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# Triad Tonnetz and Riemannian Transformations

## The Tonnetz diagram and Neo-Riemannian Theory

This page contains a link to a PDF document with *Tonnetz diagram templates*, which you may use for chord progression designs. Also there is a companion *Riemannian Transformations booklet* to the YouTube video tutorial series, and a set of *full orchestral scores* for the longer examples in the series.

Tonnetz Diagram >> | Riemannian Transformations >> | Tonnetz templates >> | Video tutorials >>

### Introduction to the Tonnetz

In **Neo-Riemannian Theory** (NRT) we may use the **Tonnetz Diagram** as a graphical tool for analysing and designing chord movement. The diagram has pitch-class labels from the chromatic scale at the nodes. Sets of three connected notes yield major and minor triads.

The example Tonnetz shown in the figure is centered around the *C major* triad. Neighbouring triads have two overlapping, common pitches, a property used in the design of triad chord progressions. Since the chromatic scale is periodic through octave transposition, there is redundancy in the diagram and we observe double (multiple) occurrence of the triads.

### Riemannian Transformations

In triadic chord progressions through the Tonnetz we use *Riemannian Transformations*. There are three **basic** transformations: *{P,R,L}*. The *P* transformation connects a triad with the *parallel* minor or major chord, e.g., going from *CM* (*M*: major triad) to *Cm*, or in the reverse direction. The *R*-transformation is the connection with the *relative* minor triad, such as in *CM-Am* (or the inverse *Am-CM*). The *L*-transformation is called the *leading tone exchange*, since when connecting *CM-Em* the latter triad contains the leading tone *b* from the *C* major diatonic 7-pitch scale.

More interesting connections are made through **compound Riemannian Transformations**, the pairwise combination of the basic transformations into *{PR,RP,PL,LP}*. These are shown in the Tonnetz diagram when starting from a major triad. The two connected triads are known as **chromatic mediants** since they do not occur in diatonic harmony. Used by classical music composers such as Schubert and Wagner, these 'operatic' chord connections imply a **narrative value** with corresponding listener emotional response, and therefore have become popular in contemporary (epic) film music.

### The Tonnetz Diagram templates

If you would like to compose music cues with Riemannian transformations, use my free template diagrams. The document with the **Tonnetz Diagram** templates may be found here:
Open the Tonnetz Diagram templates PDF document >> (PDF, landscape, 3 pages)

### Riemannian Transformations video tutorials

On my YouTube channel I uploaded a 4-episode series of video tutorials on **Riemannian Transformations** in the Tonnetz. These explain the fundamentals, all *basic* and *compound transformations* and the corresponding *root cycle* in the **Schillinger diatonic and symmetric harmony system**. All transformations are demonstrated with short composition examples.

Here are the links to the YouTube video tutorials:

- Riemannian Transformations: Part 1 Schillinger Caught in the Tonnetz (26:59) presents the three basic and the four compound transformations with composition examples.
- Riemannian Transformations: Part 2 Five Paths to Domination (18:06). Simple and compound Riemannian transformations applied to a Schillinger Dominant Pedal Point chord progression framework.
- Riemannian Transformations: Part 3 "Order, order, order!" (26:08). Third order compound transformations.
- Riemannian Transformations: Part 4 The Long and Winding Tonnetz Threads (20:17). Longer chord progressions with various transformation types.

Additional documents:

- Booklet: Riemannian Transformations and the Tonnetz Diagram >> (PDF, A4, 28 pages). Tonnetz diagrams, transformation types, chord progression and condensed score examples.
- Full score: Riemannian Transformations Part 2 Dominant Pedal Point >> (PDF, A3, 10 pages). Orchestral setting for Path 1: C to D with basic L- and R-transformations only. Path 2: From C to D with basic and compound transformations through Bb. Path 3: From C to D with mixed transformations through Cm. Path 4: From C to D with compound transformations through F#/Gb. Path 5: From C to D with compound transformatiosn through Cb/B.
- Full score: Riemannian Transformations Part 3 Third Order Compound Progressions >> (PDF, A3, 10 pages). Orchestral setting for chord progression in C major, chord progression in C minor.
- Full score: Riemannian Transformations: Longer chord progression in E major >> (PDF, A3, 7 pages). Suspense cue for woodwinds, brass, low strings and keyboards.
- Full score: Riemannian Transformations: Longer chord progression in D major >> (PDF, A3, 12 pages). Laendler dance for woodwinds, brass, strings and rhythm section.