Ground Truth · a rulebook, measured

Two Kinds of Never

Every harmony student is taught that consecutive fifths are forbidden, and every harmony student is told that Bach broke the rules anyway. Both sentences get handed over without a list. Here is the list, drawn and playable. In the 370 four-part chorales as Breitkopf printed them, 19 consecutive perfect fifths and 6 consecutive octaves survive the strictest reading, out of 8,437 moments where two voices stood on a fifth and both of them moved. The melodic augmented second, forbidden by the same books in the same breath, occurs 0 times in 82,466 melodic intervals. The direct fifth, forbidden just as flatly, occurs 5,531 times. The rulebook says never in one tone of voice and means at least three different things by it, and only counting tells them apart. Reconciled item by item against the two published counts, and against Kirnberger's own 1771 statement of the rule.

The two sentences

The first thing a harmony student learns is a prohibition. Two voices may not move from one perfect fifth to another, nor from one octave to another. The rule is old, it is stated without hedging, and a red mark costs you the same whether you wrote one or twelve.

The second thing a harmony student learns is that the rule has an asterisk. Bach breaks it. Everyone says so. What usually does not come with the claim is the list, and without the list it does no work: it could mean four times in a lifetime or four times a page, and those are different worlds. Two people have published lists, and this page is checked against both of them further down. What is added here is the part that turns out to matter most.

There is no such number as "the number of parallel fifths in Bach" until four further questions are answered, and the published authorities answer them differently. Between which voices? Does a fifth followed by a twelfth count? Does a fifth across the end of a phrase count? Does a fifth made by a note passing through count? Over the same 370 chorales the answer ranges from 0 to 89 depending on how you set those four switches, and there are 36 settings. Set them yourself.

Count them yourself

A "chance" is a moment where those two voices already stood on a perfect fifth (or an octave) and both then moved to a different pitch. That is the denominator a bare count needs: the number of times the mistake was actually available to be made. There were 8,437 such moments for the fifth and 6,829 for the octave, across all six pairs of voices in all 370 chorales. A further 1,457 moments where the pair simply held or restated the same fifth are excluded from every reading here, because no published rule counts a fifth that is merely still there.

The drawing is proportional notation, not the printed page: horizontal position is time, so a bar of four quarter notes is four equal columns and a passing eighth sits visibly off the beat. Noteheads sit at their real staff positions, hollow from a half note up. Stems and beams are omitted, and the shaded tail behind each notehead is how long the note lasts. Accidentals are drawn wherever a note differs from the key signature, which is not always where an engraver would print one. The two implicated voices are drawn in colour and the two intervals are bracketed and named; everything else is grey but still sounds.

What the strictest reading finds

Set every switch to its strictest position and the answer is 19 fifths and 6 octaves. That is 19 in 8,437 chances, one time in 444. Three things about those 19 are the sort of thing a bare number hides.

Seventeen of the nineteen are made by an ornamental note. In 17 of them at least one of the notes arriving on the second fifth is stepped into and out of, which is the shape of a note passing through rather than a note the harmony stands on. Rule those out and 19 becomes 2. Those two deserve a look. In chorale 320 (Gott sei uns gnädig und barmherzig, BWV 323) the tenor and soprano both fall a fourth on the beat, in half notes, from E and B to B and F sharp, with nothing passing anywhere. In chorale 4 (Es ist das Heil uns kommen her, BWV 86/6) the same two voices fall a fourth across the barline. Neither has an excuse available to it, and both were found independently in 2008 by Fitsioris and Conklin, who classed them as the only two "exposed" cases in their table.

Some of them are the same passage twice. Chorale 8 contributes four, which are two passages each written out twice in a repeating melody, and chorale 121 contributes two that are one passage twice. The list shows every written occurrence, which is the honest thing for a count of what is on the page. If you want distinct passages, subtract three.

Two of them are probably not Bach's. Both are known to the literature as editorial corruptions rather than compositions: the fifth in chorale 64 is absent from the modern critical edition, and the fifth in chorale 169 comes from a soprano note that Fitsioris and Conklin argue is wrong in every edition. See whose fifths are these.

