the logic of six-based minor for harmonic analyses of

Trevor de Clercq

SMT Conference November 5, 2021

Virtual

The Logic of Six-Based Minor for Harmonic Analyses of Popular Music

“Down Under” (Men at Work), Verse

i

B‹

Tra vel lin’ in a fried

bVII

A

out com bie,

i

B‹

bVI

G A

bVII i

B‹

On a hip pie trail,

bVII

A

head full of zom

i

B‹

bie.-

bVI

G A

bVII

44&

##

0:11

‰œ œ œ œœ

œ œ œ œ œŒ Ó ‰

œœ œ œœœ œ œ œ œ œœ

Œ Ó

Minor Tonic = i

(Doll 2011)


i

B‹


bVII

A

out com bie,

i

B‹

bVI

G A

bVII i

B‹

On a hip pie trail,

bVII

A

head full of zom

i

B‹

bie.-

bVI

G A

bVII

44&

##

0:11

‰œ œ œ œœ



Œ Ó

“Down Under” (Men at Work), Chorus

I

D

Do youcome from a land

V

A

downun der?

vi

B‹

IV

G A

V I

D

Wherewo menglow

V

A

andmenplunder?

vi

B‹

IV

G A

V

44&

##

0:29

‰œ œ œ œ œ

œ œ œœ œŒ Ó ‰™

œ

rœ œ œ œ

œ œ œ œ

Œ Ó

Minor Tonic = i

(Doll 2011)

vi

B‹


V

A

out com bie,

vi

B‹

IV

G A

V vi

B‹

On a hip pie trail,

V

A

head full of zom

vi

B‹

bie.-

IV

G A

V

44&

##

0:11

‰œ œ œ œœ



Œ Ó



I

D


V

A

downun der?

vi

B‹

IV

G A

V I

D

Wherewo menglow

V

A

andmenplunder?

vi

B‹

IV

G A

V

44&

##

0:29

‰œ œ œ œ œ


œ

rœ œ œ œ

œ œ œ œ

Œ Ó

Minor Tonic = vi

Nashville Chart“Down Under” (Men at Work)

“Don’t Forget Me” (Red Hot Chili Peppers)

A5

I'm an o cean-

F5

in your bed room,-

C5

make you feel warm, make

G5

you want to re sume.-

C&

‹

C=1

A=1

0:15

^

^vi

i bVI

IV

bIII

I

bVII

V

Œ œœœ

œ Œ œœ

œœ Œ œ

œœ œ

œœ

œœœœœ œ œ

A = 1 i bVI bIII bVIIC = 1 vi IV I V

Tonal Ambiguity: The “Axis” Progression

Nobile (2020)

Double-Tonic Complex

Example 7: The Aeolian progression’s kinship with IV–V–I allows the possibility of combining the two, forming a double-tonic complex (a–b reproduced from Biamonte 2010, 101).

IV V I bVI bVII i

C:a:

Double-Tonic Complexes in Rock Music

Forthcoming in Music Theory Spectrum

EXAMPLES

e - v’ry move. Pri - vate eyes, they’re watching you. Pri - vate eyes

you what my head overlooks the senses will show to my heart when it’sI’m if you’re letting me in or letting me go, don’t lie when you’re

Prechorus

gon - na knowto knowgot

Example 3: Daryl Hall and John Oates, “Private Eyes” (1981): simplified transcription of verse, prechorus, and chorus

Example 3 (cont’d):

Example 5: Possible monotonal reading of “Private Eyes” in A minor.

Bridge

Ooh, why you try to put up a front for me? I’m a spy, but on your side, you see. Slip on into

starts at 2:01

Example 4: “Private Eyes,” bridge

Example 6. The double-tonic complex plotted on Gottfried Weber’s table of key relations. C minor and A major are both one step away from the A/C complex.

f

c

g

d

F

C

G

D

d

a

e

b

D

A

E

B

DTC

DTC modal mixture in “Private Eyes”

You

neve

r gi

ve

me

your

m

oney

Sun

Kin

gM

ean

Mr.

Mus

tard

Bat

hroo

m

Win

dow

“out of college,

money spent” “one sweet

dream”“all g

ood children

go to heaven”

“here comes

the Sun……King!”

Poly

then

e Pa

m

“oh yeah!”

