Lessons · Showcase

Invention No. 1

Two voices, one subject

% Bach's whole invention grows from eight notes: the C major scale up
% to the fourth, a fold back down to the tonic, then a leap to the
% dominant. Seven sixteenths and one eighth, nine in all, so the answer
% that enters half a measure later overlaps its own tail.
 
bpm 80
instrument hollow
meter 4
unit 1/4
key c4 maj
mode diatonic
 
Subject ::= 1 2 3 4 2 3 1 5 _2
Subject
 
 
% mm.1-2: the lower voice takes the subject an octave below, half a
% measure late, and the upper voice answers itself four scale steps up
% ('@~4' is a fifth in C major). Neighbor is the eighth-note figure that
% closes each measure, and it travels with the subject it follows.
% Answer is the whole reply: the step up into it, the subject a fifth
% above, and that subject's own neighbor.
 
Neighbor ::= '1 _2 7 _2 '1 _2
Answer ::= '2 Subject@~4 Neighbor@~4
Open ::= {- Subject Neighbor Answer,
          - _9 Subject@-12 ..5 _2 - _5 Subject[0:7]@~-3}
 
Open
assert Answer == '2 5 6 7 '1 6 7 5 '2 _2 '5 _2 '4 _2 '5 _2
 
 
% '^' reflects a line around the key's axis, so the subject's four
% rising notes become four falling ones. Stated four steps up, the
% mirror begins on g (the note the subject just landed on) and the
% two lock end to end.
 
Mirror ::= Subject[0:7]^@~4
 
Subject Mirror
assert Mirror == 5 4 3 2 4 3 5
 
 
% mm.3-4: Spin is the mirror three times, each entry a third lower
% ('@~8', '@~6', '@~4'), over a lower voice walking in eighths. Fall
% answers it: four steps down from c through f# ('4#', the leading tone
% into G) and a turn back up. Both voices touch that f#.
 
Spin ::= '3 Mirror@~8 '5 Mirror@~6 '3 Mirror@~4
Fall ::= '1 7 6 5 4# 6 5 7
Episode ::= {Spin Fall,
             1 _2 .7 _2 1 _2 2 _2 3 _2 .5 _2 .6 _2 .7 _2 1 _2 .3 _2 .4# _2 .5 _2 .6 _2 .7 _2 1 _4}
 
Episode
assert Mirror@~4 == '2 '1 7 6 '1 7 '2
 
 
% mm.9-10. On the way to the relative minor the voices trade jobs:
% the LOWER voice announces the mirror and the upper answers four steps
% above it, half a measure later. The second pair climbs one scale step,
% and Bend sharpens the mirror's sixth note to c#, D minor's leading
% tone: '\5' replaces item five (the index is zero-based, so item five
% is the sixth note) and the rest of the figure stands as written.
 
Bend ::= Mirror@~5 \5 {'1#}
Minor ::= {7 _2 - _7 Mirror@~4 '1 _2 - _7 Bend,
           .5 Mirror 4 _2 3 _2 4 _2 2 _2 3 Mirror@~1 5 _2 4 _2 5 _2 3 _2}
 
Minor
assert Bend == '3 '2 '1 7 '2 '1# '3
 
 
% mm.19-20: the subject returns upright an octave above where it
% started ('@~7') and then twice more, each entry a third higher: Rise
% is those three entries, and it runs across the barline between m.19
% and m.20. Arp drops the tonic triad from c6 back to e5. Close is the
% last chord, C major from c2 up to c5.
 
Rise ::= Subject[0:7]@~7 '2 Subject[0:7]@~9 '4 Subject[0:7]@~11
Arp ::= ''1 _2 '5 _2 '3 _2
Return ::= {- Rise Arp '2 '1,
            .6 _2 .7b _2 .6 _2 .5 _2 .4 _2 2 _2 1 _2 .7b _2 .6 _2 4 _2 3 _2 2 _2 3 Subject[0:7]@~-6}
Close ::= {'1 _16, 5 _16, 3 _16, .1 _16, ..1 _16}
 
Return Close
assert Rise[:3] == Subject[0:7]@~7 '2 Subject[0:7]@~9
 
 
% THE TOP LINE, mm.1-8. From here the lesson states the upper voice of
% the whole invention, in the names built above: the subject and its
% answer (mm.1-2); Spin, then Fall (mm.3-4); the fall again a step lower
% after a leap down a sixth (m.5); rising thirds, a pair of 32nds and
% the cadence into G (m.6); then the subject twice more, each entry
% announced by a held note and half a measure of rest (mm.7-8).
%
% The fall is written out each time it returns. '@~n' moves a figure by
% steps of THIS key, and the three falls are a step apart in three
% different scales (G here, F at m.21), so no one mark carries them.
 
