Appendix - Worked Harmonic Atlas for Dm7-G7-Cmaj7

This atlas is designed for repeated use at the piano. It takes a small number of upper structures and examines them from several angles: vertical color, interval content, voice leading, melodic extraction, and ear training. The purpose is not to produce an exhaustive catalogue. It is to demonstrate a method of investigation that you can later apply to any structure.

Dm7: four ways to organize an inside sound

F major over Dm7

F-A-C / D = b3-5-b7

This is almost a reduction of Dm7 itself. Because the upper object contains the third, fifth, and seventh, it strongly states chord quality while omitting the root. Use it when you want the right hand to sound grounded.

F major over Dm7: b3-5-b7, resolving through G7 to Cmaj7.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [Bdf]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

F major over Dm7: b3-5-b7, resolving through G7 to Cmaj7. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | B2 d2 f2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

As a line, avoid always playing F-A-C. Try A-C-F, C-A-F, or F-C-A. The harmonic information is unchanged. The contour is not.

F major over Dm7 reordered as A-C-F, resolving through G7 to Cmaj7 -- harmony unchanged, contour changed.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A2 c2 f2 z2 | [Bdf]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

F major over Dm7 reordered as A-C-F, resolving through G7 to Cmaj7 -- harmony unchanged, contour changed. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A2 c2 f2 z2 | B2 d2 f2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

C major over Dm7

C-E-G / D = b7-9-11

The absence of F makes the upper layer less explicit about minor quality. If the left hand supplies F, the two layers divide responsibility elegantly: LH says ‘minor’; RH says ‘extension.’

Compare Dm7 + F major and Dm7 + C major without changing rhythm. This isolates harmonic color from phrasing.

C major over Dm7: left hand supplies the third (F), right hand the extension, resolving through G7 to Cmaj7.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[CEG]8 | [Bdf]8 | [EGB]8 |]
V:LH clef=bass
F,8 | G,,8 | C,,8 |]

C major over Dm7: left hand supplies the third (F), right hand the extension, resolving through G7 to Cmaj7. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
C2 E2 G2 z2 | B2 d2 f2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
F,8 | G,,8 | C,,8 |]

Em7 over Dm7

E-G-B-D / D = 9-11-13-1

This object places almost all of the diatonic upper extensions in one familiar shape. It is an excellent example of why superimposition can be cognitively efficient. Instead of tracking 9, 11, and 13 independently, the hand and eye can recognize Em7.

Do not let the convenience of the shape hide the interval analysis. Periodically stop and spell the degrees.

Em7 over Dm7: almost the complete diatonic upper world, resolving through G7 to Cmaj7.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[EGBd]8 | [Bdf]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Em7 over Dm7: almost the complete diatonic upper world, resolving through G7 to Cmaj7. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
E2 G2 B2 d2 | B2 d2 f2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

A minor pentatonic over Dm7

A-C-D-E-G / D = 5-b7-1-9-11

This produces an open minor sound while omitting the third. Try a left-hand shell F-C. The division is clear: the LH supplies b3-b7; the RH supplies a pentatonic melodic environment.

A minor pentatonic over Dm7, left hand supplying the missing third, resolving through G7 to Cmaj7.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A c d e g z3 | [Bdf]8 | [EGB]8 |]
V:LH clef=bass
[F,C]8 | G,,8 | C,,8 |]

A minor pentatonic over Dm7, left hand supplying the missing third, resolving through G7 to Cmaj7. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A c d e g z3 | B2 d2 f2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
[F,C]8 | G,,8 | C,,8 |]

G7: suspended and bright colors

F major over G7

F-A-C / G = b7-9-11

The presence of C, the natural 11, rubs against B if the third is sustained nearby. That friction is precisely why the sound can feel suspended. If you want a clear sus effect, temporarily omit B. If you want controlled tension, let B and C coexist and resolve C toward B or C# depending on the next color.

This example teaches an important lesson: interval labels do not tell the entire voicing story. Register and simultaneity matter.

