Workshop 82 — Symmetrical Harmony and Harmonic Geometry: Diminished Cycles, Augmented Triads, Whole-Tone Motion, Major-Third Systems, and Escaping Ordinary Tonal Gravity
Workshop 81 explored polychords and bitonal hearing: two harmonic identities can coexist if their relationship is organized clearly.
Now we move into another harmonic world where ordinary tonal gravity becomes weaker:
symmetrical harmony.
In ordinary major/minor harmony, intervals are unequal.
That inequality creates hierarchy:
- tonic feels special
- dominant points somewhere
- leading tone has a destination
- some scale degrees feel stable and others unstable
But certain harmonic structures repeat the same interval again and again.
Examples:
- diminished seventh chord: minor thirds
- augmented triad: major thirds
- whole-tone scale: whole steps
- major-third key cycles
- tritone-related structures
These systems are fascinating because:
symmetry weakens the sense that one note must be the root or one direction must be “forward.”
That makes them excellent for:
- transitions
- modulation
- suspense
- dreamlike harmony
- altered dominants
- dramatic reharmonization
- modern improvisation
- escaping one tonal center and arriving somewhere unexpected
The central lesson is:
Functional harmony creates direction through inequality; symmetrical harmony creates possibility by making several directions equally plausible.
82.1Start with the ordinary major triadC major:82.2Now augmented triadC augmented:82.3What is strange about it?The notes:82.4First principleSymmetry creates root ambiguity.82.5Play C augmentedC-E-G♯.82.6Level 1 — augmented as dominant colorTake:82.7Augmented chord inside dominant functionG-B-D♯.82.8Level 2 — augmented triad as passing chordC major:82.9PlayC:82.10Lines first againDo not think:82.11Level 3 — C → C+ → FPlay:82.12Hear the augmented chord as instability between stable chordsC:82.13Level 4 — augmented as modulation hingeBecause C+ can be reinterpreted around:82.14This is the first geometric ideaOrdinary dominant harmony gives a strong arrow:82.15Level 5 — augmented triad major-third cyclePlay:82.16Count the octave12 semitones.82.17Level 6 — major triads a major third apartPlay:82.18Listen to common tonesC major: C-E-G.82.19Major-third root movement sounds distant yet connectedWhy?82.20Level 7 — chromatic mediantsChords whose roots are a third apart but whose quality/key relation is chromatic are often discussed as:82.21Important“Chromatic mediant” is a broad descriptive category.82.22Level 8 — C→E majorC: C-E-G.82.23Level 9 — C→A♭ majorC: C-E-G.82.24Chromatic mediant lessonHarmonic distance and voice-leading distance are not the same thing.82.25Level 10 — C→E♭ majorC: C-E-G.82.26Level 11 — C→A majorC: C-E-G.82.27Practice chromatic mediants as voice leading, not giant jumpsLH roots may leap a third.82.28Level 12 — build a four-chord cinematic cycleTry:82.29Now diminished symmetryTake:82.3012 ÷ 3 = 4Minor third = 3 semitones.82.31Level 13 — inversion ambiguityB-D-F-A♭.82.32This gives diminished harmony enormous modulatory potentialEach member can act like:82.33Level 14 — resolve each diminished member upwardPlay B°7.82.34Enharmonic reinterpretationThe same notes can be spelled differently to show different function.82.35Level 15 — B°7→CClassic leading-tone diminished:82.36Level 16 — same pitches toward E♭D is now the leading tone.82.37This is why diminished chords are excellent transition devicesTheir internal structure does not strongly privilege one root.82.38Level 17 — passing diminished revisitedC → C♯°7 → Dm.82.39Level 18 — descending diminishedC/E perhaps → E♭°7 → Dm or other voice-led destinations.82.40Level 19 — dominant ♭9 connectionG7♭9:82.41Root + diminished structureG + B°7 = G7♭9.82.42Level 20 — one diminished chord, four dominant rootsTake pitch set:82.43Build the three diminished familiesBecause a diminished seventh repeats every minor third, there are only three distinct pitch collections before transposition repeats…82.44Why only three?Transpose diminished seventh by:82.45Level 21 — diminished cycle exercisePlay:82.46Level 22 — diminished chord planingMove diminished shape by semitone:82.47Whole-tone scaleNow another symmetrical system.82.48There are only two whole-tone collectionsStart C:82.49Level 23 — play whole-tone scale slowlyC-D-E-F♯-G♯-A♯-C.82.50Symmetry weakens gravity againMajor scale contains:82.51Level 24 — augmented triads inside whole toneC-E-G♯.82.52Level 25 — whole tone over dominant ♯5G7♯5.82.53Compare altered and whole-tone dominantG altered:82.54Altered sounds denser/chromaticWhole-tone often sounds:82.55Level 26 — G7♯5→CPlay:82.56Level 27 — whole-tone melodic