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Chord Cartography

Dan Nielsen 4/2019

(The previous article in this series is The Structure of Scales)

Outline of a harmonic navigation system

Negative melody and harmony

Negative harmony is a very popular concept in current music theory. Let's not elaborate here on its definition, since it's well-described by online resources. Suffice it to say that the "negative scale" begins on the fifth of the source scale and descends by the same order of whole and half steps, so that, for instance, [A B C D E F G] -> [E D C# B A G# F#]. We then reorder the result to begin with the original tonic, so it becomes [A B C# D E F# G#], or the major scale.

But there's seemingly no good reason we shouldn't give the same significance to the symmetrical operation, to map a descending scale to an ascending scale that begins down a fifth, so that [A G F E D C B] -> [D E F# G A B C]. After recentering and reordering, this becomes [A B C D E F# G], or Dorian mode.

Notice that this process is a sort of natural continuation of the thinking in "Structure of Scales", where we attached harmonic structure to a nucleus of up-a-fifth-and-down-a-fifth. Now we are attaching serial pitch structure to up-a-fifth-and-down-a-fifth.

In the chart below, when a scale may be called by various names, the names are listed and separated by commas. The term "unharmonic" is used when a leading tone is flattened, and "unmystic" when a minor second is sharpened.

By negative harmony, we've shone light on eight more scales: Lydian, Locrian, Lydian Diminished, Locrian Natural 6, Ukrainian Dorian, Dorian Flat 5, the Overtone scale, the Half-Diminished scale.

Neg1 Neg2

MF6

HarDor, MelMinAsc

MysDor

Har

HarAeo, HarMin

LocNat6

Mys

LydDim MysMix, HarMinInv

Mix

Dor

MysAeo, Phr

Har

Aeo, NatMin, MelMinDes

Loc

Mys

UkrDor, RomMin MysMF6, PhrDom, Freygish

Aeo

HarMix, Ion, Maj

Dor

Har

HarMF6, HarMaj DorFlat5

Mys

Lyd

Mix

Dor

Mix

Aeo, NatMin

Har

MF6

UnmysLoc, HalfDim

Mys

UnharLyd, Overtone

MF6

Scale spectrum

In "Stucture of Scales", a metric was given for the brightness of a scale based on its harmonic structure. Let's use that metric to order these scales by brightness.

We need to extend the metric a little, because for the original set of scales, all "prongs" were attached to a "nucleus" of tonic and a fifth in either direction. This time, we might also have prongs attached at a distance of two fifths.

For an upward prong attached at a distance of two fifths, the associated interval is the augmented fourth. If tonic is G, then it's C# attached to A. If we consider A to have pitch 9/8, then the value will be 3(8/9), or about 2.667.

For a downward prong, the associated interval is the diminished fifth. If tonic is G, then it's Db attached to F. If we consider F to have pitch 8/9, then the value will be -3(9/8), or -3.375.

Of course, we haven't given any rigorous proof for this metric, but it seems to work pretty well.

WITH G AS TONIC...

LYDIAN (w w w h w w h)                LOCRIAN (h w w h w w w)
                                                             
B      F#     C#                      Bb ---- F ---- C ---- G
|      |      |                               |      |      |
G ---- D ---- A ---- E                        Db     Ab     Eb            

3+2+2.667 = 7.667                     -2-3-3.375 = -8.375

LYDIAN DIMINISHED (w h 3h h w w h)    LOCRIAN NATURAL 6 (h w w h 3h h w)

       F#     C#                                     E
       |      |                                      |
G ---- D ---- A ---- E                Bb ---- F ---- C ---- G
       |                                      |      |
       Bb                                     Db     Ab

2+2.667-3 = 1.667                     2-3-3.375 = -4.375

UKRAINIAN DORIAN (w h 3h h w h w)     DORIAN FLAT 5 (w h w h 3h h w)

              C#                               A      E
              |                                |      |
G ---- D ---- A ---- E                 Bb ---- F ---- C ---- G
       |      |                                |
       Bb     F                                Db

2.667-3-3.375 = -3.708                2+2.667-3.375 = 1.292

OVERTONE SCALE (w w w h w h w)        HALF-DIMINISHED SCALE (w h w h w w w)

B             C#                              A
|             |                               |
G ---- D ---- A ---- E                Bb ---- F ---- C ---- G
              |                               |      |
              F                               Db     Eb

3+2.667-3.375 = 2.292                 2.667-3-3.375 = -3.708
Brightness
Lyd (7.667)
Ion (7)
Mix (5)
HarMF6 (3)
Overtone (2.292)
MysMix (2)
LydDim (1.667)
DorFlat5 (1.292)
MF6 (1)
HarDor (1)
Dor (-1)
MysMF6 (-2)
HarAeo (-3)
UkrDor (-3.708)
HalfDim (-3.708)
MysDor (-4)
LocNat6 (-4.375)
Aeo (-5)
Phr (-8)
Loc (-8.375)

Seventh chords from triads

Taking stacks of thirds, we can form seven seventh chords: augmented, major, dominant, minor-major, minor, half-diminished, diminished.

