The Schillinger System of Musical Composition
Analysing and Exploring the Theory of Rhythm
Osvaldo Glieca
2012 Rightocopy Osvaldo Glieca
Contents
3
Preface
5
Representing Rhythm and Pulse Numerically
7
The Metrical Arrangement of Pulsation
8
Generating Rhythmic Patterns Using Ratios – Type I
10
Generating Rhythmic Patterns Using Ratios – Type II
12
Applying Meter to Rhythms
14
Rhythms Made from Three or More Generators
17
Distributive Powers
20
Evolution of Rhythmic Styles
22
Further Development of the Bar-group
24
Variation Technique
25
General and Circular Permutations
28
Conclusions
37
Appendix 1: Patterns Generated Using Ratios – Type I
38
Appendix 2: Patterns Generated using Ratios – Type II
39
Appendix 3: Polynomials
40
References
2
Preface
Joseph Schillinger spent his entire life formulating the Schillinger System of Musical Composition (SSMC), in an attempt to assert a break from the conservative musical world by creating something completely new. The SSMC is essentially a discipline that aims to help creativity by allowing composers to understand how they can achieve their instinctive aims. As Schillinger maintained:
My system does not circumscribe the composer’s freedom, but merely points out the methodological way to arrive at a decision. Any decision which results in harmonic relation is acceptable. We are opposed only to vagueness and haphazard speculation. (Schillinger 1946:1356).
A prerequisite to understand the concepts of the SSMC is Schillinger’s Theory of Rhythm, and it is this subject, drawn from Book 1 of the SSMC, that I shall be considering in this dissertation. Schillinger’s theory arises out of science and mathematics. A principle that the reader must comprehend from the outset is that its aim is to provide a quantifiable outcome that is subsequently translated into music by the composer’s imagination. Rhythm is a word, however, without a generally accepted meaning. The best definition I can give is by referring to the way in which sounds of varying length and accentuation are grouped into patterns. Yet throughout history there has not been any satisfactory theoretical notion of how to explain this phenomenon. Even the Greeks who coined the term in early antiquity gave it contradictory meanings, and like so many terms, it’s connotations have changed from age to age. Schillinger even goes so far as to consider the patterns according to which instruments enter and leave an ensemble, and those formed by the pitches in a phrase, to be aspects of rhythm. Nowadays, the temptations to compare the theories contained in the SSMC with those attempted by master composers of the past are difficult, because Schillinger’s ideas are based on the concept that, primordially, music arises from rhythm. He takes this idea as the natural process of human existence since rhythm is a vital ingredient of life. He considered it to be the same in music since it too is something that belongs to and is created by nature. Thus,
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Schillinger relies on the numerical functions that regulate our lives, even unconsciously from prehistory. Personally, it seems impossible to relate his theories with previous attempts because the latter were conceived along lines completely alien from their numerical foundation. This publication falls into two broad sections. The first recounts Schillinger’s ideas about rhythm as they are presented in the SSMC. I have not, however, merely paraphrased the original text. In many places I have considered it appropriate to simplify Schillinger’s terminology with the aim of concision and intelligibility. The second section presents a comprehensive set of conclusions.
It is with pleasure that I express my gratitude to all those whose scholarship and research has given me invaluable assistance in preparing this dissertation. Foremost, I am grateful to Dr. Jeremy Arden; I have been particularly fortunate in my contact with such a distinguished exponent of Schillinger’s theories. Dr. Arden has kindly provided much source material that would otherwise be beyond my reach, and has given many valuable suggestions on a wide range of subjects.
4
Representing Rhythm and Pulse Numerically
Numbers are essential to the manipulation of rhythm through the techniques of the SSMC, and here, Schillinger uses graphs to translate rhythmic pulses into number patterns. Schillinger points out that numbers are more efficient than traditional methods of music notation because: The customary method of musical notation, which is a product of the trial and error method, is inadequate for the analysis and study of rhythmic patterns. It offers no common basis for computations. The history of creative experience in music shows that even the greatest composers have been unnecessarily limited in their rhythmic patterns because they thought in terms of ordinary musical notation. (Schillinger 1946:1) By showing numeric patterns alongside traditional notation in the SSMC, Schillinger gives the composer the best of both worlds; traditional notation reveals proportions, while graphs present clearly the connection between rhythm and number. Figure 1 illustrates the graphic representation of pulse; the horizontal lines represent the duration between attacks and the vertical lines represent the moment of attack:
Fig.1: The horizontal lines represent the duration between attacks and the vertical lines represent the moment of attack.
A monomial periodicity is a regular cyclical pattern of undifferentiated pulses of a determined length of time expressed in numbers. For example, Figure 2 shows three different monomial periodicities, 1, 2 and 3:
Fig. 2: Three different monomial periodicities; from the top 1,2, and 3.
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By assigning to these proportions an exact value, then it becomes possible to represent the foregoing using traditional music notation.
Fig. 3: The three different periodicities showing the process in musical notation.
The Metrical Arrangement of Pulsation According to Schillinger, rhythm arises through the interference of two or more sources of pulse, and the resulting pattern is called pulse interference. The interference of periodicities is the key to the organization of meter. The moments where pulses synchronize form heavy metric accents, in other words, down beats. Thus, in Figure 4, although generator a produces a pulse recurring every four units of time, and generator b a pulse recurring every one unit of time, there are moments where both pulses synchronize creating interference. This happens every time a multiple of 4 is reached, creating the strong pulse that can be interpreted as the accented downbeat of each bar:
Fig. 4: The pulse ratio of 4:1 strong accents occur where pulses collide.
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Using the ratio of 4:1 the natural meter it is a 4/4, or common time. Figure 5 shows in musical notation, the interference of pulses in the ratio of 4:1 generating accents every four beats of the bar.
Fig. 5: The pulse ratio of 4:1 in musical notation: they have same point of attack; a and b converge at every multiple of 4.
Generating Rhythmic Patterns Using Ratios – Type I
The concept of pulse interference can now extended to explain the process of how rhythms are generated. When two pulses are used whose frequencies form an irrational ratio, that is, when neither equals 1, the result is not meter, but rhythm. Schillinger assigns the larger integer to what he terms the major generator, and the lower to the minor generator. The resulting pattern created by pulse interference he terms the resultant. Figure 6 illustrates the procedure when applied to the ratio 3:2. In this example, c.d., refers to the common denominator or the basic unit of measurement, a., and b., are respectively the major and minor generators, and r., the resultant. The product of the generators, which determines the length of the cycle before it loops making a cyclical rhythmic pattern, Schillinger calls the common product, c.p. The faint vertical lines show the moments of interference. The process implies the common denominator and the common product, but they do not contribute directly to the symmetrical pattern.
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Fig. 6: The ratio of 3:2, a=3; b=2; the resultant is 2+1+1+2.
Note that the resulting pattern, in this case, 2+1+1+2, is palindromic. Figure 7 shows the same result in musical notation.