phraseOver G7♯5:82.57Symmetrical tension + tonal resolutionOne of the most useful formulas:82.58Level 28 — augmented chord passing between dominant colorsG7:82.59Level 29 — whole-tone sequenceTake motif:82.60DangerWhole-tone sequences can quickly sound like:82.61Level 30 — augmented cellMotif:82.62Now major-third harmonic cyclesWe encountered them in repertoire with Giant Steps.82.63Level 31 — C major-third cycleUse:82.64Level 32 — dominant preparation for each centerInstead of direct major triads:82.65This is an important paradoxYou can combine:82.66Level 33 — two layers of functionWhen playing a major-third cycle:82.67Level 34 — common-tone melody across major-third rootsTry high:82.68Level 35 — chromatic mediant bass, static topTry top:82.69Symmetry does not mean all voices need symmetrical motionThe roots can follow geometric cycles while upper voices use:82.70Level 36 — tritone symmetryThe tritone divides octave exactly in half.82.71G7 vs D♭7 revisitedG7 guide tones:82.72Level 37 — alternate dominant roots around same shellRH:82.73Level 38 — dominant root oscillationBefore C:82.74Level 39 — tritone axis in bassC pedal/home.82.75Symmetrical systems often overlapOne passage may simultaneously involve:82.76Level 40 — diminished dominant chainKeep one diminished RH shape.82.77Level 41 — symmetry as transitionSuppose:82.78Level 42 — tonality → symmetry → new tonalityThis is a powerful large-scale form:82.79Level 43 — symmetry → tonalityReverse.82.80Asymmetry can be resolutionAfter whole-tone scale, one simple major scale half-step:82.81Level 44 — hide tonic through symmetryC-centered piece.82.82Level 45 — diminished suspenseHold diminished chord.82.83Level 46 — diminished false endingEnd phrase:82.84Level 47 — augmented false endingTonic C.82.85Level 48 — symmetrical bass sequenceBass:82.86Level 49 — symmetrical melody sequenceTake motif:82.87Level 50 — diminished melodic sequenceMotif:82.88Symmetrical improvisation dangerBecause structures repeat perfectly, they are easy to sequence mechanically.82.89Level 51 — break symmetry deliberatelyPlay three whole-tone fragments.82.90Level 52 — asymmetrical rhythm over symmetrical pitchesUse whole-tone pitch collection.82.91Level 53 — symmetrical harmony, diatonic melodyHold a familiar melody note over:82.92Level 54 — diatonic harmony, symmetrical melodic excursionOpposite.82.93Level 55 — symmetrical dominant climaxAt major cadence:82.94Compare symmetrical dominant familiesOver G:82.95Practice Loop A — augmented transformationsPlay:82.96Practice Loop B — augmented major-third cycleC+.82.97Practice Loop C — diminished destinationsPlay one °7 chord.82.98Practice Loop D — three diminished familiesMemorize/play:82.99Practice Loop E — whole-tone dominantG whole tone.82.100Practice Loop F — tritone shellRH:82.101Practice Loop G — chromatic mediantsC→E.82.102Improvisation Challenge 1 — symmetry for exactly one barInside a tonal solo:82.103Challenge 2 — diminished connector onlyCreate 8 bars in C.82.104Challenge 3 — augmented modulationBegin C.82.105Challenge 4 — major-third tonal cycleWrite 12 bars:82.106Challenge 5 — whole-tone vs diminished dominantCreate two versions of:82.107Challenge 6 — symmetry then one simple cadenceWrite 8 bars of ambiguous/symmetrical harmony.82.108Ear Drill — augmented root ambiguityPlay:82.109Ear Drill — diminished root ambiguityPlay:82.110Ear Drill — whole-tone gravityPlay C whole-tone for several bars.82.111Ear Drill — chromatic mediant smoothnessPlay:82.112Ear Drill — tritone substitutionRH B-F.82.113Mini-composition exerciseWrite a 32-bar piece that begins tonally, enters symmetrical harmonic space, and returns to tonality.82.114Apply it to “Happy Birthday”Use the melody clearly first.82.115Apply it to “Mack the Knife”For a late solo or transition:82.116Apply it to “The Bare Necessities”This is perfect for humorous exaggeration.82.117The larger theoretical lessonWe have now encountered several kinds of harmonic organization:82.118The important question changesInstead of always asking:82.119Symmetry and uncertaintyA dominant seventh is asymmetrical enough to strongly suggest:82.120Symmetry and arrangingUse symmetrical structures when you want:82.121The mature improvisational mindsetDo not think:82.122The deepest connectionFunctional harmony says:•Workshop 82 mini-test1. Why are augmented triads harmonically ambiguous?
← PreviousWorkshop 81 — Polychords and Bitonal Hearing: Two Harmonic Worlds at Once, Upper Structure vs. True Polychord, Controlled Clashes, and Making “Wrong” Chords Sound IntentionalNext →Workshop 83 — Voice-Leading Reharmonization: Building New Chords from Melodic Lines Instead of Substitution Lists