Purpose:
To discover relationships between chord qualities (major, minor, diminished, and augmented) when a numerical designation system of key naming (and not the letter name system) is employed.

Problem:
What patterns, if any, exist in the common chord qualities under a numerically devised system of naming keys?

Introduction to the Number Systems
 Letter names are customarily designated to each key on a piano, and form a consistent pattern over and over from one end of the piano to the end:
C, C sharp, D, D sharp, E, F, F sharp, G, G sharp, A, A sharp, B…….C, C sharp, D, etc.

If a number system substitutes the letter naming system above, patterns in the extant chord qualities (major, minor, and diminished/augmented) can be easily conjured up.  In each chord quality, the same procedure of adding the individual numbers designated to the keys that make up the complete chord quality decide the final number, ‘x’.  For major keys, ‘x’ is one of three distinct numbers; for minor, ‘x’ is one of three other distinct numbers; and for diminished/augmented chords, ‘x’ is one of three other distinct numbers.

 The number system, derived directly from the letter names designation system, similarly designates a number to each key on the piano in a similar pattern:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12……1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12…1, 2, 3, 4, 5, etc.
Assign “1” to any key, and follow the same pattern as above with the key-naming system.  For example, if “C” is designated with the numerical value “1”; C sharp will be designated as “2”; D, “3”; D sharp, “4”; E, “5”; F, “6”; F sharp, “7”; G, “8”; G sharp, “9”; A, “10”; A sharp, “11”; B, “12”.  Number designations are defined by ‘x’, and are the summation of the numerical values of the three notes that make up the chord.

Patterns of the Major Quality Chords

 Major quality chords have the regular pattern of a four half-steps (four letter names or four numbers) between the bottom and middle notes; and three half-steps (three letter names or three numbers) between the middle and top notes.  C major, for example, consisting of the letters C-E-G, would have a number designation of ‘x’=2(“0” + “4” + “7” =”11”=“1”+”1”=”2”).  Assuming that “C” has a numerical value of “1”, the major keys would form the following numerical pattern of ‘x’ values:

Major Key  Summation   ‘x’
C                 1 5 8             5
C sharp        2 6 9            8
D                 3 7 10          2
D sharp       4 8 11           5
E                 5 9 12           8
F                 6 10 1           8
F sharp       7 11 2           2
G                 8 12 3          5
G sharp       9  1  4          5
A                10 2 5           8
A sharp      11 3 6           2
B                12 4 7           5
The major mode is characteristic of having ‘x’ values of “5”, “8”, and “2”.

Patterns of the Minor Quality Chords

Minor quality chords, on the other hand, have three half-step intervals (three letter names or three numbers) between the bottom and middle notes, and four half-step intervals (four letter names or four numbers) between the middle and top notes.  The C-minor chord, for example, consisting of C-E flat(or D sharp)-G, summation of “1 4 8”, has an ‘x’ value of “4”.  Assuming that C has a numerical value of “1”, the following pattern results from the minor quality chords:

Minor key  Summation  ‘x’
C               1 4 8            4
C sharp      2 5 9            7
D               3 6 10         1
D sharp     4 7 11         4
E               5 8 12         7
F               6  9 1          7
F sharp      7 10 2        1
G               8 11 3        4
G sharp     9 12 4        7
A              10 1 5         7
A sharp     11 2 6        1
B               12 3 7        4
 The ‘x’ values that characterize the minor mode, deduced from this chart are “1”, “4”, “7”.

Patterns of the Augmented Quality Chords
 Augmented chords have four half steps (four letter names or four numbers) between the low and middle notes; and four half steps (four letter names or four numbers) between the middle and top notes.  C augmented chord consists of C-E-G sharp, equivalent to the summation “1 5 9” or the ‘x’ value of 6.  The following chart portrays the numerical pattern for augmented chords, assuming that C has a value of “1”.

Augmented Key  Summation  ‘x’
C                           1 5 9           6
C sharp                 2 6 10          9
D                           3 7 11         3
D sharp                 4 8 12         6
E                           5 9 1           6
F                          6 10 2           9
F sharp                7 11 3           3
G                         8 12 4           6
G sharp               9 1 5             6
A                         10 2 6          9
A sharp               11 3 7          3
B                         12 4 8          6

Augmented chords all share ‘x’ values of 3, 6, and 9.

Patterns of the Diminished Quality Chords
 Diminished chords have intervals of three half steps (three letter names or three numbers) between the bottom and middle notes; and three half steps (three letter names Cor three numbers) between the middle and top notes.  C diminished chord, for example, is C-E flat-G flat, summation “1 4 7”, and therefore, has an ‘x’ value of “3”.  Assuming once again that C has a numerical value of “1”, diminished quality chords share the following patterns:

Diminished Key  Summation  ‘x’
C                          1 4 7           3
C sharp                  2 5 8          6
D                           3 6 9         9
D sharp                 4 7 10        3
E                           5 8 11        6
F                          6  9  12        9
F sharp                7 10 1           9
G                         8 11 2           3
G sharp               9 12 3           6
A                         10 1 4          6
A sharp               11 2 5          9
B                         12 3 6          3

Diminished chords, as with the augmented chords, all share ‘x’ values of 3, 6, and 9.

Conclusion:
 Under the number-based designation system of naming keys, the quality of the chords can be identified by the ‘x’ value.  Major chords have ‘x’ values of 2, 5, 8; minor chords, of 1, 4, or 7; and diminished and augmented chords share ‘x’ values of 3, 6, or 9.