choosing instruments and adapting the musical material within a set of rules. Nowadays with the sheet music comes Christel Stockhausen's "Introduction, Analytical Description and Performance Instructions":

"The following variations are possible: a) dynamic nuances, either within a melody or from repetition; b) change of articulation (staccato, portato, legato); c) use of various octave registers, either on the same instrument or on auxiliary instruments (for example, flute, piccolo, alto flute); d) duet interpretation: when a chordal instrument accompanies a melody instrument or a singer, variation can be achieved by the arrangement of solos and duets. e) melodic variation: by leaving out certain pitches of a melody, its structure - in terms of rhythm, interval successions, introduction of new pitches, repeated pitches (principle of 12-tone music) - can be made clear. Rhythmic variations can be made by playing / singing the central pich only at the places where it originally occurs, replacing all other pitches by rests. This last possibility can, for example, be used in LEO. f) clarification of the interval sequence: the first occurence of each interval can be especially emphasized (in certain melodies - LEO for example - all the different intervals, both rising and falling, are melodically related to the central pitch). Alternatively play only the new intervals, replacing their recurrences with exact rests (in the given rhythm of course). g) dissection of a melody: certain sections of melodies may be emphasized by leaving out others. The resulting spaces between sections can be bridged either by holding the last pitch of the previous phrase or by an exact rest during which the player remains motionless. All of these possibilities are valid both for solo and for duet versions, the latter having the additional freedom to combine several variations. The choice of variations is left to the interpreters, but they should only use them to clarify the inner structure and to bring out the distinctive characteristics of the melodies."


Over the years a family of ‘Tierkreises’ sprung up, TIERKREIS became Stockhausens most-performed work and finally the composer himself arranged ten of the twelve melodies for chamber orchestra, FÜNF STERNZEICHEN (2004) and FÜNF WEITERE STERNZEICHEN (2007), finished the night before he died. Together they present the TIERKREIS melodies from Virgo to Gemini. Very beautiful music with an ethereal quality not often found in his work. The orchestra consists of flute/piccolo, oboe, clarinet, bassoon, horn, trumpet, trombone, tuba, harp, percussion (vibraphone, glockenspiel, gong), strings: 4-4-3-3, no contrabass. Taurus has a bass tuba soloist...


TIERKREIS sets the star signs to music. The connection between the orbits of the stars and the tones of the musical scale comes from their mathematical proportions. The human brain can by nature recognize the consonances by their ratios, which is an amazing fact by itself. Based on the harmonic series the ear knows perfection in sound. In this way Macrocosmos (stars) and Microcosmos (man) can be brought together in harmony. The mathematical ordering of musical pitch, rhythm and form and it's correspondance with the human character has been an object of research since Antiquity. The TIERKREIS melodies are serial in conception. Serialism as a compositional method may be well equiped to express stars music. 


"In Western history, during the period of enlightment and rationality, almost all music was founded in what theorists call "functional harmony". Every phrase and note in a piece, no matter how long, could be heard and interpreted as a "function' of the tonic note. (...) In the twentieth century, during a time of upheaval in society and in personal understanding, Arnold Schönberg developed his "atonal" music and his system of "serial" composition. In this music, complicated rules and procedures are followed precisely to avoid giving any note dominance over the others. Schönberg's music is truly polytonal: each note is a center unto itself. (...) In the hands of some modern composres, Schönberg's twelve-tone system (...) can be eminently expressive, and that is a quality one would want to carry over analogously into the music of the psyche. " Moore, Thomas (1982) The Planets Within. The Astrological Psychology of Marsilio Ficino. Lindisfarne Books, Aurora, Colorado, 201.


Stockhausen: "So serial thinking is something that's come into our consciousness and will be there forever: it's relativity and nothing else. It just says: Use all the components of any given number of elements, don't leave out individual elements, use them all with equal importance and try to find an equidistant scale so that certain steps are no larger than others. It's a spiritual and democratic attitude toward the world. The stars are organized in a serial way. Whenever you look at a certain star sign you find a limited number of elements with different intervals. If we more thoroughly studied the distances and proportions of the stars we'd probably find certain relationships of multiples based on some logarithmic scale or whatever the scale may be." Cott, Jonathan (1973) Stockhausen; Conversations with the Composer. New York: Simon & Schuster, 101.


Stockhausen is one of the most prominent serialist composers of the 20th century. The TIERKREIS melodies are constructed on several series that rule pitch and duration. The Zentraltöne of the melodies form a chromatic scale. The tempos of the melodies are ordered like a chromatic scale. On top of that, the technicalities of writing for music boxes demand further organisation of the musical material. The number of lamella's (28/36/50) define the musical range. The rotation speed determinates the length of the melodies; they form a series that explains the rule of three or four repeats (Kohl, 1983, 164). So these simple tunes are the outcome of an extremely complicated proces of ordering sound.


