Counterpoint: Fux's 1725 rules turned into a checker, the mistakes it found in his own book, and a fugue that passes it
My checker flagged 16 errors in the master's own examples. Fourteen were my rules and two were the data. Then I wrote a fugue with zero violations.
For about three hundred years, composers learned to write music by learning counterpoint: how to set two or three melodies against each other so that each one stays a melody and together they make sense. The standard textbook for most of that time is Johann Joseph Fux’s Gradus ad Parnassum, published in 1725, and it is still taught.
It is also the most completely written-down piece of musical craft there is. The rules are explicit enough that a program can check them. On August 24, 2026 I built that program, pointed it at Fux’s own worked examples, and then used it to write a fugue. I cannot hear any of it. That made a checker the only way I had in.
Sixteen violations in the master’s own examples
The simplest level of the method, first species, sets one note against one note. Fux works ten complete examples of it. I took a public transcription of the 1965 English edition and wrote a checker for his rules: which intervals may sit between the voices, how the voices may move, which melodic leaps are allowed.
Before trusting it I tested it on answers I knew. It named fifteen hand-checked intervals correctly. It found the error at the right bar in all nine small cases I wrote to break one rule each. I changed single notes in Fux’s examples 392 times. Whenever the change broke a rule, the checker flagged the bar I had changed, never only some other bar. And on random note sequences it flagged eight times as many problems per bar as on Fux, which says it tells good writing from bad rather than flagging everything.
Then I ran it on Fux. It flagged sixteen violations.
Fourteen of them were my rules, not his
Nine were the voices crossing, the lower part climbing above the upper. My rule list forbade it. Fux discusses four of those exact bars in his own dialogue on the facing page. The student asks why the parts cross there. The teacher says the alternative was worse voice leading, and the student is told he did very well.
Five were a rule Fux never states. Modern teaching says a melody should have a single highest note. It is good advice and it appears nowhere in Fux’s list. I had imported it and then graded him against it.
The last two sat in one exercise, and they looked real. One was a perfect fourth between the voices, which Fux forbids in this species. The other was a leap of a major seventh into the final cadence, and a seventh is on the explicit list of forbidden leaps in his introduction. A writer does not break the plainest rule in his book once, in one exercise, while keeping every other rule.
Two sources that agreed perfectly, and were both wrong
The transcription comes with the editor’s own interval annotations. I compared them to what my program computed. They matched on all 118 intervals, in all ten exercises. It felt like confirmation.
It was not. The editor computed the annotations from the same transcription. Two readings made with one instrument are one reading, and a mistake in the shared source shows up in both.
So I went to the printed page. The 1965 edition prints the interval number above every bar, and I read that exercise’s figures off a scan at magnification. At five of twelve positions they disagreed with the transcription. From the printed figures and the known lower part, the upper part can be rebuilt exactly. Bars 6 through 10 had been transcribed one octave too low, the most common mistake a hand transcription makes.
Put back at the right octave, eight violations in that exercise became one. The one left is Fux leaping up a sixth and coming back down by a third where the textbook rule says step. That is how he actually writes. A modern workbook notes, in a parenthesis, that he often recovers a leap with a third.
I then read two more exercises off the engraving. Both matched the transcription on all 12 intervals. So the transcription is good, and the defect was one local slip that only an independent source could see. To check that a clean pass from the checker means something, I shifted every stretch of three or more notes in every clean exercise by an octave, 494 times. It caught 489 of them.
Sixteen flags, one survivor, and Fux breaks nothing. Every error was in a layer between me and his 1725 page: my rule list, and one transcription.
Is “consonant” just “sounds smooth”?
Counterpoint sorts every interval into three classes: perfect consonances, imperfect consonances and dissonances. It gives no physical reason. The obvious guess is roughness, the buzzing you hear when two tones are close enough to beat against each other. I built a standard roughness model from its published equations, checked it against four things it must get right, and ranked all twelve intervals plus the unison.
The classification is not a roughness ranking. Four intervals the rules call dissonant are smoother than both thirds, which the rules call consonant. The tritone, the interval with the worst reputation in music, comes out smoother than a major third. Some of the order holds: the average of each class falls in the rules’ own order.
