Pitch interval¶
The semitone separation between two octave-specified musical pitches, directed or undirected without octave reduction.
Core Idea¶
A pitch interval in chromatic musical-set analysis compares two octave-specified pitches by their semitone separation. The ordered form retains direction: y−x is positive upward and negative downward. The unordered form takes |y−x| and retains size while forgetting which endpoint came first. Both retain register; a C4-to-C5 octave is twelve semitones rather than zero. This differs from a pitch-class interval, which treats octave equivalents as the same class and reduces the comparison modulo twelve.
The chosen semitone frame matters. Chromatic distance does not automatically fix diatonic spelling: a six-semitone span can be named as an augmented fourth or diminished fifth according to the written pitches. The music21 guide explicitly distinguishes interval.semitones, a descending reverse, and intervalClass reduction. Its perfect-fifth example has seven semitones while its reduced intervalClass is five, illustrating that size, orientation, spelling, and class are separate pieces of information. A generic difference between sound frequencies without a declared musical pitch frame is not this named semitone interval.
Structural Signature¶
Sig role-phrases:
- Two octave-specified pitches — Identifies endpoints in chromatic pitch space rather than only their octave-equivalent classes. It is constitutive. Counterfactual: Replacing C4 and C5 by the same class C discards the octave interval.
- Semitone coordinate frame — Places both pitches on one chromatic scale for a meaningful difference. It is constitutive. Counterfactual: An unspecified tuning or unlike coordinate schemes make a bare integer ambiguous.
- Separation operation — Computes the difference y−x or its absolute value without collapsing octave position. It is constitutive. Counterfactual: A modulo-12 result is a pitch-class interval, not the full pitch separation.
- Direction convention — Distinguishes ordered signed displacement from unordered magnitude. It is boundary. Counterfactual: Calling −7 and +7 identical loses melodic direction even though their unordered size matches.
- Spelling and reduction qualifier — Marks when a semitone count is further interpreted as a diatonic name or reduced to a class. It is boundary. Counterfactual: Seven semitones by itself need not determine every enharmonic spelling, while interval class five is a different reduction.
What It Is Not¶
- Not a pitch class interval. Octave-sensitive endpoints cannot be replaced by equivalence classes modulo twelve.
- Not always directed. Unordered pitch intervals retain magnitude but omit sign.
- Not a note's pitch. The interval relates two specified pitches.
- Not a diatonic name alone. The same semitone size can have different written spellings.
- Closest near-miss. A pitch-class interval is the closest miss: it also compares two notes but identifies octave-equivalent endpoints and can discard a 12-semitone difference entirely.
Scope of Application¶
- Melodic analysis. Retain upward and downward displacement across successive pitches.
- Harmonic comparison. Compare unordered semitone spans while retaining register.
- Computational music theory. Keep chromatic semitone counts separate from interval-class reductions.
- Notation interpretation. Distinguish a numerical span from its diatonic spelling.
Clarity¶
State both octave-specified pitches and the semitone frame, then declare whether direction is retained. C4 to E5 is +16, the reverse −16, and both have unordered size 16. A pitch-class interval is the nearest miss because modulo-12 reduction erases the octave and yields a different object. A diatonic label additionally depends on written spellings, not only semitone count.
Manages Complexity¶
The word interval can conceal four transformations: choose a chromatic frame, retain or drop direction, retain or reduce octave, and assign a diatonic spelling. Keeping these transformations explicit explains why music21 reports a perfect fifth as seven semitones but an intervalClass of five. It also prevents a melodic contour analysis from silently becoming a pitch-class inventory.
Abstract Reasoning¶
- Specify the two pitches with octave positions and one semitone coordinate system.
- Subtract endpoint coordinates for an ordered signed interval.
- Take absolute value only if direction is intentionally discarded.
- Preserve the unreduced span before any modulo-12 pitch-class comparison.
- Keep diatonic spelling and enharmonic interpretation separate from chromatic semitone size.
Knowledge Transfer¶
The signed/absolute separation distinction transfers among melody analysis, harmony comparison, and music software when endpoints and tuning frame are preserved. music21's exact interval-class result should not be copied as an unreduced distance, and its notation spelling conventions need not transfer to another tuning system. Outside musical pitch space, a signed number difference is only a structural analogy, not this pitch interval.
Examples¶
Canonical¶
In standard semitone indexing, C4 to E5 rises 16 semitones, so its ordered pitch interval is +16 and E5 back to C4 is −16; their unordered magnitude is 16. Reducing those endpoints to pitch classes would yield a four-semitone class relation instead and lose the octave. The calculation demonstrates why directed, undirected, and octave-reduced values are not interchangeable.
Mapped back: Two octave-specified pitches → C4 and E5, retaining their registers; Semitone coordinate frame → standard chromatic semitone indexing; Separation operation → 76−60=16 before any modulo-12 reduction; Direction convention → +16 forward, −16 reverse, 16 unordered; Spelling and reduction qualifier → pitch-class reduction would return a different four-semitone relation.
Applied / In Practice¶
The official music21 Interval guide represents a perfect fifth as seven semitones, gives a descending reversal, and separately reports intervalClass five for its octave complement. This is a documented computational music-theory use of pitch-interval information: the chromatic size and direction remain available even when a reduced class is also computed. The documentation does not make intervalClass a synonym for the unreduced pitch interval or settle diatonic spelling from semitone count alone.
Mapped back: Two octave-specified pitches → music21's pitch-space interval between notes or transposed pitches; Semitone coordinate frame → music21 chromatic semitone count; Separation operation → documented P5 has seven semitones; Direction convention → reverse gives a descending P-5; Spelling and reduction qualifier → intervalClass five is reported separately as a reduced class.
Structural Tensions¶
T1 — Directional Information versus Compact Comparison. Signed intervals track melodic motion, whereas absolute magnitudes let ascending and descending spans be grouped; neither view alone answers both questions.
Diagnostic: Does the analysis require orientation or only separation size?
T2 — Register Preservation versus Octave Equivalence. Unreduced pitch intervals retain compound spans, while class reduction simplifies comparison but collapses distinct registers and directions.
Diagnostic: Was octave information intentionally discarded, or accidentally lost?
Structural–Framed Character¶
Pitch interval is mostly structural within a musical frame: endpoint comparison and sign/absolute operations are formal, but semitone coordinates and octave conventions are musical choices. Evaluative weight: +16 and 16 describe different information, not good or bad music. Human-practice-bound: acoustical frequencies exist, while pitch labels, equal semitone steps, and notated spelling come from music practice. Institutional origin: theory and software codify usage without creating the numerical difference. Vocabulary travels: ordered versus unordered separation is portable; pitch, octave, and enharmonic spelling remain musical. Import versus recognize: the same two-pitch calculation in another score is literal; calling a salary gap a pitch interval is metaphor.
The portable signed-pair comparison could be a future-prime candidate, but live Distance is nonnegative and therefore cannot strictly parent the ordered negative variant of this entry. Its character: a formal relation whose named identity depends on a declared musical pitch and octave frame.
Structural Core vs. Domain Accent¶
The relation is mathematically simple, but the named unit and endpoint interpretation are musical.
What is skeletal. Two entities in a shared coordinate space can be compared by a signed difference or an absolute magnitude. This may support a future-prime signed-separation abstraction. Live Distance covers only nonnegative separation, so it can represent the unordered result but not the full ordered identity.
What is domain-bound. These entities are octave-specified musical pitches, and the comparison counts semitones in a declared chromatic frame. Diatonic spelling and pitch-class reduction supply distinct, music-theoretic interpretations. music21's P5 seven-semitone value and intervalClass five demonstrate that reducing octave and complement information changes the object, not merely its display format.
Why this does not clear the prime bar. A signed numerical difference recurs in finance or geometry without pitch, octave, tuning, or notation. The portable arithmetic alone is thinner than this entry; treating every signed difference as a pitch interval imports musical vocabulary without its carrier. The accepted prime Distance does not absorb the ordered case, and no separate exact parent is asserted.
Instantiates / Related Primes¶
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Related — distance. The unordered nonnegative result is a distance-like measure; ordered negative spans exceed the live prime's definition.
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Related — pitch-class interval. Modulo-12 comparison deliberately drops octave position.
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Related — transposition. An interval can parameterize a pitch move; the move is not the pairwise interval itself.
Neighborhood in Abstraction Space¶
Pitch interval sits in a moderately populated region (42nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Musical & Poetic Form (11 abstractions)
Nearest neighbors
- Transposition (Music) — 0.92
- Strähle construction — 0.88
- Musical Interval — 0.87
- Boogie — 0.87
- Major Limma — 0.86
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Pitch class interval. Tell: Were octave positions retained or collapsed modulo twelve?
- Diatonic interval name. Tell: Are written letter names being inferred from semitones alone?
- Absolute pitch. Tell: Are there two endpoints or just one pitch label?
- Frequency difference. Tell: Was a musical semitone pitch frame declared?
References¶
- music21, User's Guide, Intervals: https://music21.org/music21docs/usersGuide/usersGuide_18_intervals.html
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Pitch_interval (revision 1320774604).