Lessons · Showcase

Toccata & Fugue

The whole of BWV 565 in fifty-six named figures

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% Toccata and Fugue in D minor, BWV 565 — the whole work, 143 measures, in
% one grammar. One unit is a sixteenth and the quarter is at 60, so a
% measure is sixteen units of common time.
%
% Three shapes pay for almost everything in it, and the piece sounds each
% one bare before it starts. A pair of rising steps; a pair of falling
% steps; and a TURN — a note, its upper neighbour, its lower neighbour,
% itself. The turn is the piece's signature: it occurs twenty-six times, at
% nine pitches, in every voice the piece has, and all but one of them is
% written below as this one rule and an interval — the exception straddles
% two other rules and could not be. The last of the four figures sounded
% here is simply the turn a fourth higher, and needs no rule of its own.
 
bpm 60
meter 4
unit 1/4
key d4 min
mode diatonic
 
Lift ::= .6/2 .7/2 1/2
Fall ::= .7 .6 .5
Turn ::= 6 7 5 6
 
assert Lift == bb3/2 c4/2 d4/2
assert Fall == c4 bb3 a3
assert Turn == bb4 c5 a4 bb4
assert Turn@5 == eb5 f5 d5 eb5
 
Lift
Fall
Turn
Turn@5
 
 
% mm.1-2. Three cries, each one hand doubling the other at the octave,
% each answered by three units of nothing. The first is a held A and a
% scale falling to the leading note; the second is short — A, then just
% e f c# d; the third is the first one again, two octaves down, where the
% pedals are.
 
Cry   ::= '5 _2 -/2 '4/4 '3/4 '2/4 '1/4 7#/2 '1
Reply ::= 5 _2 -/2 2/2 3/2 .7#/2 1
 
assert Cry == a5 _2 -/2 g5/4 f5/4 e5/4 d5/4 c#5/2 d5
assert Reply == a4 _2 -/2 e4/2 f4/2 c#4/2 d4
 
{Cry, Cry@-12} -*3 {Reply, Reply@-12} -*3 {Cry, Cry@-12}@-24 -*3
 
 
% mm.2-3. Under a D in the pedals, the whole organ takes a diminished
% seventh on C# across two octaves, holds it for half a measure, and
% resolves to D. Then a bare C# in octaves, and the toccata can begin.
 
{ -*4 {c#3, e3, bb3} _8 {d3, a3} _4 -*7 c#3
  , -*4 g3 _9 e3 f#3 _2 -*8
  , -*4 {c#4, e4} _8 d4 _4 -*7 c#4
  , d2 _16 -*8 }
 
 
% mm.4-7. A three-note coil in triplets — the tonic, the step above, the
% step below — turning three times and then walking away. Under the tonic
% the step below is the raised leading note; stepped up to the third and
% the fifth of the key it is a plain scale step, so the same figure comes
% twice more with no sharp in it. The two measures are then said again an
% octave higher, and both hands do the whole thing in octaves.
 
Coil ::= 1 _2/3 2 _2/3 .7 _2/3
Home ::= 1 _2/3 2 _2/3 .7# _2/3
Step ::= 1 2
 
assert Home == d4 _2/3 e4 _2/3 c#4 _2/3
assert Coil@~2 == f4 _2/3 g4 _2/3 e4 _2/3
assert Coil@~4 == a4 _2/3 bb4 _2/3 g4 _2/3
assert Step@~2 == f4 g4
 
Wave  ::= Home*3 Step Coil@~2*3 Step@~2 Coil@~4*3 5 -*8
Waves ::= Wave 7# Wave@12 '5
 
{Waves, Waves@-12}
 
 
% mm.8-10. Ten broken chords falling through the scale, each one said
% twice before the next — and each is the one before it a scale step lower,
% so nine of the ten are one figure and a step. The tenth raises its lowest
% note, because there that note is the leading note and has to lead home.
% The chord it lands on is the diminished seventh again, over the pedal D.
%
% mm.11-12. A scale down out of the held B flat, and the toccata reaches
% its first full cadence.
 
Chain ::= '4 _2/3 '6 _2/3 '2 _2/3
Sharp ::= 2 _2/3 4 _2/3 .7# _2/3
 
assert Chain == g5 _2/3 bb5 _2/3 e5 _2/3
assert Chain@~-3 == d5 _2/3 f5 _2/3 bb4 _2/3
assert Sharp == e4 _2/3 g4 _2/3 c#4 _2/3
 
Arc ::= Chain*2 Chain@~-1*2 Chain@~-2*2 Chain@~-3*2 Chain@~-4*2
  Chain@~-5*2 Chain@~-6*2 Chain@~-7*2 Chain@~-8*2 Sharp*2
 
{ Arc -*4 bb4 _17/2 a4/2 g4/2 f4/2 e4/2 d4/2 c#4/2 b3/2 c#4 a3 c#4 e4/2 g4/2 f4 _3 e4
  , Arc@-12 -*4 {c#3, e3, g3, bb3} _8 -*12
  , -*44 {c#4, e4, g4} _8 -*12
  , -*40 d2 _12 -*12 }
 
 
% mm.12-15. A repeated A starts under the cadence and keeps going for
% three measures, a note to the unit, half a unit late every time. Over
% it the right hand runs down through three turns and then says one whole
% measure twice, the second time an octave lower — and that measure has a
% fourth turn inside it, so the octave repeat brings a fifth.
 
Drone  ::= -/2 a4*15 a4/2
Tumble ::= 5 6 4 5 Turn@-5 1 4 2 3 1 2 .7#
 
assert Tumble == a4 bb4 g4 a4 f4 g4 e4 f4 d4 g4 e4 f4 d4 e4 c#4
 
{ {d2, d3, d4, f4} _4 -*3 a4 '1 '2 '3 '1 '2 Turn@7 '4 '5 '3 '4 '5 '6 '4 '5
  Turn@7 Turn@4 Tumble 1 Tumble@-12
  , - _17/2 a4*7 a4/2 Drone*3 }
 
 
% mm.16-17. The same measure twice. Fourteen units of spray in four
% voices, and the only difference between the two statements is what
% stands under the first two units: the second time the whole A major
% chord is there.
 
Cascade ::= d4/2 f4/2 bb4/2 f4/2 c4/2 e4/2 a4/2 e4/2 -/2 d4/2 g4/2 d4/2 -/2 c#4/2 e4/2 a4/2
Tread   ::= d3 _2 c3 _2 bb2 _2
 
Spray ::= { Cascade d4 bb4 - a4 - g4
  , -*9 f4 - e4 - d4
  , -*4 bb3/2 -*3/2 a3/2 -*7/2 a3 - bb3 -
  , -*8 Tread }
 
d3 _2 Spray
{a2, e3, c#4, e4, a4} _2 Spray
 
 
% mm.18-20. A scale climbing out of the A major chord, faster and faster,
% and then the cascade from the twin measures comes back an octave higher
% with two of its rests filled in. The bell between the cascades is the
% same eight units twice, and under everything the bass treads down
% d c bb a, twice.
 
Toll ::= { d5 bb5 a4 a5 bb4 g5 {e4, c#5, e5, a5} _2
  , - f5 - e5 - d5 -*2 }
 
{ a4 _9/2 g4/2 f4/2 e4/2 d4/2 c#4/2 b3/2 c#4/2 a3/2 b3/2 c#4/2 d4/2 e4/2 f4/2 g4/2 a4/2 g4/2 f4/2 e4/2 f4/2 d4/2 f4/2 a4/2 c#5/2
  d5/2 a4/2 b4/2 c#5/2 d5/2 e5/2 f5/4 g5/4 a5/2 bb5 _2
  Toll {Cascade@12, -*4 bb4/2 - _3/2 a4/2 - _3/2} Toll b4 _2
  , {a2, e3, c#4, e4} _4 -*42 {b3, d4, f4} _2
  , -*22 Tread a2 _2 -*8 Tread a2 _2 g#2 _2 }
 
% m21. One more flight up to a high B flat and a scale all the way down.
 
{ a4 _4 a4 c#5 e5/2 g5/2 bb5 a5/2 g5/2 f5/2 e5/2 f5/2 e5/2 d5/2 c#5/2 d5/2 c5/2 bb4/2 a4/2 g4/2 f4/2 e4/2 d4/2
  , c#5 _3 b4 -*12
  , {g2, a3, e4} _4 -*12 }
 
 
% mm.22-26. Five measures on one diminished seventh, and the two hands say
% the SAME figure a ninth apart the whole way through — so one voice is the
% other transposed and needs no notes of its own. The figure is a rocking
% cell in triplets: a note against the third above it, then twice against
% the third below. It sinks by tritones (C# to G to C#), then leans upward
% instead, then shrinks to a bare tremolo. One cell in the middle turns up
% at the end where its neighbours turn down, and once near the end the
% lower hand holds its tremolo where the upper has moved on.
 
Rock    ::= 7# _2/3 '2 _2/3 7# _2/3 6 _2/3 7# _2/3 6 _2/3
Bend    ::= 4 _2/3 6 _2/3 4 _2/3 2 _2/3 4 _2/3 6 _2/3
Reach   ::= .7# _2/3 2 _2/3 .7# _2/3 2 _2/3 4 _2/3 2 _2/3
Tremble ::= 4 _2/3 6 _2/3 4 _2/3 6 _2/3 4 _2/3 6 _2/3
 
assert Rock == c#5 _2/3 e5 _2/3 c#5 _2/3 bb4 _2/3 c#5 _2/3 bb4 _2/3
assert Rock@-6 == g4 _2/3 bb4 _2/3 g4 _2/3 e4 _2/3 g4 _2/3 e4 _2/3
assert Bend[:5] == Rock[:5]@-6
assert Tremble@6 == c#5 _2/3 e5 _2/3 c#5 _2/3 e5 _2/3 c#5 _2/3 e5 _2/3
 
Descent ::= c#5 _26/3 Rock[1:] Rock*3 Rock@-6
Close   ::= Rock@-6*2 Rock@-12*4 Reach*4
 
{ Descent Bend Close Tremble*2
  , {Descent Rock@-6 Close}@-9 Tremble@-15 Tremble@-9
  , {g3, c#4, g4, bb4, e5} _8 -*72 }
 
 
% mm.27-29. The tremolo narrows to C# against E, the full organ answers
% with four chords, and a long scalar descent in the left hand brings the
% toccata to a dead stop on D. m30 begins with that D held: the fugue
% starts underneath it.
 
{ 7# _2/3 6 _2/3 7# _2/3 '2 _2/3 7# _2/3 '2 _2/3 Tremble@6
  {c#4, e4, a4, c#5, e5, a5} _4 {d4, f4, a4, d5, f5, a5} _4
  {d4, g4, bb4, d5, g5} _4 -*4 {c#4, e4, a4, c#5, e5, g5} _4
  -*11 {e4, a4, c#5} {d4, f4, a4, d5} _2 d4 _4 c#4 _2 d4 _8
  , Tremble~@-6*2 a3 _4 a3 _4 -*23 a3 a3 _2 f4 _2 e4 _4 -*8
  , -*8 g3 _4 f3 _4 bb3 _6 a3 g3 a3 _3 e3 f3 Turn@-20 b2 c#3 a2 bb2 g#2 a2
  g3 f3 _2 a3 _5 g3 _3 f3 e3 f3 _4
  , -*48 {d3, a3} _8
  , -*42 d3 _2 a2 _4 d2 _8 }
 
 
% mm.30-34. The fugue, and everything in it comes from one figure. The
% subject is a line with a nail through it: a repeated A, and between every
% two A's one note of a voice that walks down g f e d c#, back up d e f,
% drops to the A an octave below, climbs a b c#, and settles d c# d e f.
% Thirty-two notes. It comes back TWELVE more times in the next ninety-five
% measures — ten of them whole or nearly, and twice as its head alone in the
% pedals — in five voices and at eight pitches. Every one of them is this
% rule and a transposition, and an entry cut short is this rule sliced.
%
% The answer comes in over the subject's own last note, a fifth higher, and
% is the same shape to the semitone. Under it the first voice spins out its
% counter-line in eighths.
 