- Subject Neighbor Answer                          % mm.1-2
  Spin Fall                                        % mm.3-4
  6 _2 2 _2 '1 _3 '2 7 6 5 4# 3 5 4# 6             % m.5
  5 7 6 '1 7 '2 '1 '3 '2 7/2 '1/2 '2 '5 7 _2 6 5   % m.6
  5 _2 - _7 Subject[0:7]@~4                        % m.7
  4# _2 - _7 Subject[0:7]@~5                       % m.8
 
 
% mm.9-19: the mirror carries the piece away from C and the subject
% brings it back. Two mirror entries in the upper voice, the second bent
% to c# (mm.9-10); eighths for two measures, the only place the
% sixteenths stop, climbing twice into d (mm.11-12); the subject four
% semitones up on e, with f# and g# ('@4' and not '@~4', because this
% entry is chromatic and not diatonic), then falling thirds and a long
% descent (mm.13-14); then four entries alternating mirror and subject,
% each handing over on a nine-unit note (mm.15-18); and Rise home to C.
 
7 _2 - _7 Mirror@~4                                % m.9
  '1 _2 - _7 Bend                                  % m.10
  '2 _2 '1# _2 '2 _2 '3 _2 '4 _2 6 _2 7 _2 '1# _2  % m.11
  '2 _2 4# _2 5# _2 6 _2 7 _2 '1 _2 '2 _5          % m.12
  Subject[0:7]@4 '3 '2 '1 '3 '2 '1 7 '2            % m.13
  '1 '6 '5# '7 '6 '3 '4 '2 5# '4 '3 '2 '1 _2 7 6   % m.14
  6 Mirror@~8 '5 _9 Subject[0:7]@~9 '4 _9          % mm.15-16
  Mirror@~7 '4 _9 Subject[0:7]@~8 '3 _9 Rise[:3]   % mm.17-19
assert Subject[0:7]@4 == 3 4# 5# 6 4# 5# 3
 
 
% mm.20-22: Rise finishes, its last entry the answer of m.2 an octave
% higher; Arp drops the tonic triad; the fall comes once more, in F now,
% with b flat where the first two had f sharp; and the upper three
% notes of the closing chord hold the measure out. 'Rise[3:]' is the
% rest of the rise the cell above began: a slice over a body of
% references counts the references, so one run can be stated across the
% section boundary it spans.
 
Rise[3:] Arp '2 '1                                 % m.20
  '1 7b 6 5 4 6 5 7b 6 7 '1 3 2 '1 4 7             % m.21
  {3 _16, 5 _16, '1 _16}                           % m.22

% Bach's whole invention grows from eight notes: the C major scale up
% to the fourth, a fold back down to the tonic, then a leap to the
% dominant. Seven sixteenths and one eighth, nine in all, so the answer
% that enters half a measure later overlaps its own tail.

bpm 80
instrument hollow
meter 4
unit 1/4
key c4 maj
mode diatonic

Subject ::= 1 2 3 4 2 3 1 5 _2
Subject

% mm.1-2: the lower voice takes the subject an octave below, half a
% measure late, and the upper voice answers itself four scale steps up
% ('@~4' is a fifth in C major). Neighbor is the eighth-note figure that
% closes each measure, and it travels with the subject it follows.
% Answer is the whole reply: the step up into it, the subject a fifth
% above, and that subject's own neighbor.

Neighbor ::= '1 _2 7 _2 '1 _2
Answer ::= '2 Subject@~4 Neighbor@~4
Open ::= {- Subject Neighbor Answer,
          - _9 Subject@-12 ..5 _2 - _5 Subject[0:7]@~-3}

Open
assert Answer == '2 5 6 7 '1 6 7 5 '2 _2 '5 _2 '4 _2 '5 _2

% '^' reflects a line around the key's axis, so the subject's four
% rising notes become four falling ones. Stated four steps up, the
% mirror begins on g (the note the subject just landed on) and the
% two lock end to end.

Mirror ::= Subject[0:7]^@~4

Subject Mirror
assert Mirror == 5 4 3 2 4 3 5

% mm.3-4: Spin is the mirror three times, each entry a third lower
% ('@~8', '@~6', '@~4'), over a lower voice walking in eighths. Fall
% answers it: four steps down from c through f# ('4#', the leading tone
% into G) and a turn back up. Both voices touch that f#.

Spin ::= '3 Mirror@~8 '5 Mirror@~6 '3 Mirror@~4
Fall ::= '1 7 6 5 4# 6 5 7
Episode ::= {Spin Fall,
             1 _2 .7 _2 1 _2 2 _2 3 _2 .5 _2 .6 _2 .7 _2 1 _2 .3 _2 .4# _2 .5 _2 .6 _2 .7 _2 1 _4}

Episode
assert Mirror@~4 == '2 '1 7 6 '1 7 '2

% mm.9-10. On the way to the relative minor the voices trade jobs:
% the LOWER voice announces the mirror and the upper answers four steps
% above it, half a measure later. The second pair climbs one scale step,
% and Bend sharpens the mirror's sixth note to c#, D minor's leading
% tone: '\5' replaces item five (the index is zero-based, so item five
% is the sixth note) and the rest of the figure stands as written.

