White Paper Cartesian Music Theory for Math and Music Integration

Abstract

Introduction: This document presents a music theory named cartesian mapping of chords and scales (Wesley). Among the many music composition applications, there is also an example of math, music, and geometric art integration. This theory combines the mathematical works of Safi al Din, Georg Cantor, Pythagoras, and DeCartes; along with the music theory of Jeff Hellmer, Arnold Schoenberg, and Josef Hauer. The music theory presented here combines traditional western harmonic theory with the serialist, atonal works of Arnold Schoenberg and Josef Hauer with some derivations of the modern jazz theory presented by Jeff Hellmer of the University of Texas. The concept is similar to the serialist theory which attributes numbers to note names (Schoenberg, Hauer). In contrast to the numeric naming of note values based on their harmonic frequency, A440 is A4; this serialist theory constrains the possible values for note values to one octave. This process limits the possibilities of notes loosely from the range of the piano to the range of 12 notes. This allows for the composer, music analyst to think of chords and scales in one octave. While this process is limited in its representation of chords and scales, this process allows for an implementation of mathematics into the practice of composition.

Full text

Introduction:
This document presents a music theory named cartesian mapping of chords and scales (Wesley).
Among the many music composition applications, there is also an example of math, music, and
geometric art integration. This theory combines the mathematical works of Safi al Din, Georg Cantor,
Pythagoras, and DeCartes; along with the music theory of Jeff Hellmer, Arnold Schoenberg, and Josef
Hauer. The music theory presented here combines traditional western harmonic theory with the
serialist, atonal works of Arnold Schoenberg and Josef Hauer with some derivations of the modern jazz
theory presented by Jeff Hellmer of the University of Texas. The concept is similar to the serialist theory
which attributes numbers to note names (Schoenberg, Hauer). In contrast to the numeric naming of note
values based on their harmonic frequency, A440 is A4; this serialist theory constrains the possible values
for note values to one octave. This process limits the possibilities of notes loosely from the range of the
piano to the range of 12 notes. This allows for the composer, music analyst to think of chords and scales
in one octave. While this process is limited in its representation of chords and scales, this process allows
for an implementation of mathematics into the practice of composition. This is achieved by the
numeration of the one chromatic octave: A->1, A#/Bb->2, B->3, C->4, C#/DB->5, D->6, D#/Eb->7, E->8,
F->9, F#/Gb->10, G->11, G#/Ab->12. A composer may use this numeration as a basis for composition.

Theory:
This numeration can be used by the music theorist as well by mapping the notes to a two-dimensional or
three-dimensional plane. Each dimension offers a different interpretation of the twelve notes of the
chromatic octave. This allows for comparison between different permutations of western harmony as
well as some serial applications to western harmony.
For the two-dimensional plane, chords and scales can be interpreted on a single line (figure 1).
If the A major scale were to be analyzed on this two-dimensional plane, one can simultaneously interpret
the Ionian scale as well as all of the modal scales for A major. In addition to the scalar interpretation, one
can also interpret the modal harmonies as well.
A Major Scale and Modes (figure 2)
A Major Triad (figure 3)

This representation is a derivative of the rhythmic theory presented by Safi al Din in the 12th century.
Safi al Din constructed the harmonic theory from a two-dimensional plane using circles to represent
divisions of an octave (Safi al Din). This representation of time works similarly to the reading of standard
notation; as each circle represents an harmonic idea. The scalar, two-dimensional representation differs
as the theory does not include time in its representation. This allows for the simultaneous interpretation
of modal scales and harmony. This allows for the application of Cantorian set theory as a composition
tool. This process allows for the full range of arrangements of possible combinations of notes and chords
in one view point. There is an algebraic component to this proposed theory of chords and scales. The
basis for the algebraic expression are as follows:
Ab/G# Major Scale (n-12 iff n>12):(12, 2, 4, 5, 7, 9, 11)
The logic of the mapping system is algebraic. The statement above refers to many concepts: twelve notes
in the western harmonic system, the numerical sequencing of the major scale, and the coordinates for
each note and subsequent chords based on the scale. The numerical sequencing of the above major
scale is as follows: n=12, therefore the major scale is (n, n+2, n+4, n+5, n+7, n+9, n+11). If the scale is Ab
Major, the theory states that Ab=12 in the twelve note reduction of western harmony; yet the values of
the resulting scale starting at 12 will yield numbers greater than 12 (12, 14, 16, 17, 19, 21, 23). This is
mitigated due to the preceding statement that any number in the sequence will subtract a value of 12
from the number if and only if the number is greater than 12. This is why the numerical values of the Ab
major scale show sums that are less than 12 in the resulting sequencing of values. In essence, the logic
operates like this:
Ab Major Scale: Ab=12
Major scales sequencing (n, n+2, n+4, n+5, n+7, n+9, n+11)
NOTE: subtract 12 from n if and only if n is greater than 12