Walk    ::= 4 3 2 1 .7# 1 2 3 .5 .6# .7# 1 .7# 1 2
Axis    ::= $1 5
Subject ::= 5 4 5 3 5 2 5 1 5 .7# 5 1 5 2 5 3 5 .5 5 .6# 5 .7# 5 1 5 .7# 5 1 5 2 5 3 _2
 
assert Subject == 5 Axis(Walk) 3 _2
assert Subject == a4 g4 a4 f4 a4 e4 a4 d4 a4 c#4 a4 d4 a4 e4 a4 f4 a4 a3 a4 b3 a4 c#4 a4 d4 a4 c#4 a4 d4 a4 e4 a4 f4 _2
assert Subject@5 == d5 c5 d5 bb4 d5 a4 d5 g4 d5 f#4 d5 g4 d5 a4 d5 bb4 d5 d4 d5 e4 d5 f#4 d5 g4 d5 f#4 d5 g4 d5 a4 d5 bb4 _2
 
{ -*33 Subject@5 d5 _2 bb4 _2 d5 _2
  , - Subject f#4 _2 g4 _2 c4 _2 bb3 _2 a3 _2 bb3 _2 c4 _2 d4 _2 f#3 _2 g3 _2 a3 _2 bb3 _2 a3 _2 bb3 _2 f#3 _2 g3 g4 g3 g4 d4 g4 d4 g4 }
 
 
% mm.35-38. Eight half-measures of two notes rocking against each other,
% the right hand in eighths and the left in sixteenths. Not one of them is
% a transposition of the last — the interval opens and closes as the
% harmony walks down — so this passage is written out, and that IS what it
% is: a chain of changing dyads between two statements of the subject.
 
{ eb5 _2 g4 _2 eb5 _2 g4 _2 c5 _2 a4 _2 c5 _2 a4 _2 d5 _2 f4 _2 d5 _2 f4 _2 bb4 _2 g4 _2 bb4 _2 g4 _2
  c#5 _2 e4 _2 c#5 _2 e4 _2 a4 _2 f4 _2 a4 _2 f4 _2 g4 _2 c#4 _2 g4 _2 c#4 _2 f4 _2 d4 _2 f4 _2 d4 _2
  , c4 eb4 c4 eb4 c4 eb4 c4 eb4 c4 f4 c4 f4 c4 f4 c4 f4 bb3 d4 bb3 d4 bb3 d4 bb3 d4 bb3 e4 bb3 e4 bb3 e4 bb3 e4
  a3 c#4 a3 c#4 a3 c#4 a3 c#4 f3 d4 f3 d4 f3 d4 f3 d4 e3 bb3 e3 bb3 e3 bb3 e3 bb3 d3 a3 d3 a3 d3 a3 d3 a3 }
 
 
% mm.39-41. The third entry, an octave above the first, over a repeated A
% in the middle of the texture. It runs thirty-one of the subject's
% thirty-two notes and then goes its own way.
 
{ e4 _2 bb3 _2 e4 _2 bb3 _2 - Subject[:31]@12 f5 a5 e5 a5 d5 a5 c5 a5
  , e3 g3 e3 g3 e3 g3 e3 g3 a3 _2*16 a3 _4 d4 _4
  , -*8 f3 _2 e3 _2 d3 _2 g3 _2 f3 _2 e3 _2 f3 _2 c#3 _2 d3 _2 c#3 _2 d3 _2 e3 _2 f3 _2 e3 _2 f3 _2 c#3 _2 d3 _4 f3 _4 }
 
 
% mm.42-44. One measure, then the same measure a step lower, then a step
% lower again. The measure is the nailed axis again — one high note repeating
% while a line climbs under it — and it needs only two high notes to do it.
% Only the third statement is touched: its ninth note is raised, because
% there that note is the leading note of A.
 
Peal ::= 6 '5 7 '5 '1 '4 6 '4 '2 '4 '1 '4 7 '4 6 '4
 
assert Peal == bb4 a5 c5 a5 d5 g5 bb4 g5 e5 g5 d5 g5 c5 g5 bb4 g5
assert Peal@~-1 == a4 g5 bb4 g5 c5 f5 a4 f5 d5 f5 c5 f5 bb4 f5 a4 f5
 
{ Peal Peal@~-1 Peal[:8]@~-2 7# Peal[9:]@~-2
  , d4 _4 -*4 c4 _4 c4 _4 c4 _4 -*4 bb3 _4 bb3 _4 bb3 _4 -*4 a3 _4 a3 _4
  , g3 _4 -*4 c3 _4 e3 _4 f3 _4 -*4 bb2 _4 d3 _4 e3 _4 -*4 a2 _4 c#3 _4 }
 
 
% mm.45-47. Two measures whose second halves are identical, over a sighing
% figure in the alto said twice without a note changed, and a bass that
% says it twice with two notes changed. Then a cadence.
 
Sigh ::= -*2 d4 _2 c#4 _2 c#4 _2 d4 _2 d4 _2 e4 _4
 
{ f4 e5 g4 e5 a4 d5 f4 d5 e4 e5 e4 e5 f4 d5 f4 d5
  bb4 c#5 bb4 c#5 a4 d5 f4 d5 e4 e5 e4 e5 f4 d5 f4 d5
  - d5 c#5 d5 b4 d5 c#5 b4 c#5 _4 e4 g4 f4 e4
  , -*41 a4 g4 a4 -*4
  , a3 _4 Sigh Sigh -*12
  , d3 _4 -*2 f3 _2 bb3 _2 bb3 _2 a3 _2 a3 _2 g3 _4 -*2 a3 _2 bb3 _2 bb3 _2 a3 _2 a3 _2 g#3 _4 -*12 }
 
 
% mm.48-55. Free counterpoint over eight measures, and halfway through it
% the FOURTH entry arrives in the pedals — thirty-one notes of the subject
% two octaves and a fifth below where it started, while the right hand
% marks eleven plain eighths on D above it. The feet cannot play fast, so
% this is the subject at its slowest and its plainest, and it is the same
% rule.
 
assert Subject[:31]@-19 == d3 c3 d3 bb2 d3 a2 d3 g2 d3 f#2 d3 g2 d3 a2 d3 bb2 d3 d2 d3 e2 d3 f#2 d3 g2 d3 f#2 d3 g2 d3 a2 d3
assert d5 _2*3 == d5 _2 d5 _2 d5 _2
 
{ f4 _4 d4 _4 c#5 _4 e5 _6 d5 _4 c#5 _2 c5 _4 bb4 _4 a4 _4 a4 _4 g4 _4 eb4 _6
  a4 _4 c5 _4 bb4 a4 bb4 _2 g5 _4 f#5 _2 g5 _4 bb4 _2 d5 _2*11
  d5 g5 Turn@7 d5 e5 c5 a5 g5 a5 Turn@7
  d5 bb5 a5 bb5 g5 a5 f5 g5 e5 c6 Turn@12 g5 a5
  , - d5 c#5 d5 f5 d5 c#5 b4 a3 a4 g4 a4 -*24 f#4 _4 f4 _4 -*30
  a4 _2 bb4 _2 f#4 _2 g4 _2 f#4 _2 g4 _2 a4 _2 bb4 _2 a4 _2 bb4 _2 f#4 _2
  g4 bb4 a4 bb4 g4 a4 f4 g4 e4 c5 Turn g4 a4
  f4 d5 c5 d5 Turn g4 e5 d5 e5 c5 d5 bb4 c5
  , -*12 c#4 g4 f4 e4 f4 _4 e4 _4 d4 _2 a4 _4 g4 _6 -*8 g4 _4 f#4 _2 d4 _2 f#4 _2 eb5 _2 d5 _4 -*2 bb4 _2 a4 _4 d5 _2 c5 _2 -*56
  , -*17 d4 c#4 d4 a3 a3 g3 a3 f#3 d4 c#4 d4 g3 f4 eb4 d4 c#4 e4 a3 c#4 d3 eb4 d4 c4 b3 d4 g3 b3 c3 d4 c4 bb3 a3 c4 f#3 a3 d3 c4 bb3 a3 bb3 a4 g4 f#4 g4 bb3 a3 g3
  d4 _4 -*2 d4 _2 d4 _2 d4 _2 d4 _2 c4 _2 bb3 _2 d4 _2 d4 _2 c4 _2 bb3 _2 d4 _2 d4 _2 c4 _2 bb3 _4 -*28
  , -*65 Subject[:31]@-19 bb2 _4 -*2 b2 _2 c3 _4 -*2 c#3 _2 d3 _4 -*2 d3 _2 e3 _4 -*2 e3 _2 }
 
 
% mm.56-59. The subject's own device turns up in the episode on a different
% note: a line nailed to a repeated C, the moving voice climbing f e f g a,
% dropping an octave to C and coming back d e. First the whole of it, then
% only its opening four steps.
 
Climb ::= .3 .2 .3 .4 .5 ..7 .1 .2
Nail  ::= $1 .7
 
assert Nail(Climb) == f3 c4 e3 c4 f3 c4 g3 c4 a3 c4 c3 c4 d3 c4 e3 c4
 
{ a4 _2 f4 _2 bb4 a4 bb4 a4 bb4 d5 bb4 a4 g4 bb4 g4 f4 e4 f4 e4 f4 g4 _2 e4 _2 a4 _4 c5 _2 bb4 _2
  , -*4 d5 c5 g4 f4 g4 bb4 g4 f4 e4 g4 e4 d4 c4 d4 g4 a4 -*4 f4 _4 -*4
  , f5 eb5 d5 c5 -*16 bb4 d5 c5 bb4 -*8
  , -*25 c4 bb3 c4 a3 c4 g3 c4
  , f3 _2 a2 _2 bb2 _2 d3 _2 g2 _4 -*2 bb2 _4 a2 _2 g2 _2 c3 _2 f2 _4 -*4 }
 
{ a4 _2 g4 _2 a4 _2 bb4 _2 c5 _2 e4 _2 f4 _2 g4 _2 a4 _2 g4 _2 a4 _2 bb4 _2 c5 bb4 a4 g4 f4 eb4 d4 c4
  , -*16 f3 c4 e3 c4 f3 c4 g3 c4 a3 _4 -*4
  , Nail(Climb) -*16 }
 
 
% mm.60-65. The clearest sequence in the fugue. Four times a held note and
% a scale falling an octave and a half out of it, each one a step higher
% than the last; over it a two-note sigh, also rising by step. Then the
% whole texture locks: a low note against a rising pair high above,
% alternating with the same pair over C, two measures of one and two of the
% other, twice.
 
Sink ::= .6 _2 6 5 4 3 2 1
Ring ::= '1 7 -*6
Ping ::= '5 '3 -*3 '3 -*2
Call ::= 3 _2 '2 '3 7 _2 '2 '3
Echo ::= 2 _2 '1 '2 7 _2 '1 '2
 
assert Sink == bb3 _2 bb4 a4 g4 f4 e4 d4
assert Sink@~3 == e4 _2 e5 d5 c5 bb4 a4 g4
assert Ring@~2 == f5 e5 -*6
assert Ping@~-1 == g5 e5 -*3 e5 -*2
assert Call == f4 _2 e5 f5 c5 _2 e5 f5
assert Echo == e4 _2 d5 e5 c5 _2 d5 e5
 
{ Sink Sink@~1 Sink@~2 Sink@~3 Call*2 Echo*2 Call*2 Echo*2
  , Ring Ring@~1 Ring@~2 Ring@~3 Ping*2 Ping@~-1*2 Ping Ping[:4] -*18
  , -*80 Ping@~-1*2 }
 
 
% mm.66-70. One gesture on four harmonies in turn, each said twice: the
% third below, back, then up through the chord and down again. The chords
% are G seventh, C seventh, B flat and A seventh, and because they are
% different chords the intervals are different too — which is why these are
% four rules and not one rule transposed.
 