Fig. 7: The ratio of 3:2 represented in a musical notation.
Schillinger suggests that these operations can be used, under the composer’s discretion, to conceive the framework of a score; the common denominator could be used as arpeggios or ostinato figures, the generators a and b could be interpreted as chords, the resultant, being the focal point of the generative process; is used for the rhythm of the principal theme. Finally the common product employed as sustained notes.
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Generating Rhythmic Patterns Using Ratios – Type II This process is slightly less straightforward then the Type I. Firstly, obtain the square of the major generator; this dictates the length of the pattern. Secondly, multiply the minor generator by the major generator in order to establish the number of repetitions of the minor generator. The minor generator now assumes an important position to establish the resultant. Both generators start from the same point of attack. Subsequently, the minor generator completes its first appearance before the major generator and in order to complete the cycle of the major generator, several groups of minor generator are required. Each minor generator group starts on a new phase of the major generator until the two rhythms synchronize.
Fig. 8: The ratio of 3:2 represented in the second type technique.
The major generator has a value 3, thus its square 9, will determine the common product. b1 and b2 are the minor generators. Note that each new minor group starts on the succeeding attack of the major generator. By multiplying the minor generator by the major establishes the number of times the minor pulse stream is required to complete the full cycle. Again, the resultant formed by the interference of the pulse streams is palindromic, 2+1+1+1+1+1+2.
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Fig. 9: The ratio of 3:2 from the score in the second type technique.
It is important to mention here that ratios sharing a common divisor, such as 4:2; 6:3; 8:4; and 9:3 for example, are omitted from the SSMC. The ratio 4:2 gives a resultant of 2222, the ratio 6:3 gives 333333, and so on. Schillinger claims that these ratios do not create rhythm, but only a metric-regular pulsation. Thus, according to Schillinger, ratios that share no common divisor other than one are the only choice available. Evidently, Schillinger’s aims were to obtain variety and syncopation, which he considered to be the spirit of rhythm, but I shall contend this view more thoroughly when I return to it in my conclusions.
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Applying Meter to Rhythms
Let’s now consider what meter is to be chosen for balancing the rhythm and situate the exact position of accenting. Again, the generators are the key for the choice of this procedure, and three different options are available, multiplying the generators by themselves, using the major generator as the common number of the time signature, or using the minor generator as the common number of the time signature. Let’s consider the first method. Taking the ratio 3:2 as our example, the product of the two integers, 6, can be represented in musical notation as any time signature with 6 single units. By using the crochet as our common denominator, it is possible to obtain two bars of 3/4, three bars of 2/4, or one bar of 6/4, as shown in Figure 10.
Fig. 10: The possible metrical arrangements in a 3:2 ratio.
As with the first method, the generators also display all the metrical options in the second type of technique. Taking again the ratio of 32:2 = (2,1,1,1,1,1,2) as an example, the meter naturally falls into bars made by 3 beats. Figure 11 shows the second type of rhythm barred by the square product of the larger generator:
Fig. 11: The second type of rhythm barred by the square product of the larger generator.
In the third method, the minor generator can also be used as metrical guide, but this results in further interference and syncopation, as the pattern will not fit into one cycle of bars. However, it is possible to calculate the cycle by dividing the square product of 3 for the minor generator. In the calculation below, the number 2 represents the minor generator:
32 = 9 9:2 =
4
1 2 11
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The length of the meter in 2/4 does not fit in the square product of the major generator, unless, as indicated by the denominator in the fraction, the rhythm is repeated twice to reach the required length. The pattern in the illustration below contains 18 beats, which is the product of 9x2. Essentially it is the pattern 2,1,1,1,1,1,2, repeated twice. In Figure 12, the second type of rhythm is accommodated. Note in bars four and five the illusion of a 3/4 time signature created by displacement of the accents which adds syncopation:
Fig. 12: The second type of rhythm accommodated in 2/4.
Rhythms Made from Three or More Generators
It is possible to create rhythmic patterns by the utilization of three or more generators to obtain the resultant. Again, Schillinger maintains that the choice of the third generator is important since, as mentioned before when using two generators, it is best to avoid ratios that share a common divisor. Ideally, the generators ought to be related in some way and for this reason they are best drawn from a number series. A series is any summation of a repeating pattern of terms.
I:
1+1+2+3+5+8+13+21+34+55+89….
II:
1+3+4+7+11+18+29+37+66….
III:
1+4+5+9+14+23+37+60+97…. Once again, large numbers tend to produce unmanageable results.
Schillinger’s
suggestions to which the composer should restrict himself:
series I:
2x3x5 or 3x5x8
series II:
3x4x7
series III:
4x5x9
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Once the series is chosen, the first step necessary is to find their common product and then all their complementary factors. The complementary factors are the quotients of the product by all the generators. The common product is a multiple of all the generators, and as in the other processes involving two generators; it is divided by itself to find the common denominator, which is always 1, in order to establish all the rest of durations that generate the rhythmic pattern. This method gives one extra resultant, created by the generators and their complementary factors. For example; choose the number’s series: 2, 3, 5, and find the common product: 2x3x5=30 complementary factors to obtain the first resultant, 30:2 = 15 (1st generator) 30:3 = 10 (2nd generator) 30:5 = 6 (3rd generator) complementary factors to obtain the second resultant, 30:15 = 2 (1st generator) 30:10 = 3 (2nd generator) 30: 6 = 5 (3rd generator)
Fig. 13: The three generators with their completed process of rhythmic origination.
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Figure 14 presents the musical score.
Fig. 14: The nucleus of the musical score grouped in the time signature of 3/4.
The first resultant gives rhythms of the shortest duration, while the second one give rhythms of longer duration, thus the two patterns may balance one another; they are related, but contrasted. But, as always, is the composer who must choose what to do. Figure 15 shows for example, how r and r1 might be arranged, either as a theme, and r1 as a counter theme.
Fig. 15: The two resultants can play solo parts showing balance and variety.
The assignment of a time signature it is again in the hands of the composer, who, evaluate which one is more suitable for his intentions, since it possible to group the music by any of the generators or any complementary factor. In this specific case of 2x3x5, the time signatures available are those with the numerators 2, 3, 5, 6, 10, or 15.
For example:
2 3 5 6 10 15 , , , , , 4 4 4 8 8 8
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Distributive Powers
According to the SSMC, fractional rhythms are those contained within the measures, while factorial rhythm is that comprised by a group of measures. Schillinger observed of Western music in general relied too heavily on the number two and its multiples: The habit of thinking in two’s and their multiples has retarded the development of our musical civilization, and again; the general field of classical music deals with division by two and multiplication by two. All Classical rhythm patterns are based on half, quarters, eights, sixteenths, etc. Measures accumulate through the same multiples. (Schillinger 1946:71) Schillinger suggested that proportions other than two could be used to control rhythms. The technique of distributive powers manifests the evolution of a number, or a rhythm, through its powers. For example, 22=4, 32=9, 42=16, 52=25, etc. Jeremy Arden, an exponent of Schillinger’s ideas suggests that: The mathematical relationship between the time signature and the bar-group as a whole might be explained by a mathematical power series. Following this principle a bar made by three beats must belong to a bar-group of three bars, rather than four. (Arden; 2006:45) To create a bar-group, the composer choose a time signature, which dictates the number of bars in the bar-group.
