# Pure Intervals

tuning
music theory

April 22, 2020

This article is the first of a series about Just Intonation. Using pure (or just) intervals to build musical tuning systems is called "Just Intonation" or short JI. In the west, just intionation systems were mostly abandoned some time ago in favor of Equal Temperment (ET), which is another way to build tuning systems, using non-pure intervals.

In this post, we will understand what pure intervals are and presents different systems to organize them. I will try to be not influenced by current musical practice, only using musical interval terms as a reference.

## Ratios on a Monochord

A pure interval is a ratio of two pitches, that can be defined as a fraction of natural numbers:

The above monochords play a pure major triad (+octave). You can activate them by hovering over the strings or clicking/tapping on the circles.

### Length Ratios

The fractions on the left of the monochords above describe the relation of the vibrating length to the total length. If we wanted to build a real instrument with those intervals, we could calculate the lengths like that:

$L_1 = 100cm$

$L_2 = \frac{4}{5}*100cm = 80cm$

$L_3 = \frac{2}{3}*100cm = 66 \frac{2}{3}cm$

$L_4 = \frac{1}{2}*100cm = 50cm$

### Frequency Ratios

It can be confusing to describe the ratios as length fractions, as the length is antiproportional to the pitch: If the length gets shorter, the pitch goes up. Because of that, the standard way of describing pitch ratios is the inversion of the length ratio, which is the ratio of the frequencies:

With those ratios, we can calculate the frequencies like that:

$f_1 = 440hz$

$f_2 = \frac{5}{4}*440hz = 550hz$

$f_3 = \frac{3}{2}*440hz = 660hz$

$f_4 = \frac{2}{1}*440hz = 880hz$

## Ordering Ratios

As a general rule, we can observe that simpler ratios result in more consonant sounds, while more complex ratios are more dissonant. Consonance is often defined as pleasant and soft, while dissonance is unpleasant and edgy.

This invites us to think about putting ratios in a specific order to get insights of their nature.

### The Harmonic Series

The harmonic series is a natural way of sorting ratios. As already shown in my last post, the harmonic series is a collection of pitches that can be generated by multiplying a base frequency by natural numbers:

$\frac{2}{1} = \text{octave}$
$\frac{3}{2} = \text{perfect fifth}$
$\frac{4}{3} = \text{perfect fourth}$
$\frac{5}{4} = \text{major third}$
$\frac{6}{5} = \text{minor third}$
$\frac{7}{6} = \text{minor third}$
$\frac{8}{7} = \text{major second}$
$\frac{9}{8} = \text{major second}$
$\frac{10}{9} = \text{major second}$
$\frac{11}{10} = \text{? second}$
$\frac{12}{11} = \text{? second}$
$\frac{13}{12} = \text{? second}$
$\frac{14}{13} = \text{minor second}$
$\frac{15}{14} = \text{minor second}$
$\frac{16}{15} = \text{minor second}$

The fractions on the right show the ratios of consecutive harmonics + the name of the closest musical interval.

As you can see, there are many different major and minor seconds, as well as seconds that are somewhere in the middle. If you have no music background, don't worry much about the names.

Additionally, we can also form ratios between non-consecutive harmonics, like 5/3. This gives us a lot of options, which are difficult to visualize in this one dimensional manner.

### The Lambdoma

The Lambdoma arranges ratios in a two dimensional grid, using one dimension for the numerator and another for the denominator:

Size 9x9

#### Observations

• Extensions of simpler fractions represent the same interval
• All extensions lay on lines that go from 0/0 to infinity
• The vertical line is the fundamental ratio
• Ratios on the left fall below, while ratios on the right rise above the fundamental.
• Each diagonal line that goes down right follows the pattern of the harmonic series
• Each diagonal line that goes down left follows the pattern of the subharmonic series, which is a mirrored version of the harmonic series
• Ratios get more complicated as we go down
• Many of the intervals further down sound "out of tune", compared to standard musical intervals

### Calculating with Ratios

Before we explore further ways of ordering ratios, we need to look at basic calculation rules:

We can add ratios together by multiplying:

$\frac{3}{2}*\frac{4}{3} = \frac{12}{6} = \frac{2}{1}$

Here we add a fifth and a fourth together, resulting in the ratio for one octave higher. In length ratios, this would be inversed:

$*\frac{3}{4}$

#### Subtracting ratios

As opposed to adding by multiplying, we can subtract ratios by dividing:

$\frac{2}{1}:\frac{4}{3} = \frac{2}{1}*\frac{3}{4} = \frac{6}{4} = \frac{3}{2}$

If we subtract a fourth from one octave, we get a fifth. The same with length ratios

$:\frac{3}{4} = *\frac{4}{3}$

#### Stacking ratios

To stack a ratio n times, we can just use the power of n

For example, $\frac{3}{2}^{n}$ stacks n fifths

On a monochord, stacking 4 fifths would look like that:

$\frac{2}{3}^{0}$
$\frac{2}{3}^{1}$
$\frac{2}{3}^{2}$
$\frac{2}{3}^{3}$
$\frac{2}{3}^{4}$

#### Octave Reduction

In most cases, we are only interested in intervals that are inside one octave. Moving ratios inside that range is called octave reduction.

We can move any ratio by n octaves if we multiply it with $2^n$

For example, we could reduce the above stacked fifths into one octave like that:

$\frac{2}{3}^{0} * 2^0$
$\frac{2}{3}^{1} * 2^0$
$\frac{2}{3}^{2} * 2^1$
$\frac{2}{3}^{3} * 2^1$
$\frac{2}{3}^{4} * 2^2$

Note that with length ratios, the movement direction is reversed (like above, positive exponents move down).

#### Sorting ratios

Ratios that have been octave reduced can be sorted, to receive a harmonious scale:

$\frac{2}{3}^{0}*2^0$
$\frac{2}{3}^{2}*2^1$
$\frac{2}{3}^{4}*2^2$
$\frac{2}{3}^{1}*2^0$
$\frac{2}{3}^{3} * 2^1$

Here we can hear that 4 octave reduced stacked fifths result in a major pentatonic scale.

This is just the beginning, as there are more possibilities to organize ratios.

In the next posts, we will look at n-Limit tunings and compare just intonation systems to equal tempered ones.

Felix Roos 2022