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### Solutions for Act III Scene 2 - Die Wohltemperierte Sequenz

This puzzle consists of excerpts of 20 fugues from Bach's Well Tempered Clavier (Das Wohltemperierte Klavier) as alluded to by the title. Each excerpt begins at the start of the piece and ends at the end of the first entry of the (first) subject. The first thing that one notices when comparing the excerpts to a score (available from IMSLP) of the Well Tempered Clavier (WTC) is that the excerpts do not match up with the fugues in either book with the same number (in the same key). Indeed, they are transposed. The next logical step is to work out the actual keys of the fugues and by how many semitones they have been transposed.

From Book I we have

Original Key Number of semitones transposed
C major 11
C sharp minor 33
D major 1
D sharp minor 2
E major 6
F minor 20
F sharp minor 1
G minor 2
B flat minor 3
B major 2

The 10 excerpts correspond to the 10 distinct notes which occur in the Inhalt (contents, from German) line before the double bar, taking enharmonic notes to be equivalent. Assigning to each note the number corresponding to the number of semitones in the interval which the excerpt was transposed up gives
1 1 2 2 1 2 3 6 2 11 2 1 20 1 33
Now adding together the values assigned to the pairs of notes joined by slurs gives
1 1 2 3 5 8 13 21 34
the first nine terms of the familiar Fibonacci sequence.

From Book II we have

Original Key Number of semitones transposed
C major 2
C sharp major 12
D major 0
D sharp minor 0
E major 1
F minor 6
F sharp minor 0
G minor 0
B flat minor 0
B major 0

The 10 excerpts also correspond to the 10 distinct notes which occur in the Inhalt (contents) line before the double bar, again taking enharmonic notes to be equivalent. Assigning to each note the number corresponding to the number of semitones in the interval which the excerpt was transposed down and adding together the values assigned to the pairs of notes joined by slurs gives
0 0 0 0 0 1 2 6 12
This sequence does not appear to be well known.

Now the fact that the Book I excerpts have all been transposed upwards and the Book I excerpts have all been transposed downwards implies subtraction. More specifically, subtracting the Book II sequence from the Book I sequence termwise gives
1 1 2 3 5 7 11 15 22
the first nine partition numbers. This can be found by searching for the sequence in the OEIS. The fact that the sequences are nine numbers long hints at the answer being nine letters long.

Design notes:
The Inhalt (contents) line is the subject of fugue 24 from the first book of the WTC. It is the only fugue subject in the WTC which contains all 12 notes of the chromatic scale. (Side note: its subject also lasts for 24 quavers. One might speculate that Bach deliberately chose these constraints for the final fugue of the book.) This, along with the ability to transpose (which we with our modern equal temperament system take for granted) was the main inspiration for this puzzle. I did not remove the portion of the subject after the double bar line so that the subject could be displayed in its entirety and for consistency with all of the other fugue subjects presented.

Stem directions were ignored throughout the excerpts to be consistent with the clef being ignored in FUGA XXII (1) as this would otherwise call for an absurd amount of ledger lines.

The fugue in C sharp minor from Book II was swapped out for the fugue in C sharp major to avoid the subject being transposed beneath the lowest note which appears in the WTC. However, the C sharp major fugue comes with its own problems---it opens with a stretto. In hindsight it would have been simpler to just use the C sharp minor fugue.

Minor errors:
FUGA XXIII (2) should be in cut common time.
FUGA XX should have its last note removed.