OnG  ::= 6# 4 6# '1 '3 '5 '3 '1
OnC  ::= 6 4 6 7 '2 '4 '2 7
OnBb ::= 5 3 5 7 '1 '3 '1 6
OnA  ::= 4 2 4 6 7# '2 7# 6
 
assert OnG == b4 g4 b4 d5 f5 a5 f5 d5
assert OnA == g4 e4 g4 bb4 c#5 e5 c#5 bb4
 
{ d4 _2 f5 e5 d5 c5 b4 a4 OnG*2 OnC*2 OnBb*2 OnA*2
  , f5 g5 -*70 }
 
 
% mm.70-72. The fifth entry, an octave above the first again, and this time
% it is broken off after twenty-seven notes and turned into a scale down.
 
{ - Subject[:27]@12 d5 _2 e5 _2 d5 _2 c5 _2 bb4 _2 a4 _2
  , -*34 a4 _2 g4 _2 f#4 _2
  , a4 _2 c#5 _2 d5 _2 g4 _2 f4 _2 a4 _2 b4 _2 c#5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 c#5 _2 - f#4 e4 f#4 - d4 c4 d4
  , -*29 a4 g4 a4 -*8 }
 
 
% mm.73-82. Ten measures with almost nothing in them but two eight-note
% arpeggios trading places — one on the A seventh, one on D — and then a
% third that climbs where they fell. The pattern of statements is the whole
% passage: curl, curl, drop, drop, curl, curl, drop; then rise, and a bell
% of three notes answered by four units of silence, over and over.
 
Curl ::= ..7# ..5 ..7# .2 .4 .6 .4 .2
Drop ::= .1 ..5 .1 .3 .5 1 .5 .3
Rise ::= .2 ..7# .2 .4 .6 .7# .6 .4
Lap  ::= .3 .1 .3 .5 -*3 .5
Bell ::= 1 3 1
 
assert Curl == c#3 a2 c#3 e3 g3 bb3 g3 e3
assert Drop == d3 a2 d3 f3 a3 d4 a3 f3
assert Rise == e3 c#3 e3 g3 bb3 c#4 bb3 g3
assert Lap@~1 == g3 e3 g3 bb3 -*3 bb3
 
{ g4 _2 g4 f4 e4 d4 c#4 b3 -*72
  , bb4 a4 -*62 Drop Rise
  , bb3 c4 bb3 a3 g3 f3 e3 d3 Curl Curl Drop Drop Curl Curl Drop -*16 }
 
{ -*12 Bell -*5 Bell -*21 Bell -*5 Bell -*5 c#4 e4 c#4 bb3 -*4 c#4 e4 c#4 bb3 -*4 Bell -
  , Rise Lap Lap Rise Rise Lap Lap Lap[:5]@~1 - Lap[:5]@~1 - a3 f3 a3 c#4 -*3 a3 }
 
 
% mm.83-90. A long fall in the left hand — one eight-note descent said
% twice over — then two scale bursts, and the sixth entry comes in a minor
% third below the first, in the tenor. Two measures later the seventh
% follows a whole tone below, in the voice above it, over a G held for
% twenty-five units: the two statements overlap, which is what a fugue is
% for.
 
Slide ::= .6 .5 .4 .3 .4 .3 .2 .1
 
assert Slide == bb3 a3 g3 f3 g3 f3 e3 d3
assert Subject@-14 == g3 f3 g3 eb3 g3 d3 g3 c3 g3 b2 g3 c3 g3 d3 g3 eb3 g3 g2 g3 a2 g3 b2 g3 c3 g3 b2 g3 c3 g3 d3 g3 eb3 _2
 
{ - d4 bb3 -*34 f4 e4 d4 a4 -*4 e5 d5 c#5
  , bb3 -*2 g3 f3 a3 f3 d3 a2 d3 a2 f2 d2 d3 c#3 b2 c#3 Slide*2
  e3/2 f3/2 g3/2 a3/2 b3/2 c#4/2 d4 -*3 a3 b3/2 c#4/2 d4/2 e4/2 f4/2 g4/2 a4 g4 f4 e4 }
 
{ d5 eb5/2 d5/2 c5/2 bb4/2 a4/2 g4/2 f#4 _2 a4 _4 g4 _2 c5 _2 b4 _2 eb5 _2 d5 _2 eb5 _2 b4 _2
  c5 _2 b4 _2 c5 _2 d5 _2 eb5 _2 d5 _2 eb5 _2 f5 _2 g5 _33 g5 f5 g5 Turn@5
  , f4 _2 -*3 eb4 d4 c4 bb3 _2 b3 _2 c4 _2 g4 _2 g4 _25 Subject[:30]@-2 d4 eb4 _2 d4 _2 g4 g#4 f4 g4
  , -*9 Subject@-14 d3 _2 c3 _2 b2 _2 c3 _2 d3 _2 eb3 _2 f3 _2 g3 _2 b2 _2 c3 _2 d3 _2 eb3 _2 d3 _2 eb3 _2 f3 _2 g3 _2 b2 _2 c3 _2 d3 _2 }
 
 
% mm.91-95. Free writing in three voices, and the turn from the very first
% cell is all over it: on E flat at m90 and m91, on B flat at m92, on E flat
% an octave down at m91 in the left hand, and on G at m93. Then the eighth
% entry, a fifth below the first, in the middle voice over a D held for
% thirty-six units.
 
{ c5 f5 Turn@5 c5 d5 bb4 eb5 d5 eb5 c5 d5 bb4 c5 a4 d5 c5 d5 Turn g4 bb4 a4 bb4 c5 bb4 a4 g4
  , eb4 _2 c4 _2 f4 g4 eb4 f4 d4 _2 bb3 _2 Turn@-7 c4 _2 f#4 _2 g4 a4 f4 g4 eb4 g4 f#4 g4 a4 g4 f#4 e4
  , eb3 _2 a2 _2 bb2 _2 c3 _2 d3 _2 g2 _2 a2 _2 bb2 _2 c3 _2 d3 _2 g3 _2 d3 _2 eb3 _2 bb2 _2 a2 _2 c3 _2 }
 
{ f#4 _2 a4 _2 d5 g4 c5 f#4 bb4 g4 d5 a4 bb4 Turn@-3 d5 f#4 d5 g4 d5 a4 d5 bb4 g4 d5 a4 bb4 g4 c5 a4 d5 c5 bb4 a4 bb4 a4 bb4 g4 a4 bb4 c5 d5 eb5 d5 c5 d5
  , - Subject@-7 a3 _2 g3 _8 f#3 _4
  , d3 _36 -*2 g2 _2 c3 _2 a2 _2 d3 _2 d2 _2 }
 
 
% mm.96-100. The turn again, twice at the top of the measure and once
% underneath, as the music works its way back to D minor and the left hand
% climbs with it.
 
{ Turn g4 _2 b4 _2 c5 eb5 f5 g5 g#5 g5 f5 g5 Turn@5 c5 _2 bb4 _2 a4 _2 bb4 _4 a4 _4 g4 _4 f#4 _2 f4 _2 eb4 _4 d4 bb4
  , g3 _2 a3 _2 bb3 a3 g3 f3 eb3 _2 eb4 _2 d4 _4 c4 _2 d4 _2 Turn@-7 c4 eb4 d4 f4 g4 f4 eb4 f4 d4 bb3 eb4 d4 c#4 a3 d4 c4 b3 g3 c4 bb3 a3 f3 bb3 d4
  , g2 _4 -*2 g2 _2 c3 _6 b2 _2 c3 _4 -*28 }
 
{ a4 bb4 a4 g4 f#4 eb5 d5 c5 bb4 a5 g5 f#5 g5 _3 f5 Turn@5 c#5 _4 d5 _4 c5 _4
  , -*24 a4 _4 f#4 _4
  , -*21 bb4 a4 g4 -*8
  , c4 d4 c4 bb3 a3 c4 bb3 a3 g3 c4 bb3 a3 bb3 c4 bb3 a3 g3 a3 f3 g3 e3 _4 f#3 _4 d3 _4
  , -*25 eb4 d4 c4 d4 c4 bb3 a3 }
 
 
% mm.101-111. Four measures of eighths turning around D, and then the
% subject comes back where it began — at pitch, in the tenor, with one A
% of its thirty-two replaced by silence. Two measures later the tenth entry
% follows an octave lower in the pedals, and that one runs thirty-one notes
% and holds its last.
 
{ bb4 _4 a4 _2 c#5 _2 d5 _2 e5 _2 f5 _4 e5 _8 d5 _6 f5 _2 g5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 g5 _2 f5 _2 d5 _2
  , -*4 c#4 _2 e4 _2 f4 _2 g4 _2 -*18 a4 _2 g4 _2 a4 _2 g4 _2 f4 _2 g4 _2 f4 _2 e4 _2 a4 _2 bb4 _2 a4 _2 g4 _2 f4 _2 a4 _2 e5 _2 d5 _2 a4 _2
  , g4 f4 eb4 d4 -*8 a4 g4 d5 _6 c#5 _4 a4 bb4 g4 a4 f4 _2 -*34
  , -*14 a4 f4 -*48
  , g3 _4 a3 bb3 a3 g3 f3 g3 e3 a3 d3 _4 g4 a4 g4 a4 bb4 a4 g4 a4 Turn@-5 d4 c4 bb3 a3 bb3 a3 g3 f3 e3 f3 e3 d3 c#3 a2 d3 a2 e3 a2 f3 a2 g3 a2 f3 a2 e3 a2 d3 a2 c#3 a3 a2 a3 d3 a3 f3 d4 }
 
{ bb4 _2 e5 _2 d5 _2 d5 _2 c#5 _4 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 c#5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 g5 _2
  a5 bb5 g5 a5 Turn@7 d5 a4 c#5 a4 d5 a4 e5 a4 f5 a4 e5 a4 f5 a4 g5 a4 a5 a4 e5 a4 f5 a4 g5 a4 a5 _4 -*28
  , g4 _2 g4 _2 a4 _2 f4 _2 -*96
  , f3 d4 e3 c#4 f3 d4 d3 d4 a3 _18 a3 _50 -*28
  , -*8 e4 Subject[:17] - Subject[18:] e4 _2 d4 _2 g4 _2 f4 _2 e4 _2 f4 _2 c#4 _2 d4 _2 c#4 _2 d4 _2 b3 _2 c#4 _2 g4 _2 f4 _2 d4 _2 c#4 _4 -*28
  , -*73 Subject[:31]@-12 }
 
{ - e5 d5 c#5 d5 _2 - f5 e5 f5 g5 a5 bb5 a5 g5 a5 f5 _2 e5 _2 d5 _2 f5 eb5
  , -*21 e5 -*2
  , - g4 f4 e4 f4 c#4 d4 _6 c#4 _4 d4 a4 b4 c#5 -*2 c5 _2
  , f3 _4 -*2 f3 _2 g3 _2 e3 _2 a3 _2 a2 _2 d3 _4 -*2 a2 _2 }
 
 
% mm.113-119. A cadence, and then the strangest passage in the fugue: a
% four-note figure, four units of silence, the same four notes again — one
% figure to a harmony, C minor then G minor then the F sharp diminished,
% each said twice, over a D tapped in the pedals every other half-measure.
% The left hand runs an eight-note pattern twice, the second time with one
% note changed, and then a second pattern twice, the second time with its
% last two changed.
 