A-C

-E-G C-m

ajor

tria

dA

-maj

or tr

iad

Alte

rnat

ion

of C

- an

dA

-maj

or tr

iads

=

com

pose

d-ou

t A-C

-E-G

four

diff

eren

t kin

ds

of to

nic

with

in D

TC

V/A

V/C

Gol

den

Slum

bers

Car

ry th

at

Wei

ght

The

End

“golden slumbers

fill your eyes”

recap of “You Never

Give Me Your M

oney”

“boy, you’re gonna

carry that weight”

clim

actic

sta

tem

ent

of A

-C-E

-G s

onor

ityA

-C-E

-G

Exam

ple

16

: The

Bea

tles,

med

ley

from

Ab

bey

Road

(196

9), r

educ

tion.

As

in B

aile

y’s

19th

-cen

tury

ex

ampl

es, b

oth

C a

nd A

are

cen

tral

for

port

ions

of t

he m

edle

y, b

ut n

eith

er e

mer

ges

as th

e gl

obal

toni

c.

One sweet dream

Example 17: Four types of tonic in “You Never Give Me Your Money”

You never give me your money You only give me your funny paper And in the middle of negotiations, you break down.

circle-of-fifths sequence in A minor

C is tonicized within sequence

Out of college, money spent See no future, pay no rent All the money’s gone, nowhere to go

descending-thirds sequence in C major

A is tonicized within sequence

a) “You never give me your money” theme (0:22): Am7 chord as tonic

b) “Out of college” theme (1:08): CM chord as tonic

c) “One sweet dream” theme (2:27): AM chord as tonic

(2:57): alternation of CM and AM chords

came true today,yes it did! Na guitar

A in melody over CM triad

Once there was a way

piano

bass

vocal

Example 18: Opening of “Golden Slumbers”

watching you, watching you, watching you, watching you You play with words

A♭

E♭

C

B♭

G

E♮

C

AThis riff continues into the next verse, but the bass note changes from A to C

Semitonal voice leading cadencing into the four-

note tonic sonority

watching for lies, wellyou can’t escape my Pri - vate eyes, they’re watching you, they see yourhurting inside, ’cause

ChorusUpper voices project C major;

bass is centered on APrechorus introduces chord structure of a major triad plus a different bass note

Chorus begins with A-C-E-G sonority

I see you, and you see me watch you blowin’ the lines, when you’re making a scene, oh girl,you play with love you can twist it around, baby that ain’t enough, cause girl,

Verse

10

Verse solidly establishes C major; most chords are triads

Chorus

a - ny disguise, I’ll still know you. Look in - to my pri - vate eyes, they’re watching you

G#Gn

Domestic animal

Pet dog

Pet cat

“I have five pets”

Petdog

Weimaraner Corgi

“I have five dogs”

Petdog

Weimaraner

“I have five Weimaraners”

???

RiverbankFinancial institution

“I am arriving at the bank”

Example 19: Cognitive engagement occurs at different conceptual levels depending on the word used in a given sentence

a) Linguistic indeterminacy: “pets” does not signify dogs and cats but rather a single abstract concept

b) Linguistic ambiguity: two potential meanings are not reconcilable into a single abstract concept

???

A minor C major

a) Tonal ambiguity

A/C complex

A minor C major

b) Double-tonic complex

Example 20: Tonal ambiguity, like linguistic ambiguity, involves two potential keys that are mutually exclusive. The double-tonic complex is analogous instead to linguistic indeterminacy, where cognitive engagement occurs at a more abstract conceptual level.

Domestic animal

Domestic animal

a) Wagner, Tristan und Isolde, “Was träumte mir von Tristans Ehre?,” Act 1, Scene 5

Example 14: Sarah McLachlan, “Building a Mystery” (1997)

Example 15: Harmonic reduction of “Building a Mystery” showing prolonged four-note chords giving rise to a double-tonic complex

Table 1: Previous tonal interpretations of “Building a Mystery”

Verse

You come out at night and the dark side’s light, and the vampires roamThat’s when the

energy comes,

etc.

0:12

Tonic

Prechorus

You’re so beautiful charm and, and so careful when I’m in your armswith anedge (and)

1:11

Subdominant Dominant

building a mys - tery Hold - ing on and hold - ing it in

Chorus

1:23Tonic

’cause you’re working,

etc.

