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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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