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The Voice-leading Matrix
Chordal and Linear Roles
The Complete Matrix
Definition of accessive and accessible
In anyOrbit
Voice-leading(Example:3-2)
- the first orbit (3) is said to be accessive to the second (2), and
- the second orbit (2) is said to be accessible from the first (3).
The orbits indicated in the top row of the matrix are accessive
- to those indicated below them in the same column, and
the orbits indicated in the left column of the matrix are accessible
- from those indicated on their right in the same row.
We will see that
-Orbit 3is the least accessive, and
-Orbit 0is the least accessible.
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3
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2
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1
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0
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-Orbit 3is only accessive toOrbit 2
-Orbit 2is accessive toOrbit 3andOrbit 1
-Orbit 1andOrbit 0are accessive to the 4 orbits
-Orbit 0is only accessible fromOrbit 1andOrbit 0
-Orbit 3,Orbit 2, andOrbit 1
are accessible from 3 orbits
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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Basic Meterials
The Primary Notes
Primary Notes (placed in the lower-right corner of the matrix)
- are theFRAMEs of chords
- as well as the birthplace of theJordan Triangle.
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3
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2
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1
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0
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-1-1is the basic linear orbit line
-0-1and1-0are the chordal Fundamentral Bass
-0-0is the chordal pivot of the swing
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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The Secondary Notes
By their origins and tunings,
the Secondary Notes might bear allegiance to different “camps”,
-Orbit 2, inShort Branch Tuningand dividing theFRAME, to the chordal camp, and
-Orbit 3, inTrunk Tuningand discordant with theFRAME, to the linear camp.
Both Secondary Notes are found in the diagonal of the Jordan Diamond and
their resolutions would probably be shared by both camps.
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3
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2
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1
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0
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-3-2and2-1are the basics of all voice-leading.
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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The Dipper
If we amalgamate both the Primary and Secondary Notes into one Matrix,
we find the complete Voice-leading from aTetradto aRealTriad,
- the Secondary Note resolutions,3-2and2-1,
- the Primary Note options,1-1and0-0, and
- the fundamental Bass possibilities,1-0and0-1.
All this is in the form of an astronomical "Dipper" (big or small, take your
choice).
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3
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2
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1
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0
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You might like to see theHistoryof Music Theory
for more comparisons with astronomy.
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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Transformations
Different Forms Of Voice-leading
Voice-leading A, with procuration0-3,
resolutions3-2and2-1,
and fundamental bass1-0.
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3
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2
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1
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0
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- Voice-leading A
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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Voice-leading B, with procuration1-3,
resolutions3-2and2-1,
and theCOMMON TONE
0-0.
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3
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2
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1
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0
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- Voice-leading B
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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Voice-leading C, with procuration2-3,
resolution3-2(2-1is aborted),
the melodic basicPROPER TONE
1-1, and theCOMMON TONE
0-0.
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3
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2
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1
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0
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- Voice-leading C
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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A Fusion Of Procurations And Substitutions
The Complete Matrix
In the section onVoice-Leading, we saw 2 procedures
- Resolutions, which diminish the tension by only 1 degree, no more (3-2,2-1),
- Procurations, which augment the tension by any number of degrees (0-3,1-3,2-3).
In the section onIncompleteness, we saw the procedure of
- Substitutions, in which a Primary Note (Orbit 1orOrbit 0)
replacedOrbit 3and resolved toOrbit 2(asOrbit 3would have done).
In other words,Orbit 0, substituting forOrbit 3,0(3), and resolving toOrbit 2,
behaved as if it had procuredOrbit 2(0-2).
Procuration = Substitution + Resolution
This equation shows us how these 2 procedures are inter-related.
Thinking in terms of substitution will be particularly convenient inJust Intonation
- because an Orbit and its substitution have the same tuning when chromaticized.
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3
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2
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1
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0
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The procuration ofOrbit 3can also be seen as
a substitution forOrbit 4
especially when0-3is chromaticized.
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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3
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2
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1
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0
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The procuration ofOrbit 2is usually seen as
a substitution ofOrbit 3
especially when1-2is chromaticized
(Augmentation)
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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3
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2
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1
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0
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The procuration ofOrbit 1can also be seen as
a substitution forOrbit 2
especially when1-1is chromaticized
(Diminution).
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3
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2-3
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1-3
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0-3
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2
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3-2
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1-2
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0-2
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1
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2-1
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1-1
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0-1
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0
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1-0
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0-0
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Conjunct Motion Required for Chromaticism
Prerequisites to Chromaticism
A. The chordal obligation of strengthening the chord, and
B. the linear obligation of having a major second movement.
The examples shown do not necessarily have this major second “space” available.
- Chromaticism is possible only if the space is sufficient
(Ex: either in3-2or2-1, not in both).
NOTE - that the procuration 2-3 is not considered “chromaticizable”
- because it would weaken the chord.
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