Checked against the people who counted first

This is not the first count. In 2008 George Fitsioris and Darrell Conklin ran a Prolog query over the Riemenschneider chorales and published a table of 18 parallel fifths with voices, measures and pitches. In 2016 Luke Dahn published a list of 46 consecutive fifths and octaves over the whole surviving corpus of about 410 chorales, having gone back to the manuscripts, and later revised it upward. Their definitions differ from each other and from this page's, and their corpora differ too, so the three counts should not agree, and the interesting question is whether they line up item by item.

They do. Of Fitsioris and Conklin's 18, 16 are settings that appear in this collection at all, and all 16 are found here, in the same chorale, at the same written measure, between the same two voices. The remaining two are chorales the Breitkopf collection does not contain. Of Dahn's 39 entries, 31 are in this collection and 30 are found here; the exception is noted below the table.

Running the comparison the other way is more interesting. Three of the 19 fall in chorales that neither published list names at all: . Each was checked against the source by hand and each is a real pair of consecutive perfect fifths between consecutive attacks in the printed edition. Each also involves an ornamental note, which is very likely why both earlier counts set it aside: Fitsioris and Conklin filtered their query's output by judgement, and Dahn separates the ornamental cases out explicitly. They are listed here rather than filtered because filtering is what the switches are for.

The rest of the rulebook

Consecutive fifths and octaves are only the famous prohibitions. The same textbooks forbid seven or eight other things in the same flat wording, and running all of them over the same 370 chorales changes the shape of the picture completely.

Every count is of written occurrences in the 370 chorales under the reading named in the third column. Where a rule's count depends on a definition, the strictest defensible reading is used and the looser one is given beside it.

The one he never once broke

One number in that table is not small but absent. In 82,466 melodic intervals across the whole collection, the augmented second, the step from the flattened sixth degree up to the raised seventh that any minor key offers a composer several times a phrase, occurs zero times. Not rarely. Not mostly avoided. Not once.

A zero is the easiest number in the world to get wrong, because a search that cannot match anything returns the same zero as a search that found nothing. So here is the whole histogram of melodic intervals in the corpus, every spelling that occurs, with the augmented second left in place so you can see the hole rather than take it on trust. The detector that reports zero augmented seconds is the same one reporting 486 augmented unisons, 120 diminished fourths and 21 diminished sevenths in the same pass, so it can plainly see spelled chromatic intervals, and its silence here is evidence rather than absence of evidence. A second program in another language, sharing no code with the first, returns the same zero, and fires correctly on a synthetic line that contains one.

Intervals are counted between consecutive notes in one voice, tied notes counted once, never across a rest. Spelling matters: a diminished fourth and a major third are the same number of semitones and are different intervals, and the whole point of the augmented second is that it is a second on the staff. The bracketed figure is how many of that interval occur only across a fermata, where a new phrase begins and the singer breathes: 25 of the 33 leaps wider than an octave are of that kind, leaving 8 real ones, every one of them in the bass.

The rule that is a tendency, not a prohibition

The direct fifth, also called the hidden or covered fifth, is the prohibition against arriving at a perfect fifth or octave with both voices moving the same way. It is stated in the same tone as the consecutive fifth, and it happens 5,531 times. That looks at first like a rule simply ignored. It is not, and the difference matters.

A raw count cannot tell an avoided thing from a rare thing. The right question is conditional: given that the two voices arrive at a perfect fifth or octave, how often did they get there by similar motion, compared with how often similar motion happens at all between those two voices? If the ratio is 1 the rule is invisible in the music. If it is below 1 the approach is being avoided.

Every pair sits well below 1. Perfect consonances are approached by similar motion at roughly half the rate the same voices use similar motion generally, and the effect is there in all six pairs. So the direct fifth is a genuine, measurable constraint in this music. It is simply not a prohibition: it is a strong preference, obeyed most of the time and broken thousands of times, which is a different kind of rule from the one that is broken 19 times in 8,437 chances.

The rules, in the words they were written in

The four switches above are not inventions. Each one is a place where the published authorities disagree with each other, and in one case where a single author disagrees with himself between two of his own books. Here is that disagreement in their own words, from editions old enough to quote in full.

Contrary motion. Ebenezer Prout's Harmony, the textbook that trained generations of English-speaking students, tolerates it:

71. Rule II. Consecutive perfect fifths by similar motion are not allowed between any two parts. They are, however, much less objectionable when taken by contrary motion, especially if one of the parts be a middle part, and the progression be between primary chords.