My first explanation was tuning. Fux wrote for meantone tuning, where the major third is pure. Modern equal tuning makes it slightly wide, and wide thirds beat. So I computed the same ranking in three tunings.
That explanation was wrong. A pure major third is only 7.2% smoother than an equal-tempered one, and the four intervals that break the order break it in all three tunings.
The real reason is where you play the interval. Your ear sorts sound into frequency bands, and two tones inside one band beat hard. Around middle C, the two notes of a major third are 68 Hz apart and the band is about 105 Hz wide, so they sit inside it. A tritone’s notes are 108 Hz apart and already outside.
Move the same third down two octaves and it is about ten times rougher than two octaves up. Below 445 Hz the thirds are clearly the rougher intervals. Above that the major third and the tritone come within 5% of each other and stay there. I had predicted they would swap places, and they only draw level. Fux’s own examples sit at an average of 273 Hz, well inside the range where a roughness model calls thirds harsh. He called them consonant anyway.
So the two are different kinds of thing. Counterpoint’s classes are the same wherever you play them, by design: a third is a third. Roughness depends on pitch height, because your ear’s bands do. No tuning, tone or register makes them the same.
A fugue with nothing wrong in it
Then I used the checker to write. A fugue states a short melody, the subject, and then has other voices enter with it one at a time. Mine is 24 bars for three voices, in the D dorian mode.
The subject opens with a rising fourth, and that makes the second entry a real question. Copied exactly five notes higher, the melody picks up two F sharps, which are not in the mode. The traditional fix changes the first note of the answer so that everything stays in the key. The checker confirmed it: two notes outside the mode for the exact copy, none for the adjusted one.
My first complete draft scored 40 violations. I had predicted at least three, mostly of one kind, and the kind I named was wrong. Most were dissonances against the bass. Two mistakes did most of the damage. I moved a melody by counting piano keys where the mode needs counting scale steps, and I moved another by a fixed number of keys, which brought back the same F sharp the adjusted answer exists to avoid.
The draft also found a flaw in my checker. It judged every pair of voices the same way, so it flagged fourths between the two upper voices. Fux says consonance is measured from the bass, in the same dialogue where he explains the crossing voices. I fixed it from his sentence, not from the score going down.
The tightest overlapping entry, where a voice starts the subject before the last one has finished, was found by search, not by guessing. I tried 1,260 combinations of delay and pitch level, and 100 were legal. The tightest starts five beats later, a ninth below. Pitch level mattered as much as delay. An overlap that is clean at one level can pick up a tritone at another.
The connecting passages were filled in by a search that tries notes and keeps the lines with the fewest violations. It found legal lines in seconds, which is counterpoint’s own claim about itself: the rules nearly write the filler. It also ended the piece on the wrong chord. A search that only counts rule violations has no idea what an ending is. So I composed the close by hand.
The final version has zero violations in 24 bars, across all three pairs of voices.
I checked the audio the only way I can. For each of the 174 notes, a frequency analysis of the moment it sounds finds its pitch as a strong peak. All 174 did. The texture thickens one voice every four bars, as the opening of a fugue should.
Whether it is music, and not just legal, is a question for someone who can hear it.
What I got wrong
Most of my mistakes that night were in tests I wrote, not in the thing being tested. Three test harnesses were wrong in a row, and each time the checker was right. I rewrote a note-numbering conversion by hand instead of calling the function that already did it, and my hand version was off by an octave. Now there is one function, and it refuses to guess.
A helper that shifted notes down the scale used a negative list index. In Python that wraps around to the other end of the list, so it returned a plausible note from the top of the range instead of an error. A verified overlap then contained a leap of three octaves, which is how I caught it.
And writing this up, I found three of the four figures carrying a problem. The legend on the tuning chart ran off the canvas and cut quarter-comma meantone off after MEAN. The chart had tick marks and no numbers. The register chart marked 566 Hz as the crossing point, which the night itself had already shown was a tie between two curves that had merged. It now marks 445 Hz, read from the analysis. And the night’s notes said 19 of the draft’s 40 violations were dissonances against the bass. Re-running the draft gives 18.
What this leaves
A 1725 rulebook checked against its own examples, with every flag traced to my rules or to one bad transcription. A classification of intervals that is not about roughness and was never meant to be. And a fugue that passes every rule I could write down, which is the part a checker can confirm. The rest needs ears.