The Overtone Series 

Why do the Octave Tests Work? To answer this question, we must first understand some basic theory about sound. When a note is played on the piano, the strings vibrate. However, the strings don’t just vibrate at one length. They vibrate at different lengths simultaneously. This produces a series of overtones.

The overtone series for most musical instruments, including the piano, is as follows:

The overtone series keeps going, but if you can commit the first six partials to memory, then you should be just fine. Let’s look at an example. The overtone series of A2 would look like this:

Coincident Partials 

If you play two notes at the same time, then each of the notes in that interval will have its own series of overtones. At a certain point, the two series will have a shared overtone. We call this overtone the coincident partial, because it is at this partial level at which the two overtone series coincide. Let’s look at an example. The overtone series of a major third from C3 to E3 would look like this. E5 (highlighted in green) is the coincident partial.

To drive this concept home even further, this diagram shows what this looks like on a keyboard.

So how does this relate to piano tuning? The answer is that beats occur at coincident partials! This video is an excerpt from On Pitch Companionship DVD Volume I: Octave Types & Test. In it, a filter is placed over the coincident partial to highlight the beating. If you think this is cool, then you should definitely buy the DVD!

Octave Types & Tests

With this foundation in place, what happens with octaves? The overtone series of the two notes in an octave are so similar. This means that octaves have many coincident partials and yes, each partial level has its own beat rate. In the image below, C2 to C3 are played (as indicated by the “x” markings). The notes highlighted in blue show the different coincident partial levels.

Each coincident partial level determines a different octave type. Thus, the Octave Types are:

2:1, 4:2, 6:3, 10:5 and 12:6

The most common being 2:1 (used in the Treble), 4:2 (used in the Midrange) and 6:3 (used in the Bass).

Each Octave Type has its own Octave Test. The most important to commit to memory are:

2:1 Octave Test: M10th-M17th
4:2 Octave Test: M3rd-M10th
6:3 Octave Test: m3rd-M6th

Why Do These Tests Work?

The answer lies in the coincident partials of the test intervals used. Let’s look at a 6:3 Octave from A2 to A3. Looking at the image below, we see that the coincident partial at the 6:3 level is E4.  

When using the 6:3 octave test (m3rd-M6th), the test note is a m3rd above the bottom note of the octave and a M6th below the top note of the octave. So in our A2 to A3 example, the test note would be C3. Notice anything about where the coincident partial for C3 lands when played as either a m3rd with A2 or a M6th with A3? Yes, the coincident partials of the test intervals coincide at the octave’s 6:3 partial level! 

The same is true for the other octave tests. In each case, the coincident partial of the test intervals is the same as the corresponding partial level for that octave type. On Pitch dedicates an entire chapter to this subject and in the book’s conclusion summarizes, “By manipulating the overtone series of three notes that have a common coincident partial, we can aurally isolate beating and tune at a particular partial level.” In other words, this is why the Octave Tests work.

-Jason Cassel, RPT