Lessons · Showcase

Caprice No. 5

Paganini — the sweep and the moto perpetuo

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% Paganini's fifth begins alone: a cadenza on the open A string, before
% the moto perpetuo it is famous for has sounded a note. Four SWEEPS up
% the a-minor triad, each reaching one rung higher than the last, and
% each answered by the scale falling the whole way back to the open A.
%
% The sweep is the triad itself, three degrees to the octave — the
% third, the fifth, the tonic above them — climbing until it runs out of
% violin. It is written once in DEGREES, eleven rungs long, and the four
% sweeps are its first eight, its first nine, its first ten, and the
% whole of it. Degrees and not intervals, because the frame comes back
% in A MAJOR at m.68 and 'i3 i4 i5' is a minor triad wherever it stands.
 
bpm 105
meter 4
unit 1/4
key a4 harm
mode diatonic
 
Sweep ::= .3 .5 1 3 5 '1 '3 '5 ''1 ''3 ''5
 
 
% Every answer is the same FALL — a harmonic minor coming down, begun
% higher each time. So it is written once at its longest, from the g
% above the top a to the b under middle C, and the three shorter ones
% are its tail: Fall[3:], Fall[5:], Fall[8:].
%
% One thing the ear catches and the spelling has to keep. A fall that
% starts from an A starts on a NATURAL g, not on the leading tone; by
% the octave below, the g# is back. That is why Fall's own head is '7b
% and why it carries '7 and .7 further down — and why the shortest
% descent, which starts from an a of its own, gets a '7b in front of it.
 
Fall ::= ''7b/2 ''6/2 ''5/2 ''4/2 ''3/2 ''2/2 ''1/2 '7/2
  '6/2 '5/2 '4/2 '3/2 '2/2 '1/2 7/2
  6/2 5/2 4/2 3/2 2/2 1/2
  .7/2 .6/2 .5/2 .4/2 .3/2 .2/2
 
 
% The cadenza entire, mm.1-10. The open A, then sweep and fall four
% times over, each pair higher than the last: the four peaks are a6,
% c7, e7, a7, which is the triad itself climbing while the figures
% climb inside it. Then a CHROMATIC run — no degrees, just semitones —
% up nearly two octaves and back down the same notes, which is how
% Paganini leaves the cadenza and starts the piece.
%
% The run is written in PLAIN notes and halved where it sounds, because
% the cadenza takes it at twice the speed the moto perpetuo will: the
% cadenza says 'Chrom/2', and m.26, which is the same run coming down,
% says 'Detach(Chrom~)' and gets the moto's own bowing for nothing. The
% top a is written once, between the two, because that is what it is —
% the turn, played and left.
 
Chrom ::= a#3 b3 c4 c#4 d4 d#4 e4 f4 f#4 g4 g#4 a4 a#4 b4 c5 c#5 d5 d#5 e5 f5 f#5 g5 g#5
 
Cadenza ::= - _4 .1 .1 _12
  Sweep[:8]/2  ''1 _6   '7b/2 Fall[8:]  .1 _6
  Sweep[:9]/2  ''3 _6   Fall[5:]        .1 _6
  Sweep[:10]/2 ''5 _6   Fall[3:]        .1 _6
  Sweep/2      '''1 _6  Fall            .1 _3
  Chrom/2 a5/2 Chrom~/2
 
Cadenza
 
 
% mm.10-13 — the MOTO PERPETUO starts. Two chords to plant the key, and
% then sixteen notes to the bar, detached, for the next sixty measures.
%
% Detached is the whole of the bowing and it is one rule. 'Detach' is a
% CELL: $1 is a hole, the line is poured into it a note at a time, and
% each note takes half its slot with a rest taking the other half. So
% the figure itself is only the sixteen pitches it walks.
 
Detach   ::= $1 /2 - /2
MotoLine ::= .1 .3 .5 .3 .2 .4 .5 .4 .3 1 .7b .6 .5 .4 .3 .2
Moto     ::= Detach(MotoLine)
 
% It starts on the second half of the bar and runs across the barline,
% so any bar of the score is the back half of one statement and the
% front half of the next. Twice as written, and then it TURNS: the same
% shape with a c# in it, pointing at the major.
 
TurnA ::= .1 .3# .5 .3# .1 .4 .6 .4 ..7b .2
 
.1 _6 {.3, .5} {3, '1} _7/2 - _9/2
  Moto Moto Detach(TurnA)
 
 
% mm.14-17 — and here the performance changes gear, from 105 to 167,
% ten notes into the turn. A tempo mark can only stand between
% statements, so the turn is written in the two pieces the mark cuts it
% into: TurnA before, TurnB after.
%
% Then the line LIFTS — a climb of two octaves on the dominant seventh,
% e g# b e g# b d e — and the moto comes back an octave up, twice,
% exactly as it was. The last ten notes are already the next figure:
% the cadence onto d that m.18 takes up.
 
TurnB    ::= .6 .2 ..7b .3 .5 .3
LiftLine ::= ..7 .2 .6 .4 .3 .1 1 .4# .5 .7 2 5 7 '2 '4 '5
Lift     ::= Detach(LiftLine)
CadA     ::= 1 3# 5 3# '1 7b 6 5 3# 4
 
bpm 166.75
Detach(TurnB) Lift Moto@12 Moto@12 Detach(CadA)
 
 
% mm.18-24 — form D, the moto's first long stretch; a minor with its
% NATURAL seventh; the line comes down what the Lift climbed, and two
% sequences carry it from there.
%
% The key mark changes, and it is a real change: 'a4 min', not the
% cadenza's 'a4 harm'. Every descending figure from here on uses g, not
% g#, and both sequences below are exact under min and wrong under harm.
% The harmonic minor was the cadenza's melodic decision, not the key the
% piece is in.
 
key a4 min
 
assert .7 == g4
 
% Both figures are poured through Detach in their own definitions, so a
% statement is the figure and nothing else. And the stretch is ONE
% statement: sixteen detached notes to the bar for seven bars, and not
% one figure in it begins on a barline, so the bars go unwritten and the
% figures are laid out a line apiece instead.
%
% Spill — five notes, a step at a time, down from the b above the staff;
% stated twice, at m.18 and at m.22, both times just after the line has
% run out of room above.
% Rung — four notes, stated five times: a step down, then up a third and
% up a fourth to the note the line is really on. Each statement is one
% scale step lower than the last, so the notes it arrives on — c5 b4 a4
% g4 f4 — spell a minor scale coming down. A ladder, one rung at a time.
% Drop — eight notes, stated four times: a leap up a fourth, then the
% scale falls a sixth, one scale step lower each time.
 
Spill ::= Detach('2 '1 7 6 5)
Rung  ::= Detach(.6 .5 .7 3)
Drop  ::= Detach(6 '2 '1 7 6 5 4 2)
 
assert Rung@~-4 == Detach(.2 .1 .3 .6)
 
% Each fall of the Drop belongs to the key it is falling through — the
% first runs down E harmonic minor (f#, d#), the second D harmonic minor
% (c#), the third is plain — and no one mark says that. So Drop stays
% the plain diatonic figure and the accidental sits with the statement
% that wants it. '\n' replaces one ITEM; Drop is bowed, so it plays a
% rest after every note and its fifth note is item 8. The replacement is
% braced because it carries a half value of its own, and it stands
% BEFORE the mark, in Drop's own degrees, since the mark moves the
% replacement too.
 
assert Drop \8 {6#/2} \12 {4#/2} == Detach(6 '2 '1 7 6# 5 4# 2)
assert Drop \14 {3/2} @~-3 == Detach(3 6 5 4 3 2 1 .7)
 
% m.18 — the spill, three more steps to the b below, then a turn under
% the open A and back.
Spill Detach(4 3 2) Detach(1 .7 .6# .7 1)
  % mm.18-20 — the ladder, five rungs, down to the f.
  Rung Rung@~-1 Rung@~-2 Rung@~-3 Rung@~-4
  % mm.20-22 — and it will not settle. The line drops under the open A
  % to g#, climbs back, and tries a rising figure twice before dropping
  % it, twice over. Two statements is a coincidence, not a rule, so
  % these two and a half measures are written out: a name here would
  % name something that is not there.
  Detach(.1 ..7# .3   .6 ..7# ..7 .2   .6 .2 .3 .5   .7 3 .4 .6)
  Detach(2 4 .5 .7   3 5 .6 1   3 6 .6# 4#   3 1 .7 7)
  Detach(5 3 .7 6   4 2 1 '3)
  % m.22 — the c above the staff, the highest note of form D, and the
  % line spills again: five notes and out, onto the open A. Not the
  % highest the moto has reached: the Lift got to e6 at m.15, a major
  % third above this c, which is why form D can climb this far and
  % still sound like it is holding something back.
  Spill Detach(1)
  % mm.22-24 — the Drop, four times. The fourth statement's override is
  % not an accidental: the sequence stops descending there and lands
  % back on the g the third landed on.
  Drop \8 {6#/2} \12 {4#/2}
  Drop \12 {4#/2} @~-1
  Drop @~-2
  Drop \14 {3/2} @~-3
  % and the last two notes are already the next figure — f#4 a4 begins
  % the diminished seventh that m.25 arpeggiates.
  Detach(.6# 1)
 