Stockhausen, 1975: "I began to busy myself with the 12 human characters of the Zodiac of which I had until then only a vague idea. In inventing each melody I thought of the characters of children, friends and acquaintances who were born under the various star signs, and I studied the human types of the star signs more thoroughly. Each melody is now composed with all its measures and proportions in keeping with the characteristics of its respective star sign, and one will discover many legitimacies when one hears a melody often, and exactly contemplates its construction ... Each melody is composed in such a way that one should play it at least 3 times. When one wishes to listen to several or all of them one after another, then each should be played 3 or 4 times in succession."

The original piece, MUSIK IM BAUCH , was written for 6 percussionist and music boxes. Now there are many versions made by interpreters in accordance with Stockhausens rules. Instruments can be chosen and melodies may be transposed. Since each melody must be played several times, the interpreter has many possibilities for variation: for instance dynamic nuances, change of articulation, use of various octave registers or instruments, solo or accompagment, melodic/rhythmic variation, emphasized intervals, dissection of the melody, pauses between melodies. The point should be to clarify the inner structure and to bring out the distinctive characteristics of the melodies.

The melodies are serial in conception and all are based on tone rows, though some have more than twelve notes—Libra, for example, has fourteen, with F and D recurring in different octaves. Because music boxes preclude any significant variation in dynamics or timbre, the structure of the TIERKREIS melodies emphasize pitch and rhythm. Each melody is centered on a different chromatic pitch, with "Leo" (Stockhausen’s own sign) = A, Virgo = A, Libra = B, Scorpio = C, etc., and each has its own distinctive tempo, chosen from the "chromatic" tempo scale first described in the composer's famous article, "... How Time Passes ..." .

Like the pitches, the rhythms are also organized serially and strive for contrast amongst the melodies rather than relatedness. Various scales of durations are employed: Fibonacci numbers (1, 2, 3, 5, 8, 13, see below), arithmetic series (1, 2, 3, 4, 5, … ), and "second order" arithmetic series, in which the difference between consecutive members increases arithmetically (2, 3, 5, 8, 12, 17, … ). One melody, "Aries", mixes all three of these scale types, at different levels of the durational organization. 

Source: Jerome Kohl on Wikipedia and elsewhere.

Analysis of LIBRA by Stockhausen, seminar on November 5th 1982 in The Hague, the Netherlands:


"Tonight, I will analyze the LIBRA melody which you have heard in the performance of SIRIUS. This melody was first composed as one of the twelve melodies of the zodiac in my work TIERKREIS (ZODIAC). There are several editions of TIERKREIS: for voice, for melody and/or chordal instrument, and for chamber orchestra.


It sounds so simple - almost like a folk melody that I could have found somewhere. Let's see what it really is - how I have composed it.


My goal is LIBRA, the balance. As I have written in several texts, I collected the birth dates of people I know, to find out if there is any meaning in the twelve human types - in all the traditional connections between the planets and these human types. People born between September 23rd and October 22nd are the LIBRA types.


LIBRA has a tempo of 71. This is a very special tempo. It is the tempo of the balance, the ideal heartbeat. Whenever you find the tempo 71 in my works, you know that the music is completely balanced and harmonious.


You should know that in all my scores since 1952, I have used a tempo scale between 60 and 120:

60, 63.5, 67, 71, 75.5, 80, 85, 90, 95, 101, 107, 113.5, 120.*


*Kohl, Jerome. 1983. "The Evolution of Macro- and Micro-Time Relations in Stockhausen’s Recent Music". Perspectives of New Music 22 (1983–84), p. 148: "Starting from the observation that pich may be understood as the microtime equivalent of rhythm, Stockhausen's truly radical solution to the "question of rhythm" proceeded to the establishment of a metronomic tempo-scale wherein the temporal "octave" (2:1 tempo-proportion) was divided logarithmically into twelve equal parts. That is to say, succesive members where related in a geometric series, the constant proportion of which was the same as that of the equal-tempered chromatic-pitch scale: 1: sqrt[12]{2} ."


When Stravinsky first saw the tempo 63.5 in my score, he joked: "... naturally .5, the German professor." Once Bruno Maderna was conducting, and I said: "Bruno, you are wrong in tempo." He said: "I know- you want your .5. Look at my metronome: there is no .5!"


In the LIBRA melody, B is the central pitch, and all of the other pitches swing out from it: up, down, up, etc. The durations of the measures areas follows: 2, 4, 6, 7, 5, 3, 1. Think of LIBRA, the balance, and how this large rhythm swings out and back over the entire melody.