Ebenezer Prout, Harmony: Its Theory and Practice, §71, p. 26. Read from the Augener printing scanned at archive.org (harmonyitstheor00prouiala).

His own Counterpoint, one year earlier, does not:

Consecutive unisons, perfect fifths, and octaves, are absolutely forbidden between any of the parts. Perfect fifths and octaves are not allowed even by contrary motion, except in counterpoint of at least seven or eight parts; and even then they should not be used unless absolutely unavoidable.

Prout, Counterpoint: Strict and Free, §25, p. 23. Scan: counterpointstri00prouuoft.

That single switch, between the same author's two books, moves the count from 19 to 52.

Ornamental notes. The default here is that a fifth is a fifth however it is made, and that is not a liberty this page took. It is the rule:

He can introduce auxiliary and passing notes wherever he thinks it advisable; but he must remember always to take care that such notes do not make consecutive fifths or octaves with the harmony notes that immediately precede or follow them.

Prout, Harmony, §332, p. 144.

A fifth simply held. The 1,457 candidates excluded from every reading are excluded on the same authority:

It is important to remember that the repetition of the same octave or unison by two parts does not make consecutives, as the parts are not then moving in octaves or unisons. The same remark applies to the consecutive fifths forbidden in Rule 2.

Prout, Harmony, §67, p. 25.

The one switch with no such authority behind it is the fermata. Prout mentions a phrase boundary once, in passing, as an excuse for a particular passage. Dahn sets his 26 phrase-boundary cases aside explicitly and gives his reasoning. It is a real question and it is unsettled, which is why it is a switch and not a decision.

Five claims, put to the corpus

The two rulebooks quoted here do not only prohibit. They make claims about what this music actually does, in wordings specific enough to be wrong. Five of those claims can be measured against the 370 chorales rather than argued about. Three hold, one holds to the letter including its own escape clause, and one fails in an interesting direction.

The one that holds to the letter

Prout does not simply permit the diminished fifth followed by a perfect one. He conditions it twice, and adds an "occasionally" that reads like the hedge of a man who has looked at a lot of music:

But a diminished fifth followed by a perfect fifth is forbidden between the bass and any upper part, but allowed between two upper or middle parts, provided the lower, or occasionally the upper part moves a semitone.

Prout, Harmony, §73, p. 27.

In the 370 chorales that succession occurs 65 times in parallel motion within a phrase. Not one of them involves the bass. 61 have the lower part moving a semitone, and every one of the remaining 4 has the upper part moving a semitone instead, which is the clause Prout wrote as an afterthought. 65 of 65. A rule written in 1889, with two conditions and a hedge, describing a corpus printed a century earlier, without a single exception.

The one that fails, and the way it fails

Kirnberger's rule for the hidden fifth is graded, in a way no modern textbook is:

[verdeckte Q]uinten gegen den Baß sorgfältig vermieden werden. Diese grosse Sorgfalt hat man in Ansehung der obern Stimmen gegen die Mittelstimmen eben nicht nöthig; wiewol einem feinen Gehör auch da durchgehende Quinten fühlbar sind. Man findet sogar, daß der selige Bach verdeckte Quinten zuweilen in den Mittelstimmen vermieden hat.

Hidden fifths against the bass are to be carefully avoided. This great care is not necessary as regards the upper voices against the middle voices, though a fine ear feels passing fifths there too. One even finds that the late Bach sometimes avoided hidden fifths in the middle voices.

Johann Philipp Kirnberger, Die Kunst des reinen Satzes in der Musik, Erster Theil (Berlin and Königsberg: Decker and Hartung, 1774 printing of the 1771 first edition), in the section on purity of part-writing, around p. 160. German transcribed from the OCR of the copy scanned at archive.org (diekunstdesreine00kirn), Fraktur normalised, one damaged word emended in brackets. English rendering ours.

Measured with the ratio in the table above, the three pairs involving the bass sit at 0.67 and the three pairs among the upper voices at 0.49. Perfect consonances are approached by similar motion more freely against the bass than among the upper voices. The prescription and the practice run opposite ways. Kirnberger's last sentence, offered as a curiosity rather than as the rule, is the part the corpus actually supports.