Ab Major = (12,12+2,12+4,12+5,12+7,12+9,12+11)
Application of Logic:
Ab Major = (12,(12+2)-12,(12+4)-12,(12+5)-12,(12+7)-12,(12+9)-12,(12+11)-12)
In this process, there is a collection of chords that derive from this scale that is the foundation of the
western harmonic practice. For instance: the I, ii, iii, IV, V, vi, and vii chords found by the following
operations:
Ab Major Scale: (12, 2, 4, 5, 7, 9, 11), (figure 4)
Ab Major Chord: (12, 4, 7), (figure 5)
The above correlation between the Ab scale and chord is the algebraic sequencing between a note’s
major scale and major chord. The sequencing for the major chord are as follows: (insert visualization)
Major Chord: (n, n+4, n+7)
n-12, iff n>12

Ab = 12
Ab Major Chord = (12, 12+4, 12+7)
Ab Major Chord = (12, (12+4)-12, (12+7)-12)
Ab Major Chord = (12, 4, 7)
Using the practice of thinking of the chord and scale from its root note, one can derive all of the possible
chords and scales relating to that key. For instance, all of the modes can be accessed through using Ab as
well through starting the numerical sequence on a different number while maintaining the structure of
the system.
Ab Major Scale = (12, 2, 4, 5, 7, 9, 11)
Bb Dorian Scale = (2, 4, 5, 7, 9, 11, 12), (figure 6)
C Phrygian Scale = (4, 5, 7, 9, 11,12, 2), (figure 7)

As seen below, this operation creates a matrix:
12,2,4,5,7,9,11
2,4,5,7,9,11,12
4,5,7,9,11,12,2
5,7,9,11,12,2,4
7,9,11,12,2,4,5
9,11,12,2,4,5,7
11,12,2,4,5,7,9
This is very similar to the lambdoma matrix presented by the Pythagoreans (Hero). This matrix organizes
the set of rational numbers in an infinite series. In this matrix, one can see that the value of one is
represented to infinity by the operation of 1/1, 2/2, 3/3, to infinity. In a corresponding manner, the
values of .5 can be seen in ½, 2/4, to infinity. If the same idea was applied to the numerical scale matrix,
then one can see how chords of the scale can be formed through the sequencing of the modal scales in
this 7x7 matrix.
Modal Scales for Ab/G#:
12,2,4,5,7,9,11
2,4,5,7,9,11,12
4,5,7,9,11,12,2
5,7,9,11,12,2,4
7,9,11,12,2,4,5
9,11,12,2,4,5,7
11,12,2,4,5,7,9
All the above algebraic representations are for the two-dimensional mapping.

For the two-dimensional mapping, the visual representation can also be applied to two note
combinations. For instance, the note A can be input into a two note sequence where all twelve notes
including A are represented as a coordinate pair: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9),
(1,10), (1,11), (1,12). This process will give a full, two-dimensional mapping of the possible two note
combinations for western harmonic theory. As with the one-dimensional representation, the
two-dimensional representation does not account for time. This allows for the user of the system to
visualize all possible combinations simultaneously. The same logic applies to the numerical values of
notes: (n-12 iff n>12). Note: the ‘v’ in the naming of the coordinates refers to ‘and’. A ‘and’ Ab for
(1,12).
Two note Combinations with A, (figure 8)

Algebra to Geometric Mapping:
There is also a three-dimensional application of the mapping system. A triad can be represented as single
coordinates in terms of x,y, and z. Here’s an algebraic representation of this phenomenon:
Major Chords
A: (1,5,8); (1,8,5); (5,8,1); (5,1,8); (8,1,5); (8,5,1)
A#/Bb: (2,6,9); (2,9,6); (6,9,2); (6,2,9); (9,2,6); (9,6,2)
B: (3,7,10); (3,10,7); (7,10,3); (7,3,10); (10,3,7); (10,7,3)
C: (4,8,11); (4,11,8); (8,11,4); (8,4,11); (11,4,8); (11,8,4)
C#/Db: (5,9,12); (5,12,9); (9,12,5); (9,5,12); (12,5,9); (12,9,5)
D: (6,10,1); (6,1,10); (10,1,6); (10,6,1); (1,6,10); (1,10,6)
D#/Eb: (7,11,2); (7,2,11); (11,2,7); (11,7,2); (2,7,11); (2,11,7)
E: (8,12,3); (8,3,12); (12,3,8); (12,8,3); (3,8,12); (3,12,8)
F: (9,1,4); (9,4,1); (1,4,9); (1,9,4); (4,9,1); (4,1,9)
F: (10,2,5); (10,5,2); (2,5,10); (2,10,5); (5,10,2); (5,2,10)
G: (11,3,6); (11,6,3); (3,6,11); (3,11,6); (6,11,3); (6,3,11)
G#/Ab: (12,4,7); (12,7,4); (4,7,12); (4,12,7); (7,4,12); (7,12,4)
(n-12 iff n>12)
Major Triad: (n, n+4, n+7)
Major Scale: (n, n+2, n+4, n+5, n+7, n+9, n+11)

