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In this section you will find my compositions that use elements of the twelve-tone technique, and other interesting methods for composition.

List of Compositions

  • Wave Propagation
  • Plasma Galaxy
  • Glaciation
  • Ninth Automation
  • Correspondence
  • Spin Echo
  • Simple Intervals
  • Local Minimum
  • Celestial Navigation
  • Cyclic Vertices
  • Platinum Resonance
  • One Point Six
  • tile

    First Three Compositions

    The following three compositions have been derived using the twelve-tone technique. These particular pieces use concepts from the twelve-tone technique to derive motifs for particular sections and at the same time loosely retain a focus around a particular key signature. These are some of the first pieces that I composed:

    Wave Propagation pdf
    Plasma Galaxy pdf
    Glaciation pdf

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    The Ninth Automation

    I have always liked the sound of ninths, for some reason I have an affinity for dissonance and ninths have always attracted my attention. I thought it would be interesting to come up with a piece that concentrated on major ninths, but toggles back and forth from minor to major to augmented ninths. Here is an example using ninths in combination with Rule 30 (a one-dimensional binary cellular automaton) and the twelve-tone technique. Rule 30 produces complex patterns from simple rules. In this example one-dimensional arrays will be calculated using the properties cellular automaton and only two states is considered. The two states are represented as the directions: up or down (1 semitone up and 1 semitone down, respectively). At specific time intervals each cell takes on a new state based on its current state and the state of its two neighbors along a one-dimensional array. Rule 30 represents a set of rules as to how these transformations take place. The following table describes the steps taken by Rule 30. Let down = 0 and up = 1.

    Tonal Pattern111110101100011010001000
    New Tonal State00011110

    For visualization of the states and pattern of tones: Let 0 = grey and 1 = blue. The following is a representation of how the state changes take place.

    Applying Rule 30 the following pattern starts to emerge, and after 12 cycles of automation the pattern looks like this.

    CA1
    CA2
    CA3
    CA4
    CA5
    CA6
    CA7
    CA8
    CA9
    CA10
    CA11
    CA12

    Start with an initial condition of a twelve-tone row populated with major ninths:

    P0 = C D E F# G# A# C# D# F G A B (0 2 4 6 8 10 1 3 5 7 9 11)

    Calculating the 12 primes, 12 retrogrades, 12 inversions, and 12 retrograde inversions, the following matrix is created:

    02468101357911
    10024681113579
    81002469111357
    68100247911135
    46810025791113
    24681003579111
    11135790246810
    91113571002468
    79111358100246
    57911136810024
    35791114681002
    13579112468100

    notice there are no invariant forms. The next step is to choose some interesting forms. Lets choose P5 and the inversion, I5.

    P5 = 5 7 9 11 1 3 6 8 10 0 2 4
    I5 = 5 3 1 11 9 7 4 2 0 10 8 6

    For a set class with "x" number of pitches, if any number n appears x times in the body of that set's normal matrix, then T(n)I will contain the same notes as the original set. For P5 and I5 have x = 12, it turns out that every number appears 12 times, meaning that T(j)I where j = 0, 11 should consist of the same pitches as the original. This property is referred to as combinatoriality.

    Now lets apply these two rows to the pattern we created with Rule 30. If we take P5 = 5 7 9 11 1 3 6 8 10 0 2 4 and assume that 5 to 7 ( or 9 to 11, etc...) represents a major ninth (14 semitones or an octave plus a second), then every blue square will generate an augmented ninth (or interval with 15 semitones) and every grey square will generate a minor ninth (13 semitones).

    Here is how it works. The following represents the first row of the pattern that was generated using cellular automation with Rule 30.

    CA1

    The series P5 original (rows 1 and 3) + I5 original (rows 2 and 4) look like this:

    P5 Original

    and after the first row of automation using Rule 30 has been applied to the first interval of each measure of P5 original + I5 original, and hence, transforming the original piece to P5 automated + I5 automated. The following represent the new rows of music, the modifications in the interval structure are shown in blue and grey:

    P5 + I5 automated

    Here is the entire piece, "Ninth Automation", where all 12 rows of cellular automation under Rule 30 have been applied to the four rows of P5 original + I5 original, respectively, creating 48 rows of musical pleasure.