 
% mm.25-32 — forms E and F, and the piece starts to travel. A
% diminished seventh laid out as an arpeggio, two triads falling from
% the top of the bow, the cadenza's chromatic run coming back — and
% then a two-bar figure that says itself three times, a fourth higher
% each time, which is what carries the moto out of C major altogether.
%
% The key is C major and it is the plain reading of mm.25-29: the
% diminished seventh on C resolves into it, both falling arpeggios are
% its own chords, and mm.28-29 rock between the tonic and its dominant
% seventh half a bar at a time.
 
key c4 maj
 
% Three figures, and each is spelled the way it is for a reason.
%
% Dim — the diminished seventh as an arpeggio: four minor thirds
% stacked, which is a chord no key owns, so it is written in plain
% notes. The LINE takes the name and the bowing stands outside it,
% because at m.62 the chord comes back FALLING and three semitones
% down. Reversing a bowed figure reverses its rests with it and puts
% every note half a unit late; 'Detach(DimLine~)@-3' turns the line
% round inside the bowing and nothing else moves.
% Cross — eight notes over C and then its dominant seventh: up a
% fourth, back a step, up a sixth, and then the line falls and lifts.
% Plain notes again, because it returns at m.37 AT THE SAME PITCH under
% another key, and once more at m.65 three semitones down.
% Wheel — two bars of broken chords, stated three times, each statement
% five semitones above the last. The sequences of m.18 moved a scale
% STEP at a time and were written @~-1; a fourth is not a scale step in
% any key, so this one is @5 and @10, and the harmony goes round the
% wheel of fourths: C at m.28, then F, B flat, E flat.
 
DimLine ::= c5 d#5 f#5 a5 c6
Dim     ::= Detach(DimLine)
Cross   ::= Detach(g3 c4 b3 g4 f4 d4 c4) Detach(e4)
Wheel   ::= Detach(e4 c4 bb3 g4 e4 bb3 a3 f4)
  Detach(c4 a3 g3 e4 bb3 g3 a3 f5)
  Detach(c5 a4 g4 e5 c#5 g4 f4 d5)
  Detach(a4 f4 e4 c#5 g4 e4 f4) Detach(d5)
 
% The Wheel is written in its four HALF-BARS, one call apiece, because a
% body of SEVERAL calls slices by the CALL: Wheel[3:] is the fourth
% half-bar, which is what the assert below asks for. That is not the only
% way to cut a bowed figure, and the difference is worth having straight,
% because a body of ONE call slices by the ITEM instead — which is what
% Moto[:18] leans on earlier in this lesson, nine notes being eighteen
% items once a rest follows each one. The same thirty-two notes written
% as a single call would give three quarters of itself as [:48]. The
% half-bars are chosen here for a musical reason, not a technical one:
% the Wheel begins in the middle of m.29, so the barline where form F
% starts falls inside it and the section has to break there too.
%
% Two of the figures are a call further divided, their seven notes and
% their eighth: where the Wheel and the Cross come back at mm.35-37 a
% chromatic run wants that very sixteenth, and a twenty-three-rung run
% has the better claim on its own head. The asserts below say the
% dividing changed nothing.
 
assert Wheel[3:] == Detach(a4 f4 e4 c#5 g4 e4 f4 d5)
 
% And the chromatic run stops at g#5 — the a above it is written on its
% own in the cadenza — which is why m.26 asks for four semitones and not
% three to start on the c above the staff.
 
assert Chrom[22:] == g#5
assert Cross \0 {a4/2} \2 {g4/2} == Detach(a4 g4 b3 g4 f4 d4 c4 e4)
 
% m.25 — the diminished seventh climbing, one more stab at its tritone,
% and then two triads falling from the top of the bow: C major, 1 5 3 1
% 5 3 1 5, and G major in the same shape from its own third.
Dim Detach(f#5)
  Detach(''1 '5 '3 '1 5 3 1 .5)
  Detach('7 '5 '2 7 5 2 .7 .5) Detach(.6)
  % mm.26-27 — the cadenza's chromatic run, note for note, coming down
  % from the c above the staff to the d below middle C in the moto's own
  % bowing. Twenty-three semitones for one name, and the name was
  % already written.
  Detach(Chrom~)@4
  Detach(1 3)
  % mm.28-29 — the Cross, three times. The middle one is the figure
  % bare; the first says a4 g4 where it says g3 c4, and the third ends a
  % third higher. Three exceptions in twenty-four notes, each riding the
  % statement that wants it.
  Cross \0 {a4/2} \2 {g4/2}
  Cross
  Cross \14 {g4/2}
  % mm.29-30 — the wheel starts on the second half of m.29: three of its
  % four half-bars, and then the section ends.
  Wheel[:3]
 
% mm.31-32 — form F. The wheel's last half-bar, and then the whole turn
% again a fourth up. Nothing else: two bars for one name and one mark.
Wheel[3:] Wheel[:3]@5
 
 
% mm.33-40 — form F prime and then form G, and the two statements below
% are those two sections. Nothing in these eight measures is new: the
% wheel finishes its climb by fourths, the cadenza's chromatic run comes
% down twice, and the bariolage of m.28 comes back at the pitch it had
% there. Only the last two measures are written out, and they are
% written out because there is nothing in them to name.
%
% Two figures give up a note at their seam, and both give it to a
% chromatic run. The wheel's last half-bar ends on the very note the
% first run starts on, and the Cross's last place is taken by the head
% of the second. The run has the better claim both times — it is
% twenty-three rungs long and the note is its top — so each figure was
% written as its seven notes and its eighth, one call apiece. A slice
% over a body of calls counts the CALLS, and that is the only way to
% divide a bowed figure.
 
assert Wheel[3:] == Wheel[3:4] Detach(d5)
assert Cross == Cross[:1] Detach(e4)
assert Chrom[22:]@4 == c6
 
% mm.33-36 — form F prime: the wheel's last two rounds. The B flat turn
% finishes in the first half of m.33, and the E flat turn — the third
% and last — runs from there to m.35. One note of it is different: on
% the last half-bar the line steps down to the c instead of repeating
% the e flat, which is how it arrives at the c above the staff. The
% override stands BEFORE the mark and in the wheel's own pitches,
% because the mark moves the replacement too.
%
% From there the run of m.26 comes again note for note — all twenty-three
% semitones, from that same c down to the same d below middle C, in the
% moto's own bowing, and the same c after it. Only the note after THAT
% differs, and only by an accidental: an e flat where m.27 had an e
% natural, which is the C minor the next run falls out of.
Wheel[3:]@5 Wheel[:3]@10
  Wheel[3:4] \12 {d4/2} @10
  Detach(Chrom~)@4
  Detach(1 3b)
 
% mm.37-40 — form G. The Cross again, at the pitch it had at m.28: its
% first two notes replaced by an a flat and the g below it — the low g
% and the c of m.28 lifted into the figure's own upper register — and
% its other five untouched. Then the run, and this one wants no
% transposition at all: g# above the staff down to the b flat below
% middle C is the cadenza's descent at the cadenza's own pitch. It ends
% as m.27's did, a whole tone under its last rung.
%
% Then the storm gathers. From the last sixteenth of m.38, two measures
% of broken chords in fours — every four falling, and the fours joined
% in pairs by a leap up an OCTAVE. Two of those pairs are the same eight
% notes a whole tone apart, the pair from m39+11 and the pair from
% m40+3, and the pair before them is close but not the same. Two
% statements is a coincidence, not a rule, so these measures are written
% out and named nothing; and they are written in plain notes because ten
% of the twelve sound in them and no key owns that.
Cross[:1] \0 {g#4/2} \2 {g4/2}
  Detach(Chrom~) Detach(g#3)
  Detach(f4 d#4 c4 g3)
  Detach(d#4 c#4 a#3 g#3) Detach(g#4 d#4 c4 b3)
  Detach(f#4 d#4 c4 a#3)  Detach(a#4 f4 c#4 b3)
  Detach(g#4 f4 d4 c4)    Detach(c5 g4 d#4 c#4)
  Detach(a#4 g4 c#4 a#3 g4)
 
 
% mm.41-47 — form H, the central storm; D FLAT from m.43 and it holds to
% the barline; two bars that belong to no key, and then one figure said
% twice.
%
% The moto has not stopped since m.14 and does not stop here — what comes
% loose is the harmony. D flat is four fourths above the F the Wheel
% turned to in m.31, the same climb the Wheel was making, and it holds:
% not one of m.47's sixteen notes is outside the key, and twenty-nine of
% the Churn's thirty-two are in it. The mark is 'c#5' rather than 'c#4'
% for the reason the cadenza's is 'a4' — the storm plays BELOW its tonic,
% so the degrees that say it are the lower-octave ones.
 
key c#5 maj
 
assert .1 == c#4
 
% Churn — the storm's unit, two whole bars, stated twice: at m.43 in D
% flat and at m.45 a whole step up, on E flat. The second statement is a
% transposition, so it needs no key mark of its own; the mark carries the
% spelling. F sharp would spell the figure with one accidental fewer and
% is still the wrong reading: only D flat puts the figure's own tonic on
% the tonic, and only then does its head read as what it is.
%
% Because the head IS THE MOTO'S OWN. MotoLine's first nine degrees, read
% in this key, are the Churn's first nine notes — the figure the piece
% has played since m.10, here for the first time in a major key. Those
% nine come three times in the caprice and no more: twice in the Churn,
% and once at m.50 in E major.
%
% Then the scale falls from the tonic; then the head's first four notes
% again; then the bass steps from the tonic to the flat second while the
% upper pair rocks between the fifth and the flat seventh, which is where
% all three of the figure's accidentals are; then the head's first four
% once more, one scale step up.
 