F major over G7: suspended, the natural 11 rubbing against the third, resolving through Cmaj7.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [FAc]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

F major over G7: suspended, the natural 11 rubbing against the third, resolving through Cmaj7. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | F2 A2 c2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

A major over G7

A-C#-E / G = 9-#11-13

This is the book’s canonical bright dominant. Practice three inversions:

A-C#-E
C#-E-A
E-A-C#

For each inversion, find a nearby Cmaj7 destination. Then create a line that touches all three notes without presenting them in arpeggio order.

The characteristic pitch is C#. If you remove it, much of the bright #11 identity disappears. This makes C# a useful ear-training target.

A major over G7 -- the canonical bright dominant, in three inversions, resolving to Cmaj7.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[A^ce]8 | [^CEA]8 | [EA^c]8 | [EGB]8 |]
V:LH clef=bass
G,,8 | G,,8 | G,,8 | C,,8 |]

A major over G7 -- the canonical bright dominant, in three inversions, resolving to Cmaj7. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A2 ^c2 e2 z2 | ^C2 E2 A2 z2 | E2 A2 ^c2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
G,,8 | G,,8 | G,,8 | C,,8 |]

G7: altered colors

Bb major over G7

Bb-D-F / G = #9-5-b7

This structure combines one strong alteration, #9, with two stabilizing dominant tones. It can therefore sound blues-inflected rather than maximally altered.

Resolve Bb to B and keep D or F as common/nearby tones. Listen to how one semitone can redefine the entire object.

Bb major over G7: blues-inflected alteration, resolving just Bb to B, within the full ii-V-I.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [_Bdf]8 | [Bdf]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | G,,8 | C,,8 |]

Bb major over G7: blues-inflected alteration, resolving just Bb to B, within the full ii-V-I. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | _B2 d2 f2 z2 | B2 d2 f2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | G,,8 | C,,8 |]

Db major over G7

Db-F-Ab / G = b5-b7-b9

All three notes participate strongly in dominant tension. The most direct resolution is downward:

Db -> C
F -> E
Ab -> G

Practice the two sonorities as slow whole notes. Then shorten the dominant until the resolution becomes a gesture rather than an event.

Db major over G7, resolving downward, within the full ii-V-I.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [_DF_A]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Db major over G7, resolving downward, within the full ii-V-I. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | _D2 F2 _A2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Eb major over G7

Eb-G-Bb / G = b13-1-#9

The root inside the upper structure stabilizes the sound while b13 and #9 create alteration. Resolve upward:

Eb -> E
Bb -> B

Compare this with G7 + Db major. One altered structure tends naturally toward downward semitone motion, the other toward upward motion. This directional distinction is more musically useful than the generic label ‘altered.’

Eb major over G7, resolving upward, within the full ii-V-I.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [_EG_B]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Eb major over G7, resolving upward, within the full ii-V-I. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | _E2 G2 _B2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Cmaj7: tonic does not mean colorless

E minor over Cmaj7

E-G-B / C = 3-5-7

This is a compact upper representation of the tonic’s defining quality. Add D to make Em7:

E-G-B-D / C = 3-5-7-9

The tonic becomes richer without losing stability.

E minor over Cmaj7, then Em7, reached through the full ii-V-I.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [Bdf]8 | [EGB]8 | [EGBd]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 | C,,8 |]

E minor over Cmaj7, then Em7, reached through the full ii-V-I. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | B2 d2 f2 z2 | E2 G2 B2 z2 | E2 G2 B2 d2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 | C,,8 |]

D major over Cmaj7

D-F#-A / C = 9-#11-13

This is a Lydian tonic color. Compare it with G7 + A major. The degree set 9-#11-13 is identical, but the underlying function changes from dominant to tonic. Similar interval color does not imply similar harmonic function.

This comparison is a powerful demonstration of the book’s layered model.

D major over Cmaj7: a Lydian tonic color, reached through the full ii-V-I.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [Bdf]8 | [D^FA]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

D major over Cmaj7: a Lydian tonic color, reached through the full ii-V-I. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | B2 d2 f2 z2 | D2 ^F2 A2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

C major plus D major

The combined triad pair gives:

C-E-G + D-F#-A

or:

1-3-5 + 9-#11-13

The six-note environment is nearly a complete C Lydian collection without B. Alternate the two triads rather than treating the result as a scale.

