Learn harmony by singing and hearing it
An experiential course through W. A. Mathieu's Harmonic Experience β plus playable labs for every diagram.
Guided lessons walk the book's path: sing sa with a drone, feel ratios as rhythm, build the five-limit lattice tone by tone, then hear what equal temperament hides β the commas. Every map from the book is also a clickable instrument.
Mode du jour
Mathieu's daily prescription: one mode, sung slowly over the drone, every day.
What's new
New here?
What is this site?
An experiential school for W. A. Mathieu's book Harmonic Experience: you learn harmony by singing over a drone and hearing pure ratios, not by memorizing rules. Guided lessons live under Learn; free-play instruments live under Labs.
Do I need the book?
It helps β every lesson cites its chapter and pages β but the lessons paraphrase the teaching and all practices work on their own.
Do I need to read music?
No. Everything here is taught by ear, by ratio, and by map. The one staff-notation view is optional.
Do I need to sing well?
No β learning to sing in tune is the practice itself, and the microphone tuner gives you honest feedback. No microphone? Every exercise also works by ear.
Diagram coverage
0 source diagrams, tables, maps, and compendia are inventoried. Each entry opens a playable model: lattice, mode, comma, cadence, cyclic sequence, key-chain, or chromatic superimposition.
Make sound
Start audio once, then click tones, ratios, chords, diagrams, cadence steps, and composer cells. The same synth engine is shared across the whole site.
Create
The composer turns selected diagram material into a 16-step sketch. Export a JSON sketch or copy a note/chord sequence for later work.
The experiential course
Learn: Harmonic Experience, chapter by chapter
Guided lessons paraphrase the book's path and turn each practice into sound: sing over the drone (with a live microphone tuner), feel ratios as cross-rhythms, grow the lattice tone by tone, compare just and tempered tunings, and hear the commas work. Start audio once in the footer, then begin.
All non-notation diagram objects found or sampled
Diagram Atlas
The atlas includes the Mathieu book's own most-referenced diagram index plus OCR-sampled chromatic-jazz diagram families from the Liebman PDF. Use the filters, then open any entry to play and transform it.
Staff notation with lattice location
Staff + Lattice View
Choose a book-derived pitch diagram, read it on a standard staff, and play it while the corresponding lattice tones or chords light up.
Single-note lattice control
Melodic Lattice Lab
Use the lattice as a single-note instrument. Cursor and mouse movement select one lattice tone at a time for melodies, scales, and interval paths.
Chordal lattice control
Harmonic Lattice Lab
Use the lattice as a chord instrument. Cursor and mouse movement select a lattice position and sound the chord generated from that location.
Major/minor triangle lattice
Triadic Lattice Lab
Use the chapter 21 note lattice where each connected triangular face is a major or minor triad. Click or arrow on a root for major; hold Shift for minor.
Modes as lattice selections
Mode Workshop
Compare church modes, mixed modes, Magic Mode, and chromatic-pair variants as sets selected from the lattice. Clicking a degree plays it; playing a mode traces its tone path.
Mode matrix β upper tetrachord over lower
Functional motion
Cadence and Sequence Lab
Cadences occur on the triadic lattice. Each chord lights its root and triangle; playback steps through the cadence as functional motion across the lattice.
Liebman-style chromaticism
Chromatic Jazz Lab
Use superimposition, side-slipping, scale-quality substitution, intervallic denial, tonal anchors, and voicing structures to pull away from and return to a key center.
Liebman structures on Mathieu's map
Voicing Lattice Lab
Pick a chromatic voicing structure, then click lattice roots to plant it anywhere on the map. Play it parallel (exact transposition, Liebman's planing) or adjusted to a mode of the tonic β the mode chosen as an upper tetrachord over a lower tetrachord, Mathieu's chapter 12 construction.
Play in just intonation
Tuned Keyboard Lab
An onscreen keyboard tuned by the lattice: in Just mode every key sounds its five-limit ratio over the current tonic; in Equal mode it plays the ordinary piano pitches. Use your computer keys (Aβ; row), click, touch, or a MIDI keyboard.