assert MotoLine[:9] == .1 .3 .5 .3 .2 .4 .5 .4 .3
assert MotoLine[:4]@~1 == .2 .4 .6 .4
 
Churn ::= Detach(.1 .3 .5 .3 .2 .4 .5 .4 .3)
  Detach(1 .7 .6 .5 .4 .3 .2)
  Detach(.1 .3 .5 .3)
  Detach(.1 .5 .7b .5 .2b .5 .7b .5)
  Detach(.2 .4 .6 .4)
 
% m.41 — six notes climbing to the b flat above the stave, then ten
% falling to the d flat above middle C. Nothing in m.41 or m.42 recurs
% anywhere else in the caprice at more than five notes, and every
% five-note match is a run of the scale, so the two bars are written
% where they stand, in plain notes: no key owns them, A flat needing six
% accidentals over these thirty-eight notes and D flat nine. What they
% are is D flat shadowed a semitone below — e under e sharp, g under g
% sharp, a under a sharp, d under d sharp, and nothing else.
Detach(e4 c5 g4 e5 c5 a#5 g#5 f5 c5 g#4 e4 f4 g#4 f4 d#4 c#4)
  % m.42 — and back up, from middle C to the g sharp above the stave. A
  % seven-note figure — a third up, then the scale falling a fifth from
  % where it started — runs from the twelfth note of m.41 across the
  % barline, and is said once more from the fourth note here, three steps
  % up the A flat scale. Then it is dropped. Two statements is a
  % coincidence, not a rule, and a name for it would name something that
  % is not there.
  Detach(c4 a#3 a4 a#4 c#5 a#4 g#4 g4 f4 d#4 d5 d#5 g5 d#5 g#5 d#5)
  % m.43 — six notes more, and the storm finds its figure.
  Detach(c5 g#4 g4 g#4 f#4 d#4)
  % mm.43-45 — the Churn, from the last quarter of m.43.
  Churn
  % mm.45-47 — and again, a whole step up, on E flat.
  Churn@2
  % m.47 — the bar's last ten notes, driving at the barline, D flat
  % throughout.
  Detach(.4 2 .7 .4 .3 3 1 .6 .5 3)
 
 
% m.48 — form I, one bar and a hinge; B MAJOR, a whole step down from the
% storm's D flat; the dominant seventh falling, and the foot of a sweep.
%
% Fifteen of the sixteen notes are in the new key. The odd one is the d
% natural the bar opens on — the last note of the old key, and the only
% accidental the bar needs. It falls from the f sharp above the stave to
% the a sharp below middle C; its third beat spells the dominant seventh
% exactly, c#5 a#4 f#4 e4; and the last two notes, b3 and d#4, are the
% tonic and its third — the foot of a sweep up a B MAJOR triad, which
% m.49 climbs.
 
key b4 maj
 
assert .1 == b3
assert 3b == d5
assert .1 .3 == b3 d#4
 
Detach(3b .6 .5 5 3 .7 .5 4 2 .7 .5 .4 .2 ..7 .1 .3)
 
 
% mm.49-55 — forms J and K, and the caprice turns for home. It has
% travelled as far as B major, and it does the plainest thing there is
% with a new key: it sweeps it. Then it sweeps the relative minor, and
% four bars later the music is standing on E, the dominant of the A
% minor it left at m.25.
%
% Two figures carry the stretch up and one carries it back down: the
% cadenza's own arpeggio, bowed; a four-note reach that climbs the E
% major scale once the moto has turned the corner; and a broken chord
% that falls three bars in six statements.
 
key b4 maj
 
% Climb — the cadenza's Sweep, bowed: the tonic below, its third and its
% fifth, then the triad twice over. Eight rungs of the eleven, stated
% twice here and once more inside the hinge at m.56.
%
% It is the Sweep and the assert says so — the same ladder from the rung
% under its first — so this is one musical object under two names, which
% the standard does not like and the language leaves no way round: a
% cell fills its hole with whatever it is handed, whole, so 'Detach' can
% never be given 'Sweep[:7]'. The eight rungs have to be written
% somewhere, and a name is the cheapest place.
%
% Reach — a note, a reach up a fourth, back down to it, and a step
% down. Stated four times, on b, c#, d#, f#: two tones and then a minor
% third, which are four rungs of the E major scale m.50 turns to.
%
% Tumble — eight notes, stated six times across mm.53-55. A broken
% chord falls, hops up, and the next chord falls from where the hop
% left it. Each of its two chords has a semitone rub in it, g against
% f# and then f against that f#, and that rub is the sound of the whole
% descent.
 
Climb  ::= Detach(.1 .3 .5 1 3 5 '1 '3)
Reach  ::= Detach(b4 e5 b4 a4)
Tumble ::= Detach(a#5 g5 d#5 f#5 b5 f#5 d5 f5)
 
assert .1 Sweep[:7] == .1 .3 .5 1 3 5 '1 '3
 
% Two scale steps down in B major and the major triad has become its
% own relative minor: the same seven notes read from a different rung.
% That is the whole modulation of m.49, in one mark.
 
assert Climb@~-2 == Detach(g#3 b3 d#4 g#4 b4 d#5 g#5 b5)
 
% And the moto's head comes back a fourth up, which is what puts the
% music in E. A slice counts ITEMS, and a bowed figure plays a rest
% after every note, so the moto's first nine notes are Moto[:18].
 
assert Moto[:18] == Detach(.1 .3 .5 .3 .2 .4 .5 .4 .3)
 
% mm.49-53 — form J. The sweep's first two rungs were m.48's, so six
% are left. Then the dominant seventh of g# minor falls twice, an
% octave apart — d# g a# c#, its third written as a plain g and not as
% f##, which is how a violinist reads it — and the second fall stops
% three notes in, where the sweep takes over. These eight notes are
% m.48's own, from the second half of its second beat, three semitones
% down, and that is the only other place in the caprice that says them;
% two statements is a coincidence, so they are written and not named.
% Then the sweep again in the relative minor, and nine notes of the moto
% on E.
Climb[4:]
  Detach(d#4 c#5 a#4 g4 d#4 c#4 a#3 g3)
  Climb@~-2
  Moto[:18]@5
  % mm.50-52 — the reach, four times, climbing. It begins a sixteenth
  % before the beat, so the reach itself lands on the beat. Between
  % each pair stands the same shape with the reach CLOSED UP, not
  % opened: the Reach spans a fourth and these span a major third, a
  % minor third, a tritone and a minor third, so three of the four are
  % NARROWER than the figure they sit between. Holding the upper note
  % while the lower keeps climbing is what shrinks them — the climb
  % eats the interval. No two of the four are alike, so each is
  % written where it stands.
  Reach Detach(c5 e5 c5 a#4) Reach@2 Detach(d#5 f#5 d#5 c5)
  Reach@4 Detach(e5 a#5 e5 d#5) Reach@7 Detach(g#5 b5 g#5 e5)
  % m.53 — the line turns over and falls, two statements to the bar.
  % One note of m.52 comes with them: the g the tumble is about to say
  % again.
  Detach(g5) Tumble Tumble@-2
 
% mm.54-55 — form K, and the sequence runs straight across the section
% break: four more statements, each a whole tone below the last except
% the fifth, which is only a semitone. No scale says that, so the marks
% count semitones.
%
% The hop notes make a line of their own, one a beat, walking down by
% semitones: f# f e d# d c# c c b a# a a. Twice it stays where it is
% instead of going on, and those two stalls are the only exceptions in
% forty-eight notes. Each rides the statement that wants it: one
% override raises the last two notes a semitone and the other only the
% last, which takes the rub out of the second chord and lands the
% figure on a plain F major and then a plain D minor — the two chords
% that lead back to A.
Tumble@-4
  Tumble \12 {d#5/2} \14 {f#5/2} @-6
  Tumble@-7
  Tumble \14 {f#5/2} @-9
 
 
% m.56 — form L, one bar, and the whole bar is a hinge: the caprice
% turns here and the reprise is on the other side of it. The open A
% rings between d, f and then d# — D minor, and then the raised fourth
% that says E is coming — and the middle eight notes of the bar are
% Climb itself, on E MAJOR, the dominant the piece has been away from
% since m.25. Two octaves and a third in eight notes. The last two,
% f and a, are the D minor that m.57 starts from.
%
% Hinge — fourteen notes, stated twice: here, and an octave up at m.57.
% It is written in plain notes and not in degrees, and that is not
% laziness. The reprise needs a minor back in force by m.57, and a
% figure spelled in degrees would be reread in THAT key. The key mark
% below is here so the claim under it can be made at all: in E, Climb's
% eight rungs ARE the middle of the bar.
 
bpm 164.75
key e5 maj
 
Hinge ::= Detach(d5 a4 f4 a4 d#5 a4 e4 g#4 b4 e5 g#5 b5 e6 g#6)
 
assert Hinge[12:] == Climb
 
Hinge Detach(f5 a5)
 
 
% mm.57-64 — form M, the long REPRISE, and it is the piece gathering up
% what it has already said. The hinge swings an octave higher; the moto's
% own line comes back two octaves above where it started; the four falls
% that closed form D say themselves again a sixth higher; the diminished
% seventh returns upside down; the cadenza's first sweep comes down
% instead of up; and the cadenza's chromatic run closes the section.
% Eight measures and not one new figure in them.
%
% The key goes back to the HARMONIC minor, and it has to: the moto's line
% spells its seventh '.7b', which is a g under 'harm' and an f# under
% 'min', and this section states that line twice.
 
key a4 harm
 
assert 7 == g#5
 
% m.57 is m.56 transposed, note for note — so the hinge is stated, not
% rewritten. Hinge is the fourteen notes the figure owns; the last two of
% m.57 are already the moto's. And it is spelled in plain notes, which is
% what lets '@12' mean the same thing under any key mark.
 
assert Hinge@12 == Detach('4 '1 6 '1 '4# '1 5 7 '2 '5 '7 ''2 ''5 ''7)
 
% The moto says its line twice. The second time it BENDS: the last two
% notes come out c# for c and a for b, which are the leading tone and the
% dominant of D — and D is the note the next figure leaps to. '\n' counts
% from 0 and MotoLine is unbowed, so its fifteenth and sixteenth notes
% are items 14 and 15.
 