    Ninth Automation pdf

    The P5 + I5 melody should be used as a foundation for building a more detailed harmonic structure. Or, you could just improvise around the foundation of the ninths, etc...

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    Correspondence

    This is a piece that is based on a simple set of notes where:

    P0 = 9 2 10 2 (i.e. A D A# D).

    Everything else in this piece is a propagation of this simple set. You see, it is not necessary to follow the exact strict rules of the twelve-tone technique, where you define P0 as twelve different tones. In fact, the twelve-tone technique should be used in a way as to guide you in a direction, the rest is up to your creative brain to decide how the composition is going to turn out. However, it is important to point out that you will have an invariant set of results if you only define a subset of the twelve tones.

    Correspondence pdf

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    Spin Echo

    This is a piece that uses some properties of the twelve-tone technique and minimalistic techniques to derive a piece that has more-or-less of a minimalistic feel to the sound. Using rotational arrays (Igor Stravinsky was know for using this technique) in combination with a form of phasing the following piece, Spin Echo, can be derived. The resulted composition can have an intense focus on melody using a relatively small group of notes.

    Lets take, for example, an augmented seventh chord; which consists of a root, major third, augmented fifth, and a minor seventh. The following represents the augmented seventh chord, the Abaug7:

    Abaug7: Ab C E Gb

    Applying rotations to this sequence of notes yield the following rows of tones:

    R0AbCEGboperation
    R1AbCDE4 semi-tones down
    R2AbBbCE4 semi-tones up
    R3AbBbDGb2 semi-tones up

    To explain what is happening here, the second line of the array takes the first line and rotates it to start on the second note (C E Gb Ab) then transposes it down 4 semi-tones to start on Ab, the same idea is used to generate lines three and four. The rows are all related by transposition. Each time the array rotates the first set of notes and transposes the rotation (or spin) to start on Ab. Therefore, each row describes the same order of sequence of intervals, allowing for wraparound, but begins one note earlier than the row directly above it. The result is similar to contrapuntal compositional styles where a melody with one or more imitations of the melody are played after a given duration, this is what is known as a canon.

    Now, lets take a look at applying the twelve-tone technique to the notes of Abaug7. The following is the twelve-tone matrix for this particular chord:

    8046
    4802
    04810
    10268

    or in other notation

    AbCEGb
    EAbCD
    CEAbBb
    BbDGbAb

    Notice that the matrix reveals an interesting property, that of rotation. If you look at the diagonal of the matrix, where Ab is populated along the following matrix elements:

    M11, M22, M33, and M44

    To complete the elements of each row, for example to produce R2, the third row:

    M31, M32, M33, and M34

    then rotate to the following:

    M33, M34, M31, and M32

    and hence, R0, R1, R2, and R3 are generated by applying this procedure to all the rows.

    An entire composition can be put together starting with only one chord, Abaug7, simply by cycling through the rows and applying some well-thoughtout creativity:

    Spin Echo pdf

    Once again, a strict adherence of the twelve-tone technique has been avoided.

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    Simple Intervals

    I believe there is a balance between atonal and tonal music. In this composition, different primes are derived using the twelve-tone technique, and using transposition of primes that possess similarity with one another keeps a more tonal feel to this piece.

    Starting with an initial row of:

    P0 = C D E F G D

    and calculating all other primes forms, in addition to P0 I decided to include use P7. I also derived another row, lets call it P'0, and it is represented by:

    P'0 = A G F E F D

    Once all primes were calculated for P'0, I decided to include P'4. Why did I end up choosing P'4? Well, I'm glad you asked, that is an excellent question. It turns out that P0 = C D E F G D produces the following additional , and distinct, prime sets:

    P5 = F G A A# C G
    P7 = G A B C D A
    P8 = G# A# C C# D# A#
    P10 = A# C D D# F C

    Next, I looked at each set to determine its Hamming distance with respect to P0. The Hamming distance is a measure of the distance between elements of a set (i.e. two strings, or two words, etc...) where the elements of the set are of equal length, it is the number of positions at which the corresponding symbols are different, or put another way, it measures the minimum number of substitutions required to change one string into the other. To calculate the Hamming distance the following equation is used:

    dHAD(i,j) = n-1k=0 [yi,k ≠ yj,k]

    where dHAD is the Hamming distance between the objects i and j. k represents the index of the respective variable reading y out of the total number of variables n. The Hamming distance, dHAD, gives the number of mismatches between the variables paired by k. For example, If I take two words "road" and "toad", there exists only one letter substitution to turn "road" in to "toad", that is, to substitute the "r" into a "t", a Hamming distance equal to 1. Likewise, if I take two strings of letters (or notes), such as P0 and P5, there are six substitutions necessary to turn P0 into P5.