Minor Chords
A: (1,4,8); (1,8,4); (4,8,1); (4,1,8); (8,1,4); (8,4,1)
A#/Bb: (2,5,9); (2,9,5); (5,9,2); (5,2,9); (9,2,5); (9,5,2)
B: (3,6,10); (3,10,6); (6,10,3); (6,3,10); (10,3,6); (10,6,3)
C: (4,7,11); (4,11,7); (7,11,4); (7,4,11); (11,4,7); (11,7,4)
C#/Db: (5,8,12); (5,12,8); (8,12,5); (8,5,12); (12,5,8); (12,8,5)
D: (6,9,1); (6,1,9); (9,1,6); (9,6,1); (1,6,9); (1,9,6)
D#/Eb: (7,10,2); (7,2,10); (10,2,7); (10,7,2); (2,7,10); (2,10,7)
E: (8,11,3); (8,3,11); (11,3,8); (11,8,3); (3,8,11); (3,11,8)
F: (9,12,4); (9,4,12); (12,4,9); (12,9,4); (4,9,12); (4,12,9)
F#/Gb: (10,1,5); (10,5,1); (1,5,10); (1,10,5); (5,10,1); (5,1,10)
G: (11,2,6); (11,6,2); (2,6,11); (2,11,6); (6,11,2); (6,2,11)
G#/Ab: (12,3,7); (12,7,3); (3,7,12); (3,12,7); (7,3,12); (7,12,3)
(n-12 iff n>12)
Minor Triad: (n, n+3, n+7)
Minor Scale: (n, n+2, n+3, n+5, n+7, n+8, n+10)

Diminished Chords
A: (1,4,7); (1,7,4); (4,7,1); (4,1,7); (7,1,4); (7,4,1)
A#/Bb: (2,5,8); (2,8,5); (5,8,2); (5,2,8); (8,2,5); (8,5,2)
B: (3,6,9); (3,9,6); (6,9,3); (6,3,9); (9,3,6); (9,6,3)
C: (4,7,10); (4,10,7); (7,10,4); (7,4,10); (10,4,7); (10,7,4)
C#/Db: (5,8,11); (5,11,8); (8,11,5); (8,5,11); (11,5,8); (11,8,5)
D: (6,9,12); (6,12,9); (9,12,6); (9,6,12); (12,6,9); (12,9,6)
D#/Eb: (7,10,1); (7,1,10); (10,1,7); (10,7,1); (1,7,10); (1,10,7)
E: (8,11,2); (8,2,11); (11,2,8); (11,8,2); (2,8,11); (2,11,8)
F: (9,12,3); (9,3,12); (12,3,9); (12,9,3); (3,9,12); (3,12,9)
F#/Gb: (10,1,4); (10,4,1); (1,4,10); (1,10,4); (4,10,1); (4,1,10)
G: (11,2,5); (11,5,2); (2,5,11); (2,11,5); (5,11,2); (5,2,11)
G#/Ab: (12,3,6); (12,6,3); (3,6,12); (3,12,6); (6,3,12); (6,12,3)
(n-12 iff n>12)
Diminished: (n, n+3, n+6)
Harmonic: (n, n+2, n+3, n+5, n+7, n+8, n+11)
Melodic: (n, n+2, n+3, n+5, n+7, n+9, n+11) Ascending
Melodic: (n, n+2, n+3, n+5, n+7, n+8, n+10) Descending

Augmented Chords
A: (1,5,9); (1,9,5); (5,9,1); (5,1,9); (9,1,5); (9,5,1)
A#/Bb: (2,6,10); (2,10,6); (6,10,2); (6,2,10); (10,2,6); (10,6,2)
B: (3,7,11); (3,11,7); (7,11,3); (7,3,11); (11,3,7); (11,7,3)
C: (4,8,12); (4,12,8); (8,12,4); (8,4,12); (12,4,8); (12,8,4)
C#/Db: (5,9,1); (5,1,9); (9,1,5); (9,5,1); (1,5,9); (1,9,5)
D: (6,10,2); (6,2,10); (10,2,6); (10,6,2); (2,6,10); (2,10,6)
D#/Eb: (7,11,3); (7,3,11); (11,3,7); (11,7,3); (3,7,11); (3,11,7)
E: (8,12,4); (8,4,12); (12,4,8); (12,8,4); (4,8,12); (4,12,8)
F: (9,1,5); (9,5,1); (1,5,9); (1,9,5); (5,9,1); (5,1,9)
F: (10,2,6); (10,6,2); (2,6,10); (2,10,6); (6,10,2); (6,2,10)
G: (11,3,7); (11,7,3); (3,7,11); (3,11,7); (7,11,3); (7,3,11)
G#/Ab: (12,4,8); (12,8,4); (4,8,12); (4,12,8); (8,4,12); (8,12,4)
(n-12 iff n>12)
Augmented: (n, n+4, n+8)
Harmonic: (n, n+2, n+3, n+5, n+7, n+8, n+11)
Melodic: (n, n+2, n+3, n+5, n+7, n+9, n+11) Ascending
Melodic: (n, n+2, n+3, n+5, n+7, n+8, n+10) Descending

The above algebraic expressions display the possible inversions for the triads of western harmony. This
system gives a simultaneous visualization of the possible triad chord values and inversions. D Major
Triads with Inversions, (figure 9)
D Major ii->V->I, (figure 10)

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