Harmony as geography over time
Analysis Lab
Positional analysis, the book's capstone: watch real phrases move across the lattice on a timeline. Below it, build key chains by a chosen ratio between tonal centers, and visit the septimal corner where the blues lives.
Positional analysis player
The folding plane
Key-chain builder
The septimal corner β a third dimension
A practice that knows you
Practice Room
Daily guided sessions, adaptive ear training that targets your weak degrees, sung phrases scored by the tuner, melodic dictation, and a blind comma-hearing test. Progress feeds the lattice heatmap below.
Make music from the maps
Composer
Build a 16- or 32-step sketch from diagrams, modes, cadences, or chromatic lines. Click a cell to fill it from the selected diagram; Shift-click cycles its length (1, 2, or Β½ beats); Alt-click clears it to a rest. Swing delays the off-beats. Playback runs in the browser synth, and MIDI export carries just-intonation pitch bends.
Look anything up, then play it
Reference: Ratios, Commas, and Glossary
Every tone of the extended lattice with its ratio, cents, and deviation from equal temperament β click a column header to sort, click a row to hear it over the tonic. Below: the four commas, and the course glossary.
Index of ratios
The commas
Glossary of singable tones
Glossary
How to use this site
Help
Everything here serves one practice: learning harmony by singing it and hearing it, following W. A. Mathieu's Harmonic Experience. This page explains every part of the site. Jump with the buttons, or read top to bottom.
Quick start
Audio unlocks on your first click anywhere. Then:
- Find your tonic. On Home, press Hum to find my tonic and hum your easiest note β the site tunes its drone to your voice. (Or pick a tonic by hand in any lab; C, B, or A suit most voices.)
- Begin the course. Learn opens the guided lessons; the three-minute first lesson samples everything. Lessons are meant to be sung, not read β leave the drone on.
- Return daily. The Practice page builds a short session for you; the π button in the footer shows what's due.
Sound & the footer
The footer is the site's mixing desk, present on every page:
- Start audio β a manual fallback; normally the first click anywhere unlocks sound.
- Tempo β beats per minute for cadences, sequences, the composer, and arpeggios.
- Sound β the melody instrument. Warm tone, Vocal ahh, and Soft e-piano are synthesized; piano, violin, oboe, French horn and the leads are sampled. Sing-along targets always use the vocal voice regardless.
- Tuning β just ratios, equal temperament, quarter-comma meantone, or Werckmeister III, applied site-wide (see Tuning systems below).
- π Music β reference pitch (A4), drone timbre (including the plucked tanpura), a separate chord instrument, and the drone / notes mix sliders.
- π Progress β weekly review, spaced-repetition reviews due, and custom phrase drills.
- β Learning β microphone sensitivity and check, singing octave, and the reset / restore controls.
- β Record output β records everything the site plays (drone included) and downloads it as an audio file. Press again to stop; auto-stops at three minutes.
The course (Learn)
About forty lessons follow the book's four parts, plus a Part Five on Liebman's chromatic approach and a starred checkpoint closing each part. Every lesson cites its chapter and pages.
- Sing cards β the heart of the course. Drone + target plays the tone to match; Drone only leaves space for your voice; Check my voice starts the live tuner (needle, scrolling pitch history, optional overtone spectrum, and a recordable ten-second take with a pitch trace).
- Rhythm trainers β feel a ratio as a looping cross-rhythm before hearing it as pitch. The tap loop runs until you stop it.
- Comparisons & comma demos β A/B buttons glow as they sound; "together" plays both tones at once so you hear the beating. These always use exact ratios, whatever the tuning switch says.
- Comma pumps β chord journeys tuned purely from chord to chord; compare the first and last tonic to hear the drift.
- Quizzes β the drone sounds, a degree plays, you name it. Questions adapt toward your weak degrees and feed the Practice heatmap.