    It turns out that all the primes generated (i.e. P5, P7, P8, and P10) from P0 all have a Hamming distance equal to 6. I then decided to take the row that did not contain any sharps, and hence, trying to stay more like to the original sequence, P0. Note, since all primes have retain the same interval structure, and only the notes (or frequencies) are different, then I ended up taking the least varied row by choosing P7.

    Simple Intervals pdf

    You will notice that I have connected each prime motif with runs of scales or variations of smaller motifs.

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    Local Minimum

    This piece of music samples areas of different octaves using a minimum of notes, and makes use of transposition of augmented sixths. Lets take for example a particular augmented sixth chord. To calculate the augmented sixth chord, we will begin by stating a particular key, for example C major. To calculate an augmented sixth chord, first determine the dominant note, and in this case that note would be a G. Then calculate the two leading tones (a note that resolve to a note one semitone higher or lower, a cornerstone of counterpoint) with respect to G, those would be an F# and and Ab, or in other words a sharped fourth scale degree or a flatted sixth scale degree with respect to the key (i.e. C major), to complete the chord structure all you have to do is add the tonic, and in this case that would be a C. So, the augmented sixth chord for C major is Ab C F#, this is also known as the ItalianAugmented sixth in C major.

    The Italian sixth (i.e. dominant seventh with no fifth) will usually resolve to a dominant chord, this is done by resolving the leading tones to the dominant, then completing the resolved chord by adding the two appropriate tones, in this case the dominant is a G and the two added tones are B and D. Here is the augmented sixth:

    Ab C F#

    and resolves to

    G D G

    This composition will take the augmented sixth and all twelve transpositions and slowly resolve the augmented sixths as they flow through sets of octaves.

    Local Minimum pdf

    This piece turned out to sound ethereal, make sure to play this piece very slowly and with a lot of connection, or flow, between the notes.

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    Celestial Navigation

    The use of a fifth, D to A, and augmented sixths are used to create a piece with a slight dissonance being created next to the fifth by adding a semi-tone in the appropriate locations, and consonance created by resolving augmented sixth chords. The inspiration of this piece came from thinking about the universe, and in particular the star cluster Pleiades. For some reason I associate this type of tune with objects in the universe. Here is a nice image of Pleiades for your own personal inspiration:

    Please note that the augmented sixths used in this piece are created from two different scales, A major and D major, but I just see it as an application of the use of the chromatic scale splashed with a bit of atonality.

    Celestial Navigation pdf

    Play this piece with intensity.

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    Cyclic Vertices

    I wanted to use perfect fourths and perfect fifths against augmented fourths to create a piece that sounds a little dissonant, to me it is like a spooky fog layer over the ocean. The structural property of a circulant matrix, which is derived from a Toeplitz matrix, is used to give this piece a cyclic scaffolding. One property of a circulant matrix is the descending diagonals of the matrix are constant. The following representation of a circulant matrix is employed:

    312
    231
    123

    The following chord vectors are used, where Mi,j are the respective matrix elements:

    Mi,jChord Vector
    1BbEb
    2BE
    3CF# G C#
    1'C#G#
    2'D A
    3'D# A A# E
    1''E A
    2''F B
    3''F# C C#G
    1'''G C
    2'''G# C#
    3'''A D# E A#

    Notice that the first element of each chord vector (CV) marches through the chromatic scale, starting on CV1 = Bb and ending on CV1''' = A. Each row of the circulant matrix represents one row of music. One a set of Mi,j's have transversed the rows of the matrix (i.e. CV1, CV2, and CV3), then the next set of Mi,j's are applied until all chord vectors have been used.

    Cyclic Vertices pdf

    Note that this piece ends on an augmented 4th for a truly unresolved sound.