Fig. 16: A bar of three beats generates a bar group of three bars.
Fig. 17: The square of four beats has given a four bars-group.
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Fig. 18: Another example, a bar comprising of five beats it is squared; 1=crotchet, 52 =25 crochet-beats.
Having established the bar-group, it is necessary to compose a rhythmic pattern of one bar that is the sum of the time signature’s numerator. The choice of these two numbers it is called factorial binomials in the SSMC, while in Arden’s Keys of the Schillinger System, it is referred to the seed of the rhythm, a term that involves the reader’s imagination to understand by analogy what this process involves: A single bar of rhythm unfolds into a multi-layered score, somewhat like the development of the mature plant that grows from a tiny seed. (Arden, 2006:46) From here on I shall following Arden, adopt the term seed to explain the process. The seed must be the duration of one bar only, and must consist of duration’s equal to or multiples of the beats indicated by the time signature. For example, Figure 19 shows all the choices available for seeds in a time signature of 3/4:
Fig. 19: Those are all the choices available of seeds based on 3/4 time signature.
The distributive power of the seed 2+1 can be obtained by the following formula. Example, I take the seed of 2+1 (a+b)2 = (a2+axb) + (bxa+b2) (2+1)2 = (22+2x1) + (1x2+12) = (4+2) + (2+1)
↑
↑↑ seed
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a b
This formula can appear simply like this; the seed it is multiplied by itself, then the first term by
€€
the second term, (22+2x1) then the second term by the first, finally the second term by itself. (1x2+12) The result 4+2+2+1 can be now transformed in a musical score,
Fig. 20:The square rhythm made by the seed of 2+1; another bar-group that can counteract with the first one illustrated above (fig.19)
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This method can be applied to seeds made out by any number of elements: Three elements: (a+b+c)2= (a2+axb+axc) + (bxa+b2+bxc) + (cxa+cxb+c2) Four elements: (a+b+c+d)2 = (a2+axb+axc+axd) + (bxa+b2+bxc+bxd) + (cxa+cxb+c2 +cxd) + (dxa+dxb+dxc+d2 ) An example to clarify this process now comprising of three elements: the seed of 2+1+1 is chosen. the squaring formula is applied: a=2 b=1 c=1 (a+b+c)2 = (a2 +axb+axc) +
(bxa+b2+bxc) + (cxa+cxb+c2)
(2+1+1)2 = (22 +2x1+2x1)
+
(1x2+12 + 1x1) + (1x2+1x1+12 )
result:
+
(2+1+1)
(4+2+2)
+ (2+1+1)
musical score:
Fig. 21: The distributive power of the seed of 2+1+1; note the sum of 4+2+2+2+1+1+2+1+1=16 is the required duration of the bargroup of the beats from a 4/4 time signature.
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Evolution of Rhythmic Styles
The Evolution of Rhythmic Styles deals with the manufacture of seed, that is, rhythms that occupy a single bar. All kinds of seed can be manufactured informally by splitting the number of beats in the bar, or formally by using a process similar to pulse interference. Both of these techniques produce and develop rhythms useful for creating accompaniment patterns. This technique is an evolution of one rhythm related with other rhythms that belong to the same family from which they have been conceived. Schillinger suggests that: All the consecutive interferences-group generated by one determinant constitutes the evolution of all rhythmic patterns in the corresponding family style. (Schillinger, 1946:84) The consecutive interferences-groups to which Schillinger refers are all the possible permutations expanded from a time signature (determinant) of the composer’s choice used to generate a pattern’s series. Let’s analyze the first type of technique: the informal splitting of beats. First, decide a time signature, then divide it in two different numbers, the sum of which is equal to the numerator, then divide the numerator again in three different number, and continue until all the possible permutations. An example would show this further process more clear:
4 4
1)
Choose a time signature of
2)
1+3 = 4
3)
1+2+1 = 4
4)
Continue all the possible divisions: (3+1); (1+3); (1+2+1); (2+1+1); (1+1+2); (1+1+1+1).
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The second type of technique works rather like the first, but it possess features similar to the first generating process of two generators. Again, the first step is to decide a time signature. Take the numerator, and subdivide it with all the two-numbers combinations. It is important that from this pair of numbers the multiple of the numerator are being excluded from the combinations, (6+2; 4+4;) since they do not produce any syncopation. The pairs of family-numbers preferred are those that share no common divisor than 1. In this case they are 7+1=8 or 5+3=8. The family of 5+3 is taken as an example; the graph shows the first line the common denominator (1) the second and the third
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lines shows respectively the family-numbers in its two only possible permutations (3+5) (5+3). The fourth line of the example below shows the resultant of this initial process, which is the pattern of 3, 2, 3. This resultant is now then permuted again until to find its final resultant; the last line of the graph. This new rhythm 2,1,2,1,2 is the mother cell, which it is permuted again in all the possibilities, until to achieve the pattern 1,1,1,1,1,1,1,1= 8 that mark the end of this generating process.
Fig. 22: The family numbers of show all the permutations and his resultant: 2+1+2+1+2=8. That is the mother cell for other related rhythm given by the further permutations of it.
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Further Development of the Bar-group
So far we have dealt with only one technique that develops the seed, and it has been possible to create one rhythmic line of the score that could be used a the motivitic material. There are three further techniques that aim to expand the bar group and may be used in a larger score if the composer so desires. Firstly, due to that fact that squares need not be symmetrical, the original and its retrograde rhythm can be synchronized to generate another resultant pattern, as shown in Figure 23, where the retrograde inversion generates a symmetric resultant pattern.
Fig. 23: The retrograde inversion generate a symmetric resultant pattern.
Secondly, each element of the seed can be multiplied by the sum of its members. Seed = 2+1+1 Sum of the elements of the seed = 4 Calculation, (2x4)+(1x4)+(1x4) = 8+4+4 =16 (beats)
Fig. 24: This second technique produce less active rhythms, useful for pedal notes.
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Lastly, symmetrical rhythms from the second-type technique can be incorporated into the score because the complementary factor of sharing the square technique, and the often asymmetrical patterns derived from the seeds, creates contrast between the parts leading naturally to rhythmic counterpoints. This point is important because it allows different techniques to be integrated into a bar group score; the bar group is an embryonic model score that exhibits structural coherence. In Figure 25, the seed 2+1+1 has been used to produce a musical score comprising of all the techniques just described.
Fig. 25: The development of the seed 2+1+1.