Veer ::= .1 .3 .5 .3 .2 .4 .5 .4 .3 1 .7b .6 .5 .4 .3# .1
 
assert Veer == MotoLine \14 {.3#} \15 {.1}
 
% The four falls are the Drop of mm.22-24, a sixth higher, and only two
% of them can say so. The first and the last fall through a NATURAL g;
% the third takes the harmonic minor's own g#, and the second touches no
% g at all. Under 'harm' a diatonic shift cannot land on a natural g —
% every '@~' into that degree gives g# — so those two are written out
% with '7b' and the middle two are the Drop itself. Under 'min' the trade
% runs the other way and the moto's line comes out wrong, which is why
% the mark above is 'harm'.
 
assert Drop@~3 == Detach('2 '5 '4 '3 '2 '1 7 5)
 
% And the diminished seventh of m.25 comes back falling, three semitones
% down. Its LINE is what carries the name — 'Dim ::= Detach(DimLine)' —
% so the line can be turned round INSIDE the bowing. 'Dim~' would not
% do: reversing a bowed figure reverses its rests with it, and every
% note would land half a unit late.
 
assert Detach(DimLine~)@-3 == Detach(a5 f#5 d#5 c5 a4)
 
% The sweep that opened the caprice comes down too. Seven of the eight
% notes of m.62 are its own first seven rungs read backwards, and now
% that the sweep is spelled in degrees the claim can be made.
 
assert Sweep[:7]~ == '3 '1 5 3 1 .5 .3
 
% m.57 — the hinge again, an octave up. Its second half is the E major
% arpeggio, and an octave up that climbs to g#7.
Hinge@12
  % mm.57-59 — the moto two octaves up, twice, the second time bent.
  Moto@24 Detach(Veer)@24
  % mm.59-61 — the four falls: the leap up a fourth and the scale
  % falling a sixth after it, one scale step lower each time.
  Detach('4 '7b '6 '5 '4 '3 '2 7b)
  Drop@~4
  Drop@~3
  Detach('1 '4 '3 '2 '1 7b 6 5)
  % mm.61-62 — the diminished seventh again, falling this time and three
  % semitones down, with one more stab at its tritone under it.
  Detach(d#5 c6) Detach(DimLine~)@-3 Detach(d#4)
  % m.62 — and the cadenza's first sweep backwards: the a minor triad
  % from the c above the staff all the way down to the open A. Seven of
  % those eight notes are the Sweep's own first seven rungs, reversed,
  % which the assert above the statement says in the language.
  Detach('3 '1 5 3 1 .5 .3 .1)
  % mm.62-63 — then the diminished seventh on the leading tone, g# b d,
  % falling an octave at a time to the g# under middle C.
  Detach('2 7 4 2 .7 .4 .2 ..7)
  % mm.63-64 — the open A, and the chromatic run of the cadenza coming
  % down from the a above the staff in the moto's own bowing: the whole
  % of Chrom reversed, one semitone up. The last two, the open A and the
  % c above it, carry into m.65 — where they are the first two notes of
  % the bariolage the Cross gives it.
  Detach(.1) Detach(Chrom~)@1 Detach(.1 .3)
 
 
% mm.65-69 — form N. The moto has been running since m.10 and here it
% runs out of road: one figure over and over around the open A, and then
% the cadenza walks back in over the top of it.
%
% The key has not moved — this is still A minor with its raised seventh,
% the key the reprise came back in — and the mark is here to say so.
 
key a4 harm
 
assert ..7 == g#3
 
% Two rising thirds a step apart, b d and c e, and then the same pair two
% scale steps lower, g# b and a c. Eight notes that rock between the
% dominant seventh and the tonic and get no further, which is the point
% of the passage: the moto has stopped going anywhere. That is why the
% figure is written as one four-note pair said twice and not as eight
% notes — the sequence IS the figure.
 
Thirds ::= .2 .4 .3 .5
Spin   ::= Detach(Thirds Thirds@~-2)
 
assert Thirds@~-2 == ..7 .2 .1 .3
assert Spin == Detach(b3 d4 c4 e4 g#3 b3 a3 c4)
 
% m.65 — two beats to arrive, and they are the bariolage of m.37 three
% semitones down: the Cross's first seven notes, with the same two
% overrides that measure wanted and one mark to carry the lot. The
% replacements are written in the Cross's own pitches — an a flat and
% the g under it — because the mark moves them too, so what sounds is
% an f and an e. Only the eighth note is not the Cross's: where the
% figure would give a c sharp the piece plays the plain c of the key,
% and the Spin takes over from there.
%
% The Spin, three times whole. The fourth breaks off six notes in —
% the pair, and two notes of the pair below — and the open A it was
% going to land on is held instead: twelve units of it, straight through
% the barline into m.68.
 
Cross[:1] \0 {g#4/2} \2 {g4/2} @-3 Detach(.3)
  Spin*3
  Detach(Thirds ..7 .2) .1 _12
  % m.68 — and the cadenza is back, at the moto's speed and in the
  % MAJOR. A major is A harmonic minor with its third and its sixth
  % raised and nothing else moved — the seventh is already g#, which is
  % why the answer's leading tone needs no mark at all. So the sweep is
  % the cadenza's own ladder with its thirds sharpened, and the answer is
  % the cadenza's own Fall with the third and the sixth of each octave
  % sharpened: six notes in nineteen, one pair an octave.
  Sweep[:8] \0 {.3#} \3 {3#} \6 {'3#} /2  ''1 _6
    '7/2 Fall[8:] \0 {'6#/2} \3 {'3#/2} \7 {6#/2} \10 {3#/2} \14 {.6#/2} \17 {.3#/2}
    .1 _6
  % m.69 — the second sweep begins before the gear change, and a tempo
  % mark can only stand between statements, so it is written in the two
  % pieces the mark cuts it into — exactly as the turn was at m.14.
  Sweep[:8] \0 {.3#} \3 {3#} \6 {'3#} /2
 
 
% mm.70-73 — form O. The tempo drops back to the cadenza's own and the
% key mark now says what the last two measures had already spelled out:
% A major. From here the cadenza's own names carry everything with no
% marks at all — the same eleven-rung ladder and the same fall, read in
% the new key.
%
% Three sweeps are left: nine rungs, ten, and all eleven, each answered
% by the fall begun one rung higher than the last, and each answer
% landing on the open A. The first of the three is already in the air —
% m.70 opens on its ninth rung and then its peak, the c# above the top A.
 
bpm 105.25
key a4 maj
 
assert Sweep[:3] == c#4 e4 a4
assert Fall[5:9] == b6/2 a6/2 g#6/2 f#6/2
 
''1/2 ''3 _6 Fall[5:] .1 _6
  Sweep[:10]/2 ''5 _6 Fall[3:] .1 _6
  % and the fourth sweep climbs all eleven rungs to the a two octaves
  % above the staff and holds it. The answer to that one begins m.74.
  Sweep/2 '''1 _6
 
 
% mm.74-77 — form P, the last of them, and the frame shuts. The fourth
% fall, the cadenza's chromatic run, and the two chords. Not one new
% name in it: mm.68-77 is the opening cadenza again in the parallel
% major, and all the major asks of it is a single note.
%
% A fall from an A begins on the note under it, and that note is the
% one place the major disagrees with the minor. Fall's head is spelled
% ''7b because the cadenza's g is natural; here it has to be the
% major's g#. Nothing below the head moves — every degree from '7 down
% is natural in A major — so the flat in Fall's head is the only
% alteration the figure carries, and the only item this return has to
% override.
 
assert Fall[0:1] == g7/2
assert Fall \0 {''7/2} == g#7/2 Fall[1:]
 
% Then the chromatic run that ended the cadenza at m.9, note for note:
% up its twenty-three rungs from the a# under middle C, the top a
% written once between the two runs, and down the same notes. It is in
% plain pitches, so the new key does not touch it, and the gear change
% at m.70 left it at the speed it had then — 105.25 against 105.
%
% And the two chords the moto perpetuo planted the key with at m.10:
% the same two degrees, .3 with .5 and then 3 with '1. Only the key
% mark has changed, so the c that was natural there is a c# here. The
% caprice opens its moto on A minor and shuts it on A major, and the
% frame needed no rewriting to say so.
 
