Lessons · Harmony

Triads & chords

Thirds stacked, chord symbols, Dia

% A TRIAD is two thirds stacked: root, third, fifth. The LOWER third
% decides the quality: a major third under a minor one is MAJOR, the
% two swapped is MINOR. Both stacks keep the same 1 and the same 5, so
% the middle note is the whole argument. Call names no third at all,
% and the triad underneath it says bright or dark.
 
bpm 50
instrument swell
meter 4
key d4 maj
 
Maj  ::= {1, 3, 5}
Min  ::= {1, 3b, 5}
Call ::= 5 _2 '1 _2 5 _4 2 _2 1 _6
 
assert 1 3 5 == 1 i4 i3
assert 1 3b 5 == 1 i3 i4
 
{Call, Maj _16}
{Call, Min _16}
 
 
% Root plus QUALITY says the same stack in one token, and the root can
% be a degree instead of a letter. Dim tightens both thirds, Aug widens
% both, and each has a job in these eight chords: the Aug lifts the
% fifth a semitone, the borrowed 4Min darkens the return, and the Dim
% on 7 leans home. Line rides an octave over the stack, one note to a
% chord, and climbs a a# b with that lifted fifth.
 
assert 1Maj == Maj
assert 1Min == Min
assert 1Dim == {1, 3b, 5b}
assert 1Aug == {1, 3, 5#}
 
Lift ::= 1Maj 1Aug .6Min .4Maj .4Min 1Maj .7Dim 1Maj
Line ::= 5 5# 6 '1 6b 5 2 1
 
Lift _4
{Line@12 _4, Lift _4}
 
 
% 'Dia' names no quality at all. It stacks the root with the tones two
% and four SCALE STEPS above it, so the key decides what comes out: in
% major, 1 4 5 come out major, 2 3 6 minor, and 7 diminished. That
% ladder is every triad the key owns, and '~' walks it back down.
 
assert 1Dia == Maj
assert 2Dia == 2Min
assert 7Dia == 7Dim
 
Ladder ::= 1Dia 2Dia 3Dia 4Dia 5Dia 6Dia 7Dia '1Dia
 
Ladder _2 Ladder~ _2
 
 
% Eight of those chords and not one quality written down: the roots run
% 1 5 6 3 4 1 4 5, Pachelbel's ground, with Ground itself two octaves
% under them. Over it Sigh is a falling second, dropped two scale steps
% and then two more; Lean walks 6 up to 7, a semitone under home. One
% note to a chord is all a tune needs with three notes under it, and
% every note Tune holds is in the triad below it.
 
Canon  ::= 1Dia .5Dia .6Dia .3Dia .4Dia 1Dia .4Dia .5Dia
Ground<bass> ::= ..1 ..5 ..6 ..3 ..4 ..1 ..4 ..5
 
Sigh ::= '3 '2
Lean ::= 6 7
Tune ::= Sigh Sigh@~-2 Sigh@~-4 Lean
 
{Tune _4, Canon _4, Ground _4}
 
 
% Three of the key's triads hold degree 1: 1Dia as its root, 4Dia as
% its fifth, 6Dia as its third. So Lean's 7 can be answered three ways
% with the tune singing the same '1 over all of them: the roof stays
% and the floor moves. 1Dia puts 5 under it and shuts. 6Dia keeps the
% other two notes of home and slides a new root beneath them. 4Dia
% shares only that 1 and owns no 7, so nothing was demanding anything.
 
assert 1Dia == {1, 3, 5}
assert .6Dia == {.6, 1, 3}
assert .4Dia == {.4, .6, 1}
 
{Lean _4 '1 _8, Canon[6:] _4 1Dia _8, Ground[6:] _4 ..1 _8}
{Lean _4 '1 _8, Canon[6:] _4 .6Dia _8, Ground[6:] _4 ..6 _8}
{Lean _4 '1 _8, Canon[6:] _4 .4Dia _8, Ground[6:] _4 ..4 _8}
 