OnCm  ::= .7 2b .7 .6
OnGm  ::= 4 6 4 1
OnFs  ::= 3# 5 3# 2b
Tap   ::= .1 _4 -*4
Wind  ::= .3# .1 .3# .4 .5 1 .5 .4
Gust  ::= .4 .1 .4 .5 .6 1 .6 .5
 
assert OnCm == c4 eb4 c4 bb3
assert OnFs == f#4 a4 f#4 eb4
assert Wind == f#3 d3 f#3 g3 a3 d4 a3 g3
assert Wind \7 {.3#} == f#3 d3 f#3 g3 a3 d4 a3 f#3
assert Gust[:6] .7 .6 == g3 d3 g3 a3 bb3 d4 c4 bb3
 
{ d5 c5 bb4 a4 g4 a4 bb4 c5 a4 _4 -*2 c5 _4 bb4 _2 -*2 g5 _2 Turn@5 c5 bb4 a4 g4
  , f4 _6 eb4 _3 c4 a3 f3 a3 c4 eb4 c4 d4 bb3 g3 d3 g3 bb3 d4 g4 c4 d4 bb3 c4 a3 g3 f#3 e3
  , bb2 _2 g2 _2 c3 _2 c2 _2 f2 _4 -*2 a2 _2 bb2 _4 -*2 bb2 _2 c3 _4 -*2 a2 _2 }
 
{ a4 _4 -*32 OnCm -*4 OnCm -*4 OnGm -*4 OnGm -*4 OnFs -*4 OnFs
  , Wind Wind \7 {.3#} Gust Gust[:6] .7 .6 -*48
  , Tap*4 a3 f#3 a3 bb3 -*4 a3 f#3 a3 bb3 -*4 a3 bb3 d4 f#4 -*4 bb3 g3 bb3 d4 -*4 c4 a3 c4 eb4 -*4 c4 a3 c4 eb4 -*4
  , -*32 Tap*6 }
 
 
% mm.120-125. The pedals take the subject's head twice over — eight notes
% at the bottom of the instrument, then its first seven a fourth higher —
% and then the twelfth and last entry arrives in the alto AT PITCH, with
% one A dropped out of its opening and its twenty-ninth note bent down to
% B so that it can turn into the cadence instead of going on.
 
assert Subject[:8]@-19 == d3 c3 d3 bb2 d3 a2 d3 g2
assert Subject[:7]@-14 == g3 f3 g3 eb3 g3 d3 g3
 
{ d4 _2 -*2 d5 _2 c5 _2 bb4 _4 c5 _2 d5 _2 eb5 _4 f5 _2 g5 _2 a5 _4 f5 _2 eb5 _2 bb4 _2
  d5 c5 bb4 c5 bb4 a4 g4 c5 bb4 a4 g4 a4 g4 f#4 g4 f4 e4 d4 c#4 _2 e4 _2 f4 g4 a4 b4 c#5 d5 e5 f5
  , -*4 g4 _2 f#4 _2 g4 _4 g4 _2 b4 _2 c5 _4 c5 _2 e5 _2 f5 _4 bb4 _2 a4 _2 -*32
  , f#3 _2 -*2 g3 _2 a3 _2 bb3 _4 eb4 _2 b3 _2 -*16 d5 eb5 -*30
  , - Subject[:8]@-19 Subject[:7]@-14 g3 _4 c4 _2 bb3 _2 a3 _4 bb3 _2 c4 _2
  d4 g4 f4 eb4 d4 eb4 d4 c4 bb3 eb4 d4 c4 bb3 c4 bb3 a3 bb3 a3 g3 f3 e3 _2 g3 _2 a3 b3 c#4 d4 e4 f4 g4 a4
  , -*16 c3 c3 bb2 c3 a2 c3 g2 c3 f2 f3 eb3 f3 d3 f3 c3 f3 bb2 _2 eb3 _2 f3 _2 f2 _2 g2 _2 c3 _2 d3 _2 d2 _2 g2 _4 - a2 g2 a2 f2 _2 d2 _2 a2 _4 }
 
{ g5 _2 f5 e5 d5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 g5 _2 a5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 d5 _2
  , bb4 _2 Subject[:2] Subject[3:29] .6# 5
  , d2 _24 -*6 d3 _2 }
 
{ e5 _2 d5 _2 d5 _2 c#5 _2 d5 _2 f5 _2 e5 _4
  , -*2 a4 _2 bb4 _2 a4 _2 a4 _2 d5 _2 -*4
  , -*12 d5 _2 c5 _2
  , c#4 _2 d4 _2 g4 _2 e4 _2 f4 _2 a4 _2 bb4 _2 a4 _2
  , a2 _2 f2 _2 g2 _2 a2 _2 f2 _2 d2 _2 g2 _2 a2 _2 }
 
 
% mm.127-129. A B flat chord, and then the fugue stops being a fugue: three
% measures of unbroken thirty-seconds in both hands, scales and broken
% chords with nothing repeated in them, ending on a held B flat. This is
% written out because that is what it is — a run.
 
{ {f4, bb4, d5} _8 -/2 f5/2 g5/2 a5/2 bb5/2 a4/2 bb4/2 c5/2 d5/2 c5/2 d5/2 eb5/2 f5/2 f4/2 g4/2 a4/2 bb4/2
  a4/2 bb4/2 c5/2 d5/2 a4/2 g4/2 f4/2 eb4/2 g4/2 a4/2 bb4/2 c5/2 g4/2 f4/2 eb4/2 d4/2 f4/2 g4/2 a4/2 bb4/2
  -*2 c4/2 d4/2 eb4/2 f4/2 -*6 bb3/2 c4/2 d4/2 eb4/2 -*2 c4/2 d4/2 e4/2 f#4/2 eb4/2 d4/2 c4/2 bb3/2 f#4/2 g4/2 a4/2 bb4 _2
  , - _93/2 c5/2 d5/2 g4/2
  , {f3, bb3, d4} _8 - _37/2 a3/2 bb3/2 c4/2 d4/2 -*2 f3/2 g3/2 a3/2 bb3/2 a3/2 bb3/2 c4/2 d4/2 c4/2 bb3/2 a3/2 g3/2 -*2 d4/2 c4/2 bb3/2 a3/2 - _15/2
  , bb2 _8 -*40 }
 
 
% mm.130-135. Held chords with nothing moving under them — a diminished
% triad on E over a C sharp, then a diminished seventh on G sharp, and the
% octaves beneath — and then the last flourish of the piece. That flourish is
% ONE figure sixteen times: a note, the step below, the note again, and a
% leap down to the third. It steps up and down the scale of C major, which
% is the key the passage is actually in, and in C major the figure needs no
% variants at all. Twice it sharpens both of its lower notes, and that is
% how the music turns toward A for the final cadence.
 
{ {e4, g4, bb4, e5} _12 d5 _12 -*8
  , -*12 a4 _4 {d4, f4, b4} _8 -*8
  , -*12 f4 _4 -*16
  , c#4 _12 d4 _4 b3 _8 -*8
  , c#3 _8 f3 _6 d3 _2 g#2 _8 - g#3 b3 f3 g#3 d3 f3 b2 }
 
key c4 maj
 
Twirl  ::= 5/2 4/2 5/2 3/2
Bright ::= g#4/2 f#4/2 g#4/2 e4/2
 
assert Twirl == g4/2 f4/2 g4/2 e4/2
assert Twirl@~-2 == e4/2 d4/2 e4/2 c4/2
assert Twirl@~4 == d5/2 c5/2 d5/2 b4/2
 
{ -*6 {e4, g#4, b4, d5} _2 e4 _16 -/2
  Twirl Twirl@~-2 Twirl@~1 Twirl@~-1 Twirl@~2 Twirl Twirl@~3 Twirl@~4 Twirl@~5
  Twirl@~6 Twirl@~7 Twirl@~3 Twirl@~4 Bright Twirl@~3 Twirl@~-1
  b4/2 a4/2 b4/2 c4/2 a4/2 g4/2 a4/2 b3/2 Bright - _3/2
  , -*8 c5 _8 c5 _8 -*40
  , -*8 a4 _6 g4 f4 -*48
  , -*6 {e3, g#3, b3} _2 c4 _16 - _77/2 c4/2 b3/2 a3/2
  , -*8 e3 _16 -*40
  , -*8 a3 _8 g3 _8 -*40
  , d3 g#2 a2 _2 e2 _20 -*40 }
 
 
% mm.136-139. Four falling three-note scraps, each a step higher than the
% last, and then the last harmony of the piece before its cadence: six
% chords in a chain of falling fifths — A, D, G, C, F, B flat — and every
% one of them is struck, spelled out from the bottom in thirty-seconds, and
% struck again.
 
Slip ::= -/2 2/2 1/2 .7/2
 
assert Slip == -/2 d4/2 c4/2 b3/2
assert Slip@~3 == -/2 g4/2 f4/2 e4/2
 
{ Slip Slip@~1 Slip@~2 Slip@~3 d4/2 a4/2 g4/2 f4/2 e4/2 g4/2 a4/2 b4/2 c5 _4
  {e4, a4, c#5} _2 -/2 c#4/2 e4/2 a4/2 c#5 _2 {e4, a4, c#5} _2
  {f4, a4, d5} _2 -/2 d4/2 f4/2 a4/2 d5 _2 {f4, a4, d5} _2
  {d4, g4, b4} _2 - d4/2 g4/2 b4 _2 {d4, g4, b4} _2
  {e4, g4, c5} _2 -/2 c4/2 e4/2 g4/2 c5 _2 {e4, g4, c5} _2
  {c4, f4, a4} _2 - c4/2 f4/2 a4 _2 {c4, f4, a4} _2
  {d4, f4, bb4} _2 - d4/2 f4/2 bb4 _2 {d4, f4, bb4} _2
  , -*14 {c4, e4, g4} _2 -*48
  , g#3/2 - _3/2 a3/2 - _3/2 b3/2 - _3/2 c4/2 - _15/2 g3 _2 -*48
  , -*14 e3 _2 a3 _2 a3/2 - _7/2 a3 _2 a3 _2 a3/2 - _7/2 a3 _2
  b3 _2 g3/2 b3/2 -*3 g3 _2 g3 _2 g3/2 - _7/2 g3 _2
  a3 _2 f3/2 a3/2 -*3 f3 _2 f3 _2 f3/2 bb3/2 -*3 f3 _2
  , -*16 g3 _2 -*4 g3 _2 f3 _2 -*4 f3 _2 f3 _2 -*4 f3 _2 e3 _2 -*4 e3 _2 e3 _2 -*4 e3 _2 d3 _2 -*4 d3 _2 }
 
 
% mm.140-143. Back in D minor for the end: the diminished seventh on C
% sharp, the D major chord after it, and then the top voice walking down
% g d c bb to a last A — over a D minor chord spread across four octaves and
% held for a whole measure, with a D in the pedals two octaves below it.
 
key d4 min
 
{ {e4, g4, bb4} _2 - _3/2 c#4/2 bb4 _2 {c#4, e4, g4, bb4} _2 {d4, f#4, a4} _2 - d4/2 f#4/2 a4 _2
  a4 _6 g4 _8 d4 _8 c4 _8 bb3 _4 a3 _16
  , -*14 f#4 _2 d4 _8 e4 _4 f4 _4 e4 _8 d4 _24
  , -*14 d4 _2 bb3 _8 c#4 _4 -*36
  , e3 _2 e3/2 g3/2 bb3/2 - _5/2 e3 _2 f#3 _2 d3/2 a3/2 -*3 f#3 _2 g3 _8 bb3 _4 a3 _4 a3 _8 f3 _4 g3 _4 f3 _16
  , c#3 _2 -*4 c#3 _2 c3 _2 -*4 c3 _2 g2 _4 bb2 _4 e2 _4 f2 _4 a2 _8 bb2 _4 g2 _4 d3 _16
  , -*48 d2 _16 }

% Toccata and Fugue in D minor, BWV 565 — the whole work, 143 measures, in
% one grammar. One unit is a sixteenth and the quarter is at 60, so a
% measure is sixteen units of common time.
%
% Three shapes pay for almost everything in it, and the piece sounds each
% one bare before it starts. A pair of rising steps; a pair of falling
% steps; and a TURN — a note, its upper neighbour, its lower neighbour,
% itself. The turn is the piece's signature: it occurs twenty-six times, at
% nine pitches, in every voice the piece has, and all but one of them is
% written below as this one rule and an interval — the exception straddles
% two other rules and could not be. The last of the four figures sounded
% here is simply the turn a fourth higher, and needs no rule of its own.

bpm 60
meter 4
unit 1/4
key d4 min
mode diatonic

Lift ::= .6/2 .7/2 1/2
Fall ::= .7 .6 .5
Turn ::= 6 7 5 6

assert Lift == bb3/2 c4/2 d4/2
assert Fall == c4 bb3 a3
assert Turn == bb4 c5 a4 bb4
assert Turn@5 == eb5 f5 d5 eb5

Lift
Fall
Turn
Turn@5

% mm.1-2. Three cries, each one hand doubling the other at the octave,
% each answered by three units of nothing. The first is a held A and a
% scale falling to the leading note; the second is short — A, then just
% e f c# d; the third is the first one again, two octaves down, where the
% pedals are.