Koozin 2008D major with “competing modal/linear pattern around B minor”

Doll 2011Verse is in B minor (although D “threatens to break through”);

choruses are in D

Richards 2017aTonal center changes from phrase to phrase in verses; choruses are in D

Attas 2011“Circular chord progression …

does not make a particular tonality clear”

de Clercq 2012,96–98

D major, with B minor an acknowledged alternative

Osborn 2017,67

D major

Example 13: The Axis progression seen as prolonging a four-note sonority with stepwise voice leading

has 6 of both A and C

has 7 and 2 of both A and C

Am F C G Am

Tonic

Subd

omin

antD

ominant

C/Am

F

Dm

(Bº)

G

(Em)

b) Mahler, Das Lied von der Erde, end

Example 1: Nineteenth-century examples of four-note sonorities functioning as tonic within a double-tonic complex. Both are built from the union of A-minor and C-major triads.

Example 2: Elliott Smith, “Everything Means Nothing to Me” (2000): a four-note sonority built from the union of two triads functions as prolonged tonic (after Schultz 2012).

melody prolongs A#

tonic sonority is union of A#m

and C#M triads

Example 8: LMFAO, “Party Rock Anthem” (2011): combination of I–V–IV and i–bVII–bVI, where the tonic chord is a four-note union of F-minor and Ab-major triads.

at 1:13

Example 9: Tina Turner, “What’s Love Got To Do With It?” (1984), prechorus into chorus: the chorus’s opening tonic contains the notes of G#m but the bass and instrumental texture hint at a B-major possibility.

Example 10: “What’s Love Got To Do With It?,” beginning of solo section (transcribed a whole step below the sounding pitch to facilitate comparison with previous example): the B-major possibility is strengthened by the solo line.

Example 11: “What’s Love Got To Do With It?,” bridge (again transcribed a whole step below the sounding pitch): this section moves away from G# minor, and when it leads back to the chorus it strongly projects B major.

Example 12: Melodic tendencies and harmonic categories within a double-tonic complex

at 0:46

Chorusend of prechorus…

Bass and guitar hint at a B chord, while e. piano plays G#m

e. guitar

e. piano

bass

After the first phrase, there is no

G# on the downbeat

at 2:02

The solo line completes aB-major triad on the first beat, forming the four-note sonority B-D#-F#-G# on the second beat

synth. harmonica

Solo

The bridge ends on a B chord, though it sounds

a bit like V of E major

etc.

When the chorus returns immediately following the bridge’s final B chord, the first chord’s B-major possibility is

especially potent

Chorus

Bridge

The bridge begins with the song’s first significant emphasis on B major

at 2:21

Example 11, cont’d

a) Non-tonic notes gravitate stepwise toward tonic notes; G and A, both members of the four-note tonic sonority, can move stepwise to one another.

b) Tonic, subdominant, and dominant functional categories are explicitly defined for diatonic triads. Songs exhibiting an Aeolian-Ionian double-tonic complex do not usually contain the chords in parentheses. (This diagram adapts Biamonte 2010’s functional diagrams, which in turn adapt those from Agmon 1996.)

iAb:

f: bVII bVII V IV

i bVII bVII V IV

at 0:29

T D S T D S

Melody outlines Ab major

(3 2 1)

Example 7: The Aeolian progression’s kinship with IV–V–I allows the possibility of combining the two, forming a double-tonic complex (a–b reproduced from Biamonte 2010, 101).

IV V I bVI bVII i

C:a:

Double-Tonic Complexes in Rock Music

Forthcoming in Music Theory Spectrum

EXAMPLES

e - v’ry move. Pri - vate eyes, they’re watching you. Pri - vate eyes

you what my head overlooks the senses will show to my heart when it’sI’m if you’re letting me in or letting me go, don’t lie when you’re

Prechorus

gon - na knowto knowgot

Example 3: Daryl Hall and John Oates, “Private Eyes” (1981): simplified transcription of verse, prechorus, and chorus

Example 3 (cont’d):

Example 5: Possible monotonal reading of “Private Eyes” in A minor.