- Mark as practiced β the checkmark drives your progress lattice, resume point, and part stars. Completed lessons offer review mode, which hides the prose and keeps the practices.
- Sharing β every lesson is a link (
#learn/lesson-id), and the π on each activity copies a link straight to that sound.
Practice Room
- Today's practice β a five-step guided session: sa, the mode du jour, your weakest degree, an ear check drawn from spaced-repetition due reviews, and a journal entry (optionally with an eight-second audio take).
- Your ear on the map β a heatmap of per-degree accuracy built from every quiz and dictation answer.
- Breath-counted sa β the book's eleven-breaths practice with a counter and drone.
- Sight-singing β a sargam phrase appears; the microphone advances you note by note as each locks. Load your own phrases from the π panel.
- Duet β hold an interval against a walking root, or hold sa against a moving voice; the tuner's target moves with the music.
- Melodic dictation β hear four notes, answer by tapping degrees.
- Transposition drill β random tonics: name the degree you hear, or the letter a degree lands on.
- Name it four ways β drill the four naming systems against each other: ratio, sargam, solfege, interval name.
- Comma JND β a blind which-was-pure test that narrows the cents gap as you succeed; thirds, fifths, and sixths each keep a best.
- Loop layer β record eight seconds of your voice, loop it, sing the other voice over yourself.
The labs
- Diagram Atlas β every non-notation figure from both books as a playable model. Pick an entry, press Play diagram (path numbers light in order), Invert path reverses it, or send it to a lattice lab or the composer.
- Staff + Lattice β read book examples in four voices while their tones light on the lattice.
- Melodic / Harmonic / Triadic lattices β click to play; arrow keys walk the map (Ctrl skips two; Shift on the triadic lattice selects minor). The status strip shows each tone's ratio and your path's drift from equal temperament. Extras: a pedal-tone hold on any degree, saved and shareable journeys, the piano strip showing what temperament folds together, and (melodic lab) Track my voice β your sung pitch as a live dot on the map. Wheel-zoom and drag to pan; double-click resets.
- Modes β church and mixed modes as lattice selections, plus the 5Γ5 tetrachord matrix: every mode is a lower tetrachord plus an upper one.
- Cadences β cadence models stepping across the triadic lattice with animated voice-leading lines; common tones get hold-rings.
- Chromatic Jazz β Liebman's devices: superimposition, side-slipping, anchors, with a tension slider.
- Voicing Lattice β plant a voicing structure (tertian, quartal, "So What", clusterβ¦) on any root; play it parallel or conformed to a tetrachord mode; chain voicings into a looping sequence. Member tones light on the map.
- Tuned Keyboard β keys sustain while held (mouse, the Aβ; computer row, or MIDI in) and sound in the active tuning; MIDI out sends tuned notes with pitch bends to hardware.
- Analysis β phrases moving across the lattice on a timeline, the folding-plane animation of the comma pump, a key-chain builder with drift readout, and the 3-D septimal corner.
- Composer β a 16/32-step sketch. Click a cell to fill it from the selected Atlas diagram; Shift-click cycles its length; Alt-click makes a rest; swing delays off-beats. Save sketches to a library, share them as links, or export MIDI with just-intonation pitch bends.
- βΆ Performance mode β on the instrument labs: fullscreen, chrome-free, for live playing.
Tuning systems
- Just ratios β every tone tuned by exact whole-number ratios from the tonic; pure, locked, position-dependent. The course's home base.
- Equal tempered β the modern piano: twelve identical semitones, every fifth 2Β’ narrow, every major third 14Β’ wide.
- Quarter-comma meantone β the Renaissance compromise: pure major thirds, gently narrowed fifths, and one unusable wolf fifth.
- Werckmeister III β a Bach-era well-temperament: all keys usable, each with its own color.
The switch retunes the keyboard, lattice chord playback, mode matrix, and course material. Comparisons, comma demos, and the sing-tuner's targets always use exact ratios β the gaps are the lesson.