Cry   ::= '5 _2 -/2 '4/4 '3/4 '2/4 '1/4 7#/2 '1
Reply ::= 5 _2 -/2 2/2 3/2 .7#/2 1

assert Cry == a5 _2 -/2 g5/4 f5/4 e5/4 d5/4 c#5/2 d5
assert Reply == a4 _2 -/2 e4/2 f4/2 c#4/2 d4

{Cry, Cry@-12} -*3 {Reply, Reply@-12} -*3 {Cry, Cry@-12}@-24 -*3

% mm.2-3. Under a D in the pedals, the whole organ takes a diminished
% seventh on C# across two octaves, holds it for half a measure, and
% resolves to D. Then a bare C# in octaves, and the toccata can begin.

{ -*4 {c#3, e3, bb3} _8 {d3, a3} _4 -*7 c#3
  , -*4 g3 _9 e3 f#3 _2 -*8
  , -*4 {c#4, e4} _8 d4 _4 -*7 c#4
  , d2 _16 -*8 }

% mm.4-7. A three-note coil in triplets — the tonic, the step above, the
% step below — turning three times and then walking away. Under the tonic
% the step below is the raised leading note; stepped up to the third and
% the fifth of the key it is a plain scale step, so the same figure comes
% twice more with no sharp in it. The two measures are then said again an
% octave higher, and both hands do the whole thing in octaves.

Coil ::= 1 _2/3 2 _2/3 .7 _2/3
Home ::= 1 _2/3 2 _2/3 .7# _2/3
Step ::= 1 2

assert Home == d4 _2/3 e4 _2/3 c#4 _2/3
assert Coil@~2 == f4 _2/3 g4 _2/3 e4 _2/3
assert Coil@~4 == a4 _2/3 bb4 _2/3 g4 _2/3
assert Step@~2 == f4 g4

Wave  ::= Home*3 Step Coil@~2*3 Step@~2 Coil@~4*3 5 -*8
Waves ::= Wave 7# Wave@12 '5

{Waves, Waves@-12}

% mm.8-10. Ten broken chords falling through the scale, each one said
% twice before the next — and each is the one before it a scale step lower,
% so nine of the ten are one figure and a step. The tenth raises its lowest
% note, because there that note is the leading note and has to lead home.
% The chord it lands on is the diminished seventh again, over the pedal D.
%
% mm.11-12. A scale down out of the held B flat, and the toccata reaches
% its first full cadence.

Chain ::= '4 _2/3 '6 _2/3 '2 _2/3
Sharp ::= 2 _2/3 4 _2/3 .7# _2/3

assert Chain == g5 _2/3 bb5 _2/3 e5 _2/3
assert Chain@~-3 == d5 _2/3 f5 _2/3 bb4 _2/3
assert Sharp == e4 _2/3 g4 _2/3 c#4 _2/3

Arc ::= Chain*2 Chain@~-1*2 Chain@~-2*2 Chain@~-3*2 Chain@~-4*2
  Chain@~-5*2 Chain@~-6*2 Chain@~-7*2 Chain@~-8*2 Sharp*2

{ Arc -*4 bb4 _17/2 a4/2 g4/2 f4/2 e4/2 d4/2 c#4/2 b3/2 c#4 a3 c#4 e4/2 g4/2 f4 _3 e4
  , Arc@-12 -*4 {c#3, e3, g3, bb3} _8 -*12
  , -*44 {c#4, e4, g4} _8 -*12
  , -*40 d2 _12 -*12 }

% mm.12-15. A repeated A starts under the cadence and keeps going for
% three measures, a note to the unit, half a unit late every time. Over
% it the right hand runs down through three turns and then says one whole
% measure twice, the second time an octave lower — and that measure has a
% fourth turn inside it, so the octave repeat brings a fifth.

Drone  ::= -/2 a4*15 a4/2
Tumble ::= 5 6 4 5 Turn@-5 1 4 2 3 1 2 .7#

assert Tumble == a4 bb4 g4 a4 f4 g4 e4 f4 d4 g4 e4 f4 d4 e4 c#4

{ {d2, d3, d4, f4} _4 -*3 a4 '1 '2 '3 '1 '2 Turn@7 '4 '5 '3 '4 '5 '6 '4 '5
  Turn@7 Turn@4 Tumble 1 Tumble@-12
  , - _17/2 a4*7 a4/2 Drone*3 }

% mm.16-17. The same measure twice. Fourteen units of spray in four
% voices, and the only difference between the two statements is what
% stands under the first two units: the second time the whole A major
% chord is there.

Cascade ::= d4/2 f4/2 bb4/2 f4/2 c4/2 e4/2 a4/2 e4/2 -/2 d4/2 g4/2 d4/2 -/2 c#4/2 e4/2 a4/2
Tread   ::= d3 _2 c3 _2 bb2 _2

Spray ::= { Cascade d4 bb4 - a4 - g4
  , -*9 f4 - e4 - d4
  , -*4 bb3/2 -*3/2 a3/2 -*7/2 a3 - bb3 -
  , -*8 Tread }

d3 _2 Spray
{a2, e3, c#4, e4, a4} _2 Spray

% mm.18-20. A scale climbing out of the A major chord, faster and faster,
% and then the cascade from the twin measures comes back an octave higher
% with two of its rests filled in. The bell between the cascades is the
% same eight units twice, and under everything the bass treads down
% d c bb a, twice.

Toll ::= { d5 bb5 a4 a5 bb4 g5 {e4, c#5, e5, a5} _2
  , - f5 - e5 - d5 -*2 }

{ a4 _9/2 g4/2 f4/2 e4/2 d4/2 c#4/2 b3/2 c#4/2 a3/2 b3/2 c#4/2 d4/2 e4/2 f4/2 g4/2 a4/2 g4/2 f4/2 e4/2 f4/2 d4/2 f4/2 a4/2 c#5/2
  d5/2 a4/2 b4/2 c#5/2 d5/2 e5/2 f5/4 g5/4 a5/2 bb5 _2
  Toll {Cascade@12, -*4 bb4/2 - _3/2 a4/2 - _3/2} Toll b4 _2
  , {a2, e3, c#4, e4} _4 -*42 {b3, d4, f4} _2
  , -*22 Tread a2 _2 -*8 Tread a2 _2 g#2 _2 }

% m21. One more flight up to a high B flat and a scale all the way down.

{ a4 _4 a4 c#5 e5/2 g5/2 bb5 a5/2 g5/2 f5/2 e5/2 f5/2 e5/2 d5/2 c#5/2 d5/2 c5/2 bb4/2 a4/2 g4/2 f4/2 e4/2 d4/2
  , c#5 _3 b4 -*12
  , {g2, a3, e4} _4 -*12 }

% mm.22-26. Five measures on one diminished seventh, and the two hands say
% the SAME figure a ninth apart the whole way through — so one voice is the
% other transposed and needs no notes of its own. The figure is a rocking
% cell in triplets: a note against the third above it, then twice against
% the third below. It sinks by tritones (C# to G to C#), then leans upward
% instead, then shrinks to a bare tremolo. One cell in the middle turns up
% at the end where its neighbours turn down, and once near the end the
% lower hand holds its tremolo where the upper has moved on.

Rock    ::= 7# _2/3 '2 _2/3 7# _2/3 6 _2/3 7# _2/3 6 _2/3
Bend    ::= 4 _2/3 6 _2/3 4 _2/3 2 _2/3 4 _2/3 6 _2/3
Reach   ::= .7# _2/3 2 _2/3 .7# _2/3 2 _2/3 4 _2/3 2 _2/3
Tremble ::= 4 _2/3 6 _2/3 4 _2/3 6 _2/3 4 _2/3 6 _2/3

assert Rock == c#5 _2/3 e5 _2/3 c#5 _2/3 bb4 _2/3 c#5 _2/3 bb4 _2/3
assert Rock@-6 == g4 _2/3 bb4 _2/3 g4 _2/3 e4 _2/3 g4 _2/3 e4 _2/3
assert Bend[:5] == Rock[:5]@-6
assert Tremble@6 == c#5 _2/3 e5 _2/3 c#5 _2/3 e5 _2/3 c#5 _2/3 e5 _2/3

Descent ::= c#5 _26/3 Rock[1:] Rock*3 Rock@-6
Close   ::= Rock@-6*2 Rock@-12*4 Reach*4

{ Descent Bend Close Tremble*2
  , {Descent Rock@-6 Close}@-9 Tremble@-15 Tremble@-9
  , {g3, c#4, g4, bb4, e5} _8 -*72 }

% mm.27-29. The tremolo narrows to C# against E, the full organ answers
% with four chords, and a long scalar descent in the left hand brings the
% toccata to a dead stop on D. m30 begins with that D held: the fugue
% starts underneath it.

{ 7# _2/3 6 _2/3 7# _2/3 '2 _2/3 7# _2/3 '2 _2/3 Tremble@6
  {c#4, e4, a4, c#5, e5, a5} _4 {d4, f4, a4, d5, f5, a5} _4
  {d4, g4, bb4, d5, g5} _4 -*4 {c#4, e4, a4, c#5, e5, g5} _4
  -*11 {e4, a4, c#5} {d4, f4, a4, d5} _2 d4 _4 c#4 _2 d4 _8
  , Tremble~@-6*2 a3 _4 a3 _4 -*23 a3 a3 _2 f4 _2 e4 _4 -*8
  , -*8 g3 _4 f3 _4 bb3 _6 a3 g3 a3 _3 e3 f3 Turn@-20 b2 c#3 a2 bb2 g#2 a2
  g3 f3 _2 a3 _5 g3 _3 f3 e3 f3 _4
  , -*48 {d3, a3} _8
  , -*42 d3 _2 a2 _4 d2 _8 }

% mm.30-34. The fugue, and everything in it comes from one figure. The
% subject is a line with a nail through it: a repeated A, and between every
% two A's one note of a voice that walks down g f e d c#, back up d e f,
% drops to the A an octave below, climbs a b c#, and settles d c# d e f.
% Thirty-two notes. It comes back TWELVE more times in the next ninety-five
% measures — ten of them whole or nearly, and twice as its head alone in the
% pedals — in five voices and at eight pitches. Every one of them is this
% rule and a transposition, and an entry cut short is this rule sliced.
%
% The answer comes in over the subject's own last note, a fifth higher, and
% is the same shape to the semitone. Under it the first voice spins out its
% counter-line in eighths.