Bridge

Ooh, why you try to put up a front for me? I’m a spy, but on your side, you see. Slip on into

starts at 2:01

Example 4: “Private Eyes,” bridge

Example 6. The double-tonic complex plotted on Gottfried Weber’s table of key relations. C minor and A major are both one step away from the A/C complex.

f

c

g

d

F

C

G

D

d

a

e

b

D

A

E

B

DTC

DTC modal mixture in “Private Eyes”

You

neve

r gi

ve

me

your

m

oney

Sun

Kin

gM

ean

Mr.

Mus

tard

Bat

hroo

m

Win

dow

“out of college,

money spent” “one sweet

dream”“all g

ood children

go to heaven”

“here comes

the Sun……King!”

Poly

then

e Pa

m

“oh yeah!”

A-C

-E-G C-m

ajor

tria

dA

-maj

or tr

iad

Alte

rnat

ion

of C

- an

dA

-maj

or tr

iads

=

com

pose

d-ou

t A-C

-E-G

four

diff

eren

t kin

ds

of to

nic

with

in D

TC

V/A

V/C

Gol

den

Slum

bers

Car

ry th

at

Wei

ght

The

End

“golden slumbers

fill your eyes”

recap of “You Never

Give Me Your M

oney”

“boy, you’re gonna

carry that weight”

clim

actic

sta

tem

ent

of A

-C-E

-G s

onor

ityA

-C-E

-G

Exam

ple

16

: The

Bea

tles,

med

ley

from

Ab

bey

Road

(196

9), r

educ

tion.

As

in B

aile

y’s

19th

-cen

tury

ex

ampl

es, b

oth

C a

nd A

are

cen

tral

for

port

ions

of t

he m

edle

y, b

ut n

eith

er e

mer

ges

as th

e gl

obal

toni

c.

One sweet dream

Example 17: Four types of tonic in “You Never Give Me Your Money”

You never give me your money You only give me your funny paper And in the middle of negotiations, you break down.

circle-of-fifths sequence in A minor

C is tonicized within sequence

Out of college, money spent See no future, pay no rent All the money’s gone, nowhere to go

descending-thirds sequence in C major

A is tonicized within sequence

a) “You never give me your money” theme (0:22): Am7 chord as tonic

b) “Out of college” theme (1:08): CM chord as tonic

c) “One sweet dream” theme (2:27): AM chord as tonic

(2:57): alternation of CM and AM chords

came true today,yes it did! Na guitar

A in melody over CM triad

Once there was a way

piano

bass

vocal

Example 18: Opening of “Golden Slumbers”

watching you, watching you, watching you, watching you You play with words

A♭

E♭

C

B♭

G

E♮

C

AThis riff continues into the next verse, but the bass note changes from A to C

Semitonal voice leading cadencing into the four-

note tonic sonority

watching for lies, wellyou can’t escape my Pri - vate eyes, they’re watching you, they see yourhurting inside, ’cause

ChorusUpper voices project C major;

bass is centered on APrechorus introduces chord structure of a major triad plus a different bass note

Chorus begins with A-C-E-G sonority

I see you, and you see me watch you blowin’ the lines, when you’re making a scene, oh girl,you play with love you can twist it around, baby that ain’t enough, cause girl,

Verse

10

Verse solidly establishes C major; most chords are triads

Chorus

a - ny disguise, I’ll still know you. Look in - to my pri - vate eyes, they’re watching you

G#Gn

Domestic animal

Pet dog

Pet cat

“I have five pets”

Petdog

Weimaraner Corgi

“I have five dogs”

Petdog

Weimaraner

“I have five Weimaraners”

???

RiverbankFinancial institution

“I am arriving at the bank”

Example 19: Cognitive engagement occurs at different conceptual levels depending on the word used in a given sentence

a) Linguistic indeterminacy: “pets” does not signify dogs and cats but rather a single abstract concept

b) Linguistic ambiguity: two potential meanings are not reconcilable into a single abstract concept

???

A minor C major

a) Tonal ambiguity

A/C complex

A minor C major

b) Double-tonic complex

Example 20: Tonal ambiguity, like linguistic ambiguity, involves two potential keys that are mutually exclusive. The double-tonic complex is analogous instead to linguistic indeterminacy, where cognitive engagement occurs at a more abstract conceptual level.