The combination of the seed, and the Type II symmetrical rhythm, needs to be chosen in regard to the number of beats in the bar group. For example in a 3/4 time signature, there are 9 beats in the bar group, so a natural choice goes to the ratio Type II 32:2 (2111112) =9 or, in a 4/4 meter, there are 16 beats in the bar group, so the choice goes to the ratio Type II 42:3 = (3,1,2,1,1,1,1,2,1,3) = 16, and so on. When the number of beats in the bar is low, such as 2, 3 or 4, there are relatively few choices. However, it is possible to accommodate a much larger Type II pattern by measuring its units using a smaller symbol to represent 1. For example, in a 4 bar group consisting of 16 crotchet beats, the rhythm 82:3 or 82:5 or 82:7 could be used. These patterns are all 64 units in length, that is, 8 squared, and will fit inside the bar group if their unit of measurement is considered to be a semiquaver.
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Variation techniques
This process may be useful to composers wishing to evolve a large amount of material. This technique produces to what Schillinger refers to as homogeneous rhythmic continuity which creates a degrees of variations, but all are based on permutation of the elements in a pattern.
This technique requires first the division of an original idea into two parts: Example: Original idea= (a+b) (a+b) has only one permutation, the retrograde form (b+a) (a+b) ; (b+a) Each form can be assigned a label, such as a1 or b1 (a+b) = a1 (b+a) = b1 a1 and b1 may be grouped as a sequence and then permutated (a1+b1) ; (b1+a1)
This form now it is a longer sequence made by (a+b) and (b+a), and it looks as follows: (a+b); (b+a); (b+a); (a+b)
The process can continue infinitely: (a1+b1) = a2 (b1+a1) = b2 (a2+b2) + (b2+a2), and again, (a2+b2) = a3 (b2+a2) = b3 (a3+b3) + (b3+a3) and so on…
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General and Circular Permutations
Various techniques can be used to create different degrees of variation. All are based on the permutation of elements within a given pattern. Permutation is a means of obtaining variety from a set of elements, such as a number of individual motifs, rhythms or chord progressions. There are two fundamental types of permutation: circular and general. Circular permutation can be considered a form of rotational variation in either the clockwise or anticlockwise direction. Variations cannot jump across the circle, but follow the ordered array by which a series of related patterns are produced. The figures below show the clock and anticlockwise circular permutation of a group containing three different elements:
1) clockwise circular permutations:
ABC
BCA
CBA
2) anticlockwise permutations:
ACB
CBA
BAC
Fig. 26: The circular permutations of the group A, B, C; courtesy from Clock and Rose Press, (Arden 2006:75)
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Fig. 27: The possible musical score of three elements permuted as show previously.
If two or more elements in a group are identical, then fewer permutations result. For example, three elements equal three variations, four elements equal four variations, and so on.
1) clockwise circular permutations:
ABBC
BBCA
BCAB
CABB
2) anticlockwise permutations:
ABBC
CBBA
BBAC
BACB
Fig. 28: The permutations with four elements, which two of them are identical.
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General permutations are the result of all the possible permutations, in other words the circular directions no longer apply and all arrangements are calculated: for example if there are three elements, to calculate all the possibilities it is necessary:
AxBxC = 1x2x3 = 6 those are all the possible variants.
ABC ACB
BAC BCA
CAB CBA
If two elements are identical the possibilities are halved: AxBxB= 1x2x3:2=3
ABB
BAB
BBA
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Post Script
Many conclusions I have drawn need to be tempered by an awareness of the circumstances surrounding the publication of Schillinger’s theories. The SSMC was published posthumously, being arranged from Schillinger’s personal manuscript by his wife and two of his close students. Had Schillinger been able to oversee the publication of the SSMC himself, it is, in my opinion, unlikely that he would have agreed to the inclusion of so much extraneous matter. The very bad reviews of the book itself, which pointed out mistakes and misinterpretations, has to be addressed directly to the publisher and editor who made no attempt to make the book more accessible. Throughout his text, Schillinger’s employment of mathematical notation contrary to established conventions, including even the most basic functions, leads to much un-necessary confusion. Neither the division sign nor the colon is employed as in ordinary arithmetic; he uses them to indicate interference, a process to which is also interpreted contrary to its established use in the physical sciences. Such a blasé attitude towards the conventions of the scientific community detracts from the credibility of his theories despite stating in the opening chapter that numerical operations will be subject to their normal mathematical operations (Schillinger 1946:1). By being couched in his own pseudo-mathematical notation, he succeeds in mystifying the reader from the outset. The SSMC opens with a description of a system of graph notation of his own devising. This notation has so little in common with the sine waves discussed in the preceding paragraphs that, as Backus rightly considers:
The material of the first two pages of the book might just as well have been omitted. (Backus 1960:223) Schillinger begins with the observation that music, being a temporal art, is broken into pulses. His idea was that rhythm arises through the interference of two or more sources of pulse. Although he called the resulting pattern pulse interference, this is a confusion on his part since his result is actually the product of the functions. Scientifically, interference occurs when two or more waves pass through a medium. The resulting disturbance at any point in the medium can be found by the addition of the individual displacements that each wave would
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have caused by itself (Loy 210:2006). Where the waves have the same sign, the sum of their displacements will be larger than either wave alone, resulting in constructive interference. Where waves have opposite signs, the sum of their displacements will be smaller, resulting in destructive interference. Where two waves of opposite sign and equal magnitude coincide, they cancel each other. The resulting periodic fluctuations in amplitude are known scientifically as beats. If we were to find the resultant of say, 3:2, using this scientific description of interference, we would obtain the following result:
By using interference by addition, instead of Schillinger’s interpretation, not only are various degrees of amplitude associated with the resultant, but also, certain combinations of periodicities will, where two waves of opposite sign and equal magnitude coincide, produce a value of zero, as in the foregoing example. While studying the SSMC, I felt that Schillinger’s ideas on rhythm needed to expanded somewhat in order to accommodate the notation of rests in the music, a subject which he mentions only briefly in his theory of rhythm (see Schillinger 1946:52). I feel that this proper interpretation of interference may be a means of logically achieving such a result since the portions where the waves cancel each other producing a value of zero, could be interpreted as silence. The varying degrees of amplitude associated with this interpretation could be used to control other aspects of composition. Schillinger then provides a list of all the necessary generators for practical purposes (Schillinger 1946:10). This is a restricted exposition of innumerable possibilities, a view shared by Backus: Schillinger's resultants do not give all possible rhythms, as claimed; in fact, the number obtainable by his method is quite small. (Backus 1960:225)
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At this stage, Schillinger fails to explain why certain ratios, such as 4:2, are excluded from the list. Schillinger claims that these ratios do not create rhythm, but only a metrically regular pulsation, maintaining that ratios that share no common divisor other than one are the only choice available. Most often, syncopation is taken to mean the shifting of an expected pattern of note-onsets to an unexpected position relative to an established pulse. However, it is arguable that, in the absence of meter, the contrast in duration between adjacent events would create such an effect in the listener. Simply because certain ratios produce more varied and syncopated rhythms does not, in my opinion, mean that those giving rise to regularity should be avoided. I feel that the student would benefit greatly here if all the possibilities were given, rather than those subjectively chosen by Schillinger (see Appendix 1). Schillinger felt that being so uniform, the resultant of 4:2 was of no practical use, but I am inclined to differ. The strongly marked rhythms of march music have been used since time immemorial for it is ideally suited to the co-ordination of large bodies of people, such as soldiers. If this is not a practical application of this resultant, then what is? Schillinger considers the processes by which rhythmic resultants are generated are not entirely satisfactory for all musical processes, feeling that the results are too rich (Schillinger 1946:15). Thus, he proceeds to elaborate a second process known as rhythmic resultants with fractioning