The 4th A4 Number
The ‘4th A4 Number’ is the width of the prime’s 2:1 after it has been tuned as a pure 4:2.
When the 2nd Partial location of A4 is known, its easy to tune the Prime Octave as a Pure 4:2. Simply tune A3 to the 2nd partial setting of A4.
After A3 has been tuned as a pure 4:2, here’s how to measure the prime as a 2:1 :
1.) set the SAT to A4
2.) play A3
3.) stop the lights using with the cents buttons.
The width of the 2:1 is the “4th A4 Number”.
The ‘4th’ A4 number is the first step in mapping the piano. I make this 4th A4 number measurement right after I have the A4 numbers. Knowing this 4th A4 number from the very beginning lets me know immediately what I’m going to be dealing with on that particular piano.
The ‘4th A4 Numbers’ vary quite a bit from piano to piano:
Pianos with 4th A4 numbers between 0.0 c. – 1.0 c.:
The prime octave on pianos in this 0.0 c. – 1.0 c. range almost always end up as wide 4:2s.
Yes, some pianos pure 4:2 prime octave ends up yielding a pure 2:1. I didn’t know that was possible until I came across one. The first one I found was a Yamaha Studio Console. I probably measured and remeasured it 3 or 4 times before I finally accepted what I was finding on that piano. Since then I have come across a few more, so they are out there.
Pianos with 4th A4 numbers between 1.1 c. – 2.4 c.:
Most pianos fall within this 1.1 c. – 2.4 c. range, and most of the time their prime 4:2s end up being either pure or wide.
Pianos with 4th A4 numbers above 2.8 c.:
This is where things get interesting. When the prime’s 2:1 is around 3.0 c., the beating starts getting too fast. If I have to, I can live with a 2.8 c. wide 2:1, but much higher than that and the beating in the 2:1 can be a bit too much.
I’ve measured pianos with 4th A4 numbers as high as 3.7 c.. Again, those octaves were all first tuned as pure 4:2s. So the offensive beating is all in the 2:1.
These high 4th A4 number pianos, can cause the prime’s 4:2 to actually end up being tuned narrow. Yes, on some pianos, the prime 4:2 needs to be narrow. Sometimes they end up being pure. On those pianos, not only can the prime 4:2 end up being narrow, but the prime 5ths will often need to be narrower as well. Whatever it takes to have a decent sounding octave and a pair of decent sounding prime 5ths.
(More on dealing with these situations will be discussed in the post on Mapping A3).
Since the rest of the tuning evolves out of the prime octave, getting the prime octave’s width correct sets up the rest of the tuning.
The A4 Numbers and the 4th A4 Number will indicate right away if the 1.) prime 2:1 less than 3.0 c. and 2.) prime 5ths @ -1.5 c. starting points will need to be modified.
4th A4 Number guidelines:
2:1 Octave width < .8 c. = Prime 5ths probably < -1.5 c. (narrow)
2:1 Octave width > .9 c. and < 2.5 c. = Prime 5ths probably @ -1.5 c.
2:1 Octave width > 3.0 c. = Prime 5ths probably > -1.5 c. (narrow)