Domestic animal

Domestic animal

a) Wagner, Tristan und Isolde, “Was träumte mir von Tristans Ehre?,” Act 1, Scene 5

Example 14: Sarah McLachlan, “Building a Mystery” (1997)

Example 15: Harmonic reduction of “Building a Mystery” showing prolonged four-note chords giving rise to a double-tonic complex

Table 1: Previous tonal interpretations of “Building a Mystery”

Verse

You come out at night and the dark side’s light, and the vampires roamThat’s when the

energy comes,

etc.

0:12

Tonic

Prechorus

You’re so beautiful charm and, and so careful when I’m in your armswith anedge (and)

1:11

Subdominant Dominant

building a mys - tery Hold - ing on and hold - ing it in

Chorus

1:23Tonic

’cause you’re working,

etc.

Koozin 2008D major with “competing modal/linear pattern around B minor”

Doll 2011Verse is in B minor (although D “threatens to break through”);

choruses are in D

Richards 2017aTonal center changes from phrase to phrase in verses; choruses are in D

Attas 2011“Circular chord progression …

does not make a particular tonality clear”

de Clercq 2012,96–98

D major, with B minor an acknowledged alternative

Osborn 2017,67

D major

Example 13: The Axis progression seen as prolonging a four-note sonority with stepwise voice leading

has 6 of both A and C

has 7 and 2 of both A and C

Am F C G Am

Tonic

Subd

omin

antD

ominant

C/Am

F

Dm

(Bº)

G

(Em)

b) Mahler, Das Lied von der Erde, end

Example 1: Nineteenth-century examples of four-note sonorities functioning as tonic within a double-tonic complex. Both are built from the union of A-minor and C-major triads.

Example 2: Elliott Smith, “Everything Means Nothing to Me” (2000): a four-note sonority built from the union of two triads functions as prolonged tonic (after Schultz 2012).

melody prolongs A#

tonic sonority is union of A#m

and C#M triads

Example 8: LMFAO, “Party Rock Anthem” (2011): combination of I–V–IV and i–bVII–bVI, where the tonic chord is a four-note union of F-minor and Ab-major triads.

at 1:13

Example 9: Tina Turner, “What’s Love Got To Do With It?” (1984), prechorus into chorus: the chorus’s opening tonic contains the notes of G#m but the bass and instrumental texture hint at a B-major possibility.

Example 10: “What’s Love Got To Do With It?,” beginning of solo section (transcribed a whole step below the sounding pitch to facilitate comparison with previous example): the B-major possibility is strengthened by the solo line.

Example 11: “What’s Love Got To Do With It?,” bridge (again transcribed a whole step below the sounding pitch): this section moves away from G# minor, and when it leads back to the chorus it strongly projects B major.

Example 12: Melodic tendencies and harmonic categories within a double-tonic complex

at 0:46

Chorusend of prechorus…

Bass and guitar hint at a B chord, while e. piano plays G#m

e. guitar

e. piano

bass

After the first phrase, there is no

G# on the downbeat

at 2:02

The solo line completes aB-major triad on the first beat, forming the four-note sonority B-D#-F#-G# on the second beat

synth. harmonica

Solo

The bridge ends on a B chord, though it sounds

a bit like V of E major

etc.

When the chorus returns immediately following the bridge’s final B chord, the first chord’s B-major possibility is

especially potent

Chorus

Bridge

The bridge begins with the song’s first significant emphasis on B major

at 2:21

Example 11, cont’d

a) Non-tonic notes gravitate stepwise toward tonic notes; G and A, both members of the four-note tonic sonority, can move stepwise to one another.

b) Tonic, subdominant, and dominant functional categories are explicitly defined for diatonic triads. Songs exhibiting an Aeolian-Ionian double-tonic complex do not usually contain the chords in parentheses. (This diagram adapts Biamonte 2010’s functional diagrams, which in turn adapt those from Agmon 1996.)

iAb:

f: bVII bVII V IV

i bVII bVII V IV

at 0:29

T D S T D S

Melody outlines Ab major

(3 2 1)

Tonal Ambiguity

Am C Dm Em F G

A minor T T S D S DC major T T S D S D

Aeolian-Ionian Complex

“Don’t Forget Me” (Red Hot Chili Peppers)