around the axis of symmetry. Whereas his first process, the interference of periodicities, espoused the principals of simplicity, elegance, and parsimony, this second process is of such a convoluted and fanciful nature that it would be more advantageous to the student if it were not included in the SSMC. Why not start the second phase at some another point in time, rather than that of Schillinger’s whim? Why not allow the second and any subsequent generators required run to completion rather than arbitrarily terminate them? As for the results, such as mysterious and complicated mathematical calculations are unnecessary in order to create long series of one’s, or for that matter, two’s or three’s (see Appendix 2). It seems strange that Schillinger should go to such lengths in order to create metrically regular pulsation, especially since his aims were to obtain variety and syncopation, which he obviously considered to be the spirit of rhythm. It must be noted that the resultants of all the methods described in the SSMC are palindromes. By using the interference of periodicities only one four-figure palindrome is generated, the remaining possibilities are not mentioned by Schillinger. Furthermore, none of Schillinger’s calculations generate any five-figure palindromes. Whereas Schillinger felt that the resultants of the interference of periodicities were too rich, I feel that they are too impoverished. Indeed, is it necessary to use palindromes in the first place, surely any old number would suffice? It would certainly negate the exposition of a great deal of dubious mathematics. Considering the number of possibilities unavailable to the composer using Schillinger’s calculations, his assertion that all the rhythmic patterns observed in music are either complete
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or incomplete resultants (Schillinger 1946:10), be it music of the past, present or future is simply untenable. Chapter 6 of the SSMC opens with a tortuous description of the Fibonacci series. Schillinger fails to mention it by name despite being known to the West for over 700 years. Alone, the numbers of this series have no relevance whatsoever to any theory of aesthetics, so I consider it doubtful that any sense of harmonic proportion will be conferred onto resultants obtained from three or more generators using these numbers. The Fibonacci series only becomes musically relevant when the ratios of two subsequent terms are considered. The corresponding sequence of quotients converges rapidly towards the value of the golden mean (Loy 2006:437). This number controls the arrangement of petals in flowers, seed clusters, and pinecones, to name but a few examples. Although composers often use this proportion to control the architecture of their works, it is debatable whether it imparts any sense of beauty. Schillinger then goes on to give a list comprising of but three number series that he considers useful for musical purposes (Schillinger 1946:24), despite in a preceding paragraph stating that mathematically, one can produce an infinite number of such series. Again, I feel that the student would benefit more from a slightly more substantive list than which Schillinger presents. Schillinger makes many references to African music. He refers to the lost art of the aboriginal African drummers (Schillinger 1946:35), prides himself with an acquaintance of the true primitive rhythm, such as the rhythm used by some African cannibalistic tribes (Schillinger 1946:11), and boasts an encyclopaedic knowledge that appears to encompass all history, even before records began, as to when he refers to the rhythms performed on the African continent thirty thousand years ago (Schillinger 1946:71). Born in Russia in 1895, he spent the last fourteen years of his life in the United States. From what is known of his life he never set foot on the African continent. His only knowledge of such lands was from the Blacks he encountered in his newly adopted home, and maybe a few recordings. Contemporary with the writing of his SSMC, Schillinger considered the United States to be undergoing a renaissance of rhythm (Schillinger 1946:35), owing to the transplantation of Africans to the American continent. As credible as this idea may have been, I feel that it is unlikely, as he then goes on to claim, that the instinct of rhythm in the present American generation surpasses anything known throughout European history. (Schillinger 1946:35), Schillinger writes of African music with as much ignorance as ever yet abased a subject in print, and with a doctoral superciliousness equal to, if not exceeding his ignorance. This is somewhat disturbing considering that he was opposed to vagueness and haphazard speculation (Schillinger; 1946:1356). According to Schillinger, Europeans have never possessed the instinct of rhythm with which the Africans are endowed (Schillinger 1946:34). Ethnomusicologist Mantle Hood, shared a similar view:
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Conventional Western training in musicianship produces a stunted growth in the perception and execution of rhythm and melody. (Hood 1982:35) Undeniably, the most outstanding feature of much African music is the complexity of its rhythmic structure, but rhythmic complexity alone does not guarantee great music. In my view, the Classical masters who Schillinger is so ready to denigrate were no less capable of the plasticity of temporal organisation (Schillinger 1946:34), to which he so aspired. Schillinger often claims that his approach is better, but what he actually avoids saying that it is better only because it is comprised of mathematical logic. This would, I feel, only provide a sort of scientific proof if he were to avoid stating that everything before his system were solely the fruit of mistakes and errors. Indeed, a compositional methodology based on a logic of numbers would give stability and coherence to the work created. In his magisterial account of the mathematical foundations of music, Musimathics, Gareth Loy provides an insight into Schillinger’s deterministic rhythmic techniques: By relying on numbers to control the compositional process Schillinger was looking to establish a scientific theory of art and to put practical methods into the hands of artists, giving them a mathematician’s vision of the nature and extent of their domain. (Loy 2006:325) The ancients attempted to explain the divine proportions they observed throughout the universe mathematically, giving rise to the concept known as the harmony of the spheres. In much the same way that these ideas were to needlessly exercise intellectuals until the middle of the 18th century, I feel that the SSMC will do likewise due to the manner in which Schillinger advanced his claims. The long debate about the connection of numbers and music will forever be a theatre of opposing factions contending contrasting theories. Some consider music a creative art that cannot be controlled or managed by numerical calculations, a solution which inevitably imprisons the composer’s imagination. Conversely, there are people that consider numbers to be the solution to formulaic music by controlling all it’s parameters. Both parties, however, refuse to admit that the sole use of both mathematical calculations or uncontrolled creativity leads to nowhere. I feel that mathematics can help you organize your ideas and control irrational thought, while irrationality can help you go beyond your imagination. When we encounter the concept of distributive powers, we meet perhaps the most bizarre pronouncement so far. Schillinger presents the student with a list of the distributive powers that have been employed to date from the beginning of music by the inhabitants of this
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planet (Schillinger 1946:71). Firstly, as with every other list presented in the SSMC, it is far from being comprehensive. Secondly, anything more than a few thousand years before the present day, can only be a matter for speculation. Unanswered questions such as these are a challenge to any would-be historian, and for want of proper evidence, their discussion could be endless. Fortunately, for Schillinger, however, no one can say that he is true or false. Lastly, in addition to his encyclopaedic knowledge of music on this planet, he obviously appears to know more than the average reader on the music of other planets too! Schillinger believed that the habit of thinking in two’s has been detrimental to the development of our musical civilisation. He claimed that three-bar phrases have been entirely untouched (Schillinger 1946:72), in Classical music, blithely unaware that three-bar phrases were a characteristic of the Branle, a group dance popular in France during the 16th and 17th centuries. The most fruitful way to approach musical composition is provided by a study of methodology. Methodologies enable information to be passed in society from one generation to another by establishing a logical set of criteria, such as rules or a sequence of operations, necessary for accomplishing a task or solving a problem (Loy 2006:288). Although the methods of generating rhythm as espoused by Schillinger may not aspire to the rigours of methodology, it does to that class of non-deterministic methodologies known as art, processes that produce variable results even when repeatedly presented with the same input. By an analysis of methodology, however, I feel that it is possible to reveal the aesthetic agenda of its creator. In developing the SSMC, Schillinger was hoping to establish a methodology of art that could be used to compose music, thus liberating composers from the stifling effect of tradition, much as Schoenberg had hoped with his development of atonality. Although it is easy to denigrate Schillinger’s theories, his impressive biography (see Brodsky 2003:47 for the most detailed account), ought to deter his opponents from their claims that he was a fraudulent charlatan. Nevertheless, a search of the professional journals and what little mention there is of Schillinger in the literature, shows that even today, he remains an insignificant footnote in the history of music. I feel that this obscures the true value of his work for he clearly demonstrated ability in several fields. The importance of the SSMC transcends its purely practical aspect, as Slonimsky says: It revitalizes musical theories that have for a long time been in a state of academic stagnation. (Slonimsky 1946:470) Before the modern age, music was considered to be one of the mathematical sciences, associated with astronomy, geometry and arithmetic (Gouk 1999:9). It was thought of as both an art and a science, a concept that is problematic to the rationale of our own times. Although a demarcation between true and false beliefs is impossible, this mode of thought potentially loosens established boundaries and enables connections to be made between things that
