From Ratios to Scales – Diatonic Major Scale

5 Limit C Major Diatonic Scale

The next logical progression in our journey to understand understand the underlying theory of microtonal guitars is creating a tuning system.  A tuning system is the way you notes or pitches on the instruments that you are playing are constructed. First we will create a tuning system using the 5 limit major scale for the 1, 4, and 5 chords to create the ratio's for all the intervals we will need. Let's look at the 5 limit diatonic major scale below:

Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.16
5/4 Major 3rd E 163.52
4/3 Perfect 4th F 174.42
3/2 Perfect 5th G 196.22
5/3 Major 6th A 218.02
15/8 Major 7th B 245.27
2/1 Octave C 261.76

F Major Diatonic Scale - C Major Ratio's

Next let's take this same scale and build it for the key of F, the 4th scale degree of the diatonic scale in the key of c.  Since our objective is to create a "combined" scale of these respective diatonic scales we need to multiply the ratio that is the root of the new scale by the ratio we want to generate for the scale, the result is the ratio in relation to the root of the parent scale (in this case the key of c).

Musical Ratio
for Diatonic Scale
Musical Interval
in Relation to Root
Interval Note Hz Value
4/3 * 1/1 4/3 Root F 174.42
4/3 * 9/8 3/2 Major 2nd G 196.22
4/3 * 5/4 5/3 Major 3rd A 218.02
4/3 * 4/3 16/9 Perfect 4th Bb 232.56
4/3 * 3/2 2/1 Perfect 5th C 290.69
4/3 * 5/3 10/9 Major 6th D 290.68
4/3 * 15/8 5/4 Major 7th E 327.03
2/1 4/3 Octave F 348.83

G Major Diatonic Scale - C Major Ratio's

Now let's do calculate the ratio's for the major diatonic scale for G and the ratio's in relationship to the root.

Musical Ratio
for Diatonic Scale
Musical Interval
in Relation to Root
Interval Note Hz Value
3/2 * 1/1 3/2 Root F 196.22
3/2 * 9/8 27/16 Major 2nd G 220.75
3/2 * 5/4 15/8 Major 3rd A 245.74
3/2 * 4/3 2/1 Perfect 4th Bb 261.62
3/2 * 3/2 9/8 Perfect 5th C 294.33
3/2 * 5/3 5/4 Major 6th D 327.03
3/2 * 15/8 45/32 Major 7th E 367.91
2/1 3/2 Octave F 392.44

C Combined Scale with F & G Major

Let's finish up this scale by adding the notes that aren't present in C Major from the F and G major diatonic scales, like we did in lesson 2, to create a "combined major scale".  Notice this is an 11 note scale and isn't a full chromatic scale.  This scale doesn't have any note/ratio's for a minor 2nd, minor 3rd or minor 6th. Also note how there are 2 different major seconds, 10/9 and 9/8, we will come back to this in feature writings.

Ratio Interval Note Hz Value
1/1 Root C 130.81
10/9 Major 2nd D 145.34
9/8 Major 2nd D 147.16
5/4 Major 3rd E 163.52
4/3 Perfect 4th F 174.42
45/32 Flatted 5th Gb 183.95
3/2 Perfect 5th G 196.22
5/3 Major 6th A 218.02
27/16 Major 6th A 220.75
16/9 Flatted 7th Bb 232.56
15/8 Major 7th B 245.74
2/1 Octave C 261.76
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