2. Motion & Forbidden Parallels

Chapter 1 classified single sonorities. This chapter adds time: how two voices move between sonorities. The page describes the validator's conservative perfect-motion policy; historical direct-fifth and direct-octave teaching often focuses on the outer voices and may allow a stepwise soprano arrival, while contrary perfect motions are treated separately (Open Music Theory).

The four types of relative motion

Between any two consecutive sonorities, a voice pair moves in one of four ways:

Motion	Definition	Risk level
Contrary	The voices move in opposite directions.	Safest — independence is audible.
Oblique	One voice holds while the other moves.	Safe — the held voice anchors the pair.
Similar	Both move in the same direction by different amounts.	Risky into perfect intervals (hidden parallels).
Parallel	Both move in the same direction while keeping the interval.	Fine for imperfect consonances; forbidden for perfects.

✓ Safe motionContrary and oblique motion

Contrary and oblique motion are the safest ways to approach any interval — even a perfect one. Independence is audible because the voices do different things. Most parallel-rule violations disappear when one voice simply moves the other way.

First the voices move apart in opposite directions, then one holds while the other moves.

✓ Allowed patternParallel sixths are fine

The parallel prohibition applies only to perfect intervals. Chains of parallel thirds, sixths, and tenths are idiomatic Baroque writing — each sonority is rich enough that the voices keep their identity even while moving together.

Both voices move in parallel, but the repeated interval is imperfect.

Bach's Goldberg Variations 12 and 15 are canons by inversion, where the second voice is the first voice upside down. They are repertoire examples of contrary motion, not a universal rule for every passage:

✓ From BachBach: Goldberg Variations — canons in contrary motion

the same mirror at two intervals —

Variation 12 of the Goldberg Variations (BWV 988) is a canon by inversion: the lower voice repeats the upper voice's line one bar later with every interval flipped — where the leader steps up, the follower steps down, mirror-exact for the whole variation. This excerpt shows how inversion produces contrary motion in this canon; it does not make contrary motion a universal rule in Bach's writing. In bar 2 the leader climbs to its peak while the follower descends beneath it. (The bass of the ground, which runs under both voices, is omitted here.)

The follower replays the leader upside down: every step up answered by a step down, one bar later.

Forbidden perfect parallels

Parallel fifths and octaves make two voices sound like a single doubled line — the classic loss of voice independence. Play the two examples below and listen for it: the octave case in particular collapses into one thick voice.

Why exactly fifths and octaves?

Perfect intervals blend so strongly (chapter 1) that when the same perfect interval is repeated by motion in both voices, the ear groups the two lines into one. A single fifth is fine; a fifth moving to another fifth is the violation.

✕ ForbiddenParallel fifths

The red verticals show the repeated perfect fifth. The arrows show that neither voice is stationary, so this is not oblique motion.

parallel_fifthBoth voices move up, and the red vertical interval remains P5 at both arrivals.

✕ ForbiddenParallel octaves

The octave frame is repeated by motion in both voices, collapsing two lines into one doubled contour.

parallel_octaveBoth voices move in the same direction while the interval remains an octave.

Implementation map:

Rule	Fires when
parallel_fifth	Two voices move in the same direction and the previous and current vertical intervals are perfect fifths.
parallel_octave	Two voices move in the same direction and the previous and current intervals are unisons or octaves.
Oblique escape	Oblique and static motion are allowed even on repeated perfect intervals — only both voices moving triggers the rule.
Cadence exemption	Notes committed as part of a declared cadence cell are exempt.
Source handling	During Generation, an all-fixed Material/Ornament pair is recorded as an exemption; FinalScore audits authored material again and the form's vertical budget decides whether observed findings close the form.

What is a "cadence cell"?

The engine writes the closing gesture of a phrase as a pre-built unit and stamps its notes CadenceCellCommitted. The strict, hidden, and contrary perfect-motion checks on this page skip a pair when the note sounding as the current lower voice carries that bit. Melodic interval and consecutive-leap loops skip a marked current note, while the weak non-chord-tone walk skips a marked current note or the next cadence note. The dedicated cadence_voice_leading rule (chapter 5) judges the cadence as a whole instead; there is no blanket exemption for every validator rule.

Contrary perfect motions

The validator keeps two contrary-motion cases separate from strict same-direction parallels. anti_parallel_perfect fires when both voices move in opposite directions and repeat the same perfect interval class — a unison, fifth, or octave — across the two sonorities. battuta is narrower: contrary motion arrives at a unison from a non-unison interval, and the voice that is upper at the arrival leaps downward by more than a major second. A stepwise arrival or an upper voice that rises is outside this battuta predicate.

Both rules are engine contracts. They are evaluated at the same union onsets as the other pair-motion checks, then routed through the Generation/FinalScore source handling and the form's vertical budget.