Walk    ::= 4 3 2 1 .7# 1 2 3 .5 .6# .7# 1 .7# 1 2
Axis    ::= $1 5
Subject ::= 5 4 5 3 5 2 5 1 5 .7# 5 1 5 2 5 3 5 .5 5 .6# 5 .7# 5 1 5 .7# 5 1 5 2 5 3 _2

assert Subject == 5 Axis(Walk) 3 _2
assert Subject == a4 g4 a4 f4 a4 e4 a4 d4 a4 c#4 a4 d4 a4 e4 a4 f4 a4 a3 a4 b3 a4 c#4 a4 d4 a4 c#4 a4 d4 a4 e4 a4 f4 _2
assert Subject@5 == d5 c5 d5 bb4 d5 a4 d5 g4 d5 f#4 d5 g4 d5 a4 d5 bb4 d5 d4 d5 e4 d5 f#4 d5 g4 d5 f#4 d5 g4 d5 a4 d5 bb4 _2

{ -*33 Subject@5 d5 _2 bb4 _2 d5 _2
  , - Subject f#4 _2 g4 _2 c4 _2 bb3 _2 a3 _2 bb3 _2 c4 _2 d4 _2 f#3 _2 g3 _2 a3 _2 bb3 _2 a3 _2 bb3 _2 f#3 _2 g3 g4 g3 g4 d4 g4 d4 g4 }

% mm.35-38. Eight half-measures of two notes rocking against each other,
% the right hand in eighths and the left in sixteenths. Not one of them is
% a transposition of the last — the interval opens and closes as the
% harmony walks down — so this passage is written out, and that IS what it
% is: a chain of changing dyads between two statements of the subject.

{ eb5 _2 g4 _2 eb5 _2 g4 _2 c5 _2 a4 _2 c5 _2 a4 _2 d5 _2 f4 _2 d5 _2 f4 _2 bb4 _2 g4 _2 bb4 _2 g4 _2
  c#5 _2 e4 _2 c#5 _2 e4 _2 a4 _2 f4 _2 a4 _2 f4 _2 g4 _2 c#4 _2 g4 _2 c#4 _2 f4 _2 d4 _2 f4 _2 d4 _2
  , c4 eb4 c4 eb4 c4 eb4 c4 eb4 c4 f4 c4 f4 c4 f4 c4 f4 bb3 d4 bb3 d4 bb3 d4 bb3 d4 bb3 e4 bb3 e4 bb3 e4 bb3 e4
  a3 c#4 a3 c#4 a3 c#4 a3 c#4 f3 d4 f3 d4 f3 d4 f3 d4 e3 bb3 e3 bb3 e3 bb3 e3 bb3 d3 a3 d3 a3 d3 a3 d3 a3 }

% mm.39-41. The third entry, an octave above the first, over a repeated A
% in the middle of the texture. It runs thirty-one of the subject's
% thirty-two notes and then goes its own way.

{ e4 _2 bb3 _2 e4 _2 bb3 _2 - Subject[:31]@12 f5 a5 e5 a5 d5 a5 c5 a5
  , e3 g3 e3 g3 e3 g3 e3 g3 a3 _2*16 a3 _4 d4 _4
  , -*8 f3 _2 e3 _2 d3 _2 g3 _2 f3 _2 e3 _2 f3 _2 c#3 _2 d3 _2 c#3 _2 d3 _2 e3 _2 f3 _2 e3 _2 f3 _2 c#3 _2 d3 _4 f3 _4 }

% mm.42-44. One measure, then the same measure a step lower, then a step
% lower again. The measure is the nailed axis again — one high note repeating
% while a line climbs under it — and it needs only two high notes to do it.
% Only the third statement is touched: its ninth note is raised, because
% there that note is the leading note of A.

Peal ::= 6 '5 7 '5 '1 '4 6 '4 '2 '4 '1 '4 7 '4 6 '4

assert Peal == bb4 a5 c5 a5 d5 g5 bb4 g5 e5 g5 d5 g5 c5 g5 bb4 g5
assert Peal@~-1 == a4 g5 bb4 g5 c5 f5 a4 f5 d5 f5 c5 f5 bb4 f5 a4 f5

{ Peal Peal@~-1 Peal[:8]@~-2 7# Peal[9:]@~-2
  , d4 _4 -*4 c4 _4 c4 _4 c4 _4 -*4 bb3 _4 bb3 _4 bb3 _4 -*4 a3 _4 a3 _4
  , g3 _4 -*4 c3 _4 e3 _4 f3 _4 -*4 bb2 _4 d3 _4 e3 _4 -*4 a2 _4 c#3 _4 }

% mm.45-47. Two measures whose second halves are identical, over a sighing
% figure in the alto said twice without a note changed, and a bass that
% says it twice with two notes changed. Then a cadence.

Sigh ::= -*2 d4 _2 c#4 _2 c#4 _2 d4 _2 d4 _2 e4 _4

{ f4 e5 g4 e5 a4 d5 f4 d5 e4 e5 e4 e5 f4 d5 f4 d5
  bb4 c#5 bb4 c#5 a4 d5 f4 d5 e4 e5 e4 e5 f4 d5 f4 d5
  - d5 c#5 d5 b4 d5 c#5 b4 c#5 _4 e4 g4 f4 e4
  , -*41 a4 g4 a4 -*4
  , a3 _4 Sigh Sigh -*12
  , d3 _4 -*2 f3 _2 bb3 _2 bb3 _2 a3 _2 a3 _2 g3 _4 -*2 a3 _2 bb3 _2 bb3 _2 a3 _2 a3 _2 g#3 _4 -*12 }

% mm.48-55. Free counterpoint over eight measures, and halfway through it
% the FOURTH entry arrives in the pedals — thirty-one notes of the subject
% two octaves and a fifth below where it started, while the right hand
% marks eleven plain eighths on D above it. The feet cannot play fast, so
% this is the subject at its slowest and its plainest, and it is the same
% rule.

assert Subject[:31]@-19 == d3 c3 d3 bb2 d3 a2 d3 g2 d3 f#2 d3 g2 d3 a2 d3 bb2 d3 d2 d3 e2 d3 f#2 d3 g2 d3 f#2 d3 g2 d3 a2 d3
assert d5 _2*3 == d5 _2 d5 _2 d5 _2

{ f4 _4 d4 _4 c#5 _4 e5 _6 d5 _4 c#5 _2 c5 _4 bb4 _4 a4 _4 a4 _4 g4 _4 eb4 _6
  a4 _4 c5 _4 bb4 a4 bb4 _2 g5 _4 f#5 _2 g5 _4 bb4 _2 d5 _2*11
  d5 g5 Turn@7 d5 e5 c5 a5 g5 a5 Turn@7
  d5 bb5 a5 bb5 g5 a5 f5 g5 e5 c6 Turn@12 g5 a5
  , - d5 c#5 d5 f5 d5 c#5 b4 a3 a4 g4 a4 -*24 f#4 _4 f4 _4 -*30
  a4 _2 bb4 _2 f#4 _2 g4 _2 f#4 _2 g4 _2 a4 _2 bb4 _2 a4 _2 bb4 _2 f#4 _2
  g4 bb4 a4 bb4 g4 a4 f4 g4 e4 c5 Turn g4 a4
  f4 d5 c5 d5 Turn g4 e5 d5 e5 c5 d5 bb4 c5
  , -*12 c#4 g4 f4 e4 f4 _4 e4 _4 d4 _2 a4 _4 g4 _6 -*8 g4 _4 f#4 _2 d4 _2 f#4 _2 eb5 _2 d5 _4 -*2 bb4 _2 a4 _4 d5 _2 c5 _2 -*56
  , -*17 d4 c#4 d4 a3 a3 g3 a3 f#3 d4 c#4 d4 g3 f4 eb4 d4 c#4 e4 a3 c#4 d3 eb4 d4 c4 b3 d4 g3 b3 c3 d4 c4 bb3 a3 c4 f#3 a3 d3 c4 bb3 a3 bb3 a4 g4 f#4 g4 bb3 a3 g3
  d4 _4 -*2 d4 _2 d4 _2 d4 _2 d4 _2 c4 _2 bb3 _2 d4 _2 d4 _2 c4 _2 bb3 _2 d4 _2 d4 _2 c4 _2 bb3 _4 -*28
  , -*65 Subject[:31]@-19 bb2 _4 -*2 b2 _2 c3 _4 -*2 c#3 _2 d3 _4 -*2 d3 _2 e3 _4 -*2 e3 _2 }

% mm.56-59. The subject's own device turns up in the episode on a different
% note: a line nailed to a repeated C, the moving voice climbing f e f g a,
% dropping an octave to C and coming back d e. First the whole of it, then
% only its opening four steps.

Climb ::= .3 .2 .3 .4 .5 ..7 .1 .2
Nail  ::= $1 .7

assert Nail(Climb) == f3 c4 e3 c4 f3 c4 g3 c4 a3 c4 c3 c4 d3 c4 e3 c4

{ a4 _2 f4 _2 bb4 a4 bb4 a4 bb4 d5 bb4 a4 g4 bb4 g4 f4 e4 f4 e4 f4 g4 _2 e4 _2 a4 _4 c5 _2 bb4 _2
  , -*4 d5 c5 g4 f4 g4 bb4 g4 f4 e4 g4 e4 d4 c4 d4 g4 a4 -*4 f4 _4 -*4
  , f5 eb5 d5 c5 -*16 bb4 d5 c5 bb4 -*8
  , -*25 c4 bb3 c4 a3 c4 g3 c4
  , f3 _2 a2 _2 bb2 _2 d3 _2 g2 _4 -*2 bb2 _4 a2 _2 g2 _2 c3 _2 f2 _4 -*4 }

{ a4 _2 g4 _2 a4 _2 bb4 _2 c5 _2 e4 _2 f4 _2 g4 _2 a4 _2 g4 _2 a4 _2 bb4 _2 c5 bb4 a4 g4 f4 eb4 d4 c4
  , -*16 f3 c4 e3 c4 f3 c4 g3 c4 a3 _4 -*4
  , Nail(Climb) -*16 }

% mm.60-65. The clearest sequence in the fugue. Four times a held note and
% a scale falling an octave and a half out of it, each one a step higher
% than the last; over it a two-note sigh, also rising by step. Then the
% whole texture locks: a low note against a rising pair high above,
% alternating with the same pair over C, two measures of one and two of the
% other, twice.

Sink ::= .6 _2 6 5 4 3 2 1
Ring ::= '1 7 -*6
Ping ::= '5 '3 -*3 '3 -*2
Call ::= 3 _2 '2 '3 7 _2 '2 '3
Echo ::= 2 _2 '1 '2 7 _2 '1 '2

assert Sink == bb3 _2 bb4 a4 g4 f4 e4 d4
assert Sink@~3 == e4 _2 e5 d5 c5 bb4 a4 g4
assert Ring@~2 == f5 e5 -*6
assert Ping@~-1 == g5 e5 -*3 e5 -*2
assert Call == f4 _2 e5 f5 c5 _2 e5 f5
assert Echo == e4 _2 d5 e5 c5 _2 d5 e5

{ Sink Sink@~1 Sink@~2 Sink@~3 Call*2 Echo*2 Call*2 Echo*2
  , Ring Ring@~1 Ring@~2 Ring@~3 Ping*2 Ping@~-1*2 Ping Ping[:4] -*18
  , -*80 Ping@~-1*2 }

% mm.66-70. One gesture on four harmonies in turn, each said twice: the
% third below, back, then up through the chord and down again. The chords
% are G seventh, C seventh, B flat and A seventh, and because they are
% different chords the intervals are different too — which is why these are
% four rules and not one rule transposed.