assert {.5, .3} {'1, 3} == {c#4, e4} {c#5, a5}
 
% m.74 — the fall from a7, and the open A under it.
Fall \0 {''7/2} .1 _3
  % mm.75-76 — the run up and back.
  Chrom/2 a5/2 Chrom~/2
  % m.76 — two stops, c# with e and then c# with the a above it: A
  % major between them, and the last of them rings through m.77.
  .1 _6 {.5, .3} {'1, 3} _11/2

% Paganini's fifth begins alone: a cadenza on the open A string, before
% the moto perpetuo it is famous for has sounded a note. Four SWEEPS up
% the a-minor triad, each reaching one rung higher than the last, and
% each answered by the scale falling the whole way back to the open A.
%
% The sweep is the triad itself, three degrees to the octave — the
% third, the fifth, the tonic above them — climbing until it runs out of
% violin. It is written once in DEGREES, eleven rungs long, and the four
% sweeps are its first eight, its first nine, its first ten, and the
% whole of it. Degrees and not intervals, because the frame comes back
% in A MAJOR at m.68 and 'i3 i4 i5' is a minor triad wherever it stands.

bpm 105
meter 4
unit 1/4
key a4 harm
mode diatonic

Sweep ::= .3 .5 1 3 5 '1 '3 '5 ''1 ''3 ''5

% Every answer is the same FALL — a harmonic minor coming down, begun
% higher each time. So it is written once at its longest, from the g
% above the top a to the b under middle C, and the three shorter ones
% are its tail: Fall[3:], Fall[5:], Fall[8:].
%
% One thing the ear catches and the spelling has to keep. A fall that
% starts from an A starts on a NATURAL g, not on the leading tone; by
% the octave below, the g# is back. That is why Fall's own head is '7b
% and why it carries '7 and .7 further down — and why the shortest
% descent, which starts from an a of its own, gets a '7b in front of it.

Fall ::= ''7b/2 ''6/2 ''5/2 ''4/2 ''3/2 ''2/2 ''1/2 '7/2
  '6/2 '5/2 '4/2 '3/2 '2/2 '1/2 7/2
  6/2 5/2 4/2 3/2 2/2 1/2
  .7/2 .6/2 .5/2 .4/2 .3/2 .2/2

% The cadenza entire, mm.1-10. The open A, then sweep and fall four
% times over, each pair higher than the last: the four peaks are a6,
% c7, e7, a7, which is the triad itself climbing while the figures
% climb inside it. Then a CHROMATIC run — no degrees, just semitones —
% up nearly two octaves and back down the same notes, which is how
% Paganini leaves the cadenza and starts the piece.
%
% The run is written in PLAIN notes and halved where it sounds, because
% the cadenza takes it at twice the speed the moto perpetuo will: the
% cadenza says 'Chrom/2', and m.26, which is the same run coming down,
% says 'Detach(Chrom~)' and gets the moto's own bowing for nothing. The
% top a is written once, between the two, because that is what it is —
% the turn, played and left.

Chrom ::= a#3 b3 c4 c#4 d4 d#4 e4 f4 f#4 g4 g#4 a4 a#4 b4 c5 c#5 d5 d#5 e5 f5 f#5 g5 g#5

Cadenza ::= - _4 .1 .1 _12
  Sweep[:8]/2  ''1 _6   '7b/2 Fall[8:]  .1 _6
  Sweep[:9]/2  ''3 _6   Fall[5:]        .1 _6
  Sweep[:10]/2 ''5 _6   Fall[3:]        .1 _6
  Sweep/2      '''1 _6  Fall            .1 _3
  Chrom/2 a5/2 Chrom~/2

Cadenza

% mm.10-13 — the MOTO PERPETUO starts. Two chords to plant the key, and
% then sixteen notes to the bar, detached, for the next sixty measures.
%
% Detached is the whole of the bowing and it is one rule. 'Detach' is a
% CELL: $1 is a hole, the line is poured into it a note at a time, and
% each note takes half its slot with a rest taking the other half. So
% the figure itself is only the sixteen pitches it walks.

Detach   ::= $1 /2 - /2
MotoLine ::= .1 .3 .5 .3 .2 .4 .5 .4 .3 1 .7b .6 .5 .4 .3 .2
Moto     ::= Detach(MotoLine)

% It starts on the second half of the bar and runs across the barline,
% so any bar of the score is the back half of one statement and the
% front half of the next. Twice as written, and then it TURNS: the same
% shape with a c# in it, pointing at the major.

TurnA ::= .1 .3# .5 .3# .1 .4 .6 .4 ..7b .2

.1 _6 {.3, .5} {3, '1} _7/2 - _9/2
  Moto Moto Detach(TurnA)

% mm.14-17 — and here the performance changes gear, from 105 to 167,
% ten notes into the turn. A tempo mark can only stand between
% statements, so the turn is written in the two pieces the mark cuts it
% into: TurnA before, TurnB after.
%
% Then the line LIFTS — a climb of two octaves on the dominant seventh,
% e g# b e g# b d e — and the moto comes back an octave up, twice,
% exactly as it was. The last ten notes are already the next figure:
% the cadence onto d that m.18 takes up.

TurnB    ::= .6 .2 ..7b .3 .5 .3
LiftLine ::= ..7 .2 .6 .4 .3 .1 1 .4# .5 .7 2 5 7 '2 '4 '5
Lift     ::= Detach(LiftLine)
CadA     ::= 1 3# 5 3# '1 7b 6 5 3# 4

bpm 166.75
Detach(TurnB) Lift Moto@12 Moto@12 Detach(CadA)

% mm.18-24 — form D, the moto's first long stretch; a minor with its
% NATURAL seventh; the line comes down what the Lift climbed, and two
% sequences carry it from there.
%
% The key mark changes, and it is a real change: 'a4 min', not the
% cadenza's 'a4 harm'. Every descending figure from here on uses g, not
% g#, and both sequences below are exact under min and wrong under harm.
% The harmonic minor was the cadenza's melodic decision, not the key the
% piece is in.

key a4 min

assert .7 == g4

% Both figures are poured through Detach in their own definitions, so a
% statement is the figure and nothing else. And the stretch is ONE
% statement: sixteen detached notes to the bar for seven bars, and not
% one figure in it begins on a barline, so the bars go unwritten and the
% figures are laid out a line apiece instead.
%
% Spill — five notes, a step at a time, down from the b above the staff;
% stated twice, at m.18 and at m.22, both times just after the line has
% run out of room above.
% Rung — four notes, stated five times: a step down, then up a third and
% up a fourth to the note the line is really on. Each statement is one
% scale step lower than the last, so the notes it arrives on — c5 b4 a4
% g4 f4 — spell a minor scale coming down. A ladder, one rung at a time.
% Drop — eight notes, stated four times: a leap up a fourth, then the
% scale falls a sixth, one scale step lower each time.

Spill ::= Detach('2 '1 7 6 5)
Rung  ::= Detach(.6 .5 .7 3)
Drop  ::= Detach(6 '2 '1 7 6 5 4 2)

assert Rung@~-4 == Detach(.2 .1 .3 .6)

% Each fall of the Drop belongs to the key it is falling through — the
% first runs down E harmonic minor (f#, d#), the second D harmonic minor
% (c#), the third is plain — and no one mark says that. So Drop stays
% the plain diatonic figure and the accidental sits with the statement
% that wants it. '\n' replaces one ITEM; Drop is bowed, so it plays a
% rest after every note and its fifth note is item 8. The replacement is
% braced because it carries a half value of its own, and it stands
% BEFORE the mark, in Drop's own degrees, since the mark moves the
% replacement too.

assert Drop \8 {6#/2} \12 {4#/2} == Detach(6 '2 '1 7 6# 5 4# 2)
assert Drop \14 {3/2} @~-3 == Detach(3 6 5 4 3 2 1 .7)

% m.18 — the spill, three more steps to the b below, then a turn under
% the open A and back.
Spill Detach(4 3 2) Detach(1 .7 .6# .7 1)
  % mm.18-20 — the ladder, five rungs, down to the f.
  Rung Rung@~-1 Rung@~-2 Rung@~-3 Rung@~-4
  % mm.20-22 — and it will not settle. The line drops under the open A
  % to g#, climbs back, and tries a rising figure twice before dropping
  % it, twice over. Two statements is a coincidence, not a rule, so
  % these two and a half measures are written out: a name here would
  % name something that is not there.
  Detach(.1 ..7# .3   .6 ..7# ..7 .2   .6 .2 .3 .5   .7 3 .4 .6)
  Detach(2 4 .5 .7   3 5 .6 1   3 6 .6# 4#   3 1 .7 7)
  Detach(5 3 .7 6   4 2 1 '3)
  % m.22 — the c above the staff, the highest note of form D, and the
  % line spills again: five notes and out, onto the open A. Not the
  % highest the moto has reached: the Lift got to e6 at m.15, a major
  % third above this c, which is why form D can climb this far and
  % still sound like it is holding something back.
  Spill Detach(1)
  % mm.22-24 — the Drop, four times. The fourth statement's override is
  % not an accidental: the sequence stops descending there and lands
  % back on the g the third landed on.
  Drop \8 {6#/2} \12 {4#/2}
  Drop \12 {4#/2} @~-1
  Drop @~-2
  Drop \14 {3/2} @~-3
  % and the last two notes are already the next figure — f#4 a4 begins
  % the diminished seventh that m.25 arpeggiates.
  Detach(.6# 1)

