Tonnetz Sequent#

Eularian Tonnetz gate-driven triad generator


Tonnetz Sequent is a triad generator that maps gate inputs to the triadic transforms of the Eularian Tonnetz allowing one to move through triadic space with rhythm. It outputs three 1 V / octave control voltages to control the pitch of oscillators. It has three gate inputs for the standard Tonnetz transforms: Parallel, Leading, and Relative. There are also inputs to reset the triad to a reference triad, perform semitone transpose and change which triad types form the transform domain (major/minor, augmented/major, diminished/minor).

  • Type: Triad Generator
  • Size: 8HP Eurorack
  • Depth: 0.8 inch
  • Power: 2x5 Eurorack
  • +12 V: 50 mA
  • -12 V: 5 mA


Power connector

To power your Noise Engineering module, turn off your case. Plug one end of your ribbon cable into your power board so that the red stripe on the ribbon cable is aligned to the side that says -12 V and each pin on the power header is plugged into the connector on the ribbon. Make sure no pins are overhanging the connector! If they are, unplug it and realign.

Line up the red stripe on the ribbon cable so that it matches the white stripe and/or -12 V indication on the board and plug in the connector.

Screw your module into your case before powering on the module. You risk bumping the module's PCB against something metallic and damaging it if it's not properly secured when powered on.

You should be good to go if you followed these instructions. Now go make some noise!

A final note. Some modules have other headers -- they may have a different number of pins or may say "not power". In general, unless a manual tells you otherwise, do not connect those to power.


illustration of Tonnetz Sequent's interface

Outputs the pitch (1 V per octave) of the root note of the triad.
Outputs the pitch (1 V per octave) of the middle note of the triad, typically a third above the root.
Outputs the pitch (1 V per octave) of the top note of the triad, typically a fifth above the root.
Parallel Transform. When in major mode, P moves the third down a semitone; when in minor mode, P moves the third up a semitone.
Tone Exchange Transform. When in major mode, L moves the root down a semitone; when in minor, L moves the fifth up a semitone.
Relative Transform. When in major mode, R moves the fifth up a tone; When in minor, R moves the root down a tone.
Mode transform; changes which pair of triad sets that the transforms act upon. It switches between minor/major, major/augmented, augmented/major, major/minor,minor/diminished, diminished/minor and then back to the start.
Home transform; changes the current triad back to the reference triad for the current Mode pair.
Transpose transform. Transforms all notes of the triad one semitone up.
The group control changes the structure of the transforms by adding a dihedral rotation to P, L, and R. The knob and CV input (0 V to 8 V) sum together. When fully counterclockwise PLR are canonical.
When enabled triads are constrained within one octave. Pitches that would fall outside of the octave are mapped to their equivalent tone class within the base octave.
The Gate output will generate a 20 ms pulse every time any transform input occurs (switch or gate). It is essentially a trigger combiner for all gate inputs.
Depressing the Tune button will cause all pitch outputs to be the same as long as it is depressed. This is used to tune connected oscillators.

Patch tutorial#

Pick three oscillators. Connect each pitch out (Root, Third, Fifth) of Tonnetz Sequent to one pitch input of each oscillator. Depress the Tune button and use the pitch controls on the oscillators to adjust their pitches to unison. Turn the Group knob fully counterclockwise.

Use the PLR buttons to explore the Eulerian Tonnetz. Each time you hit a button it will change the triad to a musically related triad.

Gates can be input to control the transforms. Connecting three rhythm outputs, perhaps from a Zularic Repetitor, to the PLR jacks will generate a triadic sequence.

To reset the sequence to the reference triad, hit the H button or trigger the H input.

Dihedral transforms#

This section will detail the model used for computing triadic transforms used in Tonnetz Sequent. Most users will not need to understand any of the following to use the module but it is provided for those who want to know more details of the implementation. Some abstract algebra knowledge is assumed.