Let's see how, within this balance of durations, the subdivisions are made.The first measure is 1+1=2 quarter notes long. The second measure is 1⅔+2⅓=4 (I could have divided it in the middle, but all natural symmetries have a slight deviation. A face is never exactly symmetrical). The next measure is 3⅓+2⅔=6. Then 3⅓+3⅔=7, 3+2=5, 1⅓+1⅔=3, and then 1.

Now let's see what makes this melody so rhythmically special. Always short, long, short, long. The durations are special. The first of the longer durations is 2 eight notes, or triplets. The progression is: 2, 2, 4, 5, [8], 6, 8, 10, 8, 5, 2, 2, 2. The shorter note is always the repeated note. One way of interpreting this for a melody instrument would be to first play only the repeated central pitches in the original positions leaving out all the others.



Now let's look at the 'new' pitches. They have the following progression of intervals in respect to the central note B: +1, -2, +3, -3, +4, -6, +7, -8, +5, -5, +2, -4, -1, 0. Plus (+) means that it is below it. Notice that I've only counted a new pitch when it has a long duration; sometimes they appear as passing notes. The principle is that in each limb there are two new pitches. This aspect is also like a balance, as are all aspects of this melody. No two of the melodies of TIERKREIS are composed in the same way.If I were to analyze another melody you would see a completely different method of composition. " ...


(source: Weiland, F. e.a. eds: Stockhausen in Den Haag: Documentatie van het Karlheinz Stockhausen Project in het Koninklijk Conservatorium te Den Haag, 27 oktober tot en met 1 december 1982; Zeist, 1982)


Fibonacci (Leonardo da Pisa, c. 1170 – c. 1250) spread the Hindu–Arabic numeral system in Europe. He also introduced a sequence of numbers, related to the divine proportion (golden section), the spiral and the irrational number φ. This mysterious proportion is found in nature, painting, architecture and star nebulae. Some composers (i.e. Debussy, Bartok, Stockhausen) have employed golden section proportions in rhythm and form.

Leonardo Pisano Bigollo (c. 1170 – c. 1250), known as Fibonacci, and also Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci – was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages." Fibonacci is best known to the modern world for spreading the Hindu–Arabic numeral system in Europe, primarily through the publication in 1202 of his Liber Abaci (Book of Calculation), and for a number sequence named the Fibonacci numbers after him, which he did not discover but used as an example in the Liber Abaci.

In Chapter 12 of the Liber Abaci Fibonacci states the problem which involves the famous sequence with which his name is linked (Quot paria coniculorum in uno anno ex uno paro germinentur): 'A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?' This leads to a series which in the 1870's has been called the Fibonacci series: 


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 , ...


Each number is the addition of the preceding two; the ratio of the last two progressively approaches the irrational number φ ( 1.6180339887...), called the Golden Section or Divine Proportion. In words: two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. When φ is employed in geometry you can get the Golden Spiral as depicted above. The golden section goes back at least as far as 300 B.C., when Euclid described it in his major work, the Elements. Moreover, the Pythagoreans apparently knew about the golden section around 500 B.C. The ancient Egyptians used the golden section in the construction of the great pyramids and in the design of hieroglyphs found on tomb walls. φ stands for Phidias, believed to have made frequent use of the proportion in sculpture. Plato in his Timaeus considered it the most binding of all mathematical relations and makes it the key to the physics of the cosmos. φ proportions are found extensively in visual arts and nature.  φ relationships are also found in the timing of musical compositions, for instance when the climax is at the φ point (61.8%) of the piece. But the most common form of application of φ to music is the conscious use of the Fibonacci series by composers, as is the case in some of the melodies of TIERKREIS.



“Virgo” uses Fibonacci numbers to govern both duration and pitch. (…) The example shows the pitches used for “Virgo”, which are deployed according to proportions defined by Fibonacci numbers: thirteen notes for the melody (the basic range plus an F above), eight pitches for the accompaniment, with intervals of 1, 2, 3, 5, 8, etc. (…) The example outlines the pitch and duration structures of “Virgo”. The duration series of “Virgo” is compound, made upof six progressively larger sets of Fibonacci numbers: 1 / 1, 2 / 1, 2, 3 / 1, 2, 3, 5 / 1, 2, 3, 5, 8 / 1, 2, 3, 5, 8, 13 (though within each set the elements do not usually appear in scale order). This results in a scale for the larger group-durations with differences which are the Fibonacci numbers from 2 to 13, which I call a “second-order” Fibonacci scale."


Kohl, Jerome. 1983. "The Evolution of Macro- and Micro-Time Relations in Stockhausen’s Recent Music". Perspectives of New Music 22 (1983–84): 158.

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