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would normally be considered unrelated. Such matters are conventionally excluded from the realm of science, but it does enable practitioners, such as composers, to playfully indulge in creativity, which, as Arden recalls, was Schillinger’s intention from the very outset: The SSMC is a practical tool and a creative stimulant that can enable the practitioner to move beyond habitual perspectives and cultural barriers. (Arden 2006:4) Such correlations, however, also threaten established norms. Schillinger proposed that the SSMC would make the composition of music available to the masses. This concept would probably have been met with hostility from the established network of composers and arrangers who had spent many years learning their craft. Disregarding the failures of his text, such feelings of jealously among the musical establishment may also have played a role in preventing the dissemination Schillinger’s ideas. Brodsky considers that: Ultimately, his failure to succeed in academic circles can be traced to his unconventional terminology and methods, his ridicule of the academic establishment, and his attachment to the commercial side of music through his ties to Tin Pan Alley and Hollywood. (Brodsky 2003:51) The SSMC is not an effect of the music industry, but rather an aspect of it. He devised the method in an effort to coordinate the activities of the composer so that they could work more efficiently. The music industry has a role in culture too, determining what is consumed, but it does not control it since the audiences drive the market. Actually, it needs to continually respond to the changes within it, meaning that music has to accommodate styles and trends continuously, as Arden suggests: There is more music manufactured now than at any other time; style and fashions are proclaimed and then pass in an instant, therefore the study and the practice of music needs to be undertaken ever more objectively and accurately in order to penetrate to the heart of a musical context. (Arden 2006:3) The real mistake is to rely entirely on the SSMC to control the outcome. If this is allowed, the composer ends up with a huge amorphous mass of material. Thus, the composer must select from among the generated compositions those that are good enough to be performed. Obviously, editing is a necessary step since there would be little audience for every one of the possible compositions Schillinger’s techniques could create. This procedure, however, becomes problematic in that the selected compositions become official specimens, thus proving the effectiveness of Schillinger’s methods.
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In my opinion, it would be absolutely unthinkable to compose without your own aural response, personal judgments and the actual playing of the music in order to control and achieve the result you actually looking for. I would like to stress this point since I do believe that the SSMC is a set of hypothetical techniques that aim purely to organize the composer’s ideas, enabling them to be developed and organized by their creator; the composer must control the music. In fact, as it stands, Book 1 of the SSMC is not so much a theory of rhythm, but is really a supposition or proposed explanation made on limited evidence, that can be used as a starting point for further creative musical investigations. At the beginning of the 20th century, revolutionary discoveries in the sciences gave rise to new approaches and concepts of how the Arts should be both conceived and perceived. In music, this revolution initiated a new school of thought that had a profound influence on intensifying and improving the mathematical approaches used in musical composition. These phenomena have been incorporated into compositional methodology, changing the way in which composers formulate new ideas: The triumph of the machine in the twentieth century, including most particularly the revolution that began in the late 1940s, must have had a part in changing ideas about the brain and therefore about creativity, which could now be seen as dependent on processes of selection and arrangements, and so able to profit from systematic methods. (Griffith 2001:234) Certainly, Griffith points out the attention towards the approach of music in general rather than to particular styles of composing. This is also similar to the circumstances that surrounded the ideas behind the creation of the SSMC, a view supported by Seeger: The external conditions for the creation of Schillinger’s system are found first in the confusion of the 20th century music which suggested the end of one historical epoch and the beginning of another, and second in the pattern of over-all systematizing that has been found recently all the way from theoretical physics through the social sciences, lastly, the arts. (Seeger 1947:183) Today’s composers must be impartial and should not deny that music can be conceived through different techniques and mediums. The SSMC may blend with traditional ideas and elevate human composition to another stage. Today’s composers and music performers are struggling with the competition of computers that can compose faster and in any style. Computer software has drastically changed the compositional process; musical patterns can be edited, copied and pasted. The use of music software to compose and produce music is to me, enough to think that new techniques are necessary in order to allow human creativity on an industrial scale. The SSMC should be seen as a complementary activity to composition:
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A pedagogical method meant to improve students’ intuitive musical knowledge and augment their individual styles. (Taxier 2003 see ref. list). However, the system does not give any answer to your music if you have not been trained in the traditional field of composition. The art of composing cannot be undertaken by a series of mathematical processes. It should not be forgotten that even though the great success of SSMC, which was due the influence of students that had achieved commercial success (Gershwin above all), were all highly trained. It would be misguided to take the SSMC as a base for composition, as Carter states: The basic philosophical fallacy of the Schillinger point of view is of course the assumption that the “correspondences” between patterns of arts and patterns of the natural world can be mechanically translated from one to the other by the use of geometry or numbers. Wherever this system has been successfully used, it has been by composers who were already well trained to distinguish the musical result from the nonmusical ones (Carter 1985:240). Schillinger’s ideas may give to composers the opportunity to devise their own material, without historical reference to Master scores. It has always been difficult for students of composition to understand the teachings methods capable in making the distinction between the objective and subjective factors in music. The danger is that after spending a long time studying a particular methodology, your music can end up sounding like somebody else’s. I feel that the techniques described by Schillinger in the field of rhythm have much to offer contemporary composers, they also compensate for the long perceived imbalance in the literature on composition, which is still largely dominated by the considerations of melody and harmony. I feel that some of Schillinger’s tools can be incorporated into my compositional vocabulary. I find the derivation of rhythm from ratios, and the methods of permutation inspiring. However, I feel that creativity should flow and not be circumscribed by rules. The SSMC provides the opportunity to exercise your own taste and judgment. Moreover, I feel that it will enable me to invent and develop my own rules. Dr. Arden’s efforts in making the texts more accessible have much to do with the present renaissance of interest in the SSMC. Pedagogically, much work still needs to be done to make the basic ideas contained in the text more understandable to the student. The SSMC needs to be completely dismantled. Schillinger’s eccentric and inexact terminology needs to be modernized and replaced by terms understood not only to the scientific community of the 21st century, but to any average student who has undergone a comprehensive education. Much of what Schillinger has written simply needs to be discarded.
