Even the smoothest set of software-created numbers for the prime octave’s tuning ‘curve’, will often not be a good fit for the piano.
A low 4th A4 Number just means that when the prime octave is tuned as a pure 4:2, the resultant 2:1 is also pure or very close to pure.
This procedure is for pianos whose 4th A4 Number is generally less than 2.0 c. When the 4th A4 number is less than 2.0 c., the prime octave’s 2:1 can be either expanded or contracted – based on what the prime 5ths dictate – and still remain less than 3.0 c. wide. Most of the time on these pianos, the prime 4:2 octave will be either wide or pure.
My approach to midrange tuning is minimal beating. I don’t want to stretch any more than necessary to get a pair of good sounding prime 5ths contained within the prime octave.
Though not absolute by any means, there is a useful ‘relationship’ between the prime octave width and the widths of the prime 5ths.
This system uses the relationships of the prime octave (A3/A4) to the prime 5ths (A3/E4 & D4/A4) to determine the ideal widths for both the prime octave and the prime 5ths. This relationship will be found using the A4 Numbers the 4th A4 Number, and ‘Templates’.
The “A4 numbers” are the locations of the 2nd, 4th, and 8th partials of A4 when A4 is tuned to A440 (1st Partial).
Even though the width of the prime octave’s may be perfect, the piano’s scaling within that octave often results in a pair of ‘unbalanced’ prime 5ths: (A3/E4 & D4/A4)
The ‘4th A4 Number’ is the width of the prime’s 2:1 after it has been tuned as a pure 4:2. 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.
Balancing the prime 5ths involves tweaking the shape of the prime octave ‘curve’ by slightly raising or lowering the mid point of the prime octave curve, while leaving A3 and A4 unmoved. Since using D#4 is not practical to be used as a mapping note, D4 and E4 are used instead. D#4 can easily be placed between the settings of D4 and E4. Being able to apply this tweak to the tuning is a unique feature of this system of mapping with templates and then using the LC Spreadsheet to create the tuning.
The prime 5ths are used to determine the best prime octave width for the particular piano being mapped and eventually tuned. When the widths of the prime 5ths are added together, when that total is -3.0, the prime octave width is about right.
For years and years, I have used the triple octave location for A7. It’s not a bad spot either. But since March 2020, I’ve begun tuning A7 higher than that. I’ve started using the triple octave +5th as a starting point for my A7 location.
Once the location for A2 is known, mapping A1 is very straightforward. A good location for A1 is as a 6:3 octave (from A1), 3.0 cents wide.
Mapping A0 can be done any number of ways. It can be mapped however you like, but just like A1, once you have it where you want it, it’s location must be measured using it’s 4th partial, since the LC Spreadsheet uses the 4th partials for A0 – A4.
When talking or writing a lot about tuning, descriptive shortcuts are inevitable. Referring to the A3/A4 octave as the “Prime” octave came about as one of those shortcuts. Soon after that, the term ‘Sub-Prime’ was used to describe or identify the octave below the prime octave. The A2-A3 octave.
Mapping (and tuning) notes during the mapping process is the equivalent of setting the temperament in aural-only tuning.