Since the augmented triad appears as a root triad only in the augmented seventh chord, we can think of both chords as harmonically equivalent in traditional harmony, with the seventh being an embellishment of the triad.

But the story's not so simple for the other seventh-chord types. The minor, major, and diminished triads each appear in two types of seventh chords. We might directly associate the major triad with the major seventh chord, the minor triad with the minor seventh chord, and the diminished triad with the fully diminished seventh chord. But besides the matching monikers already given to these chords, what's the justification?

Using traditional theory to "naturally" extend triads, we could argue against the candidacy of minor-major, dominant, and half-diminished as alternative interpretations...

It seems these general conclusions are acceptable. Regarding the dominant seventh, we could amend that only a half-step modification is required to reach either a minor seventh chord or a major seventh chord; likewise, we could simply move away from the seventh, leaving the major triad. Therefore, taken without the context of a major scale, the dominant seventh is not necessarily so filled with the need to resolve down by a fifth, especially when doubling octaves of the root note. When the dominant seventh chord is viewed as an approximation of the first 8 harmonics, empowering the root tone is like restoring the harmonics 1:2:4:8.

However, the major seventh contains two major thirds, one built on perfect one, and another built on the perfect fifth, so it makes more sense to consider it the natural extension of the major triad.

Pivot-chord modulation

The previous article mentioned "secret passages", pairs of modes with a perfect-fifth relationship that share exactly the same harmonic structure, with only a different choice of tonic placement in the harmonic lattice. My interpretation is this: whereas we are used to thinking of a dominant relationship as being a heavy one, seeking resolution in the initial tonic, this perfect-fifth relation is weightless, and tonic can tranquilly float between either mode.

But let's consider more traditional modulation now. The politest form of this modulation employs a pivot chord, meaning a chord that has a harmonic function in both the progressions of the original key and the target key.

The table below gives us the seventh chord built on each scale degree in our new set of canonical scales.

Just for example, it shows that we could pivot from MF6 into HarDor when we consider VIIV7 in MF6 to be VV7 in HarDor.

I+7

I7

IV7

imM7

i7

iϕ7

io7

HarMix Mix HarAeo Aeo
HarMF6 MF6 HarDor Dor
MysMix MysAeo
MysMF6 MysDor

II+7

II7

IIV7

iimM7

ii7

iiϕ7

iio7

MysMix MysMF6 Mix MF6
MysDor MysAeo Dor Aeo
HarMix HarMF6
HarDor HarAeo

III+7

III7

IIIV7

iiimM7

iii7

iiiϕ7

iiio7

HarAeo Aeo MysAeo HarMix Mix MysMix
HarDor Dor MysDor HarMF6 MF6 MysMF6

IV+7

IV7

IVV7

ivmM7

iv7

ivϕ7

ivo7

Mix Dor MF6 Aeo
MysMix MysDor MysMF6 MysAeo
HarMix HarDor HarMF6 HarAeo

AugV+7

MajV7

VV7

vmM7

v7

vϕ7

vo7

HarMix Mix MysMix
HarMF6 MF6 MysMF6
HarAeo Aeo MysAeo
HarDor Dor MysDor

VI+7

VI7

VIV7

vimM7

vi7

viϕ7

vio7

MF6 Aeo Mix Dor
MysMF6 MysAeo MysMix MysDor
HarMF6 HarAeo HarMix HarDor

VII+7

VII7

VIIV7

viimM7

vii7

viiϕ7

viio7

Mix MF6 HarMix HarMF6
Dor Aeo HarDor HarAeo
MysMix MysMF6
MysDor MysAeo

Voice-leading map

Voice-leading: triadic on the left, and tetradic on the right (R Cohn, based on Douthett, Boretz, et al)

The figures above show the triadic and tetradic possibilities of half-step voice-leading changes. The figure on the left gives movement of tetrads as extensions of triads, while the other firgure shows purely tetradic movement. The minor seventh chords (shown as triangles on the righthand side, and as lowercase letters on the lefthand side) are equivalent chords connecting both maps, meaning the two figures actually represent a single unified map of tetradic voice leading.

Suspensions

Of course, not included in these figures are non-tertiary triads like C-F-G (suspensions), but they are easy enough to account for, since they just attach to the major and minor triads.

Relative major and minor

The typical core theory of harmonic progression is PLR: parallel, leading-tone exchange, relative. The relative-chord movement requires two half-step moves, while the others require only one.

In "A Generalized Intervallic Approach to Triads", Aziz & Haughton state, "In summary, we assert that sonorities in voice-leading space can be contextualized as having dual functional identities: one related to its derivation from a scalar collection (Hexatonic or Octatonic), and one related to minimal displacements of an equal division of the octave (an augmented triad or a diminished seventh chord)."