If, this is not done, Schillinger’s
monumental work, which ought to lead to a total conviction of his theories, will continue to be absent from the syllabuses of the leading universities.
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Appendix 1: Patterns Generated Using Ratios – Type I
The following table shows all the resultants obtained from the Type I method of pulse interference, using the integers 2 through to 9.
3:2 2112 4:2 2222 4:3 312213 5:2 221122 5:3 3213123 5:4 41322314 6:2 222222 6:3 333333 6:4 42244224 6:5 5142332415 7:2 22211222 7:3 331232133 7:4 4314224134 7:5 52341514325 7:6 615243342516 8:2 22222222 8:3 3321331233 8:4 44444444 8:5 532514415235 8:6 624426624426 8:7 71625344352617 9:2 2222112222 9:3 333333333 9:4 441342243144 9:5 5415325235145 9:6 633663366336 9:7 725436171634527 9:8 817263544536271
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Appendix 2: Patterns Generated Using Ratios – Type II
The following table shows all the resultants obtained from the Type II method of pulse interference, using the integers 2 through to 9.
3:2 2111112 4:2 22222222 4:3 3121111213 5:2 221111111111111111122 5:3 32121111111112123 5:4 4131121211314 6:2 222222222222222222 6:3 333333333333 6:4 4222222222222224 6:5 5141131221311415 7:2 2221111111111111111111111111111111111111222 7:3 3312121111111111111111111111111212133 7:4 4313121121111111111111211213134 7:5 5232212211121211122122325 7:6 6151141231321411516 8:2 22222222222222222222222222222222 8:3 33212121111111111111111111111111111111111112121233 8:4 4444444444444444 8:5 532321221211121111111121112122123235 8:6 62422222222222222222222426 8:7 7161151241331421511617 9:2 2222111111111111111111111111111111111111111111111111111111111111111112222 9:3 333333333333333333333333333 9:4 441313112112111111111111111111111111111111111211211313144 9:5 5414131131121112111111111111111112111211311314145 9:6 6333333333333333333333336 9:7 725223222122311232113221222322527 9:8 8171161251341431521611718
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Appendix 3: Polynomials
This table provides a comprehensive list of all the polynomials, that is, groups consisting of more than one element, employing the numbers 2 through to 8.
2: (1+1).
3: (1+1+1); (1+2); (2+1).
4: (1+1+1+1); (1+1+2); (1+2+1); (1+3); (2+1+1); (2+2); (3+1).
5: (1+1+1+1+1); (1+1+1+2); (1+1+2+1); (1+1+3); (1+2+1+1); (1+2+2); (1+3+1); (1+4); (2+1+1+1); (2+1+2); (2+2+1); (2+3); (3+1+1); (3+2); (4+1).
6: (1+1+1+1+1+1); (1+1+1+1+2); (1+1+1+2+1); (1+1+1+3); (1+1+2+1+1); (1+1+2+2); (1+1+3+1); (1+1+4); (1+2+1+1+1); (1+2+1+2); (1+2+2+1); (1+2+3); (1+3+1+1); (1+3+2); (1+4+1); (1+5); (2+1+1+1+1); (2+1+1+2); (2+1+2+1); (2+1+3); (2+2+1+1); (2+2+2); (2+3+1); (2+4); (3+1+1+1); (3+1+2); (3+2+1); (3+3); (4+1+1); (4+2); (5+1).
7: (1+1+1+1+1+1+1); (1+1+1+1+1+2); (1+1+1+1+2+1); (1+1+1+1+3); (1+1+1+2+1+1); (1+1+1+2+2); (1+1+1+3+1); (1+1+1+4); (1+1+2+1+1+1); (1+1+2+1+2); (1+1+2+2+1); (1+1+2+3); (1+1+3+1+1); (1+1+3+2); (1+1+4+1); (1+1+5); (1+2+1+1+1+1); (1+2+1+1+2); (1+2+1+2+1); (1+2+1+3); (1+2+2+1+1); (1+2+2+2); (1+2+3+1); (1+2+4); (1+3+1+1+1); (1+3+1+2); (1+3+2+1); (1+3+3); (1+4+1+1); (1+4+2); (1+5+1); (1+6); (2+1+1+1+1+1); (2+1+1+1+2); (2+1+1+2+1); (2+1+1+3); (2+1+2+1+1); (2+1+2+2); (2+1+3+1); (2+1+4); (2+2+1+1+1); (2+2+1+2); (2+2+2+1); (2+2+3); (2+3+1+1); (2+3+2); (2+4+1); (2+5); (3+1+1+1+1); (3+1+1+2); (3+1+2+1); (3+1+3); (3+2+1+1); (3+2+2); (3+3+1); (3+4); (4+1+1+1); (4+1+2); (4+2+1); (4+3); (5+1+1); (5+2); (6+1).