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    Platinum Resonance

    This composition uses a sequence of numbers that can be interpreted as musical intervals: Fibonacci Numbers and the Golden Ratio. Composers such as Chopin, Mozart, Beethoven, and Bartok have been known to use these numbers and their properties of symmetry to compose music. I will derive the set of Fibonacci numbers, then create the Pisano Period (the period where the sequence of Fibonacci numbers, modulo n, repeats). I will use this sequence of numbers to create the intervals between pitch frequencies to derive a composition.

    The sequence of Fibonacci numbers is defined as follows:

    Fn = Fn-1 + Fn-2

    where F0 = 0 and F1 = 1. So, the first few numbers of this sequence are as follows:

    Fn = 0,1 ,1 ,2, 3, 5, 8, 13, 21, 34, 55, 89, ...

    There are many interesting properties of Fibonacci numbers. One very interesting property of this sequence is that if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 13/8 = 1.6250, which is an approximation of the Golden Ratio. I will concentrate on a certain property of the Fibonacci sequence that involves transforming the sequence, modulo n, to obtain a periodic structure that will be used in creating intervals. This periodic sequence is called the Pisano Period. For example: for n = 3, that is modulo 3, the above Fibonacci numbers translate to:

    P3 = 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, ...

    It can been seen that a repeating pattern starts to emerge: 0, 1, 1, 2, 0, 2, 2, 1. It will be this periodic sequence that will define a set of intervals to be used in construction of a composition.

    Platinum Resonance pdf

    Play this piece with speed and energy.

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    One Point Six

    The observation of a particular ratio, the Golden Ratio, in nature (and duplicated by the human) is not just coincidental. The Golden Ratio is equal to 1.618... and is used frequently in geometry. Phidias (circa 450 BC) the great painter and sculptor used the golden ratio in his work. Fibonacci (1170–1250) created the numerical series now named after him; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically. The golden ratio is used in architecture (Parthenon's facade), painting (Leonardo da Vinci's illustrations of polyhedra in De divina proportione), book design, finance (trading algorithms), music (Bartok, Chopin, Wolfe, etc...), Egyptian pyramids, and observed in nature (atomic scale in the magnetic resonance of spins in cobalt niobate crystals).

    I thought it might be interesting to relate the Golden Ratio to the tonal frequency, not just the ratio of the intervals between notes or regions of a motif in a piece. The following table displays the note and its associated frequency:

    Scientific NameFrequency (Hz)F*1.618 (Hz)Golden Note
    G3195.998317.125D#4
    G#3207.652335.980E4
    A3220.000355.960F4
    A#3233.082377.127 F#4
    B3246.942399.552G4
    C4261.626423.311G#4
    C#4277.183448.482A4
    D4293.665475.150A#4
    D#4311.127503.404B4
    E4329.628533.338C5
    F4349.228565.051C#5
    F#4369.994598.650D5

    Notice that the "Golden Note" does not exactly match the calculation, for example: (195.998 Hz)(1.618) = 317.125, yet D#4 that has a frequency of 311.127, is associated with G3. Since the human ear can not detect a difference of frequency between the interval of about 5 - 10 Hz, then this calculation is still deemed valid.

    The next step is to determine some golden-chords, or chords made of minor sixth intervals, so that I can create some three-note chords to manipulate using the twelve-tone technique. These particular three-note chords will be derived by constraining the intervals betweens tone to be that of the Golden Ratio. The chord is derived by taking the ratio 1.618 and multiplying it by the tone frequency. For example A is associated with F, which in turn is associated with C#. The following table displays these chords:

    Chord
    A F C#
    B G D#
    C G# E
    D A# F#
    E C G#
    F C# A
    G D# B

    Note that every interval is now the Golden Ratio equivalent and can now be used to compose a piece of music. If you take any one of these chords and try to build each chord by continuing the 1.6 rule, it will become apparent that the chord actually has a periodic structure to it. For example, if the chord, A F C#, is to be made into a tetra-chord by adding another tone, it will eventually wrap around on itself: "A" is to "F" and "F" is to "C#", but "C#" is to "A", and "A" is to "F". As you can see, this is the case for all these chords. This is quite interesting, and represents a sort-of symmetry structure to these chords.

    I decided to calculate the primes, inverses, retrogrades, and retrograde inverses from the following set:

    A F C# B G D# C G# E D A# F#

    I choose to use only the prime and retrograde inverse to derive the following piece. There are obviously many variations that can be produced.

    One Point Six pdf

    Play this piece slow and gracefully, yet with some intensity.

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