Lessons · Harmony

Sevenths and where they go

The chord that has to resolve

% A triad is two thirds stacked. Stack one more and it is a SEVENTH
% CHORD: four notes, so four fields in the brace. Three qualities on
% one root: the major seventh keeps 3 and 7, the minor seventh lowers
% both, and the dominant seventh lowers only the 7. The language names
% all three, so the stack appears here to say what they are and the
% name does the work from here on.
 
bpm 25
instrument swell
meter 4
key f maj
 
assert 1Maj7 == {1Dia, 7}
assert 1Min7 == {1, 3b, 5, 7b}
assert 1Dom7 == {1Dia, 7b}
 
1Dia _4 1Maj7 _4 1Min7 _4 1Dom7 _4 1Dia _8
 
 
% The dominant seventh is the odd one of the three, and one interval is
% why. 'Dia' stacks out of the SCALE, so '5Dia7' is the seventh on
% degree 5 built from notes the key already owns (in F that is
% C E G Bb) and between its 7 and its '4 sits the one TRITONE the
% scale has. '7@6' says how wide that is: six semitones, half an
% octave. Hear it bare, then packed inside the chord, then closing in:
% 7 rises a semitone to '1 and '4 falls a semitone to '3.
 
assert 5Dia7 == 5Dom7
assert 5Dia7 == {5, 7, '2, '4}
assert {7, '4} == {7, 7@6}
 
1Maj7 _4 5Dia7 _4 1Maj7 _4
{7, '4} _4 -*2 5Dia7 _4 -*2 {7, '4} _2 {'1, '3} _6
 
 
% Give it four voices and the resolution is forced. The seventh 4
% FALLS to 3, the leading tone .7 RISES to 1, the bass leaps ..5 up to
% .1, and the doubled 5 on top never moves: root of the dominant,
% fifth of the tonic. 'Voices' is those two chords read across instead
% of down. What V7 has not got is its own fifth: '$' lists a chord's
% members bottom to top, and '..5Dia7$1-'3-7-1' is root, third,
% seventh, root. The fifth is what a dominant seventh can spare: the
% tritone is the part with somewhere to be.
 
V7     ::= {..5, .7, 4, 5}
I      ::= {.1, 1, 3, 5}
Voices ::= {..5 .1, .7 1, 4 3, 5 5}
 
assert V7 == ..5Dia7$1-'3-7-1
assert I == .1Dia$1-1-3-5
assert Voices _4 == V7 _4 I _4
 
V7 _8 I _8
Voices _4
 
 
% ii-V-I is the sentence common practice is made of. The chord on 2 is
% a minor seventh the key spells without an accidental, and its seventh
% is 1 (the tonic itself) falling to .7, the third of V7. So the
% sevenths fall in turn, 4 to 3 last. The tune doubles that fall an
% octave up, and on the way it sings the G that V7 gave away.
 
II7  ::= {..2, 1, 4, 6}
Turn ::= II7 _4 V7 _4 I _4
Tune ::= 6 '1 _2 6 '2 _2 '4 _2 '3 _2 '1 _2
 
assert 2Dia7 == 2Min7
assert 2Dia7 == {2, 4, 6, '1}
assert II7 == ..2Dia7$1-'7-3-5
 
Turn
{Tune, Turn}
 
 
% Sevenths in a row. At every change the seventh steps down and lands
% as the next chord's third, which is what drops the roots by fifths:
% F Bb E A D, then the Turn's G and C, and one of those fifths is the
% diminished one, Bb to E, the tritone again. The three upper voices do
% nothing else: they walk down the scale and take turns, so the chords
% alternate, a root doubled with the fifth left out and then all four
% notes. Five links, and the chain arrives at the Turn it was aimed at.
 
