The Width of the A3/A4 2:1, After Tuning A3/A4 as a Pure 4:2
Knowing the width of the A3/A4 2:1 after it has been tuned as a pure 4:2 is a useful and interesting piece of information about the piano being tuned.
Once the A4 numbers are known, simply tune A3 to the 2nd Partial of A4, and then measure the width of the A3/A4 as a 2:1.
It’s interesting how many different A3/A4 2:1 widths a pure A3/A4 4:2 will create. I’ve measured the A3/A4 2:1 width on a small Knabe grand (1982) to actually be narrow when it was tuned as a pure 4:2! And on other pianos I’ve measured that 2:1 to be 3.3 c. wide! And on some pianos the 2:1 will be pure.
I believe Dan Levitan mentions this in his book and I think he refers to this phenomena as something that happens on a ‘well scaled’ piano.
The first time I ran into it, I didn’t know anything about Dan’s comments and so I ended up measuring and re-measuring, tuning and re-tuned everything 3 or 4 times.
But there it was. The piano was a 2008 or so Yamaha P660S (a studio sized piano with a very nice cabinet).
A pure 2:1 here can be a bit of a problem, because it can cause A3 to need to be lowered more in order to get the prime 5ths to fit within the prime octave. After lowering A3 to accommodate the prime 5ths, both A3/A4 4:2 and A3/A4 2:1 can end up beating similarly, which may be more 4:2 beating than what might be considered the ‘usual’. But of course this is very piano dependent.
Most of the time, on Steinway Grands in particular, I find the A3/A4 2:1’s (when tuned as a pure 4:2) to be less wide than on most other brands. And on those pianos, this only slightly wide or even pure 2:1 is not a problem. So again, it is very piano dependent.
On the other side of the scale, I’ve measured 2:1’s as wide as 3.7 c. when the 4:2 was pure. What is very interesting to me about those pianos is that most of the time on those, the prime 5ths wouldn’t let me contract the 2:1 much at all – maybe only .5 c. or so. If I try to contract the A3/A4 much at all, the prime 5ths (A3/E4 & D4/A4) would start to get too narrow and start beating too fast.
I might only be able to contract it by .5 c. or so. On those pianos the A3/A4 4:2 ends up actually being narrow.
In the more normal situations, where the pure 4:2 creates a .9 c. – 1.7 c. wide 2:1, most of the time the location of A3 won’t need to be moved too far away from it’s pure 4:2 location to get the 5ths to sound good and fit within the A3/A4 prime octave. And the 4:2 will not beat as much as the 2:1.
When the pure A3/A4 :2 creates a pure A3/A4 2:1, in order to get the prime 5ths to work and sound good, A3 needs to be lowered more than usual and creates similarly wide 4:2 and 2:1 A3/A4 octaves.
Sometimes on some pianos, a pure A3/A4 4:2 resulting in a pure A3/A4 2:1 requires some larger compromises when mapping both A2 and A5.
And it’s all related to A3 and the width of the A3/A4 2:1 when A3/A4 is tuned as a pure 4:2.
Extra lowering of A3, will usually necessitate a larger lowering or more stretch needed in the A2/A3 octave.
A5: Extra or excessive lowering of A3 can require a bigger compromise for placing A5.
Both the A4/A5 2:1 and the A3/A5 4:1 should be wide. But with A3 already being a bit on the low side, the A3/A5 4:1 can easily be wide even when the A4/A5 is almost pure!. In order to get both the A4/A5 and the A3/A5 wide, sometimes A5 needs to be raised up quite a bit, which can make the A3/A5 4:1 beating almost too fast.
For the A4/E5, and the D5/A5 5ths to sound good, A5 needed to be sharper which can put pressure on the A3/A5 double octave.
Most of the time a compromise can be found but it is going more of a compromise than if A3 didn’t need to be lowered so much to accommodate good 5ths in the prime octave.
So is this a well scaled piano?
But I’ve found most Steinway’s do have a less wide A3/A4 2:1’s when their A3/A4 4:2 is pure. Some Steinway’s A3/A4 2:1’s will measure pure, but most of them I’ve tuned are in the 0.0 – .8 c. range, which is definitely in the less wide range as far as the pianos I’ve tuned.
I remember years and years ago being taught that the prime 4:2 should always be tuned wide, period. I think that’s true for Steinway, since many Steinway’s A3/A4 4:2’s will measure only .3 to .8 c. wide as a 2:1. And, for some reason they are not hard to deal with at all.
Not at all like that Yamaha P660S. I don’t know why.
Having learned about the 4:2 and the 2:1 widths of the Prime octave, I sympathize for those trying to learn to tune a temperament without being able to check this. Most of the time, they are taught to tune A3/A4 s a wide 4:2, with the F3/A3 M3rd beating slightly slower than the F3/A4 10th.
That will work on Steinways, but it may not work on every piano. Much of the time on smaller scales, the A3/A4 4:2 needs to be narrow. Which means the F3/A3 M3rd should beat slightly faster than the F3/A4 10th. And on some pianos the F3/A3 M3rd and the F3/A4 10th should beat the same!
Many guys just starting will end up learning to tune on a spinet or some other short scale. Trying to tune a wide A3/A4 4:2 on a scale that really needs a slightly narrow 4:2 or or even a pure 4:2, can make that task even more difficult than it already is.