 
% One 'key' rewrites every chord in both lines and Tune does not move a
% note: its degrees were chord tones, and Dia rebuilds the triads under
% them off the same ladder. In D minor Canon reads i v VI III iv i iv v
% and the chord on 5 comes back minor, so the walk home that answered
% Lean in the major answers nothing here: Lean's 7 is a whole step
% under home, and the same three chords just walk in. 'harm' raises
% that 7 and the identical bar pulls the door shut: 5Dia is major
% again, the lean is back, and that raised note makes Canon's fourth
% chord AUGMENTED: two major thirds, the quality the scale never
% owned. Under every one of those answers the bass is the same 4 5 1,
% the one line no mode touches.
 
key d min
 
{Tune _4, Canon _4, Ground _4}
{Lean _4 '1 _8, Canon[6:] _4 1Dia _8, Ground[6:] _4 ..1 _8}
 
key d harm
 
assert 5Dia == 5Maj
assert 3Dia == 3Aug
 
{Tune _4, Canon _4, Ground _4}
{Lean _4 '1 _8, Canon[6:] _4 1Dia _8, Ground[6:] _4 ..1 _8}
 
 
% One harmony, three SPACINGS. CLOSE position is Maj itself, all three
% notes inside an octave; Open drops the root a register and spreads
% the rest; Wide doubles the root at both ends for weight. Three pitch
% classes every time, and moved up the ladder the hand-built stack IS
% the chord the scale builds there, so all three walk the same 4 5 1
% home, thin, wide, and wider.
 
key d4 maj
 
Open ::= {..1, .5, 3, '1}
Wide ::= {..1, .1, 3, 5, '1}
 
assert Maj@~3 == 4Dia
assert Maj@~4 == 5Dia
 
Maj@~3 _4 Maj@~4 _4 Maj _8
Open@~3 _4 Open@~4 _4 Open _8
Wide@~3 _4 Wide@~4 _4 Wide _8
 
 
% All of it in order: the ground bright, then the same eight chords
% with one word changed and one triad more. Canon's own last two are 4
% and 5, so a ninth chord of home makes the oldest ending there is:
% 4 5 1, a CADENCE, which the next lesson takes apart. The key is minor
% and that last triad is not: Home spells 1Maj where Dia would have
% said 1Min, and with Ground under it the close lands Wide.
 
key d4 maj
 
{Tune _4, Canon _4, Ground _4}
 
key d harm
 
Home ::= {1Maj, '1}
 
{Tune _4 '1 _16, Canon _4 Home _16, Ground _4 ..1 _16}

% A TRIAD is two thirds stacked: root, third, fifth. The LOWER third
% decides the quality: a major third under a minor one is MAJOR, the
% two swapped is MINOR. Both stacks keep the same 1 and the same 5, so
% the middle note is the whole argument. Call names no third at all,
% and the triad underneath it says bright or dark.

bpm 50
instrument swell
meter 4
key d4 maj

Maj  ::= {1, 3, 5}
Min  ::= {1, 3b, 5}
Call ::= 5 _2 '1 _2 5 _4 2 _2 1 _6

assert 1 3 5 == 1 i4 i3
assert 1 3b 5 == 1 i3 i4

{Call, Maj _16}
{Call, Min _16}

% Root plus QUALITY says the same stack in one token, and the root can
% be a degree instead of a letter. Dim tightens both thirds, Aug widens
% both, and each has a job in these eight chords: the Aug lifts the
% fifth a semitone, the borrowed 4Min darkens the return, and the Dim
% on 7 leans home. Line rides an octave over the stack, one note to a
% chord, and climbs a a# b with that lifted fifth.

assert 1Maj == Maj
assert 1Min == Min
assert 1Dim == {1, 3b, 5b}
assert 1Aug == {1, 3, 5#}

Lift ::= 1Maj 1Aug .6Min .4Maj .4Min 1Maj .7Dim 1Maj
Line ::= 5 5# 6 '1 6b 5 2 1

Lift _4
{Line@12 _4, Lift _4}

% 'Dia' names no quality at all. It stacks the root with the tones two
% and four SCALE STEPS above it, so the key decides what comes out: in
% major, 1 4 5 come out major, 2 3 6 minor, and 7 diminished. That
% ladder is every triad the key owns, and '~' walks it back down.

assert 1Dia == Maj
assert 2Dia == 2Min
assert 7Dia == 7Dim

Ladder ::= 1Dia 2Dia 3Dia 4Dia 5Dia 6Dia 7Dia '1Dia

Ladder _2 Ladder~ _2

% Eight of those chords and not one quality written down: the roots run
% 1 5 6 3 4 1 4 5, Pachelbel's ground, with Ground itself two octaves
% under them. Over it Sigh is a falling second, dropped two scale steps
% and then two more; Lean walks 6 up to 7, a semitone under home. One
% note to a chord is all a tune needs with three notes under it, and
% every note Tune holds is in the triad below it.