Chain ::= {.1 .4 ..7 ..3 ..6,
           3 3 2 2 1,
           7 6 6 5 5,
           '1 '1 7 7 6}
 
Chain _4 Turn
 
 
% Nothing so far is spelled as a pitch, degrees the whole way, so all
% of it travels. In D minor the five links come out i7, iv7, VII7,
% IIImaj7 and VImaj7, the chord on 2 turns half-diminished, and only
% the key line changed. But the Turn goes slack, and ONE note is why:
% natural minor's seventh degree is c, a WHOLE step under home, so 7
% and '4 have opened from a tritone to a fifth and 5Dia7 comes out
% A C E G, a minor seventh with nothing in it that has to move.
% 'harm' raises that one degree to c#. Same passage, one note
% different, and it closes.
 
key d min
 
assert 2Dia7 == 2Min7b5
assert 5Dia7 == 5Min7
assert {7, '4} == {7, 7@7}
 
Chain _2 {Tune, Turn}
 
key d harm
 
assert 5Dia7 == 5Dom7
assert {7, '4} == {7, 7@6}
 
{Tune, Turn}
 
 
% The raise pays twice. The seventh standing ON the leading tone is
% C# E G Bb, four notes each a minor third above the last (two
% tritones, the second one the first moved up a third) and not a note
% of it can stay. The bass rises a semitone into .1 while the three
% voices above it fall a step, so 'Close' reads across what the last
% assert reads down, and nothing moves further than a whole tone.
 
Close ::= {..7 .1, 2 1, 4 3, 6 5}
 
assert 7Dia7 == 7Dim7
assert 7Dia7 == {7, '2, '4, '6}
assert {7, '4}@3 == {'2, '6}
assert Close _4 == ..7Dia7$1-'3-5-7 _4 I _4
 
{7, '4} _4 {7, '4}@3 _4 7Dia7 _8
Close _4
 
 
% All of it as a piece. The chain falls by fifths in natural minor with
% a three-note Fall over each link, '@~' putting the next one a scale
% step lower; then the key line raises the seventh, the diminished
% seventh steps into i under a leading tone two octaves above the bass,
% and ii-V-i shuts the door. The last chord raises one degree more:
% 3 to 3#, I with a major third, which is how a piece in minor ended
% for two hundred years.
 
key d min
 
Fall  ::= '5 _2 '4 '3
Steps ::= Fall Fall@~-1 Fall@~-2 Fall@~-3 Fall@~-4
 
assert {.1, 1, 3#, 5} == .1Maj$1-1-3-5
 
{Steps, Chain _4}
 
key d harm
 
{7 _4 '1 _4, Close _4}
{Tune, Turn}
{.1, 1, 3#, 5} _8

% A triad is two thirds stacked. Stack one more and it is a SEVENTH
% CHORD: four notes, so four fields in the brace. Three qualities on
% one root: the major seventh keeps 3 and 7, the minor seventh lowers
% both, and the dominant seventh lowers only the 7. The language names
% all three, so the stack appears here to say what they are and the
% name does the work from here on.

bpm 25
instrument swell
meter 4
key f maj

assert 1Maj7 == {1Dia, 7}
assert 1Min7 == {1, 3b, 5, 7b}
assert 1Dom7 == {1Dia, 7b}

1Dia _4 1Maj7 _4 1Min7 _4 1Dom7 _4 1Dia _8

% The dominant seventh is the odd one of the three, and one interval is
% why. 'Dia' stacks out of the SCALE, so '5Dia7' is the seventh on
% degree 5 built from notes the key already owns (in F that is
% C E G Bb) and between its 7 and its '4 sits the one TRITONE the
% scale has. '7@6' says how wide that is: six semitones, half an
% octave. Hear it bare, then packed inside the chord, then closing in:
% 7 rises a semitone to '1 and '4 falls a semitone to '3.

assert 5Dia7 == 5Dom7
assert 5Dia7 == {5, 7, '2, '4}
assert {7, '4} == {7, 7@6}

1Maj7 _4 5Dia7 _4 1Maj7 _4
{7, '4} _4 -*2 5Dia7 _4 -*2 {7, '4} _2 {'1, '3} _6

% Give it four voices and the resolution is forced. The seventh 4
% FALLS to 3, the leading tone .7 RISES to 1, the bass leaps ..5 up to
% .1, and the doubled 5 on top never moves: root of the dominant,
% fifth of the tonic. 'Voices' is those two chords read across instead
% of down. What V7 has not got is its own fifth: '$' lists a chord's
% members bottom to top, and '..5Dia7$1-'3-7-1' is root, third,
% seventh, root. The fifth is what a dominant seventh can spare: the
% tritone is the part with somewhere to be.

