Tonnetz Chord Builder: Neo-Riemannian Analysis and Progressions

The slide transformation is a third order compound Riemannian transformation involving three moves in the Tönens diagram. This transformation connects triads with a single common note, and the chord quality changes from minor to major or vice versa. The example shows the connection between A flat minor and G major triads.

The slide transformation is the inverse of the parallel transformation. Instead of moving only the third, both the root and fifth move in the same direction by half steps while keeping the third stationary. From C major, moving C and G up a half step each produces C# minor. From C minor, moving C and G down a half step each produces Bb major. This creates a sliding motion between triads rather than the direct parallel motion.

The S transformation involves two voices moving simultaneously rather than one. It relates a major triad and a minor triad that share a common tone, specifically the third of the chord. For example, C major (C-E-G) and C-sharp minor (C#-E-G) share E while the other voices move by semitones (C to C#, G to G#).

Instead of representing chords with multiple points on a one-dimensional circle, mathematicians represent chords with one point on a two-dimensional space called the configuration space. For two-note chords, this is the two-dimensional torus. The space of two-note chords is a Mobius strip with mirror-like properties. The space of three-note chords is a three-dimensional triangular prism with its top edge glued to its bottom edge. The space of four-note chords is a four-dimensional tesseract. In Chopin's E minor Prelude, Chopin moves between notes that form a circle of fifths, represented on this space by a rigorous pattern. Structures that are totally difficult to understand in notation become much more accessible when viewed from a geometrical point of view. This approach reveals deep structures linking different tonal genres from traditional Bach Harmony to romantic trickery to scale manipulations in Shostakovich or Steve Reich to contemporary rock music.

The category of manifolds contains perfectly smooth objects, but this very regularity creates fundamental algebraic problems: limits and quotients typically fail to produce manifolds. This crisis motivates orbifolds as a more flexible framework that retains much of the manifold formalism while allowing better behavior under operations like taking quotients. An orbifold chart consists of an open Euclidean set, a finite group acting smoothly and effectively, and an invariant map to the underlying space whose quotient is a homeomorphism. Charts are compatible via embeddings respecting group homomorphisms. Manifolds are trivial orbifolds with trivial local groups. Quotients of manifolds by properly discontinuous Lie group actions yield orbifolds called 'good' or 'developable.' The simplest non-manifold orbifold is a cone: quotient of the plane by cyclic rotation group, featuring a single singular vertex lacking a well-defined tangent space. Musical chords provide a beautiful application: modeling tones as frequencies on a circle, chords as points in tori Tⁿ, and quotienting by symmetric groups yields orbifolds where Möbius strips represent two-note chords and Möbius prisms represent three-note chords, with singularities encoding repeated-note configurations.

Dr. Dmitri Tymoczko explains how core concepts in music theory can be translated into contemporary geometric language. He introduces the "OPTIC transformations" — five types of musical transformations that musicians use to form equivalence classes of musical objects, such as chord, chord type, chord progression, voice leading, and pitch class. These equivalence classes are represented as points in singular quotient spaces known as orbifolds. For instance, two-note chords are modeled on a Möbius strip, where the boundary functions as a mirror, reflecting symmetries in musical structure. Four-note chord types are represented on a cone over the real projective plane, illustrating higher-dimensional musical relationships. The geometric representation reveals structural constraints that govern musical style and specific compositions. By visualizing music in these spaces, listeners and analysts can better understand how musical elements relate across transformations. The talk emphasizes the utility of interactive 3D computer models that allow simultaneous visualization and auditory experience of music, making abstract theoretical concepts tangible. This approach bridges music theory and geometry without requiring prior musical expertise, offering a novel framework for analyzing musical patterns through spatial reasoning.

All possible harmonies can be visualized as points in a 3D space (chord space), where chords that take fewer steps to reach each other are clustered together, while those requiring many steps are further apart. This geometric representation allows musicians to understand relationships between chords and find efficient paths between harmonic states, similar to assembly theory in chemistry.

There exists a mathematical isomorphism between the repetition group (transformations for large musical chunks) and the voice leading group (chord-to-chord moves). This corresponds to symmetry groups and fundamental groups in topology—every repetition is essentially a chord progression. Geometrically, music can be represented through two-dimensional configuration spaces for chords and scales, revealing self-similar motion across time scales. Beethoven's Ninth and McCoy Tyner's jazz show remarkably similar geometric patterns—systematic navigation through chordal and scalar domains at multiple levels simultaneously. The quality of music derives primarily from systemic quality (the hierarchical framework) rather than individual taste. A 25-line algorithm can generate structures that five years of training achieves. Voice leading can be translated into sequences through multiplication techniques.

Generative music is created by establishing rules that systems follow to produce music, using Ableton Live's built-in devices like probability settings (chance), arpeggiators, LFOs, drum racks, and samplers to create evolving, unpredictable musical patterns where events may or may not happen, and sounds can be randomized and modulated to create unique compositions each time they are played.

Spacewalk integrates with major DAWs through specific routing methods: in Ableton Live, place it on a MIDI track and route to a synthesizer; in Logic Pro, add it as a MIDI effect; in Bitwig, place it before plugins. Users select keys (G minor, C major, etc.) and scales (major, Dorian, Phrygian, Lydian, minor, harmonic minor, melodic minor) to establish harmonic foundations. Progression length ranges from 8 to 128 bars, with chord duration controls determining how long each chord lasts. The 'new pattern' function generates fresh progressions while preserving settings, while 'variation' completely reconfigures all parameters. Generated MIDI can be dragged directly into compositions for further refinement.

This segment covers the complete workflow of using AI music tools within a Digital Audio Workstation. Chord Gio generates chord progressions based on user input and selected musical styles. Superstition provides randomization features for infinite sound presets, audio processing effects like looping and reversal, and BPM synchronization. Users can load custom samples and manipulate individual sampler sections. The workflow demonstrates how AI tools integrate with any DAW, allowing musicians to drag and drop generated content directly into their production environment while maintaining full creative control.

Self-generative music is music generated by computer systems using MIDI devices that create endless, non-repeating musical compositions through randomization of notes, velocity, and other parameters, combined with scale plugins to ensure musical coherence. In Ableton Live, this is achieved using MIDI devices like random plugins, LFO devices for velocity and length variation, chord generators, and scale plugins that correct random notes to fit a specific musical scale (such as D minor), while audio effects like delay, auto pan, and filters add depth and movement to the generative output.

This comprehensive segment covers Devin Townsend's core philosophy on music creation and essential DAW workflow fundamentals. He emphasizes that learning comes through failure and trial and error, advising musicians to 'go out and fail' because failure is where learning begins. He discusses maintaining passion through casual practice rather than disciplined obligation, explaining that playing while watching TV keeps instruments interesting. His vision-based approach means having clear objectives and avoiding ego-driven conflicts. The segment covers DAW fundamentals including setting tempo in Pro Tools by playing a section and letting the software detect BPM, and his preference for metronomes without accents for flexibility across time signatures. He demonstrates drum programming using Superior Drummer, creating custom kits with specific sounds and selecting loops from different presets. He demonstrates multi-track recording with his template setup including click, drum software, guitar tracks, bass, clean track, ambient presets, 12-string, acoustic, fuzz, echo, piano, and vocal tracks. He explains that presets serve as creative inspiration and help hear the final sound immediately, preventing the tendency to say 'we'll fix it later.' The key principle is that harmony vocals add depth and richness to arrangements, and recording them separately allows for precise alignment and processing.