% mm.25-32 — forms E and F, and the piece starts to travel. A
% diminished seventh laid out as an arpeggio, two triads falling from
% the top of the bow, the cadenza's chromatic run coming back — and
% then a two-bar figure that says itself three times, a fourth higher
% each time, which is what carries the moto out of C major altogether.
%
% The key is C major and it is the plain reading of mm.25-29: the
% diminished seventh on C resolves into it, both falling arpeggios are
% its own chords, and mm.28-29 rock between the tonic and its dominant
% seventh half a bar at a time.

key c4 maj

% Three figures, and each is spelled the way it is for a reason.
%
% Dim — the diminished seventh as an arpeggio: four minor thirds
% stacked, which is a chord no key owns, so it is written in plain
% notes. The LINE takes the name and the bowing stands outside it,
% because at m.62 the chord comes back FALLING and three semitones
% down. Reversing a bowed figure reverses its rests with it and puts
% every note half a unit late; 'Detach(DimLine~)@-3' turns the line
% round inside the bowing and nothing else moves.
% Cross — eight notes over C and then its dominant seventh: up a
% fourth, back a step, up a sixth, and then the line falls and lifts.
% Plain notes again, because it returns at m.37 AT THE SAME PITCH under
% another key, and once more at m.65 three semitones down.
% Wheel — two bars of broken chords, stated three times, each statement
% five semitones above the last. The sequences of m.18 moved a scale
% STEP at a time and were written @~-1; a fourth is not a scale step in
% any key, so this one is @5 and @10, and the harmony goes round the
% wheel of fourths: C at m.28, then F, B flat, E flat.

DimLine ::= c5 d#5 f#5 a5 c6
Dim     ::= Detach(DimLine)
Cross   ::= Detach(g3 c4 b3 g4 f4 d4 c4) Detach(e4)
Wheel   ::= Detach(e4 c4 bb3 g4 e4 bb3 a3 f4)
  Detach(c4 a3 g3 e4 bb3 g3 a3 f5)
  Detach(c5 a4 g4 e5 c#5 g4 f4 d5)
  Detach(a4 f4 e4 c#5 g4 e4 f4) Detach(d5)

% The Wheel is written in its four HALF-BARS, one call apiece, because a
% body of SEVERAL calls slices by the CALL: Wheel[3:] is the fourth
% half-bar, which is what the assert below asks for. That is not the only
% way to cut a bowed figure, and the difference is worth having straight,
% because a body of ONE call slices by the ITEM instead — which is what
% Moto[:18] leans on earlier in this lesson, nine notes being eighteen
% items once a rest follows each one. The same thirty-two notes written
% as a single call would give three quarters of itself as [:48]. The
% half-bars are chosen here for a musical reason, not a technical one:
% the Wheel begins in the middle of m.29, so the barline where form F
% starts falls inside it and the section has to break there too.
%
% Two of the figures are a call further divided, their seven notes and
% their eighth: where the Wheel and the Cross come back at mm.35-37 a
% chromatic run wants that very sixteenth, and a twenty-three-rung run
% has the better claim on its own head. The asserts below say the
% dividing changed nothing.

assert Wheel[3:] == Detach(a4 f4 e4 c#5 g4 e4 f4 d5)

% And the chromatic run stops at g#5 — the a above it is written on its
% own in the cadenza — which is why m.26 asks for four semitones and not
% three to start on the c above the staff.

assert Chrom[22:] == g#5
assert Cross \0 {a4/2} \2 {g4/2} == Detach(a4 g4 b3 g4 f4 d4 c4 e4)

% m.25 — the diminished seventh climbing, one more stab at its tritone,
% and then two triads falling from the top of the bow: C major, 1 5 3 1
% 5 3 1 5, and G major in the same shape from its own third.
Dim Detach(f#5)
  Detach(''1 '5 '3 '1 5 3 1 .5)
  Detach('7 '5 '2 7 5 2 .7 .5) Detach(.6)
  % mm.26-27 — the cadenza's chromatic run, note for note, coming down
  % from the c above the staff to the d below middle C in the moto's own
  % bowing. Twenty-three semitones for one name, and the name was
  % already written.
  Detach(Chrom~)@4
  Detach(1 3)
  % mm.28-29 — the Cross, three times. The middle one is the figure
  % bare; the first says a4 g4 where it says g3 c4, and the third ends a
  % third higher. Three exceptions in twenty-four notes, each riding the
  % statement that wants it.
  Cross \0 {a4/2} \2 {g4/2}
  Cross
  Cross \14 {g4/2}
  % mm.29-30 — the wheel starts on the second half of m.29: three of its
  % four half-bars, and then the section ends.
  Wheel[:3]

% mm.31-32 — form F. The wheel's last half-bar, and then the whole turn
% again a fourth up. Nothing else: two bars for one name and one mark.
Wheel[3:] Wheel[:3]@5

% mm.33-40 — form F prime and then form G, and the two statements below
% are those two sections. Nothing in these eight measures is new: the
% wheel finishes its climb by fourths, the cadenza's chromatic run comes
% down twice, and the bariolage of m.28 comes back at the pitch it had
% there. Only the last two measures are written out, and they are
% written out because there is nothing in them to name.
%
% Two figures give up a note at their seam, and both give it to a
% chromatic run. The wheel's last half-bar ends on the very note the
% first run starts on, and the Cross's last place is taken by the head
% of the second. The run has the better claim both times — it is
% twenty-three rungs long and the note is its top — so each figure was
% written as its seven notes and its eighth, one call apiece. A slice
% over a body of calls counts the CALLS, and that is the only way to
% divide a bowed figure.

assert Wheel[3:] == Wheel[3:4] Detach(d5)
assert Cross == Cross[:1] Detach(e4)
assert Chrom[22:]@4 == c6

% mm.33-36 — form F prime: the wheel's last two rounds. The B flat turn
% finishes in the first half of m.33, and the E flat turn — the third
% and last — runs from there to m.35. One note of it is different: on
% the last half-bar the line steps down to the c instead of repeating
% the e flat, which is how it arrives at the c above the staff. The
% override stands BEFORE the mark and in the wheel's own pitches,
% because the mark moves the replacement too.
%
% From there the run of m.26 comes again note for note — all twenty-three
% semitones, from that same c down to the same d below middle C, in the
% moto's own bowing, and the same c after it. Only the note after THAT
% differs, and only by an accidental: an e flat where m.27 had an e
% natural, which is the C minor the next run falls out of.
Wheel[3:]@5 Wheel[:3]@10
  Wheel[3:4] \12 {d4/2} @10
  Detach(Chrom~)@4
  Detach(1 3b)

% mm.37-40 — form G. The Cross again, at the pitch it had at m.28: its
% first two notes replaced by an a flat and the g below it — the low g
% and the c of m.28 lifted into the figure's own upper register — and
% its other five untouched. Then the run, and this one wants no
% transposition at all: g# above the staff down to the b flat below
% middle C is the cadenza's descent at the cadenza's own pitch. It ends
% as m.27's did, a whole tone under its last rung.
%
% Then the storm gathers. From the last sixteenth of m.38, two measures
% of broken chords in fours — every four falling, and the fours joined
% in pairs by a leap up an OCTAVE. Two of those pairs are the same eight
% notes a whole tone apart, the pair from m39+11 and the pair from
% m40+3, and the pair before them is close but not the same. Two
% statements is a coincidence, not a rule, so these measures are written
% out and named nothing; and they are written in plain notes because ten
% of the twelve sound in them and no key owns that.
Cross[:1] \0 {g#4/2} \2 {g4/2}
  Detach(Chrom~) Detach(g#3)
  Detach(f4 d#4 c4 g3)
  Detach(d#4 c#4 a#3 g#3) Detach(g#4 d#4 c4 b3)
  Detach(f#4 d#4 c4 a#3)  Detach(a#4 f4 c#4 b3)
  Detach(g#4 f4 d4 c4)    Detach(c5 g4 d#4 c#4)
  Detach(a#4 g4 c#4 a#3 g4)

% mm.41-47 — form H, the central storm; D FLAT from m.43 and it holds to
% the barline; two bars that belong to no key, and then one figure said
% twice.
%
% The moto has not stopped since m.14 and does not stop here — what comes
% loose is the harmony. D flat is four fourths above the F the Wheel
% turned to in m.31, the same climb the Wheel was making, and it holds:
% not one of m.47's sixteen notes is outside the key, and twenty-nine of
% the Churn's thirty-two are in it. The mark is 'c#5' rather than 'c#4'
% for the reason the cadenza's is 'a4' — the storm plays BELOW its tonic,
% so the degrees that say it are the lower-octave ones.

key c#5 maj

assert .1 == c#4

% Churn — the storm's unit, two whole bars, stated twice: at m.43 in D
% flat and at m.45 a whole step up, on E flat. The second statement is a
% transposition, so it needs no key mark of its own; the mark carries the
% spelling. F sharp would spell the figure with one accidental fewer and
% is still the wrong reading: only D flat puts the figure's own tonic on
% the tonic, and only then does its head read as what it is.
%
% Because the head IS THE MOTO'S OWN. MotoLine's first nine degrees, read
% in this key, are the Churn's first nine notes — the figure the piece
% has played since m.10, here for the first time in a major key. Those
% nine come three times in the caprice and no more: twice in the Churn,
% and once at m.50 in E major.
%
% Then the scale falls from the tonic; then the head's first four notes
% again; then the bass steps from the tonic to the flat second while the
% upper pair rocks between the fifth and the flat seventh, which is where
% all three of the figure's accidentals are; then the head's first four
% once more, one scale step up.

