Why Does This System Work?
Tune – the Octave
After a temperament has been set, you can then use this system to aurally tune octaves up and down the keyboard. The first step is to simply tune the octave so that it sounds acceptable. This is important because if not done first, then the test intervals used in the Refine Step will not work. For example, in the midrange, if the top note of the octave is not relatively close, then both the fourth and the fifth will beat too much. When this happens, these test intervals lose all meaning. So before anything else can be done, tune the octave so that it sounds acceptable.
Refine – with Slow Beating Intervals
When the octave sounds acceptable, this does not mean that the octave is necessarily tuned correctly. The octave could be narrow, or it could be that other intervals in the piano require the octave to be tuned with more beating or less beating. This is what the Refine and Confirm Steps are for. To understand why the test intervals used in the Refine Step work, we must first acknowledge that the demands of Equal Temperament require some intervals to be tuned wide of pure and other intervals to be tuned narrow of pure. In the Refine Step, one of the test intervals is tuned narrow (the fifth, or the octave and a fifth) and the other is tuned wide (the fourth or the double octave). Because of this, the two intervals can work together to narrow in on a target.
Let’s look at an example. Let’s say we are tuning the octave from A3 to A4. Because we are in the midrange, we would be using a fourth (from E4 to A4) and a fifth (from D4 to A4) to refine this octave. To sound acceptable, the fourth would need to beat around 1 beat per second and the fifth would need to beat close to pure. The measurements in the charts below were taken on a mid-sized grand piano. The exact cent deviations will differ from one piano to the next, but the concepts will remain the same.
In tuning this piano, I determined that the best placement for A4 would be what is highlighted green in the chart below. At that target, the fifth would beat 0.36 times per second and the fourth would beat 0.92 times per second.
Now, if A4 was raised 0.7 cents from the target, then the fifth would become pure. This means that fifth would sound nice, but let’s look at the fourth. At 0.7 cents above the target, the fourth would beat 1.45 times per second, around a half a beat more than the ideal. Thus, the beating in the fourth would tell us to lower A4. If A4 was lowered 1.2 cents from the target, then the fourth would become pure. Because we are used to listening to fourths tuned around 1 beat per second in the midrange, this pure fourth wouldn’t sound quite right. What’s more, is that the fifth is now beating 0.97 times per second, which is not acceptable either. In this case, it would be the beating in the fifth that would tell us to raise A4.
Can you see how the fourth and the fifth are pushing us toward a center? If we tune the note too sharp, then the fourth will start to beat too much. If we tune the note too flat, then the fifth will start to beat too much. When we arrive at the center, then both intervals will be satisfied at the same time and we have successfully refined the placement of the top note of the octave. In the completed chart below, I highlighted in yellow an area of acceptability. I did so by seeing how sharp I could tune A4 before the fourth started to become offensive and how flat I could tune A4 before the fifth started to become offensive. The range ended up being around ±0.4 cents from the target.
The challenge of course, is learning to hear these Refine intervals correctly. Slow beating intervals can be difficult to distinguish at times. This is especially true of the fifth as it often has two beat rates that are audible. Fortunately, if you are listening to the wrong beat rate in the fifth, then the fourth can reign you back in, as it will start to beat too much. Once the octave, the fourth and the fifth all sound acceptable, then the top note of the octave has most likely been placed correctly. To confirm this, we will use fast beating intervals.
Confirm – with Fast Beating Intervals
There are two types of fast beating interval checks used in this system. The first is the corresponding octave test for the region of the piano in which you are tuning. 4:2 octaves are used in the midrange, 2:1 octaves are used in the treble and 6:3 octaves are used in the bass. I have a whole article on why these octave tests work (click here to view), so we won’t go into those tests here. The second test is a series of parallel intervals, such as M6ths, M10ths or M17ths. The beat rate of each should increase or decrease evenly when played chromatically.
For example, I measured the beat rate of the M10th from E3 to G#4 at a rate of 6.55 beats per second, and I measured the beat rate of the M10th from G3 to B4 at a rate of 7.35 beats per second. For the progression to be nice and even, the beat rate of the M10th from F3 to A4 would need to fit perfectly in between those two beat rates, which comes to 6.94 beats per second. That is our target.
As you can see in the chart below. If A4 was tuned 0.8 cents above the target, then the resulting beat rate of the M10th from F3 to A4 would be 7.35. Which is the same as the beat rate of the M10th from G3 to B4. The even progression would be disturbed. Similarly, if A4 was tuned 0.8 cents below the target, then the M10th from F3 to A4 would be 6.53. Which is essentially the same as the bear rate of the M10th from E3 to G#4.
Once again, I created an area of acceptability by seeing how sharp and how flat I could tune A4 before the progression no longer sounded even. To my surprise, the tolerance was again around ±0.4 cents from the target. Of course, Refining with slow beats and Confirming with fast beats will ensure that your tuning is as close to the target as possible.
I hope that you have found this explanation to be helpful. With the DVD presentation, I intentionally took a practical approach and avoided including the theoretical reasoning behind the tests. This was done to simplify the system and to keep the DVD from being too long. Even still, I wanted to ensure that access to the theoretical reasoning was made available through this website.
All the best,
Jason Cassel, RPT