Bend ::= Mirror@~5 \5 {'1#}
Minor ::= {7 _2 - _7 Mirror@~4 '1 _2 - _7 Bend,
           .5 Mirror 4 _2 3 _2 4 _2 2 _2 3 Mirror@~1 5 _2 4 _2 5 _2 3 _2}

Minor
assert Bend == '3 '2 '1 7 '2 '1# '3

% mm.19-20: the subject returns upright an octave above where it
% started ('@~7') and then twice more, each entry a third higher: Rise
% is those three entries, and it runs across the barline between m.19
% and m.20. Arp drops the tonic triad from c6 back to e5. Close is the
% last chord, C major from c2 up to c5.

Rise ::= Subject[0:7]@~7 '2 Subject[0:7]@~9 '4 Subject[0:7]@~11
Arp ::= ''1 _2 '5 _2 '3 _2
Return ::= {- Rise Arp '2 '1,
            .6 _2 .7b _2 .6 _2 .5 _2 .4 _2 2 _2 1 _2 .7b _2 .6 _2 4 _2 3 _2 2 _2 3 Subject[0:7]@~-6}
Close ::= {'1 _16, 5 _16, 3 _16, .1 _16, ..1 _16}

Return Close
assert Rise[:3] == Subject[0:7]@~7 '2 Subject[0:7]@~9

% THE TOP LINE, mm.1-8. From here the lesson states the upper voice of
% the whole invention, in the names built above: the subject and its
% answer (mm.1-2); Spin, then Fall (mm.3-4); the fall again a step lower
% after a leap down a sixth (m.5); rising thirds, a pair of 32nds and
% the cadence into G (m.6); then the subject twice more, each entry
% announced by a held note and half a measure of rest (mm.7-8).
%
% The fall is written out each time it returns. '@~n' moves a figure by
% steps of THIS key, and the three falls are a step apart in three
% different scales (G here, F at m.21), so no one mark carries them.

- Subject Neighbor Answer                          % mm.1-2
  Spin Fall                                        % mm.3-4
  6 _2 2 _2 '1 _3 '2 7 6 5 4# 3 5 4# 6             % m.5
  5 7 6 '1 7 '2 '1 '3 '2 7/2 '1/2 '2 '5 7 _2 6 5   % m.6
  5 _2 - _7 Subject[0:7]@~4                        % m.7
  4# _2 - _7 Subject[0:7]@~5                       % m.8

% mm.9-19: the mirror carries the piece away from C and the subject
% brings it back. Two mirror entries in the upper voice, the second bent
% to c# (mm.9-10); eighths for two measures, the only place the
% sixteenths stop, climbing twice into d (mm.11-12); the subject four
% semitones up on e, with f# and g# ('@4' and not '@~4', because this
% entry is chromatic and not diatonic), then falling thirds and a long
% descent (mm.13-14); then four entries alternating mirror and subject,
% each handing over on a nine-unit note (mm.15-18); and Rise home to C.

7 _2 - _7 Mirror@~4                                % m.9
  '1 _2 - _7 Bend                                  % m.10
  '2 _2 '1# _2 '2 _2 '3 _2 '4 _2 6 _2 7 _2 '1# _2  % m.11
  '2 _2 4# _2 5# _2 6 _2 7 _2 '1 _2 '2 _5          % m.12
  Subject[0:7]@4 '3 '2 '1 '3 '2 '1 7 '2            % m.13
  '1 '6 '5# '7 '6 '3 '4 '2 5# '4 '3 '2 '1 _2 7 6   % m.14
  6 Mirror@~8 '5 _9 Subject[0:7]@~9 '4 _9          % mm.15-16
  Mirror@~7 '4 _9 Subject[0:7]@~8 '3 _9 Rise[:3]   % mm.17-19
assert Subject[0:7]@4 == 3 4# 5# 6 4# 5# 3

% mm.20-22: Rise finishes, its last entry the answer of m.2 an octave
% higher; Arp drops the tonic triad; the fall comes once more, in F now,
% with b flat where the first two had f sharp; and the upper three
% notes of the closing chord hold the measure out. 'Rise[3:]' is the
% rest of the rise the cell above began: a slice over a body of
% references counts the references, so one run can be stated across the
% section boundary it spans.

Rise[3:] Arp '2 '1                                 % m.20
  '1 7b 6 5 4 6 5 7b 6 7 '1 3 2 '1 4 7             % m.21
  {3 _16, 5 _16, '1 _16}                           % m.22

running…edit it and it re-runs as you type

Invention No. 1 · Showcase