Microphone features
All microphone audio is analyzed locally in your browser β never recorded or uploaded (your own journal takes and loop-layer recordings stay on your device). Octave doesn't matter: sing any register and the tuner folds it.
- Run Mic check (β Learning) once: hum for three seconds and it reports your level and how well pitch locks, with advice.
- Microphone sensitivity sets the gate: High for a quiet voice, Low for a noisy room.
- Singing octave sets where targets sound (around C3, C4, or C5) β match it to your voice.
- No microphone? Every exercise also works by ear against the drone.
Progress & data
Everything is stored in your browser (localStorage): lesson checkmarks, quiz statistics, spaced-repetition schedule, journal, settings, saved journeys and sketches.
- Export / Import progress (Learn sidebar) moves it between browsers as a JSON file.
- Reset lesson progress (β Learning) stashes an automatic backup first; Restore last backup undoes it. Reset user experience only replays the welcome and lab tours.
- The site is an installable PWA and works offline after the first visit.
Keyboard & mouse controls
- A S D F G H J K L ; (with W E T Y U O P for the black keys) β play the piano row anywhere; in the Tuned Keyboard lab, keys sustain until released.
- Arrow keys β walk the lattice in the lattice labs (diagonals work); Ctrl+arrow skips two; Shift selects minor on the triadic lattice.
- Mouse wheel / drag / double-click β zoom, pan, and reset the lattice maps.
- Shift-click / Alt-click β in the composer: cycle a cell's length / clear it to a rest.
- Esc β closes the search overlay and confirmation dialogs; leaving fullscreen exits performance mode.
Troubleshooting
- No sound? Click anywhere once (browsers require a gesture), check the footer says audio is ready, and check the Tuning/Sound selects. The Notes & chords slider in π Music may be low.
- Tuner won't hear me? Run Mic check; allow microphone access in the browser's site permissions; raise sensitivity; sing longer, steadier notes.
- Drone too loud / quiet? The Drone level slider in π Music; each play button balances against it.
- MIDI not appearing? Press Enable MIDI in/out in the Tuned Keyboard lab and allow the permission; for MIDI out, set your synth's pitch-bend range to Β±2 semitones.
- Something looks broken? Reload β your progress is saved. A β notice in the footer means an error was caught; reloading clears it.
Local review notes
Sources and Coverage
This site is grounded in the two PDFs present in the folder on May 11, 2026. The source text is paraphrased and operationalized rather than reproduced.
The course
The Learn section is a guided, experiential walk through the book's four parts β 29 lessons that paraphrase its teaching prose (with chapter and page citations) and turn each of its practices into interactive sound β plus a five-lesson mini-course on Liebman's chromatic approach. The microphone tuner checks sung tones against just-intonation targets, records ten-second takes with a pitch trace, and runs interval-singing drills from random pitches.
Harmonic Experience.pdf
Text-native PDF with 582 pages and a detailed outline. The diagram atlas uses the book's own index of most-referenced examples, figures, and tables, including lattices, comma families, cadence models, key-chain diagrams, and cyclic-sequence summaries.
A chromatic approach to jazz harmony and melody.pdf
Image-scanned PDF with 166 pages. OCR samples identified the table of contents, chromatic-superimposition chapters, voicing and line compendia, complex-chord analysis, and practical improvisation devices.
Coverage rule
Staff notation examples are not reproduced. Non-notation diagrams and diagram-like models are represented as interactive systems: rhythm ratios, tone lattices, chord lattices, mode charts, comma maps, cadence paths, key chains, chromatic devices, voicing structures, and line-analysis templates.
Instrument samples
Piano, violin, oboe, French horn, square lead, calliope lead, and voice lead samples are local MusyngKite General MIDI soundfont exports from MIDI.js Soundfonts, released under Creative Commons Attribution 3.0. The default Warm tone, Soft e-piano, and Vocal ahh voices are synthesized in the browser (custom wavetables and a formant filter bank); sing-along targets always use the vocal voice so the reference is easy to match.