The Architecture of Music

Organizing All The Scales

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How Many Scales Are There?

According to Joe Walker, there are 2,048 possible modes expressed by the simple and elegant equation 2^11. Joe further breaks down the total number of chords and scales possible based on the number of notes in each scale. When you add them all up you again get 2,048. We already know a couple of the numbers. We know there is (1) one note scale with one mode. And we know there is one 12-note scale with 12 identical modes (the Chromatic Scale). So again, there is just one unique 12-note scale. We know there are 11 possible two note chords because there are only 11 possible notes in between an octave to choose from and each one of those notes represents a unique dyad. We also know there is one symmetrical 11-note scale with 11 unique modes. Calculating any further than there though takes some pretty interesting and complicated math (from Joe Walker's website).

2^11 = 2,048

1-note scales = 11! / (11! * 0!) = 1
2-note scales = 11! / (10! * 1!) = 11
3-note scales = 11! / (9! * 2!) = 11 * 10 / 2 = 55
4-note scales = 11! / (8! * 3!) = 11 * 10 * 9 / (3 * 2) = 165
5-note scales = 11! / (7! * 4!) = 11 * 10 * 9 * 8 / (4 * 3 * 2) = 330
6-note scales = 11! / (6! * 5!) = 11 * 10 * 9 * 8 * 7 / (5 * 4 * 3 * 2) = 462
7-note scales = 11! / (5! * 6!) = 462
8-note scales = 11! / (4! * 7!) = 330
9-note scales = 11! / (3! * 8!) = 165
10-note scales = 11! / (2! * 9!) = 55
11-note scales = 11! / (1! * 10!) = 11
12-note scales = 11! / (0! * 11!) = 1

Total Possible Chords and Scales: 2,048

 

Verifying the Math

To verify Joe Walker's math I created a 3-note chord matrix. Although (110) 3-note chords were generated. Upon further inspection of the matrix, it became clear there was an axis of symmetry that runs diagonally through the matrix with identical chords on either side of it meaning their formulas matched exactly. When all the unique 3-note chords are tallied we end up with (55) total unique 3-note chords matching exactly with Joe Walker's math above. The holes in the matrix were 2-note chords that were generated and were removed because they were not 3-note chords. What is interesting is almost half of all the possible 3-note chords are the root position chords and their inversions indicated by the grey boxes. The remaining 28 chords are fifth-less chords.

3-Tone Chord Matrix

Documenting and Organizing 350-400 Scales

2,048 represents total possible unique modes. However, because of modes, we can divide the numbers above by the number of notes in each scale to get an approximate number of scales we would need to document and organize that generate the total 2,048 modes.

1-note: 1/1 = 1
2-note: 11/2 = 5.5 (6)
3-note: 55/3 = 18.333 (19)
4-note: 165/4 = 41.25 (42)
5-note: 330/5 = 66
6-note: 462/6 = 77
7-note: 462/7 = 66
8-note: 330/8 = 41.25 (42)
9-note: 165/9 = 18.333 (19)
10-note: 55/10 = 5.5 (6)
11-note: 11/11 = 1 (Symmetrical 11-Tone)
12-note: 12/12 = 1 (Chromatic [12 identical modes])

Total Number of Root Chords and Scales: 337.313 (346)

Documenting and organizing 2,048 modes is a somewhat impossible task. However, documenting and organizing 350-400 scales is more manageable and accomplish-able if one has the time and inclination to do it.

 

Fractional Chords and Scales?

As can be seen above, when we divide the number of modes by the number of notes we sometimes get fractional numbers. How is this possible? How is it possible to have a fraction of a scale? The fractional numbers have to do with symmetry in scales that have modes that are identical to the original scale. To show how fractional scales work I diagrammed the eleven 2-tone chords (dyads) and their inversions. As can be seen below, all the 2-tone chords have unique inversions except for the 6 6 chord which would have an identical inversion. When added up, there are five chords with five unique inversions and one chord without any unique inversions making a total of (11) 2-tone dyads and six chords that generate eleven total unique inversions. What is really interesting about the 2-tone dyads and their inversions is the amount of symmetry that exists. Every dyad's inversion is the negative chord of the original and vice versa. As can be seen below, the 6 6 chord sits on the mirror line because it is symmetrical and its negative chord would be itself.

To understand how (55) 3-tone chords divides down to 18.333 generating chords I again diagrammed and organized all the 3-tone chords by root chords, inversions, and symmetry. As can be seen in the chord inversions diagrams in the chords section of this tutorial, all the 3-tone chords have two unique inversions except for the augmented triad (4 4 4) which would have two identical 4 4 4 inversions. If you count them all up, there are 18 groups of inversions and 1 chord without any unique inversions giving us 19 chords that generate 55 unique chords. What fractional numbers indicate is there are symmetrical scales that have modes that are are identical to the original scale.

However, scales that have identical modes isn't always indicated by a fractional number of generating scales. The Whole-Step Scale is a 6-tone scale with six identical modes but the total number of generating six tone scales is not a fractional number. This most likely means there are other symmetrical 6-tone scales where all the modes within the scale are not unique and when added together gives us a whole number of generating scales. This, of course, makes things more complicated if you are trying to document and organize all 2,048 modes.

Musically wise symmetry doesn't mean much. We don't listen to music and describe it as being symmetrical. However, when you are trying to document, diagram, organize, and explore every possible scale, symmetry plays a big part as will soon be seen. Ian Ring has done some phenomenal work on documenting symmetry in scales and his work will prove to be an invaluable tool in trying to figure out exactly how many scales should be on the greater organization of all scales chart that generate all 2,048 modes.

2-toned Chords (Dyads) and Inversions