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8: (1+1+1+1+1+1+1+1); (1+1+1+1+1+1+2); (1+1+1+1+1+2+1); (1+1+1+1+1+3); (1+1+1+1+2+1+1); (1+1+1+1+2+2); (1+1+1+1+3+1); (1+1+1+1+4); (1+1+1+2+1+1+1); (1+1+1+2+1+2); (1+1+1+2+2+1); (1+1+1+2+3); (1+1+1+3+1+1); (1+1+1+3+2); (1+1+1+4+1); (1+1+1+5); (1+1+2+1+1+1+1); (1+1+2+1+1+2); (1+1+2+1+2+1); (1+1+2+1+3); (1+1+2+2+1+1); (1+1+2+2+2); (1+1+2+3+1); (1+1+2+4); (1+1+3+1+1+1); (1+1+3+1+2); (1+1+3+2+1); (1+1+3+3); (1+1+4+1+1); (1+1+4+2); (1+1+5+1); (1+1+6); (1+2+1+1+1+1+1); (1+2+1+1+1+2); (1+2+1+1+2+1); (1+2+1+1+3); (1+2+1+2+1+1); (1+2+1+2+2); (1+2+1+3+1); (1+2+1+4); (1+2+2+1+1+1); (1+2+2+1+2); (1+2+2+2+1); (1+2+2+3); (1+2+3+1+1); (1+2+3+2); (1+2+4+1); (1+2+5); (1+3+1+1+1+1); (1+3+1+1+2); (1+3+1+2+1); (1+3+1+3); (1+3+2+1+1); (1+3+2+2); (1+3+3+1); (1+3+4); (1+4+1+1+1); (1+4+1+2); (1+4+2+1); (1+4+3); (1+5+1+1); (1+5+2); (1+6+1); (1+7); (2+1+1+1+1+1+1); (2+1+1+1+1+2); (2+1+1+1+2+1); (2+1+1+1+3); (2+1+1+2+1+1); (2+1+1+2+2); (2+1+1+3+1); (2+1+1+4); (2+1+2+1+1+1); (2+1+2+1+2); (2+1+2+2+1); (2+1+2+3); (2+1+3+1+1); (2+1+3+2); (2+1+4+1); (2+1+5); (2+2+1+1+1+1); (2+2+1+1+2); (2+2+1+2+1); (2+2+1+3); (2+2+2+1+1); (2+2+2+2); (2+2+3+1); (2+2+4); (2+3+1+1+1); (2+3+1+2); (2+3+2+1); (2+3+3); (2+4+1+1); (2+4+2); (2+5+1); (2+6); (3+1+1+1+1+1); (3+1+1+1+2); (3+1+1+2+1); (3+1+1+3); (3+1+2+1+1); (3+1+2+2); (3+1+3+1); (3+1+4); (3+2+1+1+1); (3+2+1+2); (3+2+2+1); (3+2+3); (3+3+1+1); (3+3+2); (3+4+1); (3+5); (4+1+1+1+1); (4+1+1+2); (4+1+2+1); (4+1+3); (4+2+1+1); (4+2+2); (4+3+1); (4+4); (5+1+1+1); (5+1+2); (5+2+1); (5+3); (6+1+1); (6+2); (7+1).
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References
Apel, Willi, Harvard Dictionary of Music, Harvard University Press, Massachusetts, (1972).
Arden, Jeremy, Keys of the Schillinger System, Clock&Rose Press (2006)
Arden, Jeremy, Focusing the Musical Imagination: Exploring in Composition the Ideas and Technique of Joseph Schillinger, Submission for the degree of PhD in music, City University of London(1996)
Backus, John, Pseudo-Science in Music, Journal of Music Theory, Vol. 4, No. 2 (Nov., 1960), pp. 221-232.
Barbour, J. Murray, The Schillinger System of Musical Composition by Joseph Schillinger, Notes, Second Series, Vol. 3, No. 3 (Jun., 1946), pp. 274-283.
Brodsky, Warren, Joseph Schillinger (1895-1943): Music Science Promethean, American Music, Vol. 21, No. 1 (Spring, 2003), pp. 45-73.
Cowell, Henry, Five Pieces for Piano, Op. 12; Excentriade, Three Pieces for Piano, Op. 14 by Joseph Schillinger, Notes, Second Series, Vol. 5, No. 1 (Dec., 1947), pp. 128-129.
Duke, Vernon, Gershwin, Schillinger, and Dukelsky, The Musical Quarterly, Vol. 75, No. 4, Anniversary Issue: Highlights from the First 75 Years (Winter, 1991), pp. 119-124.
Gouk, Penelope, Music, Science and Natural Magic in Seventeenth-Century England, Yale University Press, London, (1999).
Griffith, Paul, ‘Numbers and Music’ in Sadie, Stanley, and Tyrrell, John (eds,) The New Grove Dictionary of Music and Musicians, Grove Publisher (2001).
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Carter, Elliot, in ‘Clotho and Atropos’ in Lewis Rowell, Eugene, Thinking about Music: An Introduction to the Philosophy of Music, University of Massachusetts Press, (1985).
Hood, Mantle, The Ethnomusicologist, Second Edition, Kent State University Press, Ohio, (1982).
James, Jamie, The Music of the Spheres, Abacus, London, (1993).
Huron, David, Sweet Anticipation, The Massachusetts Institute of Technology Press, London, (2006).
Loy, Gareth, Musimathics: Volume 1, The Massachusetts Institute of Technology Press Press, London, (2006).
Moekle, Janet, Graph Method of Dance Notation by Joseph Schillinger, Dance Chronicle, Vol. 9, No. 3 (1986), pp. 403-404.
Nauert, Paul, Theory and Practice in "Porgy and Bess": The Gershwin-Schillinger Connection, The Musical Quarterly, Vol. 78, No. 1 (Spring, 1994), pp. 9-33.
Quist, Ned, Toward a Reconstruction of the Legacy of Joseph Schillinger, Notes, Second Series, Vol. 58, No. 4 (Jun., 2002), pp. 765-786.
Sachs, Curt, Rhythm & Tempo, W. W. Norton & Co., New York, (1953).
Sadie, Stanley (editor), The Grove Concise Dictionary of Music, Macmillan Press Ltd., London, (1988).
Scholes, Percy A., The Oxford Companion to Music, Oxford University Press, London, (1970).
Schillinger, Joseph, The Schillinger System of Musical Composition, Clock & Rose Press, Clock & Rose ed. (2003)
40
Seeger, Charles, Joseph Schillinger by Frances Schillinger, Notes, Second Series, Vol. 7, No. 3 (Jun., 1950), pp. 476-477.
Seeger, Charles, The Schillinger System of Musical Composition by Joseph Schillinger; Arnold Shaw, Notes, Second Series, Vol. 2, No. 4 (Sep., 1945), p. 299.
Slonimsky, Nicolas, The Schillinger System of Musical Composition by Joseph Schillinger, The Musical Quarterly, Vol. 32, No. 3 (Jul., 1946), pp. 465-470.
Strunk, Oliver (editor), Source Readings in Music History, W. W. Norton, New York, (1950).
Taxier, Eric, Review of Jeremy Arden’s thesis on Schillinger, [online] available from: http:// www.schillingersystem.com/ericarden.htm [accessed 16th December 2008].
Weiss, Piero & Taruskin, Richard, Music in the Western World, Schirmer Books, London, (1984).
Yasser, Joseph, The Mathematical Basis of the Arts by Joseph Schillinger, Notes, Second Series, Vol. 6, No. 3 (Jun., 1949), pp. 465-468
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