In the later part of Igor Stravinsky’s career (1882 – 1971), in the 1950s, he started writing serial and twelve-tone music by creating a hexachord array and rotating the notes in this array to be used for both melodic lines and structural harmony. This technique became Stravinsky’s most characteristic element in his compositions. The hexachord is acted upon by transposition, either up or down to get back to the first note of the original hexachord, which is repeated for each new hexachord until the original melodic line is obtained. This creates an array of lines that are all related by transposition, and the intervals of the transpositions are complements of mod (12) of the interval within the original hexachord. For example, if we take the following melody:

A A# D C E F

then rotate the A note to the end of the array, then transpose the array to begin on the note A, it would resemble the following diagram:

where the blue arrays represent the new transposed values, and the black arrays represent the precursor to the transposition. By the way, you can use any size array you want to, it does not have to be a hexachord, the array could consist of any number of notes. The following piece of music represent each blue array as sets that are used one after another. When used in this manner, the frequency of a repeated pitch class is useful for situations requiring pitch symmetry. Please have a look at the following piece of music called, Rotational Translation V, which used the above theory of rotational arrays:

There are other ways to compose using rotational arrays. For example, you could take the array and apply the retrograde, inverse, and retrograde-inverse and extend the piece of music into more dimensions.

The axis system involves harmonic and tonal substitution, and creates a functional relationship between tones and chords. This system shows how chords and tones relate by intervals of a minor third and a tritone, which function as tonal substitutes for one another. The axis system was created by Ernő Lendvaí, who was one of the first music theorists to explain the music of Béla Bartók in terms of the golden rule and Fibonacci numbers.

In classical harmony certain chord substitutions are rather common, such as the submediant (sixth scale degree) chord being replaced with the tonic (first scale degree), such as Am chord being replaced with a CM. However, Lendvaí discovered a novel set of tonal substitutions that relates chords and notes in a flat mediant (third scale degree) manner, and also chorus and notes related to the tritone, this tonal characteristic is normally regarded as being the farthest from the tonic.

To explain the axis system, group the 12 chromatic tones into three sets, where each set contains notes found at intervals of a minor third and a tritone (i.e. the notes that make up a diminished seventh chord). These sets are called axes, the three axes are referred to as the tonic, subdominant, and the dominant. The tones in the axis system relate to one another by the interval of a tritone as part of a brach (C/F#, A/Eb), so each of the axes contains two branches; the principal and secondary branch. The members of each branch are referred to as either poles or counter poles. In other words, each tone is a member of a four note axis, and each branch consists of a pole and a counter pole.

The axes group together substitutable key areas, and assign strength and appropriateness of their inter-substitutability. The counter poles that form on the branch of an axis are more closely related than the counter poles of the other branch of that axis. Intersubstitutability within a branch is a stronger relationship than between the two branches of an axis. However, each axis possesses a two-fold affinity, one being a relationship between pole and counter pole the other being between principal branch and secondary branch. It is important to remember that a pole is always interchangeable with its counter pole without any change in its function.

The particular axes should not be considered as chords of the diminished seventh, but as the functional relationships of four different tonalities. A good comparison to this would be the major-minor relations of classical music (i.e. CM and Am, EbM and Cm).

For example, the axis system is none other than recognizing the fact that the tonic A and Eb, not only have C as a common relative, but also the F#. Likewise the common relative between the subdominant D and Ab is not only F, but it is also B. The following diagram illustrate this concept:

The opposite poles-counter poles (C and F#) are more directly attached than the relative keys of classical harmony. The following diagram shows how each of the poles are related to their respective counter pole:

The counter pole tension becomes the most important and fundamental constructional principle in Bartók’s work.

Tonal Reflections

In the axis system, the same function can be attained by a major second step in one direction or a minor second step in the other direction:

An extension to this rule is that by moving a major second, fourth, minor sixth, or major seventh interval in one direction, or by a minor second, major third, fifth, or minor seventh in the other direction, you will arrive at the same function in the axis system.