A note on notes: Numeric notation is used for note classes. To map this to standard musical notation let the number 0 be the note class A, the number 1 be the note class A# and so on.

The dihedral group is constructed from two integer modulo groups \(X\) and \(M\). \(M\) is the group of two elements. \(X\) is any additive integer modulo group. The operation of the group is defined by simply composing the elements separately: \((X_{0}, M_{0}) * (X_{1}, M_{1}) = (X_{0} * X_{1}, M_{0} * M_{1})\) for all \(X_{0}, X_{1}\) in \(X\) and all \(M_{1}, M_{0}\) in \(M\). A common interpretation of the dihedral group is rotating and flipping a coin with the orientation of the coin being modeled by \(X\) and which side is up being represented by \(M\).

For our discussion \(M\) will represent if the triad is major or minor and \(X\) is the group of order 12 representing the triads by their root note. For example the element \((0, 0)\) is the major triad \(\{0, 4, 7\}\); the element \((1, 1)\) is the minor triad \(\{1, 4, 8\}\).

Using the coin representation our coin might look like this:

Illustration of a dihedral coin

Though it takes a bit of work to verify the three parsimonious transforms P, L and R can be represented with this operation by a single number. P is 0, L is 4 and R is 9. This is quite easy to see with P as P takes a triad and maps it to its relative major or minor. L and R are left as an exercise to the reader.

When you hit P, L or R on Tonnetz Sequent internally it does exactly what was previously described. If the current triad is major than the next triad will be minor and rotated according to the number defining the particular transform. Internally subtraction is used for clockwise rotation, addition for counterclockwise. Major triads are denoted \(0 \to 11\) and minor \(12 \to 23\) so transforming between major and minor is an offset by 12.

The Group knob modifies this transform by adding its position as an offset; for example if the Group knob is at position 2, then the L transform is now \(4 + 2 = 6\).

This same structure is extended to augmented and diminished triads. At any point in time, Tonnetz Sequent's dihedral coin has either major/minor, major/augmented or minor/diminished on each side.

This structure has many different representations; with perhaps the most famous being the Eulerian Tonnetz. In 1739, humanity's greatest mathematician, Leonhard Euler, arranged the major and minor triads as a triangular grid on the surface of a torus in such a way that motion across the edges of the triangles produces the parsimonious transforms P, L and R. This beautiful structure is, of course, where the name of this module comes from.

Tonnetz Sequent generates triads using gates to determine the direction of each step about the surface of this torus.

Design notes#

This module came directly out of my interest in Neo-Riemanian music theory. The first design was based around the Eularian Tonnetz (PLR) but as I read more, my design became more abstract, focusing on universal triadic transforms. At some point, I realized that this would make a much less intuitive module and went back to PLR as a basis and added a very simple way of modifying the PLR to produce other transform sets. Similar transform sets to PLR were designed for the diminished and augmented triads.

The goal of Tonnetz Sequent is to encapsulate triadic transforms into a module that requires no understanding of the math but just allows the user to intuitively explore the Tonnetz.

On the following page are some references that influenced this module. They are in no means required reading to be able to use this module; they are provided for the curious explorers of music theory.



We will repair or replace (at our discretion) any product that we manufactured as long as we are in business and are able to get the parts to do so. We aim to support modules that have been discontinued for as long as possible. This warranty does not apply to normal wear and tear, including art/panel wear, or any products that have been modified, abused, or misused. Our warranty is limited to manufacturing defects.

Warranty repairs/replacements are free. Repairs due to user modification or other damage are charged at an affordable rate. Customers are responsible for the cost of shipping to Noise Engineering for repair.

All returns must be coordinated through Noise Engineering; returns without a Return Authorization will be refused and returned to sender.

Please contact us if you think one of your modules needs a repair.

Special thanks#

  • Leonhard Euler
  • Kris Kaiser
  • Shawn Jimmerson
  • William Mathewson
  • Mickey Bakas
  • Tyler Thompson
  • Alex Anderson