OnG  ::= 6# 4 6# '1 '3 '5 '3 '1
OnC  ::= 6 4 6 7 '2 '4 '2 7
OnBb ::= 5 3 5 7 '1 '3 '1 6
OnA  ::= 4 2 4 6 7# '2 7# 6

assert OnG == b4 g4 b4 d5 f5 a5 f5 d5
assert OnA == g4 e4 g4 bb4 c#5 e5 c#5 bb4

{ d4 _2 f5 e5 d5 c5 b4 a4 OnG*2 OnC*2 OnBb*2 OnA*2
  , f5 g5 -*70 }

% mm.70-72. The fifth entry, an octave above the first again, and this time
% it is broken off after twenty-seven notes and turned into a scale down.

{ - Subject[:27]@12 d5 _2 e5 _2 d5 _2 c5 _2 bb4 _2 a4 _2
  , -*34 a4 _2 g4 _2 f#4 _2
  , a4 _2 c#5 _2 d5 _2 g4 _2 f4 _2 a4 _2 b4 _2 c#5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 c#5 _2 - f#4 e4 f#4 - d4 c4 d4
  , -*29 a4 g4 a4 -*8 }

% mm.73-82. Ten measures with almost nothing in them but two eight-note
% arpeggios trading places — one on the A seventh, one on D — and then a
% third that climbs where they fell. The pattern of statements is the whole
% passage: curl, curl, drop, drop, curl, curl, drop; then rise, and a bell
% of three notes answered by four units of silence, over and over.

Curl ::= ..7# ..5 ..7# .2 .4 .6 .4 .2
Drop ::= .1 ..5 .1 .3 .5 1 .5 .3
Rise ::= .2 ..7# .2 .4 .6 .7# .6 .4
Lap  ::= .3 .1 .3 .5 -*3 .5
Bell ::= 1 3 1

assert Curl == c#3 a2 c#3 e3 g3 bb3 g3 e3
assert Drop == d3 a2 d3 f3 a3 d4 a3 f3
assert Rise == e3 c#3 e3 g3 bb3 c#4 bb3 g3
assert Lap@~1 == g3 e3 g3 bb3 -*3 bb3

{ g4 _2 g4 f4 e4 d4 c#4 b3 -*72
  , bb4 a4 -*62 Drop Rise
  , bb3 c4 bb3 a3 g3 f3 e3 d3 Curl Curl Drop Drop Curl Curl Drop -*16 }

{ -*12 Bell -*5 Bell -*21 Bell -*5 Bell -*5 c#4 e4 c#4 bb3 -*4 c#4 e4 c#4 bb3 -*4 Bell -
  , Rise Lap Lap Rise Rise Lap Lap Lap[:5]@~1 - Lap[:5]@~1 - a3 f3 a3 c#4 -*3 a3 }

% mm.83-90. A long fall in the left hand — one eight-note descent said
% twice over — then two scale bursts, and the sixth entry comes in a minor
% third below the first, in the tenor. Two measures later the seventh
% follows a whole tone below, in the voice above it, over a G held for
% twenty-five units: the two statements overlap, which is what a fugue is
% for.

Slide ::= .6 .5 .4 .3 .4 .3 .2 .1

assert Slide == bb3 a3 g3 f3 g3 f3 e3 d3
assert Subject@-14 == g3 f3 g3 eb3 g3 d3 g3 c3 g3 b2 g3 c3 g3 d3 g3 eb3 g3 g2 g3 a2 g3 b2 g3 c3 g3 b2 g3 c3 g3 d3 g3 eb3 _2

{ - d4 bb3 -*34 f4 e4 d4 a4 -*4 e5 d5 c#5
  , bb3 -*2 g3 f3 a3 f3 d3 a2 d3 a2 f2 d2 d3 c#3 b2 c#3 Slide*2
  e3/2 f3/2 g3/2 a3/2 b3/2 c#4/2 d4 -*3 a3 b3/2 c#4/2 d4/2 e4/2 f4/2 g4/2 a4 g4 f4 e4 }

{ d5 eb5/2 d5/2 c5/2 bb4/2 a4/2 g4/2 f#4 _2 a4 _4 g4 _2 c5 _2 b4 _2 eb5 _2 d5 _2 eb5 _2 b4 _2
  c5 _2 b4 _2 c5 _2 d5 _2 eb5 _2 d5 _2 eb5 _2 f5 _2 g5 _33 g5 f5 g5 Turn@5
  , f4 _2 -*3 eb4 d4 c4 bb3 _2 b3 _2 c4 _2 g4 _2 g4 _25 Subject[:30]@-2 d4 eb4 _2 d4 _2 g4 g#4 f4 g4
  , -*9 Subject@-14 d3 _2 c3 _2 b2 _2 c3 _2 d3 _2 eb3 _2 f3 _2 g3 _2 b2 _2 c3 _2 d3 _2 eb3 _2 d3 _2 eb3 _2 f3 _2 g3 _2 b2 _2 c3 _2 d3 _2 }

% mm.91-95. Free writing in three voices, and the turn from the very first
% cell is all over it: on E flat at m90 and m91, on B flat at m92, on E flat
% an octave down at m91 in the left hand, and on G at m93. Then the eighth
% entry, a fifth below the first, in the middle voice over a D held for
% thirty-six units.

{ c5 f5 Turn@5 c5 d5 bb4 eb5 d5 eb5 c5 d5 bb4 c5 a4 d5 c5 d5 Turn g4 bb4 a4 bb4 c5 bb4 a4 g4
  , eb4 _2 c4 _2 f4 g4 eb4 f4 d4 _2 bb3 _2 Turn@-7 c4 _2 f#4 _2 g4 a4 f4 g4 eb4 g4 f#4 g4 a4 g4 f#4 e4
  , eb3 _2 a2 _2 bb2 _2 c3 _2 d3 _2 g2 _2 a2 _2 bb2 _2 c3 _2 d3 _2 g3 _2 d3 _2 eb3 _2 bb2 _2 a2 _2 c3 _2 }

{ f#4 _2 a4 _2 d5 g4 c5 f#4 bb4 g4 d5 a4 bb4 Turn@-3 d5 f#4 d5 g4 d5 a4 d5 bb4 g4 d5 a4 bb4 g4 c5 a4 d5 c5 bb4 a4 bb4 a4 bb4 g4 a4 bb4 c5 d5 eb5 d5 c5 d5
  , - Subject@-7 a3 _2 g3 _8 f#3 _4
  , d3 _36 -*2 g2 _2 c3 _2 a2 _2 d3 _2 d2 _2 }

% mm.96-100. The turn again, twice at the top of the measure and once
% underneath, as the music works its way back to D minor and the left hand
% climbs with it.

{ Turn g4 _2 b4 _2 c5 eb5 f5 g5 g#5 g5 f5 g5 Turn@5 c5 _2 bb4 _2 a4 _2 bb4 _4 a4 _4 g4 _4 f#4 _2 f4 _2 eb4 _4 d4 bb4
  , g3 _2 a3 _2 bb3 a3 g3 f3 eb3 _2 eb4 _2 d4 _4 c4 _2 d4 _2 Turn@-7 c4 eb4 d4 f4 g4 f4 eb4 f4 d4 bb3 eb4 d4 c#4 a3 d4 c4 b3 g3 c4 bb3 a3 f3 bb3 d4
  , g2 _4 -*2 g2 _2 c3 _6 b2 _2 c3 _4 -*28 }

{ a4 bb4 a4 g4 f#4 eb5 d5 c5 bb4 a5 g5 f#5 g5 _3 f5 Turn@5 c#5 _4 d5 _4 c5 _4
  , -*24 a4 _4 f#4 _4
  , -*21 bb4 a4 g4 -*8
  , c4 d4 c4 bb3 a3 c4 bb3 a3 g3 c4 bb3 a3 bb3 c4 bb3 a3 g3 a3 f3 g3 e3 _4 f#3 _4 d3 _4
  , -*25 eb4 d4 c4 d4 c4 bb3 a3 }

% mm.101-111. Four measures of eighths turning around D, and then the
% subject comes back where it began — at pitch, in the tenor, with one A
% of its thirty-two replaced by silence. Two measures later the tenth entry
% follows an octave lower in the pedals, and that one runs thirty-one notes
% and holds its last.

{ bb4 _4 a4 _2 c#5 _2 d5 _2 e5 _2 f5 _4 e5 _8 d5 _6 f5 _2 g5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 g5 _2 f5 _2 d5 _2
  , -*4 c#4 _2 e4 _2 f4 _2 g4 _2 -*18 a4 _2 g4 _2 a4 _2 g4 _2 f4 _2 g4 _2 f4 _2 e4 _2 a4 _2 bb4 _2 a4 _2 g4 _2 f4 _2 a4 _2 e5 _2 d5 _2 a4 _2
  , g4 f4 eb4 d4 -*8 a4 g4 d5 _6 c#5 _4 a4 bb4 g4 a4 f4 _2 -*34
  , -*14 a4 f4 -*48
  , g3 _4 a3 bb3 a3 g3 f3 g3 e3 a3 d3 _4 g4 a4 g4 a4 bb4 a4 g4 a4 Turn@-5 d4 c4 bb3 a3 bb3 a3 g3 f3 e3 f3 e3 d3 c#3 a2 d3 a2 e3 a2 f3 a2 g3 a2 f3 a2 e3 a2 d3 a2 c#3 a3 a2 a3 d3 a3 f3 d4 }

{ bb4 _2 e5 _2 d5 _2 d5 _2 c#5 _4 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 c#5 _2 d5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 g5 _2
  a5 bb5 g5 a5 Turn@7 d5 a4 c#5 a4 d5 a4 e5 a4 f5 a4 e5 a4 f5 a4 g5 a4 a5 a4 e5 a4 f5 a4 g5 a4 a5 _4 -*28
  , g4 _2 g4 _2 a4 _2 f4 _2 -*96
  , f3 d4 e3 c#4 f3 d4 d3 d4 a3 _18 a3 _50 -*28
  , -*8 e4 Subject[:17] - Subject[18:] e4 _2 d4 _2 g4 _2 f4 _2 e4 _2 f4 _2 c#4 _2 d4 _2 c#4 _2 d4 _2 b3 _2 c#4 _2 g4 _2 f4 _2 d4 _2 c#4 _4 -*28
  , -*73 Subject[:31]@-12 }

{ - e5 d5 c#5 d5 _2 - f5 e5 f5 g5 a5 bb5 a5 g5 a5 f5 _2 e5 _2 d5 _2 f5 eb5
  , -*21 e5 -*2
  , - g4 f4 e4 f4 c#4 d4 _6 c#4 _4 d4 a4 b4 c#5 -*2 c5 _2
  , f3 _4 -*2 f3 _2 g3 _2 e3 _2 a3 _2 a2 _2 d3 _4 -*2 a2 _2 }

% mm.113-119. A cadence, and then the strangest passage in the fugue: a
% four-note figure, four units of silence, the same four notes again — one
% figure to a harmony, C minor then G minor then the F sharp diminished,
% each said twice, over a D tapped in the pedals every other half-measure.
% The left hand runs an eight-note pattern twice, the second time with one
% note changed, and then a second pattern twice, the second time with its
% last two changed.

