Lessons · Grammars

Enumerating pattern spaces

enum, bounds, optionals — the divisions of a ground

% A rule with alternatives does not name a pattern — it names a SPACE
% of patterns, and 'enum' walks all of it: every combination of every
% choice, each result with its own ▶. Two two-way choices make four
% divisions, one per note of the ground below. Enumeration order is
% lexicographic — the first result takes the first alternative
% everywhere — and the assert makes the engine check that.
 
bpm 80
meter 4
unit 1/4
key d harm
 
Ground ::= | ..1 |
           | ..3 |
           | ..6 |
           | ..5 |
Run    ::= 1 2 3 2 3 4 3 2
         = 1 2 3 4 5 4 3 2
Arp    ::= 5 3 '1 5 3 5 3 1
         = '1 5 3 5 3 1 2 1
Div    ::= Run Arp
 
enum Div
assert Div#1 == Run#1 Arp#1
 
Ground
{Div#1 Div#2 Div#3 Div#4, Ground}
 
 
% A space can be endless, so the search is bounded and always answers:
% '--max' caps the results, '--depth' the recursion, '--reps' the
% unrolling of '&' (zero-or-more). Each unrolling is one more turn of
% the figure, so the enumeration IS a written-out expansion: the closes
% grow four units at a time, shrink back to the bare tonic, then the
% longest close answers itself backwards. The listing asks for four
% turns; a reference outside it, like every one played here, still
% uses the default cap of three.
 
Turn  ::= 2 1 7 1
Chain ::= (Turn)& 1 _4
 
enum Chain --reps 4
 
Chain#1 Chain#2 Chain#3 Chain#4
Chain#4 Chain#3 Chain#2 Chain#1
Chain#4~ Chain#4
 
 
% Timed optionals breed a space without moving a beat: 'x?' derives x
% or a rest of exactly x's length, and 'x?y' substitutes y scaled to
% x's length — wrap '(x?)' or the '?' greedily eats the next sound as
% its substitute. So all four cadences run the same sixteen units, the
% assert spells the first of them out, and they stack over the ground
% with no bar shifting underneath.
 
Cadence ::= 5 _2 (4?) _2 3 _2 2?7 _2 1 _8
 
enum Cadence
assert Cadence#1 == 5 _2 - _2 3 _2 7 _2 1 _8
 
Cadence#1 Cadence#2
{Cadence#3 Cadence#4, Ground/2}
 
 
% Why a grammar has MANY derivations: they are the piece's variations,
% written once. A chaconne is exactly that — one ground, the divisions
% the rule allows, in the order the composer picks. So the grammar is
% the material and the form is the choosing: the four divisions stated,
% run backwards, then mirrored — 'axis 3' swaps 1 and 5, which keeps
% the mirror in the same octave — and a cadence onto the tonic triad.
 
Chaconne ::= Div#1 Div#3 Div#2 Div#4
 
{Chaconne, Ground}
{Chaconne~, Ground}
 
axis 3
{Chaconne^, Ground}
 
{Cadence#2 Cadence#4, Ground/2}
{1Dia _16, ..1 _16}

% A rule with alternatives does not name a pattern — it names a SPACE
% of patterns, and 'enum' walks all of it: every combination of every
% choice, each result with its own ▶. Two two-way choices make four
% divisions, one per note of the ground below. Enumeration order is
% lexicographic — the first result takes the first alternative
% everywhere — and the assert makes the engine check that.

bpm 80
meter 4
unit 1/4
key d harm

Ground ::= | ..1 |
           | ..3 |
           | ..6 |
           | ..5 |
Run    ::= 1 2 3 2 3 4 3 2
         = 1 2 3 4 5 4 3 2
Arp    ::= 5 3 '1 5 3 5 3 1
         = '1 5 3 5 3 1 2 1
Div    ::= Run Arp

enum Div
assert Div#1 == Run#1 Arp#1

Ground
{Div#1 Div#2 Div#3 Div#4, Ground}

% A space can be endless, so the search is bounded and always answers:
% '--max' caps the results, '--depth' the recursion, '--reps' the
% unrolling of '&' (zero-or-more). Each unrolling is one more turn of
% the figure, so the enumeration IS a written-out expansion: the closes
% grow four units at a time, shrink back to the bare tonic, then the
% longest close answers itself backwards. The listing asks for four
% turns; a reference outside it, like every one played here, still
% uses the default cap of three.

Turn  ::= 2 1 7 1
Chain ::= (Turn)& 1 _4

enum Chain --reps 4

Chain#1 Chain#2 Chain#3 Chain#4
Chain#4 Chain#3 Chain#2 Chain#1
Chain#4~ Chain#4

% Timed optionals breed a space without moving a beat: 'x?' derives x
% or a rest of exactly x's length, and 'x?y' substitutes y scaled to
% x's length — wrap '(x?)' or the '?' greedily eats the next sound as
% its substitute. So all four cadences run the same sixteen units, the
% assert spells the first of them out, and they stack over the ground
% with no bar shifting underneath.

Cadence ::= 5 _2 (4?) _2 3 _2 2?7 _2 1 _8

enum Cadence
assert Cadence#1 == 5 _2 - _2 3 _2 7 _2 1 _8

Cadence#1 Cadence#2
{Cadence#3 Cadence#4, Ground/2}

% Why a grammar has MANY derivations: they are the piece's variations,
% written once. A chaconne is exactly that — one ground, the divisions
% the rule allows, in the order the composer picks. So the grammar is
% the material and the form is the choosing: the four divisions stated,
% run backwards, then mirrored — 'axis 3' swaps 1 and 5, which keeps
% the mirror in the same octave — and a cadence onto the tonic triad.

Chaconne ::= Div#1 Div#3 Div#2 Div#4

{Chaconne, Ground}
{Chaconne~, Ground}

axis 3
{Chaconne^, Ground}

{Cadence#2 Cadence#4, Ground/2}
{1Dia _16, ..1 _16}

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

Opening…

Enumerating pattern spaces · Pattern Engine