Hidden (direct) fifths and octaves

A hidden, or direct, perfect interval occurs when voices move in the same direction into a perfect fifth or octave. The parallel never literally happens — it is implied by the similar-motion arrival, which makes the hollow interval jump out of the texture. In this engine, the arrival is classified as hidden only when the upper voice leaps by more than a major second (more than 2 semitones); a stepwise upper voice is allowed by this particular predicate.

Hidden means "arrived by similar motion"

In a hidden fifth, the first vertical interval is not the same perfect fifth. The previous interval can be another class — for example, a fifth moving to an octave is classified as a hidden octave when the upper voice makes the required leap. The engine applies this rule to every voice pair and every union onset, a deliberately conservative scope.

✕ Forbidden arrivalHidden fifth

The first sonority is imperfect; the second is the red perfect fifth. Similar motion makes the perfect arrival sound exposed.

hidden_parallel_fifthSimilar motion moves from an imperfect interval directly into P5.

✕ Forbidden arrivalHidden octave

Landing on an octave by similar motion makes the empty interval jump out of the texture. The engine checks every voice pair at every onset, with the same exemptions as the hidden fifth: cadence cells and both-material pairs.

hidden_parallel_octaveBoth voices move upward into a perfect octave by similar motion.

Rule	Scope
hidden_parallel_fifth	Similar motion lands on a perfect fifth, the previous perfect class differs, and the upper voice leaps more than 2 semitones. Cadence cells are skipped; source handling follows the Generation/FinalScore lifecycle above.
hidden_parallel_octave	Similar motion lands on a unison or octave under the same previous-class and upper-leap conditions. The engine checks every voice pair at every union onset, including weak positions.

Two scoping terms used in this chapter

A structural accent is a Strong or Medium position in the meter: the downbeat, beat 3 in 4/4, each dotted pulse in compound meter, or beat 2 in a Sarabande (primer). An upper-voice pair is a pair of adjacent voices that does not include the lowest voice: in a three-voice texture, the top two. The bass is excluded from invertible_at_octave because that check protects material whose upper voices may later exchange registers.

Voice crossing and spacing

Vertical writing must remain readable as separate voices even before any interval question arises. Two layout rules guarantee it:

✕ ForbiddenVoice crossing

Voice order is part of the texture contract. Crossing makes the lines ambiguous.

voice_crossingThe lower voice rises above the upper voice at the red arrival.

Bach himself crosses voices on purpose when a stricter contract demands it. The third Goldberg variation is a canon at the unison — the follower repeats the leader at the very same pitch, so both voices share one register and tangle the moment the second enters. The engine has no page of slurs and stems to keep tangled lines readable, so its ordinary crossing check refuses that trade; the explicit upper-trio exception appears below:

From BachBach: Goldberg Variation 3 — a unison canon crosses at the entry

Variation 3 of the Goldbergs is a canon at the unison: the follower repeats the leader's line one bar later at the very same pitch. Same pitch means same register — and when the follower enters in bar 2 on the B the leader sang a bar ago, the leader has already stepped down to G, a third below it. The voices cross from the follower's first note in this excerpt; half a bar later the leader vaults an octave to G5 and order is restored. Bach accepts the tangle as the price of strict imitation — on a page, slurs and stems keep the lines legible. The engine's generic voice_crossing rule rejects this Goldberg crossing. Its one-onset allowance is limited to the Trio Sonata's upper-voice policy and does not apply here. The ground bass below the two canonic voices is omitted here.

voice_crossingBoth canonic voices occupy an overlapping register; in this excerpt, the leader drops below the follower as it enters.

✕ SpacingUpper voices too far apart

For three or more voices, upper adjacent pairs stay within an octave. The bottom pair may be wider in four-voice writing.

spacing_adjacent_voices_within_octaveThe adjacent upper voices exceed an octave.

In a real three-voice texture the rule splits cleanly in two — tight above, free below:

✓ SpacingThree-voice spacing: tight on top, free below

This is the rule as it actually applies — to real three-voice texture. Upper adjacent pairs are kept within an octave so the chord sounds connected; the bottom pair is allowed to spread because a wide gap above the bass is exactly how Bach voices keyboard music. The validator checks the spacing at each chord start.

spacing_adjacent_voices_within_octaveThe soprano-alto gap stays within an octave; the alto-bass gap may exceed it.

Rule	Scope
voice_crossing	By convention, a lower voice index means a higher pitch (voice 0 is the soprano). A negative interval between any pair — every combination is checked, not just neighbors — means the voices have swapped; the momentary upper-trio exception below is the only policy allowance.
spacing_adjacent_voices_within_octave	In textures of three or more voices, adjacent upper pairs must stay within an octave (12 semitones). The bottom pair (tenor–bass) may be wider, matching four-part Bach practice. Checked at chord starts, only where the harmonic plan declares the chord (has_degree).

The trio-sonata policy has one explicit crossing allowance. When both upper notes are TrioVoiceCarrier material and the harmonic plan enables AllowTrioUpperMomentary, one union-onset exchange may cross the upper pair if the pair was ordered at the previous onset. A crossing that persists into the next onset still fails voice_crossing; the policy also requires the two upper register ranges to overlap, otherwise trio_upper_register_overlap is reported.

Invertible counterpoint at the octave

Fugues reuse two-voice combinations with the voices swapped: what was on top moves below. For that to work, every interval must survive being replaced by its octave complement — thirds become sixths, fifths become fourths, octaves become unisons. Toggle the example below: lift the lower line an octave and every third turns into a sixth, both versions equally consonant.

✓ Invertible counterpointLift one line an octave — and it still works

raise line B an octave —

Two short lines written to be invertible. Line A on top holds still in both views; only line B (blue) rises a whole octave when you switch to “inverted,” so the voices trade places and every interval maps to its octave complement — thirds become sixths — yet both versions stay consonant. That benign mapping is what invertible_at_octave relies on; the next example shows where it turns hostile.

invertible_at_octaveToggle the two: line A on top never moves; only line B (blue) rises an octave, and every third becomes a sixth.

Not every interval survives so kindly. The perfect fifth — a stable consonance — inverts to a perfect fourth, which between two voices alone counts as a dissonance:

! Invertible counterpointInversion at the octave: 5 becomes 4

Invertible counterpoint means two lines still work after swapping registers. Every interval maps to its complement: 5ths become 4ths, octaves become unisons. The engine specifically rejects parallel octaves in upper-voice pairs at structural accents because inversion turns them into parallel unisons. A fourth is judged separately against the actual bass.

invertible_at_octaveThe same two pitch classes form a fifth, then a fourth once the lower note moves up an octave.

Bach cashes in on the benign cases constantly. In the C minor fugue of WTC I, the subject and its countersubject are written so that either can sit on top. Toggle between bar 7 and bar 20 and follow the blue countersubject as it moves from above the subject to below it:

✓ From BachBach: BWV 847 — the same pair, top and bottom

follow the blue countersubject —

The same subject and countersubject from the C minor fugue (WTC I, BWV 847), at two points in the piece. Toggle between them and follow the blue line: at bar 7 the countersubject sits above the subject; at bar 20 the pair is turned over and the same countersubject runs below — the two voices trade octaves note for note. The engine checks parallel octaves with invertible_at_octave; vertical_dissonance judges any fourth against the actual bass of the complete texture. (A third voice is dropped at each spot for clarity.)

invertible_at_octavevertical_dissonanceAt bar 7 the countersubject (blue) sits above the subject; at bar 20 Bach turns the pair over and the same countersubject runs below — every interval still holds.

One scoped rule tests one failure mode in that exchange:

Rule	What it prevents
invertible_at_octave	Parallel octaves in an upper-voice pair at structural accents — after inversion they would become parallel unisons, the most extreme fusion possible. Oblique motion and weak positions are exempt. This is not a proof of complete octave-invertible counterpoint: fifth/fourth treatment, suspensions, and the bass context are checked by other rules.

A fourth is not intrinsically illegal in an upper-voice pair. The vertical validator finds the actual lowest sounding pitch: a fourth above that bass is dissonant unless a declared suspension or cadential 6/4 licenses it, while a fourth between upper voices is legal when both notes are consonant above the bass.

✓ Bass-sensitive consonanceAn upper-voice fourth can be legal

The validator classifies the whole sonority from its lowest sounding pitch. A fourth above the bass is dissonant unless a declared suspension or cadential 6/4 licenses it. This fourth lies between upper voices, so the F-major first-inversion chord is legal even on a structural accent.

vertical_dissonanceC and F form a fourth, but both are consonant above the actual bass A.

How the validator sees this chapter

Rule	FailKind	Key exemptions
parallel_fifth, parallel_octave	MusicalFail	oblique/static motion, cadence cells; fixed-source handling follows the validation lifecycle
anti_parallel_perfect	MusicalFail	contrary motion is checked separately from strict parallels; cadence cells and fixed-source handling follow the lifecycle
battuta	MusicalFail	contrary arrival to a unison with a downward leap in the arriving upper voice; cadence cells and fixed-source handling follow the lifecycle
hidden_parallel_fifth	MusicalFail	cadence cells; all-pair/all-onset scope is an engine policy
hidden_parallel_octave	MusicalFail	cadence cells; all-pair/all-onset scope is an engine policy
voice_crossing	MusicalFail	one momentary upper-trio exchange may be allowed by policy
trio_upper_register_overlap	MusicalFail	only emitted when the momentary upper-trio policy is enabled
spacing_adjacent_voices_within_octave	MusicalFail	bottom pair may exceed an octave; ≥3 voices only
invertible_at_octave	MusicalFail	only adjacent upper pairs at structural accents; this rule checks parallel octaves, not complete invertibility

Continue with Chapter 3 — Dissonance Treatment.