assert MotoLine[:9] == .1 .3 .5 .3 .2 .4 .5 .4 .3
assert MotoLine[:4]@~1 == .2 .4 .6 .4

Churn ::= Detach(.1 .3 .5 .3 .2 .4 .5 .4 .3)
  Detach(1 .7 .6 .5 .4 .3 .2)
  Detach(.1 .3 .5 .3)
  Detach(.1 .5 .7b .5 .2b .5 .7b .5)
  Detach(.2 .4 .6 .4)

% m.41 — six notes climbing to the b flat above the stave, then ten
% falling to the d flat above middle C. Nothing in m.41 or m.42 recurs
% anywhere else in the caprice at more than five notes, and every
% five-note match is a run of the scale, so the two bars are written
% where they stand, in plain notes: no key owns them, A flat needing six
% accidentals over these thirty-eight notes and D flat nine. What they
% are is D flat shadowed a semitone below — e under e sharp, g under g
% sharp, a under a sharp, d under d sharp, and nothing else.
Detach(e4 c5 g4 e5 c5 a#5 g#5 f5 c5 g#4 e4 f4 g#4 f4 d#4 c#4)
  % m.42 — and back up, from middle C to the g sharp above the stave. A
  % seven-note figure — a third up, then the scale falling a fifth from
  % where it started — runs from the twelfth note of m.41 across the
  % barline, and is said once more from the fourth note here, three steps
  % up the A flat scale. Then it is dropped. Two statements is a
  % coincidence, not a rule, and a name for it would name something that
  % is not there.
  Detach(c4 a#3 a4 a#4 c#5 a#4 g#4 g4 f4 d#4 d5 d#5 g5 d#5 g#5 d#5)
  % m.43 — six notes more, and the storm finds its figure.
  Detach(c5 g#4 g4 g#4 f#4 d#4)
  % mm.43-45 — the Churn, from the last quarter of m.43.
  Churn
  % mm.45-47 — and again, a whole step up, on E flat.
  Churn@2
  % m.47 — the bar's last ten notes, driving at the barline, D flat
  % throughout.
  Detach(.4 2 .7 .4 .3 3 1 .6 .5 3)

% m.48 — form I, one bar and a hinge; B MAJOR, a whole step down from the
% storm's D flat; the dominant seventh falling, and the foot of a sweep.
%
% Fifteen of the sixteen notes are in the new key. The odd one is the d
% natural the bar opens on — the last note of the old key, and the only
% accidental the bar needs. It falls from the f sharp above the stave to
% the a sharp below middle C; its third beat spells the dominant seventh
% exactly, c#5 a#4 f#4 e4; and the last two notes, b3 and d#4, are the
% tonic and its third — the foot of a sweep up a B MAJOR triad, which
% m.49 climbs.

key b4 maj

assert .1 == b3
assert 3b == d5
assert .1 .3 == b3 d#4

Detach(3b .6 .5 5 3 .7 .5 4 2 .7 .5 .4 .2 ..7 .1 .3)

% mm.49-55 — forms J and K, and the caprice turns for home. It has
% travelled as far as B major, and it does the plainest thing there is
% with a new key: it sweeps it. Then it sweeps the relative minor, and
% four bars later the music is standing on E, the dominant of the A
% minor it left at m.25.
%
% Two figures carry the stretch up and one carries it back down: the
% cadenza's own arpeggio, bowed; a four-note reach that climbs the E
% major scale once the moto has turned the corner; and a broken chord
% that falls three bars in six statements.

key b4 maj

% Climb — the cadenza's Sweep, bowed: the tonic below, its third and its
% fifth, then the triad twice over. Eight rungs of the eleven, stated
% twice here and once more inside the hinge at m.56.
%
% It is the Sweep and the assert says so — the same ladder from the rung
% under its first — so this is one musical object under two names, which
% the standard does not like and the language leaves no way round: a
% cell fills its hole with whatever it is handed, whole, so 'Detach' can
% never be given 'Sweep[:7]'. The eight rungs have to be written
% somewhere, and a name is the cheapest place.
%
% Reach — a note, a reach up a fourth, back down to it, and a step
% down. Stated four times, on b, c#, d#, f#: two tones and then a minor
% third, which are four rungs of the E major scale m.50 turns to.
%
% Tumble — eight notes, stated six times across mm.53-55. A broken
% chord falls, hops up, and the next chord falls from where the hop
% left it. Each of its two chords has a semitone rub in it, g against
% f# and then f against that f#, and that rub is the sound of the whole
% descent.

Climb  ::= Detach(.1 .3 .5 1 3 5 '1 '3)
Reach  ::= Detach(b4 e5 b4 a4)
Tumble ::= Detach(a#5 g5 d#5 f#5 b5 f#5 d5 f5)

assert .1 Sweep[:7] == .1 .3 .5 1 3 5 '1 '3

% Two scale steps down in B major and the major triad has become its
% own relative minor: the same seven notes read from a different rung.
% That is the whole modulation of m.49, in one mark.

assert Climb@~-2 == Detach(g#3 b3 d#4 g#4 b4 d#5 g#5 b5)

% And the moto's head comes back a fourth up, which is what puts the
% music in E. A slice counts ITEMS, and a bowed figure plays a rest
% after every note, so the moto's first nine notes are Moto[:18].

assert Moto[:18] == Detach(.1 .3 .5 .3 .2 .4 .5 .4 .3)

% mm.49-53 — form J. The sweep's first two rungs were m.48's, so six
% are left. Then the dominant seventh of g# minor falls twice, an
% octave apart — d# g a# c#, its third written as a plain g and not as
% f##, which is how a violinist reads it — and the second fall stops
% three notes in, where the sweep takes over. These eight notes are
% m.48's own, from the second half of its second beat, three semitones
% down, and that is the only other place in the caprice that says them;
% two statements is a coincidence, so they are written and not named.
% Then the sweep again in the relative minor, and nine notes of the moto
% on E.
Climb[4:]
  Detach(d#4 c#5 a#4 g4 d#4 c#4 a#3 g3)
  Climb@~-2
  Moto[:18]@5
  % mm.50-52 — the reach, four times, climbing. It begins a sixteenth
  % before the beat, so the reach itself lands on the beat. Between
  % each pair stands the same shape with the reach CLOSED UP, not
  % opened: the Reach spans a fourth and these span a major third, a
  % minor third, a tritone and a minor third, so three of the four are
  % NARROWER than the figure they sit between. Holding the upper note
  % while the lower keeps climbing is what shrinks them — the climb
  % eats the interval. No two of the four are alike, so each is
  % written where it stands.
  Reach Detach(c5 e5 c5 a#4) Reach@2 Detach(d#5 f#5 d#5 c5)
  Reach@4 Detach(e5 a#5 e5 d#5) Reach@7 Detach(g#5 b5 g#5 e5)
  % m.53 — the line turns over and falls, two statements to the bar.
  % One note of m.52 comes with them: the g the tumble is about to say
  % again.
  Detach(g5) Tumble Tumble@-2

% mm.54-55 — form K, and the sequence runs straight across the section
% break: four more statements, each a whole tone below the last except
% the fifth, which is only a semitone. No scale says that, so the marks
% count semitones.
%
% The hop notes make a line of their own, one a beat, walking down by
% semitones: f# f e d# d c# c c b a# a a. Twice it stays where it is
% instead of going on, and those two stalls are the only exceptions in
% forty-eight notes. Each rides the statement that wants it: one
% override raises the last two notes a semitone and the other only the
% last, which takes the rub out of the second chord and lands the
% figure on a plain F major and then a plain D minor — the two chords
% that lead back to A.
Tumble@-4
  Tumble \12 {d#5/2} \14 {f#5/2} @-6
  Tumble@-7
  Tumble \14 {f#5/2} @-9

% m.56 — form L, one bar, and the whole bar is a hinge: the caprice
% turns here and the reprise is on the other side of it. The open A
% rings between d, f and then d# — D minor, and then the raised fourth
% that says E is coming — and the middle eight notes of the bar are
% Climb itself, on E MAJOR, the dominant the piece has been away from
% since m.25. Two octaves and a third in eight notes. The last two,
% f and a, are the D minor that m.57 starts from.
%
% Hinge — fourteen notes, stated twice: here, and an octave up at m.57.
% It is written in plain notes and not in degrees, and that is not
% laziness. The reprise needs a minor back in force by m.57, and a
% figure spelled in degrees would be reread in THAT key. The key mark
% below is here so the claim under it can be made at all: in E, Climb's
% eight rungs ARE the middle of the bar.

bpm 164.75
key e5 maj

Hinge ::= Detach(d5 a4 f4 a4 d#5 a4 e4 g#4 b4 e5 g#5 b5 e6 g#6)

assert Hinge[12:] == Climb

Hinge Detach(f5 a5)

% mm.57-64 — form M, the long REPRISE, and it is the piece gathering up
% what it has already said. The hinge swings an octave higher; the moto's
% own line comes back two octaves above where it started; the four falls
% that closed form D say themselves again a sixth higher; the diminished
% seventh returns upside down; the cadenza's first sweep comes down
% instead of up; and the cadenza's chromatic run closes the section.
% Eight measures and not one new figure in them.
%
% The key goes back to the HARMONIC minor, and it has to: the moto's line
% spells its seventh '.7b', which is a g under 'harm' and an f# under
% 'min', and this section states that line twice.

key a4 harm

assert 7 == g#5

% m.57 is m.56 transposed, note for note — so the hinge is stated, not
% rewritten. Hinge is the fourteen notes the figure owns; the last two of
% m.57 are already the moto's. And it is spelled in plain notes, which is
% what lets '@12' mean the same thing under any key mark.

assert Hinge@12 == Detach('4 '1 6 '1 '4# '1 5 7 '2 '5 '7 ''2 ''5 ''7)

% The moto says its line twice. The second time it BENDS: the last two
% notes come out c# for c and a for b, which are the leading tone and the
% dominant of D — and D is the note the next figure leaps to. '\n' counts
% from 0 and MotoLine is unbowed, so its fifteenth and sixteenth notes
% are items 14 and 15.