Relationship of Three Functions

Dividing the circle of fifths into three equal parts, for example, C-E-Ab, in the sense of the tonic, dominant, and subdominant, it turns out that the tonal system created by the equal division of the circle of fifths matches the model of the axis system. The following diagram represents this relationship:

The following represents the relationship between the pole and counter pole with respect to dimension:

where T = Tonic, D = Dominant, and S = Subdominant. Now that we have covered the basics of the axis system, the next step is to compose a piece of music. Lets start with a progression that looks like this: ii – V – I, where ii = supertonic V = dominant, and I = tonic. We will let I = C major, then V = G major, and ii = D minor. However, from what we have learned above, we can substitute G major for Db major (tritone substitution) and retain the same functionality. We now have a progression that looks like this: Dm – GM – CM, and along the way we will substitute the GM7 with DbM7, turning the progression into Dm – DbM – C. As can be observed, this create a downward movement of thirds and sevenths, which you will distinctly hear in the piece we have composed. We are also going to give this piece the minimalistic feel and with a little added flair of a retrograde-inversion (highlighted in green below) which is set in slightly before going into the final ii – V – I progression and tritone substitution (highlighted in blue below). The following represents our piece, appropriately named, Cartesian V:

Binary numbers allow for the representation of any number using only two digits, 0 and 1. This piece of music is derived from taking a set of integers, in this case, multiples of 6, and converting these integers into their corresponding binary representation. For each binary number the 1′s are summed to create an integer which represents its binary counterpart. For example, the binary number for 12 is, 1100, then this number is summed to the integer 2. The next step is to generate a corresponding note for each summed integer.

There are many ways to match the number with a musical note. In this case we take a D major scale and let 1 = D, 2 = E, 3 = F#, 4 = G, 5 = A, 6 = B, and 7 = C#. Notice, in our D major scale, we did not go higher than the integer 7. To accommodate for numbers greater than 7, the use of mod (7) is incorporated into the program. For example, if the 1′s summed for the binary number is 8, then 1 ≡ 8 mod (7) , and this binary number would be assigned the note D. The following diagram demonstrates the first 200 binary numbers represented as summed 1′s ranging from 1 to 7:

Interestingly, it seems that for the first 200 numbers generated by intervals of 6, the summed binary value of 3 (i.e. F#) dominates. The following represent the distribution of the 200 binary numbers as they are produced from 6 on up to 1200 (i.e. 6*200):

This graph is quite informative, it tells us the envelope of the piece of music.

The note duration is calculated by calculating if adjacent notes are the same. For example if the binary sequences are producing two E’s in a row then we will assign the E with a 1/8 note duration. Here is the corresponding piece called, Machine Language V:

Please enjoy listening to this piece. There is quite a bit you can do to modify this piece of music, for instance, it would be interesting to add some chord structure into this envelope to build up a more complex harmony.

Transformational music theory involves the study of how musical objects, such as chords and rhythm, can transform from one to another. The music theorist Hugo Riemann (1849 – 1919) studied transformations of chords through operators that formed a group, these operators were defined by their voice leading properties. Voice leading is essentially how individual notes move from one chord to another. The Riemann group is generated by maximally smooth voice leading operations on the set class of major and minor triads. However, this breaks down and is not easily generalized to include all set classes. In 2002 Julian Hook published a seminal on what are called, Uniform Triadic Transformations (UTT), which is an algebraic structure for transformational theory with specific operations for major and minor triads (1). We will be creating a chord progression using this theory and notation.