C major + D major combined over Cmaj7, reached through the full ii-V-I.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[FAc]8 | [Bdf]8 | [CDE^FGA]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

C major + D major combined over Cmaj7, reached through the full ii-V-I. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
F2 A2 c2 z2 | B2 d2 f2 z2 | C1 D1 E1 ^F1 G1 A1 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Voice-leading atlas

Now connect selected objects through the entire progression.

Path A: diatonic to bright to stable

Dm7 + C major -> G7 + A major -> Cmaj7 + E minor

Pitch sets:

C-E-G -> A-C#-E -> E-G-B

Search for the nearest inversion rather than using root positions mechanically. The theoretical objects are fixed; the actual voicing is a voice-leading decision.

Path A -- diatonic to bright to stable: Dm7 + C major, G7 + A major, Cmaj7 + E minor.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[CEG]8 | [A^ce]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Path A -- diatonic to bright to stable: Dm7 + C major, G7 + A major, Cmaj7 + E minor. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
C2 E2 G2 z2 | A2 ^c2 e2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Path B: extension to altered to stable

Dm7 + Em7 -> G7 + Eb major -> Cmaj7 + E minor

Notice the visual and auditory semitone relationship between Eb major and E minor. The names of the structures themselves reveal the resolution.

Path B -- extension to altered to stable: Dm7 + Em7, G7 + Eb major, Cmaj7 + E minor.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
[EGBd]8 | [_EG_B]8 | [EGB]8 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Path B -- extension to altered to stable: Dm7 + Em7, G7 + Eb major, Cmaj7 + E minor. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
E2 G2 B2 d2 | _E2 G2 _B2 z2 | E2 G2 B2 z2 |]
V:LH clef=bass
D,,8 | G,,8 | C,,8 |]

Path C: pentatonic to altered to Lydian tonic

Dm7 + A minor pentatonic -> G7 + Db major -> Cmaj7 + D major

This path deliberately changes organizational systems: five-note object, triad, triad. Preserve a rhythmic motif so the phrase remains coherent.

Path C -- pentatonic to altered to Lydian tonic: Dm7 + A minor pentatonic, G7 + Db major, Cmaj7 + D major.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A c d e g z3 | [_DF_A]8 | [D^FA]8 |]
V:LH clef=bass
[F,C]8 | G,,8 | C,,8 |]

Path C -- pentatonic to altered to Lydian tonic: Dm7 + A minor pentatonic, G7 + Db major, Cmaj7 + D major. Same voicing, arpeggiated instead of stacked.

X:1
M:4/4
L:1/8
K:C
V:RH clef=treble
A c d e g z3 | _D2 F2 _A2 z2 | D2 ^F2 A2 z2 |]
V:LH clef=bass
[F,C]8 | G,,8 | C,,8 |]

Melodic extraction protocol

For any object in the atlas, create five lines.

Line 1: pure object. Use only notes from the structure.

Line 2: one chromatic approach. Choose the most characteristic tension as the target.

Line 3: enclosure. Surround one structural tone chromatically.

Line 4: rhythmic displacement. Preserve pitches but begin on an offbeat.

Line 5: voice-led connection. Make the last note of the object lead by semitone or common tone into the next harmony.

If the five lines sound like variations of one idea, the object has become vocabulary. If they sound unrelated, simplify and preserve more contour or rhythm.

Ear-training protocol

Record a sustained G7 shell and randomly play A, Bb, Db, and Eb major above it. Identify the color before naming the triad.

Then reverse the task. Decide on one of four perceptual categories:

• bright;
• blues-altered;
• dark/downward altered;
• altered/upward resolving.

Sing one characteristic tension and find the corresponding structure.

Finally, remove the bass. Play the upper triad alone, imagine G underneath it, and name the resulting degrees. This trains the ability to carry harmonic context internally.