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 twelvetone row populated with major ninths:
P_{0} = 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:
0  2  4  6  8  10  1  3  5  7  9  11 
10  0  2  4  6  8  11  1  3  5  7  9 
8  10  0  2  4  6  9  11  1  3  5  7 
6  8  10  0  2  4  7  9  11  1  3  5 
4  6  8  10  0  2  5  7  9  11  1  3 
2  4  6  8  10  0  3  5  7  9  11  1 
11  1  3  5  7  9  0  2  4  6  8  10 
9  11  1  3  5  7  10  0  2  4  6  8 
7  9  11  1  3  5  8  10  0  2  4  6 
5  7  9  11  1  3  6  8  10  0  2  4 
3  5  7  9  11  1  4  6  8  10  0  2 
1  3  5  7  9  11  2  4  6  8  10  0 
notice there are no invariant forms. The next step is to choose some interesting forms. Lets choose P_{5} and the inversion, I_{5}.
P_{5} = 5 7 9 11 1 3 6 8 10 0 2 4

I_{5} = 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 P_{5} and I_{5} 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 P_{5} = 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.
The series P_{5 original} (rows 1 and 3) + I_{5 original} (rows 2 and 4) look like this:
and after the first row of automation using Rule 30 has been applied to the first interval of each measure of P_{5 original} + I_{5 original}, and hence, transforming the original piece to P_{5 automated} + I_{5 automated}. The following represent the new rows of music, the modifications in the interval structure are shown in blue and grey:
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 P_{5 original} + I_{5 original}, respectively, creating 48 rows of musical pleasure.
The P_{5} + I_{5} 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...
Applying rotations to this sequence of notes yield the following rows of tones:
R_{0}  Ab  C  E  Gb  operation 
R_{1}  Ab  C  D  E  4 semitones down 
R_{2}  Ab  Bb  C  E  4 semitones up 
R_{3}  Ab  Bb  D  Gb  2 semitones 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 semitones 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 twelvetone technique to the notes of Abaug7. The following is the twelvetone matrix for this particular chord:
or in other notation
Ab  C  E  Gb 
E  Ab  C  D 
C  E  Ab  Bb 
Bb  D  Gb  Ab 
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:
M_{11}, M_{22}, M_{33}, and M_{44}

To complete the elements of each row, for example to produce R_{2}, the third row:
M_{31}, M_{32}, M_{33}, and M_{34}

then rotate to the following:
M_{33}, M_{34}, M_{31}, and M_{32}

and hence, R_{0}, R_{1}, R_{2}, and R_{3} 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 wellthoughtout creativity:
Once again, a strict adherence of the twelvetone technique has been avoided.
and calculating all other primes forms, in addition to P_{0} I decided to include use P_{7}. I also derived another row, lets call it
P^{'}_{0}, and it is represented by:
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 P_{0} = C D E F G D produces the following additional , and distinct, prime sets:
P_{5} = F G A A# C G

P_{7} = G A B C D A

P_{8} = G# A# C C# D# A#

P_{10} = A# C D D# F C

Next, I looked at each set to determine its Hamming distance with respect to P_{0}. 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:
d^{HAD}(i,j) = ∑^{n1}_{k=0} [y_{i,k} ≠ y_{j,k}]
where d^{HAD} 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, d^{HAD}, 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 P_{0} and P_{5}, there are six substitutions necessary to turn P_{0} into P_{5}.
It turns out that all the primes generated (i.e. P_{5}, P_{7}, P_{8}, and P_{10}) from P_{0} 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, P_{0}. 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 P_{7}.
You will notice that I have connected each prime motif with runs of scales or variations of smaller motifs.
The following chord vectors are used, where M_{i,j} are the respective matrix elements:
M_{i,j}  Chord Vector 
1  Bb  Eb 
2  B  E 
3  C  F#  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 CV_{1} = Bb and ending on CV_{1'''} = A. Each row of the circulant matrix represents one row of music. One a set of M_{i,j}'s have transversed the rows of the matrix (i.e. CV1, CV2, and CV3), then the next set of M_{i,j}'s are applied until all chord vectors have been used.
Note that this piece ends on an augmented 4^{th} for a truly unresolved sound.
where F_{0} = 0 and F_{1} = 1. So, the first few numbers of this sequence are as follows:
F_{n} = 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:
P_{3} = 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.
Play this piece with speed and energy.
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 goldenchords, or chords made of minor sixth intervals, so that I can create some threenote chords to manipulate using the twelvetone technique. These particular threenote 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 tetrachord 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 sortof 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.
Play this piece slow and gracefully, yet with some intensity.