To start, we will define a triad as an ordered pair: Δ ≡ (r, σ) where:

r ≡ Root of the triad expressed as an integer

σ ≡ Sign of the integer which represents its mode; + for major, – for minor

For example, consider the UTT’s U = <+, 4, 7> and W = <-, 5, 10>, and calculate the product UW =

Δ = (0, +) which represents a C major triad

Δ = (5, -) which represents an F minor triad

The set of all 24 major and minor triads form a group called an abelian group, which has the property of being commutative. In other words, applying this group operation on two group elements does not depend on their order. For the set of 24 triads a group multiplication is defined by:

Therefore, the interval from (0, +) to (5, -) is (5, -). Any UTT is completely defined by the following parameters:

t^{+} ≡ Transposition level for the major triad

t^{-} ≡ Transposition level for the minor triad

σ ≡ The sign of the transposition, positive implies no change in mode, negative implies switching the opposite mode

σ^{+} = -σ^{-}

Now you can determine any UTT, by the ordered triplet:

U = <σ, ,t^{+}, t^{-}>

For example, let U = <+, 4, 7> and W = <-, 5, 10>, then calculate the product UW when U and W acts on a C major triad:

What this says is that U transformed a C major triad to a E major triad, this is accomplished by adding 0 + 4 to get 4, and multiplying + times + to get +, resulting in (4, +). Then for W transformed the E major triad to an A minor triad.

We will now create a two-dimensional network of pitch classes, where each dimension is generated by a different interval. Let the x-axis = <-, 1, 2> and the y-axis = <-, 7, 7>. The following represents the partial layout of the grid:

We will now construct the gris with the given composite UTT definitions:

<+, a, b><+, c, d> = <+, a+c, b+d>

<-, a, b><+, c, d> = <-, a+d, b+c>

<-, a, b><-, c, d> = <+, a+d, b+d>

When traveling in the negative X or Y direction, the UTT multiplication get a bit more tricky. The inverse of a mode preserving UTT <+, t^{+}, t^{-}> is <+, -t^{+}, -t^{-}>, and the inverse of a mode preserving UTT <-, t^{+}, t^{-}> is <-, -t^{-}, -t^{+}>. The following represent the construction of the gris in the X direction, where the blue UTT indicates the grid point:

and here is the diagram for the Y direction:

Now, we can label the grid with these UTT’s, which are represented by the light blue nodes:

This will be out triadic transformational map, we will be sampling these blue points to create our composition. The dark blue nodes represent connecting points, and are very important in creating our progression. We will start out at our initial reference point <+, 0, 0>, and act on (+,0). We will set 0 = D major. The following maps out all dotter states:

Starting at (0,+), or D major, then transform to G major, then onto F major, etc… The following diagram represents the dark blue connecting points:

We now have all the information necessary for creating our composition. The composition is constructed by stepping through the UTTs with steps corresponding to the first few Fibonacci numbers: 1, 1, 2, 3.

where the large red circles represent the first steps of the Fibonacci sequence; 1, 1, 2, 3 steps, and the large green circles represent the second step of Fibonacci numbers, in 3,2,1,1 order. Another path is also created, which contains the same Fibonacci sequence, except in the counterclockwise manner.

The piece is split into two time signatures. The first time signature described above and a second time signature describes in the following diagram:

The following represents the composition called, Sinusoidal V, resulting from mapping out the path generated by the UTT’s, where the blue notes depict a node in the UTT map:

Please enjoy studying and listening to this piece of music.

1. Hook, Julian. 2002. “Uniform Triadic Transformations.” Journal of Music Theory 46: 57–126.

Here is a short piece called, Fractionals V, which obeys a power law with respect to note duration, and hence, could be considered a fractal piece of music. For details on how to create this type of music please see my post dated February 25, 2014 titled, Using the Properties of Fractals to Compose Music.

I’m not sure if they knew it at the time, but composers such as Johann Sebastian Bach and Ludwig van Beethoven composed music notes that obeyed the principals of fractals, and in particular, a power law function. Hsü et al (1) reported in 1990 that log-log linear plots can be used to determine if a particular piece of music obeys the power law, and suggesting a fractal nature to music composition. When discussing the properties of fractals, it is the dimension of the object that reveals the existence of a fractal. In general, fractals are characterized by three concepts:

Self similarity

Response to measure of scale

Recursive subdivision of space

Self Similarity: A self similar object is exactly or similar to part of itself, essentially the whole object has the same shape as its smaller parts.

