As this is the third post of a series of a series about Just Intonation, let's talk about thirds.

Stacking Thirds

Let's start by stacking major thirds on top of each other. The most simple ratio that represents a major third is 5/4:

Note that the ratios are frequency ratios instead of length ratios.

Like in the last post, we are multiplying powers of harmonic partials, in this case 5 and 2, which is the same powering thirds:

I prefer the combination of powers of natural numbers, as this expresses more clearly what we are doing here: combining harmonic partials.

Observation:

We can clearly see and hear a pattern: There are 3 distinct pitches, which repeat in a detuned way. On strings 4 and 7, we arrive at a pitches that are really close to octaves of the fundamental frequency.

This makes sense if we compare that to standard western tuning, where 3 major thirds fit inside one octave.

Diesis

This little "error" between 3 stacked thirds and 1 octave is called Diesis:

This is similar to the pythagorean comma, which is the little "error" between 12 stacked fifths and 7 octaves (see last post).

5 Limit Tuning

Now that we have a little knowledge of thirds, we can look at 5 limit tuning, which combines pure fifths and thirds (powers of 3 and powers of 5) to generate ratios. This means that for each number of fifths, we can add any number of thirds. As a formula, this means:

Like in the last post, we will use octave reduction to automatically set the exponent of 2 to keep the end result between 1 and 2. Having that out of the way, we have free control over the two exponents b and c.

So 5 limit tuning brings a whole new dimension to our system: The first dimension is the 3 (fifths) and the second, new dimension is the 5 (thirds).

Naming collision

It may sound wrong that the 3 stands for a fifth, while the 5 stands for a third, but this just stems from the naming convention of musical intervals, based on 7 notes per octave. Funny enough, the next prime numbers are 7, which adds the minor seventh, and 9, which adds the major second (or 9th). But let's stay with 3 and 5 for now.

Tuning Circle

In contrast to the Tuning Circle of the last post, where we had a one dimensional table of ratios, here we have a two dimensional grid.

• The fifths stack in vertical while thirds stack in horizontal direction.
• From each field, the fields below and above are pure fifths apart, while the fields to the left and right are pure major thirds apart.
• The bounds for both grid dimensions can be adjusted with the sliders on the left
• The rotation for each dimension can be adjusted with the sliders on the right. It controls the offset of the exponents in that direction.
• You can switch between different units to change the representation type of each ratio
• Each ratio can be disabled/enabled by clicking. This will gray it out and also hide it in the circle.
• You can pick a preset to load different 5 limit scales

5 limit vs 3 limit

With 5 limit, we can simplify many of the complex ratios of 3 limit. This results in much purer intervals. The downside is, that it adds complexity by introducing another dimension.

Also, some intervals can be reached in multiple ways, and the right choice depends on the use case. For example, the the sixth can either be 27/16 or 5/3. The first one seems more complex, but if we wanted to build a chord on the second degree of the scale (9/8), we should choose that one. But this is a topic for another post!