On the rule itself he is unambiguous, and it is worth seeing how a prohibition looked before the classroom flattened it. Page 143, three numbered clauses:

Von diesen Bewegungen sind folgende drey Regeln zu beobachten, wodurch in den meisten Fällen die verbothenen Octaven- und Quintenfortschreitungen vermieden werden. 1. Von einer vollkommenen Consonanz, nämlich einer Octave oder Quinte, zur andern, muß man durch die [Gegen]bewegung, oder durch die Seitenbewegung gehen.

Of these motions the following three rules are to be observed, by which in most cases the forbidden successions of octaves and fifths are avoided. 1. From one perfect consonance, that is an octave or a fifth, to another, one must proceed by contrary motion or by oblique motion.

Kirnberger, same volume, p. 143.

Note what that permits. Requiring contrary or oblique motion between perfect consonances licenses a fifth moving to a twelfth in contrary motion, which is exactly what the third switch above turns on, and exactly what Prout's Counterpoint forbids absolutely. Kirnberger was called a Quinten- und Oktavenjäger, a fifth and octave hunter, by his own pupil J. A. P. Schulz, and his rule is more permissive than the one a modern harmony class states.

The one that inverts

The rule Bach keeps most absolutely is not one of the famous prohibitions. It is the melodic augmented second, which he never writes once in 82,466 intervals. And Prout, in the same book that forbids consecutive fifths in bold capitals, lets it through:

60. An augmented interval should seldom be used in melody unless both the notes belong to the same harmony. But the interval of the augmented second, which we find in the minor scale (Ex. 5) between the sixth and seventh degrees may be used more freely.

Prout, Harmony, §60, p. 23.

So the flat prohibition is broken 19 times and the relaxed permission is declined 100 times out of 100. Whatever the rulebook is tracking, on this point it is not tracking the corpus. Kirnberger, a century earlier, is closer: he calls the melodic augmented second wiedrige, repellent, and that is the word the count supports.

Whose fifths are these?

Every note counted on this page is a note in the Breitkopf print of 1784 to 1787, in Alfred Dörffel's later re-engraving of it, read through Craig Sapp's digital edition. That is the right corpus for this question, because it is the collection the rules were drawn from and the one nearly every student has been graded against. It is not the same thing as Bach's manuscript, and the difference is not academic.

The editors removed parallels Bach wrote. Luke Dahn compared the print against the surviving originals for those chorales that came from larger works, and found that 8 of 11 checkable instances of consecutives had been quietly altered away: a passing note delayed by a sixteenth here, an anticipation removed there, an alto line rewritten as a suspension. Comparing the print against the c.1735 Dietel manuscript turned up five more.

The editions also added parallels Bach did not write. Fitsioris and Conklin checked their own findings against Frieder Rempp's critical edition and showed that two of their 18 are artefacts of the print: the fifth in the G major chorale filed as BWV 194 (number 64 here) does not appear in Rempp at all, and the fifth in BWV 355 (number 169 here) comes from a soprano note that they argue is wrong in every edition and should be a step lower. Both of those are in the 19 counted above, and they are counted because this page counts the print.

The collection contains its own control group. Eleven pairs of numbers in the 370 are the same setting printed twice, note for note, in the same key: 22 of the 370 numbers are one of a printed pair, leaving 359 distinct settings by that strict test. The scholarly literature counts more duplicates than this, including transposed ones and ones that differ in a note or two; the test here is deliberately unforgiving and only reports pairs that match exactly. None of the 19 falls in a duplicated number.

Two kinds of never

Line the numbers up and the rulebook splits, and the split is not the one the rulebook announces.

Absolute. The melodic augmented second: never, in 82,466 intervals. Consecutive octaves: 6, in 6,829 chances. Consecutive fifths: 19, in 8,437 chances, of which 17 have an ornamental note in them and at least 2 are probably the printer's rather than the composer's. These are not tendencies. At those rates the exceptions fit on one screen, and a student who never wrote one would be imitating the corpus accurately.

A strong preference. The direct fifth is avoided, measurably, in every pair of voices, at roughly half the rate chance would give. And it happens 5,531 times. Both of those are true at once, and a rule stated as never cannot express that.

Not a constraint you can see here. Voices cross in 788 places, across 205 chorales. A voice moves past the note its neighbour has just left in 1,874 places, across 329 of the 370. The gap between two upper voices opens past an octave in 477 places. A student handing in work at those rates would come back covered in red pen.