A5

I'm an o cean-

F5

in your bed room,-

C5

make you feel warm, make

G5

you want to re sume.-

C&

‹

C=1

A=1

0:15

^

^vi

i bVI

IV

bIII

I

bVII

V

Œ œœœ

œ Œ œœ

œœ Œ œ

œœ œ

œœ

œœœœœ œ œ

A = 1 i bVI bIII bVIIC = 1 vi IV I V

Double-Tonic Complex (Nobile 2020)

||: vi I V IV :||||: vi V I IV :||||: vi I IV V :||||: vi IV I iii :||||: vi I V iii :||||: vi I V ii :||||: ii vi I V :||

A Few Other Double-Tonic Complex Loops

A Few Other Double-Tonic Complex Loops||: vi I V IV :||||: vi V I IV :||||: vi I IV V :||||: vi IV I iii :||||: vi I V iii :||||: vi I V ii :||||: ii vi I V :||

• “Jar of Hearts” (Christina Perri) • “Set Fire to the Rain” (Adele) • “Prayer in C” (Lilly Wood and the Prick)

vi

B‹


V

A

out com bie,

vi

B‹

IV

G A

V vi

B‹

On a hip pie trail,

V

A

head full of zom

vi

B‹

bie.-

IV

G A

V

44&

##

0:11

‰œ œ œ œœ



Œ Ó



I

D


V

A

downun der?

vi

B‹

IV

G A

V I

D

Wherewo menglow

V

A

andmenplunder?

vi

B‹

IV

G A

V

44&

##

0:29

‰œ œ œ œ œ


œ

rœ œ œ œ

œ œ œ œ

Œ Ó

Shifting Between Relative Tonal Centers

From Minor (Verse) to Relative Major (Chorus)

“Innocence” (Avril Lavigne)



C#m



C#m

E



C#m

AB

E

A mer- i- can- glo-

ry

B‹

fa ded- be fore- me, now I'm feel ing- hope

A

less,- ripped up my prom dress run ning- through rose

1

G

thorns, I saw the score board- and ran for my life.

A5

G

No cam era's- catch my pag eant- smile;

B‹

I coun ted- days,

A

I coun ted- miles

9

G

to see you there, to see you there. It's been a long time com ing,- but. . . .

13

G

It's you and me, that's

B‹

my whole world. They whis

A

per- in the hall way,- "She's a bad, bad girl."

B‹17

G

The whole school is roll

B‹

ing- fake dice. You play

A

stu pid- games, you win stu pid- pri zes.-

21

22&

‹

##

0:24

&

‹

##Vr)

&

‹

## ∑

&

‹

##Pre)

&

‹

##

&

‹

##Ch)

* * *

&

‹

##

œ

J

œ œ œ œ

œœ Œ œ œ œ œ œ

œ ‰ œ

J

œ œ œ œ œœ Œ

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œœ

Œ œ œ œ œ œœ

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(Taylor Swift)

Vocal

Night

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Double–Parallel Tonics

I / i bVII IV6 I / i bVII IV6

The Triple-Tonic Complex

C major

A minor C minor

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D

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“Gorilla” (Bruno Mars), Verse

Three Harmonic PalettesSt

ephe

nson

(200

2)

Three Harmonic PalettesSt

ephe

nson

(200

2)Triple-Tonic C

omplex

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“Radioactive” (Imagine Dragons), Chorus

B = 1 i bIII bVII IV6

D = 1 vi I V II6

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B‹

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D

my bones, e nough

A

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E

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œ

“Radioactive” (Imagine Dragons), Chorus

B = 1 i bIII bVII IV6

D = 1 vi I V II6

Dorian–Lydian Complex

The Three Modal ComplexesAeolian-Ionian: A, B, C, D, E, F, G

6 7 1 2 3 4 5

Dorian-Lydian: A, B, C, D, E, F#, G 6 7 1 2 3 #4 5

Phrygian-Mixolydian: A, Bb, C, D, E, F, G 6 b7 1 2 3 4 5

Vr)_ Am Am D/A D/A

F F G G

Pre)_ Bb Bb F F

Am C/G F F

Ch)_ G Am F G G Am F G

G Am F G G Am F G

“Save Me” (Fleetwood Mac)

Phrygian-Mixolydian Complex

“Save Me” (Fleetwood Mac)

Chas Williams (Session Musician)

Nashville Chart

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the logic of six-based minor for harmonic analyses of

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