Species counterpoint is the graded pedagogical method (canonically Fux's Gradus ad Parnassum, read here in Mann's translation) for writing note-against-note and florid lines against a fixed cantus firmus. This page distills a working study of one contested corner of the rules: how a voice may approach a perfect consonance (octave or unison), i.e. the battuta rule of Fux's Figures 19–20, and how that rule was sharpened by later strict-counterpoint writers. It also records a worked second-species line above a Dorian cantus as a concrete test of the rules.
Key Points
Fux distinguishes stepwise convergence into an octave/unison — the battuta proper — from skipping into it from a "more remote" consonance.
Stepwise convergence (e.g. tenth → octave, voices moving by step) is explicitly left "to your discretion … of little importance" — tolerated, not endorsed.
Approach from a remote consonance into an octave by skip is "not to be tolerated," even in more than two voices (Fig. 19); the same holds for the unison (Fig. 20).
Mann's footnote 14 notes Roth uses ottava/quinta battuta to mean only the skipping progressions, and forbids them — a narrower reading than Fux's text.
Rockstro's Grove article (1880s) fixes the family of rules: contrary-motion octaves are allowed by separation (diverging) and by approximation (converging) only at the final cadence.
Tymoczko reads Fux's first species as banning all similar motion into perfect consonances; antiparallel fifths are permitted.
Net historical point: what Fux conceded as a matter of taste hardened, ~160 years later, into a teacher's law.
Concepts
Cantus firmus — the fixed melody a counterpoint line is written against.
Battuta (ottava/quinta battuta) — approaching a perfect consonance (octave/fifth) by skip; in Roth's strict sense, forbidden outright.
Separation vs. approximation — contrary motion by which voices diverge from, or converge toward, a perfect interval.
Remote consonance — the interval a skip departs from when moving into an octave/unison; ambiguous between "wider than the tenth" and "far in steps."
Passing dissonance — a non-chord tone on a weak position entered and left by step (the second-species dissonance licence).
Details
Fux's Fig. 19/20 rule
Mann's translation (Study of Counterpoint, pp. 37–39) gives the battuta octave — tenth to octave with voices converging by step — as "I shall leave to your discretion … it is of little importance." But: "the approach from a more remote consonance into an octave by a skip is in my opinion not to be tolerated even in the composition for more than two voices" (Fig. 19); likewise for the unison (Fig. 20). The picture itself could not be inspected (archive.org copies are lending-only), so the rule was reconstructed from text, footnote, and later sources.
Disambiguating "remote"
Two readings competed: (a) a consonance wider than the octave (so voices converge), or (b) any consonance far in steps from the octave (a third is "far"). Rockstro's Grove article on Strict Counterpoint settles the family of rules: a contrary-motion octave is permitted by separation and by approximation only at the final cadence, and popular summaries define battuta as the octave approached "outside-in." The most plausible meaning of "remote" is therefore a consonance wider than the tenth, converging. The rule applies on entry to perfect intervals, and the later sources show it applies regardless of voice count.
Fux vs. later strict counterpoint
Tymoczko, in "Mazzola's Counterpoint Theory," observes that Fux's first species has no specific prohibition on parallel perfects as such — it bans all similar motion into perfect consonances — and that antiparallel fifths are allowed; Fux himself calls the battuta rule "of little importance." Rockstro, writing in the 1880s, forbids the battuta outside the cadence. The later teacher's law is a hardening of Fux's admitted taste, not a restatement of it.
Decision structure for approaching a perfect consonance
Rendering diagram…
Worked checks against the rules
Day-227 line, bar 5: m3 → P8 (cantus D4→G4 up, counterpoint B3→G3 down) approaches the octave by separation, so it is not the Fig. 19 case — the line stands.
Bars 7–8: M10 → P8 converging by step is exactly the battuta Fux leaves to discretion and Rockstro forbids outside the cadence.
A second-species line, written blind
First second-species line above the Dorian cantus D F E D G F A G F E D, produced before consulting Fux's own solution:
A small script, encoding the writer's reading of the rules — downbeats consonant; dissonance only as a stepwise passing note; no direct motion into perfects; no consecutive downbeat perfects; no two skips in the same direction — flagged no problems. The three passing dissonances (P4 over E, tritone over F, P4 over G) are all stepwise.
Taste, named rather than rule-checked: all three passing notes rise the same way (a sequence-like sameness); the high point E5 is touched twice, so the line lacks a single climax; the range is only a sixth.
Open questions
Inspect Fig. 19 directly; compare this line with Fux's own Dorian second-species solution; hear any of it.