OnCm  ::= .7 2b .7 .6
OnGm  ::= 4 6 4 1
OnFs  ::= 3# 5 3# 2b
Tap   ::= .1 _4 -*4
Wind  ::= .3# .1 .3# .4 .5 1 .5 .4
Gust  ::= .4 .1 .4 .5 .6 1 .6 .5

assert OnCm == c4 eb4 c4 bb3
assert OnFs == f#4 a4 f#4 eb4
assert Wind == f#3 d3 f#3 g3 a3 d4 a3 g3
assert Wind \7 {.3#} == f#3 d3 f#3 g3 a3 d4 a3 f#3
assert Gust[:6] .7 .6 == g3 d3 g3 a3 bb3 d4 c4 bb3

{ d5 c5 bb4 a4 g4 a4 bb4 c5 a4 _4 -*2 c5 _4 bb4 _2 -*2 g5 _2 Turn@5 c5 bb4 a4 g4
  , f4 _6 eb4 _3 c4 a3 f3 a3 c4 eb4 c4 d4 bb3 g3 d3 g3 bb3 d4 g4 c4 d4 bb3 c4 a3 g3 f#3 e3
  , bb2 _2 g2 _2 c3 _2 c2 _2 f2 _4 -*2 a2 _2 bb2 _4 -*2 bb2 _2 c3 _4 -*2 a2 _2 }

{ a4 _4 -*32 OnCm -*4 OnCm -*4 OnGm -*4 OnGm -*4 OnFs -*4 OnFs
  , Wind Wind \7 {.3#} Gust Gust[:6] .7 .6 -*48
  , Tap*4 a3 f#3 a3 bb3 -*4 a3 f#3 a3 bb3 -*4 a3 bb3 d4 f#4 -*4 bb3 g3 bb3 d4 -*4 c4 a3 c4 eb4 -*4 c4 a3 c4 eb4 -*4
  , -*32 Tap*6 }

% mm.120-125. The pedals take the subject's head twice over — eight notes
% at the bottom of the instrument, then its first seven a fourth higher —
% and then the twelfth and last entry arrives in the alto AT PITCH, with
% one A dropped out of its opening and its twenty-ninth note bent down to
% B so that it can turn into the cadence instead of going on.

assert Subject[:8]@-19 == d3 c3 d3 bb2 d3 a2 d3 g2
assert Subject[:7]@-14 == g3 f3 g3 eb3 g3 d3 g3

{ d4 _2 -*2 d5 _2 c5 _2 bb4 _4 c5 _2 d5 _2 eb5 _4 f5 _2 g5 _2 a5 _4 f5 _2 eb5 _2 bb4 _2
  d5 c5 bb4 c5 bb4 a4 g4 c5 bb4 a4 g4 a4 g4 f#4 g4 f4 e4 d4 c#4 _2 e4 _2 f4 g4 a4 b4 c#5 d5 e5 f5
  , -*4 g4 _2 f#4 _2 g4 _4 g4 _2 b4 _2 c5 _4 c5 _2 e5 _2 f5 _4 bb4 _2 a4 _2 -*32
  , f#3 _2 -*2 g3 _2 a3 _2 bb3 _4 eb4 _2 b3 _2 -*16 d5 eb5 -*30
  , - Subject[:8]@-19 Subject[:7]@-14 g3 _4 c4 _2 bb3 _2 a3 _4 bb3 _2 c4 _2
  d4 g4 f4 eb4 d4 eb4 d4 c4 bb3 eb4 d4 c4 bb3 c4 bb3 a3 bb3 a3 g3 f3 e3 _2 g3 _2 a3 b3 c#4 d4 e4 f4 g4 a4
  , -*16 c3 c3 bb2 c3 a2 c3 g2 c3 f2 f3 eb3 f3 d3 f3 c3 f3 bb2 _2 eb3 _2 f3 _2 f2 _2 g2 _2 c3 _2 d3 _2 d2 _2 g2 _4 - a2 g2 a2 f2 _2 d2 _2 a2 _4 }

{ g5 _2 f5 e5 d5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 g5 _2 a5 _2 c#5 _2 d5 _2 e5 _2 f5 _2 e5 _2 f5 _2 d5 _2
  , bb4 _2 Subject[:2] Subject[3:29] .6# 5
  , d2 _24 -*6 d3 _2 }

{ e5 _2 d5 _2 d5 _2 c#5 _2 d5 _2 f5 _2 e5 _4
  , -*2 a4 _2 bb4 _2 a4 _2 a4 _2 d5 _2 -*4
  , -*12 d5 _2 c5 _2
  , c#4 _2 d4 _2 g4 _2 e4 _2 f4 _2 a4 _2 bb4 _2 a4 _2
  , a2 _2 f2 _2 g2 _2 a2 _2 f2 _2 d2 _2 g2 _2 a2 _2 }

% mm.127-129. A B flat chord, and then the fugue stops being a fugue: three
% measures of unbroken thirty-seconds in both hands, scales and broken
% chords with nothing repeated in them, ending on a held B flat. This is
% written out because that is what it is — a run.

{ {f4, bb4, d5} _8 -/2 f5/2 g5/2 a5/2 bb5/2 a4/2 bb4/2 c5/2 d5/2 c5/2 d5/2 eb5/2 f5/2 f4/2 g4/2 a4/2 bb4/2
  a4/2 bb4/2 c5/2 d5/2 a4/2 g4/2 f4/2 eb4/2 g4/2 a4/2 bb4/2 c5/2 g4/2 f4/2 eb4/2 d4/2 f4/2 g4/2 a4/2 bb4/2
  -*2 c4/2 d4/2 eb4/2 f4/2 -*6 bb3/2 c4/2 d4/2 eb4/2 -*2 c4/2 d4/2 e4/2 f#4/2 eb4/2 d4/2 c4/2 bb3/2 f#4/2 g4/2 a4/2 bb4 _2
  , - _93/2 c5/2 d5/2 g4/2
  , {f3, bb3, d4} _8 - _37/2 a3/2 bb3/2 c4/2 d4/2 -*2 f3/2 g3/2 a3/2 bb3/2 a3/2 bb3/2 c4/2 d4/2 c4/2 bb3/2 a3/2 g3/2 -*2 d4/2 c4/2 bb3/2 a3/2 - _15/2
  , bb2 _8 -*40 }

% mm.130-135. Held chords with nothing moving under them — a diminished
% triad on E over a C sharp, then a diminished seventh on G sharp, and the
% octaves beneath — and then the last flourish of the piece. That flourish is
% ONE figure sixteen times: a note, the step below, the note again, and a
% leap down to the third. It steps up and down the scale of C major, which
% is the key the passage is actually in, and in C major the figure needs no
% variants at all. Twice it sharpens both of its lower notes, and that is
% how the music turns toward A for the final cadence.

{ {e4, g4, bb4, e5} _12 d5 _12 -*8
  , -*12 a4 _4 {d4, f4, b4} _8 -*8
  , -*12 f4 _4 -*16
  , c#4 _12 d4 _4 b3 _8 -*8
  , c#3 _8 f3 _6 d3 _2 g#2 _8 - g#3 b3 f3 g#3 d3 f3 b2 }

key c4 maj

Twirl  ::= 5/2 4/2 5/2 3/2
Bright ::= g#4/2 f#4/2 g#4/2 e4/2

assert Twirl == g4/2 f4/2 g4/2 e4/2
assert Twirl@~-2 == e4/2 d4/2 e4/2 c4/2
assert Twirl@~4 == d5/2 c5/2 d5/2 b4/2

{ -*6 {e4, g#4, b4, d5} _2 e4 _16 -/2
  Twirl Twirl@~-2 Twirl@~1 Twirl@~-1 Twirl@~2 Twirl Twirl@~3 Twirl@~4 Twirl@~5
  Twirl@~6 Twirl@~7 Twirl@~3 Twirl@~4 Bright Twirl@~3 Twirl@~-1
  b4/2 a4/2 b4/2 c4/2 a4/2 g4/2 a4/2 b3/2 Bright - _3/2
  , -*8 c5 _8 c5 _8 -*40
  , -*8 a4 _6 g4 f4 -*48
  , -*6 {e3, g#3, b3} _2 c4 _16 - _77/2 c4/2 b3/2 a3/2
  , -*8 e3 _16 -*40
  , -*8 a3 _8 g3 _8 -*40
  , d3 g#2 a2 _2 e2 _20 -*40 }

% mm.136-139. Four falling three-note scraps, each a step higher than the
% last, and then the last harmony of the piece before its cadence: six
% chords in a chain of falling fifths — A, D, G, C, F, B flat — and every
% one of them is struck, spelled out from the bottom in thirty-seconds, and
% struck again.

Slip ::= -/2 2/2 1/2 .7/2

assert Slip == -/2 d4/2 c4/2 b3/2
assert Slip@~3 == -/2 g4/2 f4/2 e4/2

{ Slip Slip@~1 Slip@~2 Slip@~3 d4/2 a4/2 g4/2 f4/2 e4/2 g4/2 a4/2 b4/2 c5 _4
  {e4, a4, c#5} _2 -/2 c#4/2 e4/2 a4/2 c#5 _2 {e4, a4, c#5} _2
  {f4, a4, d5} _2 -/2 d4/2 f4/2 a4/2 d5 _2 {f4, a4, d5} _2
  {d4, g4, b4} _2 - d4/2 g4/2 b4 _2 {d4, g4, b4} _2
  {e4, g4, c5} _2 -/2 c4/2 e4/2 g4/2 c5 _2 {e4, g4, c5} _2
  {c4, f4, a4} _2 - c4/2 f4/2 a4 _2 {c4, f4, a4} _2
  {d4, f4, bb4} _2 - d4/2 f4/2 bb4 _2 {d4, f4, bb4} _2
  , -*14 {c4, e4, g4} _2 -*48
  , g#3/2 - _3/2 a3/2 - _3/2 b3/2 - _3/2 c4/2 - _15/2 g3 _2 -*48
  , -*14 e3 _2 a3 _2 a3/2 - _7/2 a3 _2 a3 _2 a3/2 - _7/2 a3 _2
  b3 _2 g3/2 b3/2 -*3 g3 _2 g3 _2 g3/2 - _7/2 g3 _2
  a3 _2 f3/2 a3/2 -*3 f3 _2 f3 _2 f3/2 bb3/2 -*3 f3 _2
  , -*16 g3 _2 -*4 g3 _2 f3 _2 -*4 f3 _2 f3 _2 -*4 f3 _2 e3 _2 -*4 e3 _2 e3 _2 -*4 e3 _2 d3 _2 -*4 d3 _2 }

% mm.140-143. Back in D minor for the end: the diminished seventh on C
% sharp, the D major chord after it, and then the top voice walking down
% g d c bb to a last A — over a D minor chord spread across four octaves and
% held for a whole measure, with a D in the pedals two octaves below it.

key d4 min

{ {e4, g4, bb4} _2 - _3/2 c#4/2 bb4 _2 {c#4, e4, g4, bb4} _2 {d4, f#4, a4} _2 - d4/2 f#4/2 a4 _2
  a4 _6 g4 _8 d4 _8 c4 _8 bb3 _4 a3 _16
  , -*14 f#4 _2 d4 _8 e4 _4 f4 _4 e4 _8 d4 _24
  , -*14 d4 _2 bb3 _8 c#4 _4 -*36
  , e3 _2 e3/2 g3/2 bb3/2 - _5/2 e3 _2 f#3 _2 d3/2 a3/2 -*3 f#3 _2 g3 _8 bb3 _4 a3 _4 a3 _8 f3 _4 g3 _4 f3 _16
  , c#3 _2 -*4 c#3 _2 c3 _2 -*4 c3 _2 g2 _4 bb2 _4 e2 _4 f2 _4 a2 _8 bb2 _4 g2 _4 d3 _16
  , -*48 d2 _16 }

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Toccata & Fugue · Pattern Engine