# From Ratios to Scales

The next step as we continue our journey towards understanding the world of Microtonal music and Microtonal Guitar is understanding how musical scales are generated using the musical ratios derived from the harmonic series. One important concept to clarify before we proceed is the prime numbers and prime limits. A prime number is a number that is only evenly divisible by itself and 1.  A prime limit is the highest prime number that is being used in the generation of a musical scale derived from the harmonic series.  Let's look at the scale interval and ratios in a table:

Musical Ratio Musical Interval
1/1 Root
2/1 Octave

Just Intonation is the term used for a system of tuning when creating scales using the intervals generated from the harmonic series.  We'll get into a more definitive understanding of Just Intonation later but for now let's just stick with this basic definition.

## 3 Limit Diatonic Major Scale

A 3 limit tuning system is a tuning system that used the prime number of 3 as it's basis.  A typical diatonic scale in the three limit system is based on the first 5 notes based on the circle fifths, the first note from the circle of 4ths and the octave.

Note for the 9/8 interval we have changed the ratio from 9/4 (3/2 * 3/2) to 9/8 based on the concept of octave reduction discussed earlier. View the table below to see this converted to a diatonic scale (with the added octave as the end of the scale) starting with C 130.8 hz.

Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.15
81/64 Major 3rd E 165.56
4/3 Perfect 4th F 174.41
3/2 Perfect 5th G 196.22
27/16 Major 6th A 220.74
243/128 Major 7th B 248.33
2/1 Octave C 261.62

## 5 Limit Diatonic Major Scale

The most common 5 limit major diatonic scale shares the major 2nd, perfect 4th, and perfect 5th from the 3 limit major diatonic scale and replaces the Major 3rd, Major 6th, and Major 7th with notes based on the 5 limit ratios.

Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.15
5/4 Major 3rd E 163.51
4/3 Perfect 4th F 174.41
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 261.62

## Nature's Scale

Another common scale you will come across when being introduced to scales derived from the harmonic series is "Natures's Scale". This scale is the first full scale that is encountered in the harmonic series.  It consists of ratio's and notes based on the 8th through the 16th harmonics.  This scale features ratio's derived from prime numbers 2,3,5,7, 11, and 13, so is considered a 13 limit scale. Note there isn't a perfect 4th in the scale. The scale is charted out below:

Musical Ratio Musical Interval Note Hz Value
1/1 Root C 130.81
9/8 Major 2nd D 147.15
5/4 Major 3rd E 163.51
11/8 Flatted 5th Gb 179.86
3/2 Perfect 5th G 196.22
13/8 Nuetral 6th AB 212.56
7/4 Flatted 7th Bb 228.92
15/8 Major 7th B 245.27
2/1 Octave C 261.62
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