Veer ::= .1 .3 .5 .3 .2 .4 .5 .4 .3 1 .7b .6 .5 .4 .3# .1

assert Veer == MotoLine \14 {.3#} \15 {.1}

% The four falls are the Drop of mm.22-24, a sixth higher, and only two
% of them can say so. The first and the last fall through a NATURAL g;
% the third takes the harmonic minor's own g#, and the second touches no
% g at all. Under 'harm' a diatonic shift cannot land on a natural g —
% every '@~' into that degree gives g# — so those two are written out
% with '7b' and the middle two are the Drop itself. Under 'min' the trade
% runs the other way and the moto's line comes out wrong, which is why
% the mark above is 'harm'.

assert Drop@~3 == Detach('2 '5 '4 '3 '2 '1 7 5)

% And the diminished seventh of m.25 comes back falling, three semitones
% down. Its LINE is what carries the name — 'Dim ::= Detach(DimLine)' —
% so the line can be turned round INSIDE the bowing. 'Dim~' would not
% do: reversing a bowed figure reverses its rests with it, and every
% note would land half a unit late.

assert Detach(DimLine~)@-3 == Detach(a5 f#5 d#5 c5 a4)

% The sweep that opened the caprice comes down too. Seven of the eight
% notes of m.62 are its own first seven rungs read backwards, and now
% that the sweep is spelled in degrees the claim can be made.

assert Sweep[:7]~ == '3 '1 5 3 1 .5 .3

% m.57 — the hinge again, an octave up. Its second half is the E major
% arpeggio, and an octave up that climbs to g#7.
Hinge@12
  % mm.57-59 — the moto two octaves up, twice, the second time bent.
  Moto@24 Detach(Veer)@24
  % mm.59-61 — the four falls: the leap up a fourth and the scale
  % falling a sixth after it, one scale step lower each time.
  Detach('4 '7b '6 '5 '4 '3 '2 7b)
  Drop@~4
  Drop@~3
  Detach('1 '4 '3 '2 '1 7b 6 5)
  % mm.61-62 — the diminished seventh again, falling this time and three
  % semitones down, with one more stab at its tritone under it.
  Detach(d#5 c6) Detach(DimLine~)@-3 Detach(d#4)
  % m.62 — and the cadenza's first sweep backwards: the a minor triad
  % from the c above the staff all the way down to the open A. Seven of
  % those eight notes are the Sweep's own first seven rungs, reversed,
  % which the assert above the statement says in the language.
  Detach('3 '1 5 3 1 .5 .3 .1)
  % mm.62-63 — then the diminished seventh on the leading tone, g# b d,
  % falling an octave at a time to the g# under middle C.
  Detach('2 7 4 2 .7 .4 .2 ..7)
  % mm.63-64 — the open A, and the chromatic run of the cadenza coming
  % down from the a above the staff in the moto's own bowing: the whole
  % of Chrom reversed, one semitone up. The last two, the open A and the
  % c above it, carry into m.65 — where they are the first two notes of
  % the bariolage the Cross gives it.
  Detach(.1) Detach(Chrom~)@1 Detach(.1 .3)

% mm.65-69 — form N. The moto has been running since m.10 and here it
% runs out of road: one figure over and over around the open A, and then
% the cadenza walks back in over the top of it.
%
% The key has not moved — this is still A minor with its raised seventh,
% the key the reprise came back in — and the mark is here to say so.

key a4 harm

assert ..7 == g#3

% Two rising thirds a step apart, b d and c e, and then the same pair two
% scale steps lower, g# b and a c. Eight notes that rock between the
% dominant seventh and the tonic and get no further, which is the point
% of the passage: the moto has stopped going anywhere. That is why the
% figure is written as one four-note pair said twice and not as eight
% notes — the sequence IS the figure.

Thirds ::= .2 .4 .3 .5
Spin   ::= Detach(Thirds Thirds@~-2)

assert Thirds@~-2 == ..7 .2 .1 .3
assert Spin == Detach(b3 d4 c4 e4 g#3 b3 a3 c4)

% m.65 — two beats to arrive, and they are the bariolage of m.37 three
% semitones down: the Cross's first seven notes, with the same two
% overrides that measure wanted and one mark to carry the lot. The
% replacements are written in the Cross's own pitches — an a flat and
% the g under it — because the mark moves them too, so what sounds is
% an f and an e. Only the eighth note is not the Cross's: where the
% figure would give a c sharp the piece plays the plain c of the key,
% and the Spin takes over from there.
%
% The Spin, three times whole. The fourth breaks off six notes in —
% the pair, and two notes of the pair below — and the open A it was
% going to land on is held instead: twelve units of it, straight through
% the barline into m.68.

Cross[:1] \0 {g#4/2} \2 {g4/2} @-3 Detach(.3)
  Spin*3
  Detach(Thirds ..7 .2) .1 _12
  % m.68 — and the cadenza is back, at the moto's speed and in the
  % MAJOR. A major is A harmonic minor with its third and its sixth
  % raised and nothing else moved — the seventh is already g#, which is
  % why the answer's leading tone needs no mark at all. So the sweep is
  % the cadenza's own ladder with its thirds sharpened, and the answer is
  % the cadenza's own Fall with the third and the sixth of each octave
  % sharpened: six notes in nineteen, one pair an octave.
  Sweep[:8] \0 {.3#} \3 {3#} \6 {'3#} /2  ''1 _6
    '7/2 Fall[8:] \0 {'6#/2} \3 {'3#/2} \7 {6#/2} \10 {3#/2} \14 {.6#/2} \17 {.3#/2}
    .1 _6
  % m.69 — the second sweep begins before the gear change, and a tempo
  % mark can only stand between statements, so it is written in the two
  % pieces the mark cuts it into — exactly as the turn was at m.14.
  Sweep[:8] \0 {.3#} \3 {3#} \6 {'3#} /2

% mm.70-73 — form O. The tempo drops back to the cadenza's own and the
% key mark now says what the last two measures had already spelled out:
% A major. From here the cadenza's own names carry everything with no
% marks at all — the same eleven-rung ladder and the same fall, read in
% the new key.
%
% Three sweeps are left: nine rungs, ten, and all eleven, each answered
% by the fall begun one rung higher than the last, and each answer
% landing on the open A. The first of the three is already in the air —
% m.70 opens on its ninth rung and then its peak, the c# above the top A.

bpm 105.25
key a4 maj

assert Sweep[:3] == c#4 e4 a4
assert Fall[5:9] == b6/2 a6/2 g#6/2 f#6/2

''1/2 ''3 _6 Fall[5:] .1 _6
  Sweep[:10]/2 ''5 _6 Fall[3:] .1 _6
  % and the fourth sweep climbs all eleven rungs to the a two octaves
  % above the staff and holds it. The answer to that one begins m.74.
  Sweep/2 '''1 _6

% mm.74-77 — form P, the last of them, and the frame shuts. The fourth
% fall, the cadenza's chromatic run, and the two chords. Not one new
% name in it: mm.68-77 is the opening cadenza again in the parallel
% major, and all the major asks of it is a single note.
%
% A fall from an A begins on the note under it, and that note is the
% one place the major disagrees with the minor. Fall's head is spelled
% ''7b because the cadenza's g is natural; here it has to be the
% major's g#. Nothing below the head moves — every degree from '7 down
% is natural in A major — so the flat in Fall's head is the only
% alteration the figure carries, and the only item this return has to
% override.

assert Fall[0:1] == g7/2
assert Fall \0 {''7/2} == g#7/2 Fall[1:]

% Then the chromatic run that ended the cadenza at m.9, note for note:
% up its twenty-three rungs from the a# under middle C, the top a
% written once between the two runs, and down the same notes. It is in
% plain pitches, so the new key does not touch it, and the gear change
% at m.70 left it at the speed it had then — 105.25 against 105.
%
% And the two chords the moto perpetuo planted the key with at m.10:
% the same two degrees, .3 with .5 and then 3 with '1. Only the key
% mark has changed, so the c that was natural there is a c# here. The
% caprice opens its moto on A minor and shuts it on A major, and the
% frame needed no rewriting to say so.

assert {.5, .3} {'1, 3} == {c#4, e4} {c#5, a5}

% m.74 — the fall from a7, and the open A under it.
Fall \0 {''7/2} .1 _3
  % mm.75-76 — the run up and back.
  Chrom/2 a5/2 Chrom~/2
  % m.76 — two stops, c# with e and then c# with the a above it: A
  % major between them, and the last of them rings through m.77.
  .1 _6 {.5, .3} {'1, 3} _11/2

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Caprice No. 5 · Pattern Engine