V7     ::= {..5, .7, 4, 5}
I      ::= {.1, 1, 3, 5}
Voices ::= {..5 .1, .7 1, 4 3, 5 5}

assert V7 == ..5Dia7$1-'3-7-1
assert I == .1Dia$1-1-3-5
assert Voices _4 == V7 _4 I _4

V7 _8 I _8
Voices _4

% ii-V-I is the sentence common practice is made of. The chord on 2 is
% a minor seventh the key spells without an accidental, and its seventh
% is 1 (the tonic itself) falling to .7, the third of V7. So the
% sevenths fall in turn, 4 to 3 last. The tune doubles that fall an
% octave up, and on the way it sings the G that V7 gave away.

II7  ::= {..2, 1, 4, 6}
Turn ::= II7 _4 V7 _4 I _4
Tune ::= 6 '1 _2 6 '2 _2 '4 _2 '3 _2 '1 _2

assert 2Dia7 == 2Min7
assert 2Dia7 == {2, 4, 6, '1}
assert II7 == ..2Dia7$1-'7-3-5

Turn
{Tune, Turn}

% Sevenths in a row. At every change the seventh steps down and lands
% as the next chord's third, which is what drops the roots by fifths:
% F Bb E A D, then the Turn's G and C, and one of those fifths is the
% diminished one, Bb to E, the tritone again. The three upper voices do
% nothing else: they walk down the scale and take turns, so the chords
% alternate, a root doubled with the fifth left out and then all four
% notes. Five links, and the chain arrives at the Turn it was aimed at.

Chain ::= {.1 .4 ..7 ..3 ..6,
           3 3 2 2 1,
           7 6 6 5 5,
           '1 '1 7 7 6}

Chain _4 Turn

% Nothing so far is spelled as a pitch, degrees the whole way, so all
% of it travels. In D minor the five links come out i7, iv7, VII7,
% IIImaj7 and VImaj7, the chord on 2 turns half-diminished, and only
% the key line changed. But the Turn goes slack, and ONE note is why:
% natural minor's seventh degree is c, a WHOLE step under home, so 7
% and '4 have opened from a tritone to a fifth and 5Dia7 comes out
% A C E G, a minor seventh with nothing in it that has to move.
% 'harm' raises that one degree to c#. Same passage, one note
% different, and it closes.

key d min

assert 2Dia7 == 2Min7b5
assert 5Dia7 == 5Min7
assert {7, '4} == {7, 7@7}

Chain _2 {Tune, Turn}

key d harm

assert 5Dia7 == 5Dom7
assert {7, '4} == {7, 7@6}

{Tune, Turn}

% The raise pays twice. The seventh standing ON the leading tone is
% C# E G Bb, four notes each a minor third above the last (two
% tritones, the second one the first moved up a third) and not a note
% of it can stay. The bass rises a semitone into .1 while the three
% voices above it fall a step, so 'Close' reads across what the last
% assert reads down, and nothing moves further than a whole tone.

Close ::= {..7 .1, 2 1, 4 3, 6 5}

assert 7Dia7 == 7Dim7
assert 7Dia7 == {7, '2, '4, '6}
assert {7, '4}@3 == {'2, '6}
assert Close _4 == ..7Dia7$1-'3-5-7 _4 I _4

{7, '4} _4 {7, '4}@3 _4 7Dia7 _8
Close _4

% All of it as a piece. The chain falls by fifths in natural minor with
% a three-note Fall over each link, '@~' putting the next one a scale
% step lower; then the key line raises the seventh, the diminished
% seventh steps into i under a leading tone two octaves above the bass,
% and ii-V-i shuts the door. The last chord raises one degree more:
% 3 to 3#, I with a major third, which is how a piece in minor ended
% for two hundred years.

key d min

Fall  ::= '5 _2 '4 '3
Steps ::= Fall Fall@~-1 Fall@~-2 Fall@~-3 Fall@~-4

assert {.1, 1, 3#, 5} == .1Maj$1-1-3-5

{Steps, Chain _4}

key d harm

{7 _4 '1 _4, Close _4}
{Tune, Turn}
{.1, 1, 3#, 5} _8

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Sevenths and where they go · Harmony