So the interesting fact about the rulebook is not that Bach broke it. It is that the rulebook is at least three rulebooks wearing one voice. Some of its nevers are load-bearing descriptions of a practice. Some are strong preferences that the practice keeps most of the time. Some are teaching devices, useful for getting a beginner to hear four independent lines, that the practice never obeyed. Nothing in the flat wording tells them apart.

And there is a second pattern, which is the one that surprised this page most. The statements that survive contact with the corpus are the specific ones. Prout's rule for the unequal fifth carries two conditions and a hedge, and it comes out 65 for 65. His remark that the fifth rule is broken more often than the octave rule comes out at 2.6 times as often. Kirnberger's distinction between the equal and the unequal fifth comes out at a factor of twelve. What fails is what has been flattened: the bare word never, applied identically to a thing done 19 times, a thing done 5,531 times, and a thing never done at all. The tradition knew more than the classroom kept, and the part it dropped was the part that was true.

What this cannot tell you

• It is an edition, not a manuscript. Where the print differs from Bach's own hand, this page counts the print, in both directions, as the section above sets out.
• Formalising a rule is an interpretation. "Consecutive fifths" is not a well-defined predicate until someone decides about phrase endings, ornamental notes, voice crossing and the twelfth. That is why the decisions are handed over rather than made. But the switches are the ones this page thought of, and another reader could want a fifth.
• The ornamental-note test is a shape, not an analysis. A note is called passing here when it is stepped into and out of in the same direction, and a neighbour when it steps away and back. That is objective and needs no theory, and it will also catch chord tones that happen to sit between two steps. A real harmonic analysis would classify some of them differently and would itself be arguable.
• Written occurrences, not heard ones. The score is read once through as printed. Where a chorale carries a repeat, and 181 of the 370 do, the material inside it is counted once and the junction at the repeat sign is not examined. Dahn's measure numbers count repeats and this page's do not, which is why the reconciliation above matches on chorale rather than on bar for his list.
• One discrepancy is unresolved. Dahn lists a consecutive in BWV 190/7, number 327 here, among his fermata cases. Nothing in this edition of that chorale meets any reading on this page. It may be a difference of edition, of definition, or of measure numbering, and it is left standing rather than explained away.
• A count is not a judgement. "Bach does this 5,531 times" is not an argument that you should. It is only an argument that the rule as stated is not a description of this music.

Sources, and how to redo this

• The scores. Craig Stuart Sapp, 371 Bach chorales in the Humdrum **kern format, referenced against 371 vierstimmige Choralgesänge von Johann Sebastian Bach, 4th ed. by Alfred Dörffel (Leipzig: Breitkopf und Härtel, c.1875). 370 files: the source edition's number 150 is a five-part setting and is omitted, while the numbering is kept. github.com/craigsapp/bach-370-chorales
• The earlier counts. George Fitsioris and Darrell Conklin, "Parallel successions of perfect fifths in the Bach chorales", Proceedings of the Fourth Conference on Interdisciplinary Musicology (CIM08), Thessaloniki, 2008, PDF; Luke Dahn, "Consecutive 5ths and Octaves in Bach Chorales", For M is Musick, 15 April 2016, and "'Correcting' Bach's Parallel Fifths: Breitkopf Editorial Practices", 22 July 2016. Dahn's bach-chorales.com is the fuller and later version of his figures.
• The rulebook. Johann Philipp Kirnberger, Die Kunst des reinen Satzes in der Musik, Erster Theil (Berlin: Voss, 1771; the printing read here is Berlin and Königsberg: Decker and Hartung, 1774), scanned at archive.org and on IMSLP.
• The reading and the counting. research/bach-chorale-rulebook/ in this project's repository: kern.mjs (the score reader), analyze.mjs (every motion between every pair of voices), reduce.mjs (the 36 readings), reconcile.mjs (the comparison above), duplicates.mjs (the printed pairs), independent-check.py (the second opinion).
• The corpus, on this page. Every note of all 370 chorales is loaded here, not just the passages discussed, so a reader who doubts a count can open any bar of any chorale and look at it. That is what scores.json is: 85,448 notes in 431 KB, one note to four characters.