Using Uniform Triadic Transformations to Map Out Chord Progressions

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:

(t1, σ1)(t2, σ2) = (t1 + t2, σ1σ2)

σ is multiplied as follows:

+ + = +

+ – = –

– + = –

– – = +

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:

UTT_progression_diagrams_UW

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:

UTT_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:

UTT_progression_diagrams_Xaxis

and here is the diagram for the Y direction:

UTT_progression_diagrams_Yaxis

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

UTT_progression_diagrams_utt_map

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:

UTT_progression_map

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:

UTT_progression_diagrams_dkblue

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.

UTT_progression_diagrams_UTT_green

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:

UTT_progression_diagrams_UTT_map_2

 As you can see, it is apparent that UTTs can be effectively used to generate very interesting chord progressions.

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:

UTT_v2

      UTT_audio

Please enjoy studying and listening to this piece of music.

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

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