Canon  ::= 1Dia .5Dia .6Dia .3Dia .4Dia 1Dia .4Dia .5Dia
Ground<bass> ::= ..1 ..5 ..6 ..3 ..4 ..1 ..4 ..5

Sigh ::= '3 '2
Lean ::= 6 7
Tune ::= Sigh Sigh@~-2 Sigh@~-4 Lean

{Tune _4, Canon _4, Ground _4}

% Three of the key's triads hold degree 1: 1Dia as its root, 4Dia as
% its fifth, 6Dia as its third. So Lean's 7 can be answered three ways
% with the tune singing the same '1 over all of them: the roof stays
% and the floor moves. 1Dia puts 5 under it and shuts. 6Dia keeps the
% other two notes of home and slides a new root beneath them. 4Dia
% shares only that 1 and owns no 7, so nothing was demanding anything.

assert 1Dia == {1, 3, 5}
assert .6Dia == {.6, 1, 3}
assert .4Dia == {.4, .6, 1}

{Lean _4 '1 _8, Canon[6:] _4 1Dia _8, Ground[6:] _4 ..1 _8}
{Lean _4 '1 _8, Canon[6:] _4 .6Dia _8, Ground[6:] _4 ..6 _8}
{Lean _4 '1 _8, Canon[6:] _4 .4Dia _8, Ground[6:] _4 ..4 _8}

% One 'key' rewrites every chord in both lines and Tune does not move a
% note: its degrees were chord tones, and Dia rebuilds the triads under
% them off the same ladder. In D minor Canon reads i v VI III iv i iv v
% and the chord on 5 comes back minor, so the walk home that answered
% Lean in the major answers nothing here: Lean's 7 is a whole step
% under home, and the same three chords just walk in. 'harm' raises
% that 7 and the identical bar pulls the door shut: 5Dia is major
% again, the lean is back, and that raised note makes Canon's fourth
% chord AUGMENTED: two major thirds, the quality the scale never
% owned. Under every one of those answers the bass is the same 4 5 1,
% the one line no mode touches.

key d min

{Tune _4, Canon _4, Ground _4}
{Lean _4 '1 _8, Canon[6:] _4 1Dia _8, Ground[6:] _4 ..1 _8}

key d harm

assert 5Dia == 5Maj
assert 3Dia == 3Aug

{Tune _4, Canon _4, Ground _4}
{Lean _4 '1 _8, Canon[6:] _4 1Dia _8, Ground[6:] _4 ..1 _8}

% One harmony, three SPACINGS. CLOSE position is Maj itself, all three
% notes inside an octave; Open drops the root a register and spreads
% the rest; Wide doubles the root at both ends for weight. Three pitch
% classes every time, and moved up the ladder the hand-built stack IS
% the chord the scale builds there, so all three walk the same 4 5 1
% home, thin, wide, and wider.

key d4 maj

Open ::= {..1, .5, 3, '1}
Wide ::= {..1, .1, 3, 5, '1}

assert Maj@~3 == 4Dia
assert Maj@~4 == 5Dia

Maj@~3 _4 Maj@~4 _4 Maj _8
Open@~3 _4 Open@~4 _4 Open _8
Wide@~3 _4 Wide@~4 _4 Wide _8

% All of it in order: the ground bright, then the same eight chords
% with one word changed and one triad more. Canon's own last two are 4
% and 5, so a ninth chord of home makes the oldest ending there is:
% 4 5 1, a CADENCE, which the next lesson takes apart. The key is minor
% and that last triad is not: Home spells 1Maj where Dia would have
% said 1Min, and with Ground under it the close lands Wide.

key d4 maj

{Tune _4, Canon _4, Ground _4}

key d harm

Home ::= {1Maj, '1}

{Tune _4 '1 _16, Canon _4 Home _16, Ground _4 ..1 _16}

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Triads & chords · Harmony