Response to measure of scale: One concept of self similarity is scale invariance, which states that at any magnification there is a smaller piece of the object that is the same as the whole object. For example, the Thue-Morse sequence exhibits scale invariance.

Recursive subdivision of space: Known as a finite subdivision rule, which is a recursive way (i.e. a process of repeating items in a self similar way) of dividing a polygon or a two-dimensional shape into smaller and smaller pieces. Subdivision rules can be applied to the study of hyperbolic surfaces, this is important to me because I think of the layout of notes in a composition as a hyperbolic surface. The patterns exhibited by quasicrystals represent another example of a hyperbolic structure. Also, the Fibonacci sequence, which is defined by the following recursive relationship:

F_{n} = F_{n-1} + F_{n-2}

which translate to the following infinite sequence:

0, 1, 1, 2, 3, 5, 8, 13, …

has fractal properties. It turns out that the Fibonacci sequence is related to many other interesting quantities, such as the golden ratio, Pascal’s triangle, and right triangles, just to name a few.

Now, onward to our discussion of fractal dimension. Fractal dimension is defined as:

D = -log_{10}N/log_{10}(1/R)

N ≡ number of segments created when dividing an object

R ≡ length of each segment

this is called the Hausdorff dimension. The Hausdorff dimension is related to the power law:

N = R^{-D}

As a fundamental starting point to appreciate the dimension of fractals, lets calculate the dimensions of other shapes in Euclidean space. We will use examples from one, two, and three dimensional space. To start with, lets calculate the dimension (D) of a line. All that is necessary is to figure out how many parts to divide (N) the line into and the scale (R) of the line. The following digram explains how the line is segmented up and the dimension calculated:

It is no surprise that the dimension of a line is equal to 1. Next, lets calculate the dimension of a square and a cube. The flowing diagrams provide for a visual representation of how these two dimensions are calculated:

Now, we will show you why fractals are called fractals. The following diagram explains how the dimension of a Koch snowflake is calculated:

As you can see, the dimension is a fraction, hence, fractal. Finally, lets create a piece of music and use the power law to calculate if the structure of the composition can potentially be called fractal. We will use the note duration as a criteria for producing fractal music. To do this we will create 1/16, 1/8, 1/4, and 1/2 notes, then design a pattern of ratios of the notes that obey the power law. The following table explains how many notes are used for each duration:

To see if these values do indeed obey the power law, a log-log plot is created from these values, and if this plot shows a straight line with a fractional slope then the music created has the criteria met for obeying the power law of duration. The following is a representation of the log-log plot:

and as can be seen, the points map out a straight line. Therefore, with a straight line as a result we assume that this particular piece of music satisfies a power law distribution. However, the idea that this pieces of music, based solely on the note duration, has a fractal structure could be premature. This is due to the length of the piece and how evenly distributed the notes are. By inspection, this piece of music may not contain the correct distribution of notes where any given subregion would contain the same distribution as the whole (i.e. self similarity). The power law property can also be applied to other aspects of music, such as pitch, but I will leave this for another post. Here is the piece of music called, Power V, which satisfies the power law for note duration:

Please enjoy listening to this piece.

1. Hsü & Hsü (1990) Fractal Geometry of Music (Physics of Melody) Proc. Natl. Acad. Sci. Vol. 87 pp. 938-941.

The Thue-Morse sequence is a binary sequence and was first used by Eugene Prouhet in 1851 by applying this sequence to the study of integers. However, it was Axel Thue in 1906 that used this sequence to create the field of study called, “combinatorics on words”. In 1921 Marston Morse applied the sequence to differential geometry. Morse used this sequence to prove that geodesics (mapping of straight lines to curved space) are recurrent but non-periodic on certain surfaces of negative curvature (i.e. hyperboloid). Here, the sequence will be used to construct a rhythm pattern for composing music. The Thue-Morse sequence can be generated in several ways. A very simple way of generating the sequence is to let 0 -> 01 and let 1 -> 10. The first eight digits of the sequence can be built up in the following manner:

0 -> 01 -> 0110 -> 01101001

this is just taking the initial state 0 -> 01 and replacing the 0 with 01 and and replacing the 1 with 10 to get 0110, and so forth. Other ways to generate the sequence is with using bitwise negation, this is an operation that yields the bitwise complement of the operand. Every bit has an opposite. For example the bit 0 has the complement (opposite) equal to 1. Applying this technique to the above operator 0 -> 01, complement of 01 is 10, this is then added to the sequence to create 0110, moving forward, the complement of 0110 is 1001, and this is added to the sequence to create 01101001, which yields the same result as above. Remember that 0 and 1 could represent anything. For example, 0 could represent a word such as “hello” and 1 could represent another word such as “goodbye”.

The Thue-Morse sequence has some unique properties. The sequence is self-similar as represented in the following diagram:

as shown above in the diagram, the 0′s and 1′s are replaced with colored square to get an idea of how this sequence can be visualized. The property of self-similarity is apparent, by striking out every other number the original sequence is produced. The self-similarity property of this sequence leads to its fractal nature. Another interesting property of the Thue-Morse sequence is that it contains no cube sequences. For example, the sequences 000 or 111 will never be found. However, the square sequences such as 00 or 11 are present. Now we will focus on creating a rhythmic structure for composing music.

Lets choose a trichord as a first example, we will use a C major chord; CEG. Then we will apply the Thue-Morse sequence as the rhythm pattern for performing arpeggios of the this chord. If we let 0 = 1/16 notes and 1 = 1/4 notes, and every time a square sequence is encounters, such as 00, or 1/16 1/16, we will add these two together to get an 1/8 note. The following diagram represent the rhythm of just the single not C:

this is interesting to look as an educational tool and can by itself produce a pleasant sound. Moving onto the trichord, here is the representation of this same rhythm patter applied to the major chord CEG:

This is starting to look more like a piece of music. Now lets try something slightly more complicated. We will use the diminished chord, Cdim7, and apply this rhythm sequence then at the end of the piece the diminished chord will resolved to the dominant form. Here is the rhythm pattern applied to Cdim7:

and there you have it. There are many ways to use the Thue-Morse sequence as applied to music composition, one could imagine that instead of using the sequence for rhythm, it would be possible to substitute notes for 0 and 1, or maybe short squares could dictate what note is used.

Other properties of the Thue-Morse sequence are very interesting. For example, this sequence can be used to construct the Koch snowflake. The following diagram expelling how this is accomplished:

One could imagine a composition where the Thue-Morse sequence is used in combination with the Koch snowflake pattern to produce a piece of music. Stay tuned for my next blog entry.

Here, I apply the structure of the Cantor ternary set to act as the main building block for the composition I call: Cantor V. V for victory! Cantor V is composed for solo violin. I chose a particular set class, 3-12 [048], essentially an augmented triad, which has an interval class vector of <000300>, which states that the major third is the only interval this symmetric set contains. A 12-tone-matrix is built up by using the general serial techniques for constructing the prime, retrograde, inversion, and retrograde inversion. This piece has build into it a soft dissonance due to the major third. However, the way in which the prime form is constructed, there are minor seconds, such as G-G# and F-F#, which I use to create tension and from the tonal structure of the major thirds. Groups of rows are chosen to construct the Cantor set, I chose rows Po, Ro, RIo, and RI9. I can build up major chords, such as B-D-F when constructing the Cantor set, then I can use the sharper dissonant intervals, such as G-G# and F-# to create a tense environment. I hope this doesn’t make you too up tight.

When listening to this piece, you will hear an overall atonal character, eventually resolving in a quasi-tonal manner. The following diagram represents the 12-tone matrix:

The following rows, except for RI2, were used to construct the Cantor set for Cantor V:

The next diagram shoes how I use each row from the 12-tone matrix to construct the details of this piece: