This page is disputed and low-confidence — its sources disagree. A reconciliation is open on the discussion. Read with care.
Summary
First species counterpoint (first-species counterpoint) is the first and simplest exercise in species counterpoint: note-against-note counterpoint, in which the added voice writes exactly one note for each note of a given melody, the cantus firmus (abbreviated c.f. or CF), and every measure must form a consonant vertical interval with it. The added voice sits either above or below the cantus firmus. It is a foundational exercise in which a note-against-note melody in whole notes is written against the cantus firmus, and it is the first exercise of the Fuxian species method taught in Gradus ad Parnassum (Book II, ch. 1). The rules constrain both the harmonic intervals formed with the cantus firmus and the melodic shape (motion, leaps, and contour) of the added line, as well as the beginning and ending. They are slightly modified and simplified here — a full semester course in counterpoint would apply stricter ones. The exercise teaches consonance, mode, and melodic motion, and Fux distinguishes rules he can justify from rules he leaves to taste.
Several related strands are documented on this page. The first is a JavaScript generator that produces all valid first-species lines for a given cantus firmus according to the hard rules from Johann Joseph Fux's Gradus ad Parnassum. It was written to answer a combinatorial question — how many counterpoints could be written for Fux's first exercise cantus firmus — and reports 17,988 valid lines above the cantus firmus d f e d g f a g f e d (an 11-note Dorian line, D4 F4 E4 D4 G4 F4 A4 G4 F4 E4 D4, middle C = C4). The generator implements only the hard rules; the soft rules that make lines musical and singable were deliberately omitted. The second is a practitioner's report of writing one such exercise above that same cantus firmus and hand-checking it against Fux's stated rules, whose recurring finding is that the hard rules are easy to satisfy and that nearly all the real compositional work is done by rule 6, "good Melody" — the one rule that openly admits it is taste. A third, more recent strand is a close reading of Fux's own approved first-species solutions (preserved in Mark Gotham's encoded species corpus) above and below this same Dorian cantus: those solutions contain repeated notes and successive leaps in the same direction, showing that simplified modern rules — especially "avoid repeated notes" and "no successive leaps in the same direction" — are not absolute. Fux himself distinguishes rules he can justify from rules he cannot justify and leaves to the student's discretion.
Disputed framing of the rules' origins. This page describes the rules as codified by Fux but derived from observation of sixteenth-century Renaissance practice (the style of Palestrina), with Palestrina as the model composer. A previously ingested article reports scholarship (Mattias Lundberg, quoting Jen-Yen Chen) arguing instead that Fux "subtly exploits the powerful mystique associated with Palestrina" to lend his own theory authority, that some Fux principles have no analogy in sixteenth-century repertoire and are mutually exclusive with Palestrina's practice, and that the rules are a pedagogue's constructed system rather than a neutral transcription of the old style. Both positions are retained here.
Key Points
The counterpoint is written in whole notes, one note against each note of the cantus firmus; the added voice may be written above or below it.
Every measure must form a consonant harmonic interval with the cantus firmus; dissonances are forbidden entirely in this species.
The exercise begins and ends on a perfect consonance — in the sources here given as an octave or unison (the newer reading generalizes the opening to unison, fifth, or octave).
The final octave/unison is approached by contrary motion, with one voice moving 7̂–8̂ and the other 2̂–1̂.
"Imperfect" consonances (3rds and 6ths) may be approached in any manner, but no more than three consecutive 3rds or 6ths are allowed.
"Perfect" consonances (unisons, 5ths, octaves, 12ths) may only be approached by contrary motion, to avoid direct (parallel) 5ths and 8ves. (Fux's own stated rule permits motion into a perfect consonance that is contrary or oblique; the stricter contrary-only formulation on this page is a modern textbook simplification.)
The penultimate interval must be a third or a sixth; when a sixth resolves by step to the octave it must be the major sixth.
Fux's stated rules (abridged English translation of Gradus ad Parnassum II.1) are six general rules — motion, no skip into an octave/unison even by contrary motion, prefer imperfect consonances, stay in the key, avoid mi contra fa, and "Regularity, Variety and good Melody" — plus begin and end on a perfect consonance and a prescribed penultimate.
The added line is restricted to a small set of allowed leaps and must avoid repeated notes and melodic tritone outlines — but Fux's own solutions repeat notes and use successive leaps in the same direction, so these are not absolute rules.
The rules underconstrain. A generator README (dhf-fullstack/counterpoint) reports 17,988 valid lines above this cantus satisfying only the hard rules (allowed intervals and motions), so nearly all selection is done by rule 6, "good Melody."
Rule vs. taste. Modern textbooks add constraints Fux does not state (no more than three consecutive 3rds/6ths, a single climax, avoid repeated notes, avoid an outlined tritone, no successive leaps in the same direction) and drop one Fux states (no skip into an octave even by contrary motion). What counts as a rule is edition-dependent.
Fux's approved solutions contradict the simplified absolutes. Above the cantus (Fig. 5) the line repeats notes at bars 1–2 and 6–7; below (Fig. 6) it repeats a note at bars 1–2 and leaps twice in the same direction (D3–F3–C4). The repeated note below is the master's remedy for a direct-motion fifth, not a lapse.
The battuta octave is prohibited but unjustified. Fux forbids a tenth moving to an octave (upper voice down a step, lower voice up a step), but says he has long searched for the reason and found none; he leaves its use or avoidance to the student's discretion and calls the matter "of little importance." Modern rule lists often omit it.
Some Fux rules are obscure. Fig. 19 declares "The approach from a more remote consonance into an octave by a skip is in my opinion not to be tolerated," but the exact shape this forbids is unclear from the available figure.
Worked examples can be wrong. A published "counterpoint skill" example above the same cantus contains nine parallel thirds in a row, claims all motion is contrary or oblique, and ends its penultimate on a minor third rather than the required major sixth.
Fux ≠ Palestrina. The disputed-origins issue above: scholarship argues Fux's authority rests partly on Palestrina's mystique rather than demonstrated continuity.
Separate reasoned rules from taste. Mode identity and the treatment of direct perfect intervals are rules Fux can justify; the battuta prohibition, "avoid repeated notes," and "avoid successive leaps in the same direction" are stylistic preferences or pedagogical simplifications. A modern manual allows at most one repeated note, while Fux's Fig. 5 has two.
Although the rules may seem dry, they arise from careful observation of note-by-note details in Renaissance compositions; practicing them develops awareness of counterpoint and can improve any composer's writing.
Counterpoint codifies early Renaissance practice for writing multi-voice music in which every part is independently tuneful yet consonant with the others; the ideas remain relevant to modern music writing.
The project follows Fux's Gradus ad Parnassum, the venerable textbook used by Haydn, Mozart, and Beethoven.
Only the hard rules were implemented — allowed intervals and allowed motions. The soft rules (no large skips, pleasing rising and falling contour) that make lines musical and singable were deliberately omitted.
The original motivation was combinatorial: at the first Fux exercise the author wondered how many possible counterpoints exist, could not solve the combinatorics mathematically, and wrote a script to enumerate them instead.
The count produced for the example cantus firmus was 17,988 valid first-species lines above it.
The generator was later wrapped in a Fullstack Academy teaching project: a divs-and-text UI to motivate learning DOM manipulation. A real app would render music on a canvas or via SVG and add audio output.
Concepts
Cantus firmus (c.f. / CF) — the given melody the counterpoint is written against; here an 11-note Dorian line (D4 F4 E4 D4 G4 F4 A4 G4 F4 E4 D4, middle C = C4).
Species counterpoint — the graded set of counterpoint exercises; first species is note-against-note.
First species (1:1) — note-against-note counterpoint (one note of the added voice per note of the cantus firmus).
Consonance and dissonance — consonant intervals are unisons, 3rds, 5ths, 6ths (and their compounds); dissonances are 2nds, 4ths, and 7ths.
Fux — Johann Joseph Fux, author of the rule set for writing in proper sixteenth-century style (Gradus ad Parnassum, 1725). In the project's code, "Fux" also names the JavaScript library that generates the music.
Aloysius and Joseph — the teacher and student of Fux's dialogue; it is Aloysius who admits, on the battuta octave, that he has searched for the reason and found neither it nor the nature of the mistake.
Palestrina — the model composer whose works the rules are said to be observed from; the target of the scholarly caveat noted above.
Stile antico / Gradus ad Parnassum — the sixteenth-century style and the 1725 pedagogical treatise that codified it.
Perfect consonances — unisons, 5ths, 8ves (octaves), 12ths; motion into them must be contrary (or, in Fux's own wording, contrary or oblique), and parallel perfects are forbidden.
Imperfect consonances — 3rds and 6ths (and their compounds, such as 10ths and 13ths).
Direct 5ths / direct 8ves — the prohibited result of approaching a perfect consonance by similar motion (also called hidden 5ths and 8ves). Fux's Fig. 6 corrects a direct-motion fifth by keeping the lower voice on D, producing oblique motion.
Types of motion — similar, parallel, contrary, and oblique; contrary motion is required for approaching perfect consonances and for the final cadence.
Battuta octave — a tenth moving to an octave, upper voice down a step and lower voice up a step; prohibited by Fux, who admits he cannot justify the prohibition and leaves it to the student's discretion.
Leading tone — the raised seventh scale degree used in cadences for certain modes.
Mi contra fa — the harmonic tritone (B against F) that Fux forbids as an interval and that can also be outlined melodically across a line's turning points.
Penultimate interval — must be a sixth or third; a sixth resolving by step to the octave must be the major sixth (C♯5 over E4 in the reported exercise; the minor third in the below-cantus examples).
Climax — a single melodic peak; a constraint modern texts add, but which Fux's stated rules here do not.
Modes — G mixolydian, D dorian, and A aeolian require a raised leading tone (F♯, C♯, and G♯ respectively), raised for the cadence in modes whose seventh degree lies a whole step below the tonic. The cantus on this page is Dorian.
Melodic leaps — the permitted intervals (3rds, 4ths, 5ths, ascending minor 6ths, octaves) and the need for direction changes after leaps.
Melodic outline of a tritone / outlined tritone — avoided by changing direction at two turning points; treated in the newer report as a craft extension of mi contra fa "stretched across time."
Turning points — changes of direction in a melody; the places at which a tritone outline is prevented or created.
Hard rules — the permitted intervals and motions that define the valid counterpoint lines; the only rules the generator enforces.
Soft rules — stylistic guidance for musicality (avoid large skips, shape an appealing contour); ignored by the generator.
DOM.ui — an experiment in displaying music using DOM elements alone.
VEX.ui — a music display layer based on VexFlow.
Mark Gotham's encoded species corpus — the source preserving Fux's own first-species solutions above and below the Dorian cantus (Figs. 5 and 6).
Ian Stoner's Fux workbook — a modern workbook that flags Fux's own exceptions ("Avoid repeated notes (though Fux makes many exceptions in Species One)"; "Fux contradicts this [no successive skips same direction] all the time").
Alfred Mann — translator of the edition of Gradus ad Parnassum referenced by the newer material.
Puget Sound music theory (Hutchinson) — one of the modern textbooks whose added rules (e.g. avoid repeated notes) go beyond Fux.
Details
Harmonic intervals (vertical rules)
Each measure must contain a consonant harmonic interval. Allowed: unisons, 3rds, 5ths, 6ths, plus their compound versions (10ths, 12ths, 13ths, etc.).
Dissonances — 2nds, 4ths, and 7ths — are not allowed at all in first species.
Imperfect consonances (3rds and 6ths) can be approached in any manner, but no more than three consecutive 3rds or 6ths.
Perfect consonances (unisons, 5ths, 8ves, 12ths) may be approached only in contrary motion, which avoids direct (hidden) 5ths and direct 8ves. Fux's own stated rule is that motion into a perfect consonance must be contrary or oblique; the contrary-only reading is the modern textbook hardening, and Fux's own below-cantus solution uses the oblique option.
When the cantus firmus is in the upper part, the counterpoint below it must form a unison, 3rd, 5th, or 6th.
Melodic rules (horizontal)
Allowed leaps: 3rds, 4ths, 5ths, ascending minor sixths, and 8ves. Direction changes after leaps follow the Rules of Melody (e.g., stepwise motion in the opposite direction to balance the leap).
Avoid melodically outlining a tritone — do so by changing direction at two turning points in the melody.
Avoid repeating notes, so that the melody flows and is not hindered — a guideline Fux's own solutions repeatedly violate (see below), and which Stoner's workbook qualifies as "though Fux makes many exceptions in Species One."
Fux's rule 6, "Regularity, Variety and good Melody," is the catch-all that governs melodic quality.
Opening, closing, and mode
Begin with a perfect consonance; in the exercise above this cantus, and in Fux's own solutions, an octave or unison. The newer reading states the opening may be a unison, fifth, or octave.
End on an octave or unison.
Approach the final octave/unison by contrary motion: one part ends 7̂–8̂ while the other ends 2̂–1̂. This creates a proper cadential formula.
The penultimate interval must be a sixth or third; a sixth resolving to the octave must be major (in the worked exercise, C♯5 over E4).
In G mixolydian, D dorian, or A aeolian, raise the leading tone with a sharp — F♯, C♯, and G♯ respectively — so that the cadence has a proper leading tone.
Rendering diagram…
Fux's six stated rules
Per the abridged English translation of Gradus ad Parnassum II.1, the general rules are:
Motion (avoid parallel perfect intervals; avoid similar motion into a perfect consonance).
Do not skip into an octave or unison, even by contrary motion.
Prefer imperfect consonances.
Stay in the key.
Avoid mi contra fa.
"Regularity, Variety and good Melody."
To these are added the conventions of beginning and ending on a perfect consonance and the prescribed penultimate. Notably, the translation does attach a reason to the motion rules — avoid open/hidden fifths and eighths — which corrects an earlier over-strong claim that the motion rules "reached English without the reasoning."
Fux's own first-species solutions
Fux's first-species solutions for this Dorian cantus are preserved in Mark Gotham's encoded corpus, both above and below the cantus.
Above (Fig. 5): A4 A4 G4 A4 B4 C5 C5 B4 D5 C♯5 D5. The intervals are P5 M3 m3 P5 M3 P5 m3 M3 M6 M6 P8. Two repeated notes appear (bars 1–2 and 6–7); the range is a fifth.
Below (corrected Fig. 6): D3 D3 A3 F3 E3 D3 F3 C4 D4 C♯4 D4. The intervals are P8 m10 P5 M6 m10 m10 M10 P5 m3 m3 P1. It repeats a note at bars 1–2 and contains two leaps in the same direction (D3–F3–C4).
The below example contains the master's correction of a direct-motion fifth. Rather than allowing both voices to move into a fifth, the lower voice stays on D, producing oblique motion. The repeated note is the remedy, not a lapse. These solutions confirm that the below-cantus exercise begins on a perfect consonance and ends with the minor third in the penultimate interval.
The battuta octave
In Fig. 15–18, Fux treats the battuta octave: a tenth moving to an octave, with the upper voice moving down by step and the lower voice moving up by step. He prohibits it, but Aloysius says he has long searched for the reason and found neither it nor the nature of the mistake. He leaves its use or avoidance to the student's discretion and calls the matter "of little importance." Modern simplified rule lists often omit this prohibition entirely.
Contested and unclear rules
Fig. 19 states: "The approach from a more remote consonance into an octave by a skip is in my opinion not to be tolerated." The exact shape this forbids remains unclear from the available figure. Modern teaching frequently states absolute rules — no repeated notes, no successive skips in the same direction — that Fux's own examples violate. Ian Stoner's workbook notes both: "Avoid repeated notes (though Fux makes many exceptions in Species One)" and "Fux contradicts this [no successive skips same direction] all the time." A modern manual allows at most one repeated note, while Fux's Fig. 5 has two.
The exercise as performed
An 11-note first-species line written above Fux's Dorian cantus firmus and checked by hand:
8 of the 11 intervals are imperfect; the line runs A4–F5.
Penultimate is the greater (major) sixth, C♯5 over E4, as the rule requires.
Climax F5 occurs once (bar 4); the cantus climax A4 falls at bar 7.
Motion alternates similar/parallel/contrary passages, with contrary motion used into each perfect consonance (e.g. into the closing P8 by step).
Mistakes kept, and what they teach
Bar 8 first drafted as B4 (M3 over G4). The line F5 (peak) descending to B4 would outline a tritone between turning points; revised to D5 (P5, contrary). The writer treats this as a craft rule — the same B-against-F worry Fux states harmonically, applied melodically over time.
Bar 9 nearly written as C5 — P5 following P5 with both voices falling is parallel fifths; Fux's motion rule catches it.
Bar 9 then had two legal options: leap A4 (D5–A4–C♯5, two leaps in a row) or repeat D5. Neither is forbidden by Fux's six stated rules, yet modern textbooks forbid one or the other (Puget Sound: avoid repeated notes; others: avoid consecutive leaps). The choice of A4 is named explicitly as taste, not rule. Fux's own Fig. 5 (two repeated notes) and Fig. 6 (two leaps in the same direction) show that he permitted both.
How selection actually happens
The hard constraints (consonant intervals + motion rules) leave an enormous space — hence the 17,988 admissible lines. Filtering to a single line is dominated by rule 6, "good Melody," which is precisely the rule that admits it is judgment. This makes the boundary between rule and taste the central working problem of the exercise. A rule-clean line can still be hollow; taste enters exactly where the rules leave room.
Rendering diagram…
The rules are not stable across editions
Fux states six general rules plus the begin/end and penultimate conventions.
Modern textbooks add: no more than three consecutive 3rds/6ths, a single climax, no repeated notes, no outlined tritone,