Chordal and Linear Roles

The Complete Matrix

Definition of accessive and accessible
In any OrbitVoice-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 3 is the least accessive, and
- Orbit 0 is the least accessible.

 3 2 1 0 - Orbit 3 is only accessive to Orbit 2- Orbit 2 is accessive to Orbit 3 and Orbit 1- Orbit 1 and Orbit 0 are accessive to the 4 orbits- Orbit 0 is only accessible from Orbit 1 and Orbit 0- Orbit 3, Orbit 2, and Orbit 1     are accessible from 3 orbits 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

Basic Meterials

The Primary Notes
Primary Notes (placed in the lower-right corner of the matrix)
- are the FRAMEs of chords
- as well as the birthplace of the Jordan Triangle.

 3 2 1 0 - 1-1 is the basic linear orbit line- 0-1 and 1-0 are the chordal Fundamentral Bass- 0-0 is the chordal pivot of the swing 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

The Secondary Notes
By their origins and tunings,
the Secondary Notes might bear allegiance to different “camps”,
- Orbit 2, in Short Branch Tuning and dividing the FRAME, to the chordal camp, and
- Orbit 3, in Trunk Tuning and discordant with the FRAME, 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.

 3 2 1 0 - 3-2 and 2-1 are the basics of all voice-leading. 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

The Dipper
If we amalgamate both the Primary and Secondary Notes into one Matrix,
we find the complete Voice-leading from a Tetrad to a Real Triad,
- the Secondary Note resolutions, 3-2 and 2-1,
- the Primary Note options, 1-1 and 0-0, and
- the fundamental Bass possibilities, 1-0 and 0-1.
All this is in the form of an astronomical "Dipper" (big or small, take your choice).

 3 2 1 0 You might like to see the History of Music Theoryfor more comparisons with astronomy. 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

Transformations

Different Forms Of Voice-leading
Voice-leading A, with procuration 0-3,
resolutions 3-2 and 2-1,
and fundamental bass 1-0.

 3 2 1 0 - Voice-leading A 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

Voice-leading B, with procuration 1-3,
resolutions 3-2 and 2-1,
and the COMMON TONE0-0.

 3 2 1 0 - Voice-leading B 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

Voice-leading C, with procuration 2-3,
resolution 3-2 (2-1 is aborted),
the melodic basic PROPER TONE1-1, and the COMMON TONE0-0.

 3 2 1 0 - Voice-leading C 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

A Fusion Of Procurations And Substitutions

The Complete Matrix
In the section on Voice-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 on Incompleteness, we saw the procedure of
- Substitutions, in which a Primary Note (Orbit 1 or Orbit 0)
replaced Orbit 3 and resolved to Orbit 2 (as Orbit 3 would have done).
In other words, Orbit 0, substituting for Orbit 3, 0(3), and resolving to Orbit 2,
behaved as if it had procured Orbit 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 in Just Intonation
- because an Orbit and its substitution have the same tuning when chromaticized.

 3 2 1 0 The procuration of Orbit 3 can also be seen as     a substitution for Orbit 4          especially when 0-3 is chromaticized. 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0
 3 2 1 0 The procuration of Orbit 2 is usually seen as     a substitution of Orbit 3          especially when 1-2 is chromaticized 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0
 3 2 1 0 The procuration of Orbit 1 can also be seen as     a substitution for Orbit 2          especially when 1-1 is chromaticized               (Diminution). 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0

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 in 3-2 or 2-1, not in both).
NOTE - that the procuration 2-3 is not considered “chromaticizable”
- because it would weaken the chord.

 3 2 1 0 0-3 - See Orbit 43-2 and 1-1 - See Diminution1-2 - See Augmentation2-1 - See Dominantization 3 2-3 1-3 0-3 2 3-2 1-2 0-2 1 2-1 1-1 0-1 0 1-0 0-0