Overview of Basic Mathematical Concepts For Just Intonation

Just Intonation Defined

Just intonation  is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. Any interval tuned in this way is called a just interval. Just intervals (and chords created by combining them) consist of members of a single harmonic series of a (lower) implied fundamental.

While the below is common explanations of these terms I’m paraphrasing (or in a few cases directly copying) these definitions from David Doty’s Just Intonation Primer.

Musical Ratios

A music ratio is often used as musical intervals, the distance between two notes and to represent a specific pitch. For instance when talking octaves we use the ratio 2/1, when talking about a specific octave that ratio would then represent the pitch correlating to what it is an octave of.

Complementary Intervals

A complementary intervals is when you find what distance an interval from a given interval would equal an octave. Let’s review basic intervals real quick, to find a complement when thinking of intervals by scale degrees in equal temperament you take the quality of the interval, subtract from 9 and flip the interval. For instance if you have a major 2nd, it’s complementary interval is a Minor 7th, if you had a C to D that would be a major second and D to C is a minor 7th.

Using that same example with ratio’s, if you had a 9/8 major 2nd it’s complementary interval is 16/9.

Prime Numbers

A number that can only be divided by itself and 1
Examples: 2, 3 , 5 , 7 , 11, 13


The highest prime number that is used in the ratios that determine a tuning system. For instance if our tuning was just octaves it would be a 2 limit system, Pythagorean tuning which is based on the perfect fifth ratio of 3/2 is a 3 limit system and can have pitches from based on 2 and 3.

Primary Intervals

Primary intervals are of the form p:2n where p is the prime and 2n is greatest power of 2 less than p.

Primary intervals for the first 8 primes:


Adding Ratio’s

If you wanted to know what the interval added to another interval is you would multiply the two ratio’s. For example if we wanted to know what the ratio for the 5 Limit major 3rd, (5/4 ) above a perfect 5th of 3/2 we would multiply them together.


Subtracting Ratio’s

To figure out the distance between existing ratio’s take the first ratio (representing the lower frequency) and flip it upside down, then multiply by the ratio you are looking to find the distance of. Let’s look at an example, say we’re trying to determine the interval between 4/3 and 3/2
here’s how we would do that.

Tuning Lattice

A tuning lattice is a multidimensional diagram that represents a given tuning system. Each dimension represents one of the prime limits the system uses.

For example if we were to diagram a 5 limit system we can use the 1st, 3rd, and 5th harmonics (corresponding to the root, Perfect 5th and Major 3rds.

5/4 15/8 45/32 135/128
1/1 3/2 9/8 27/16
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