From Ratios to Scales to Tuning Systems – Combined Nature’s Scale

 C Nature's 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:
1/1 Root C 130.81 0
9/8 Major 2nd D 147.16 203.91
5/4 Major 3rd E 163.51 386.314
11/8 Neutral4th F# 179.56 551.318
3/2 Perfect 5th G 196.22 701.96
13/8 Neutral 6th Ab 212.57 840.53
7/4 Flatted 7th Bb 228.92 968.83
15/8 Major 7th B 245.27 1088.27
2/1 Octave C 261.62 1200

F Nature's Scale - C Nature's Scale 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).
4/3 * 1 4/3 Root F 174.41 0
4/3 * 9/8 3/2 Major 2nd G 196.22 203.91
4/3 * 5/4 5/3 Major 3rd Ab 218.01 386.31
4/3 * 11/8 11/8 Neutral 4th Bb 239.81 551.318
4/3 * 3/2 3/2 Perfect 5th C 261.62 701.96
4/3 * 13/8 13/12 Neutral 6th Db 283.42 540.53
4/3 * 7/4 7/6 Flatted 7th Eb 305.22 968.83
4/3 * 15/8 6/5 Major 7th E 327.02 1088.27
4/3 * 2/1 5/4 Octave F 348.82 1200

G Nature's 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.
3/2 * 1 3/2 Root G 196.22 0
3/2 * 9/8 27/16 Major 2nd A 220.75 203.91
3/2 * 5/4 15/8 Major 3rd B 245.275 386.31
3/2 * 11/8 33/32 Neutral 4th C 269.8 551.318
3/2 * 3/2 9/8 Perfect 5th D 294.33 701.96
3/2 * 13/8 39/32 Neutral 6th E 315.858 540.53
3/2 * 7/4 21/16 Flatted 7th Fb 343.39 968.83
3/2 * 15/8 45/32 Major 7th F 367.91 1088.27
3/2 * 2/1 2/1 Octave G 392.44 1200

C Combined Nature's 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.
1/1 Unison, Perfect Prime C 130.81 0
33/32 Undecimal Comma 134.90 53.27
16/15 Minor Diatonic Semitone C# 139.53 111.73
13/12 Tridecimal 2/3-Tone 141.71 138.57
9/8 Major Whole Tone D 147.17 203.91
7/6 Septimal Minor Third 152.62 266.87
6/5 Minor Third Eb 156.98 315.64
39/32 39Th Harmonic 159.43 342.48
5/4 Major Third E 163.52 386.31
81/64 Pythagorean Major Third 165.56 407.82
21/16 Narrow Fourth 171.69 470.78
4/3 Perfect Fourth F# 174.42 498.05
11/8 Undecimal Semi-Augmented Fourth F# 179.87 551.32
45/32 Tritone 183.96 590.22
3/2 Perfect Fifth G 196.22 701.96
14/9 Septimal Minor Sixth 203.49 764.92
8/5 Minor Sixth Ab 209.30 813.69
13/8 Tridecimal Neutral Sixth 212.57 840.53
5/3 Major Sixth B 218.02 884.36
27/16 Pythagorean Major Sixth 220.75 905.87
7/4 Harmonic Seventh Bb 228.92 968.83
16/9 Pythagorean Minor Seventh 232.56 996.09
11/6 Undecimal Neutral Seventh 239.82 1049.36
15/8 Classic Major Seventh B 245.27 1088.27
2/1 Octave C 261.62 1200

C Combined Nature's Scale with Fretboard

Below is a diagram of how these intervals are mapped out on a guitar fretboard tuned to Open D tuning using the above intervals as a map for the neck.

Scroll to Top