WHEN we have practiced good actions awhile, they become easy; when they become easy, we take pleasure in them; when they please us, we do them frequently; and then, by frequency of act, they grow into a habit. –Tillotson
Mapping A3 establishes the width of the prime octave.
Like setting the temperament, finding the best position for A3 is the basis for the tuning.
A good way to determine the width of the prime octave is by using the widths of the prime 5ths (A3/E4 & D4/A4). The prime octave width will vary from piano to piano but the widths of the prime 5ths are fairly consistent.
The prime octave needs to be wide enough for the 5ths to sound good, but not so wide as to create excessive beating in the prime 4ths, or in the prime octave itself.
The tuning will sound better too, if the prime 5ths are ‘balanced’, which means the widths of those prime 5ths are the same width.
As in other posts on this subject, my suggested width for the prime 5ths is -1.5 c. (1.5 c. narrow). When the prime 5ths are that width, the A3/A4 octave and the resultant prime 4ths (A3/E4 & E4/A4) will sound pretty good.
Being able to map A3 using the prime 5ths requires a template with a reasonably consistent curve between A3 and A4. As long as there are no partial changes within that A3/A4 octave, almost any ‘template’ can be used.
An FAC tuning or other tuning curve can be used to tune A3, D4, E4, and A4. Once those notes are tuned the prime 5ths can be measured. (When using an FAC tuning, DOB could be used to shrink or contract the A3/A4 octave).
At this point, it’s the combined total of the widths of the two prime 5ths, that is used to determine the placement for A3. For a pair of prime 5ths, each being -1.5 c. narrow, the combined total should be -3.0 c.
If the combined widths of the two prime 5ths equals -2.0, the octave is too wide – A3 is too low by about 1.0 c.
If the combined widths of the two prime 5ths equals -4.0, the octave is too narrow – A3 is too high by about 1.0 c.
Now that we know the prime octave is too wide or too narrow, and by about how much, it is easy to select a different template with a higher or lower A3 setting.
That’s really all there is to mapping A3 using templates. Using the prime 5ths and a template in this way will result in a nice sounding prime octave for that piano.
For the tuning, the A3/A4 2:1 should always be wide. But sometimes the A3/A4 4:2 will end up being a pure 2:1. And, there are times when the A3/A4 4:2 needs to be narrow!
Here are some guidelines for what to do based on the width of the A3/A4 2:1 when tuned as a pure 4:2:
1. If the A3/A4 2:1, is between 0.0 c. and .8 c., the A3/A4 octave will probably need to be widened ( by lowering A3).
2. If the prime octave 2:1 is between 0.8 c. and 2.5 c., the pure 4:2 might be a good width for the prime octave. Measuring the 5ths will indicate any adjustment that needs to be made to the width of the A3/A4 octave..
3. When the prime octave 2:1 gets close to 3.0 c. wide, the 2:1 may start beating too much. On those pianos, A3 may need to be raised, thus resulting in a slightly narrow A3/A4 4:2.
Most of the time on those pianos, the 5ths just won’t let me contract the octave much at all. Maybe only .5 c. or so, which means nothing to the 4:2 but does help a little with the 2:1. The 5ths in there just seem to need a wider octave to maintain their -1.5 c. width. Go figure. But the prime 5ths remain the best guide.
Balancing the prime 5ths involves moving the mid point of the prime octave up or down.
When the prime 5ths are ‘balanced’, they are the same width. Balancing the prime 5ths is possible using the LC spreadsheet.
When mapping the prime octave, the prime 5ths can be measured and those settings for D4 and E4 that give us a pair of ‘balanced’ 5ths can be written down. Even thought the mid point in the prime octave is D#4, when creating the tuning using the LC spreadsheet, the settings for D4 and E4 are used to tweak the position of D#4, thus raising or lowering the midpoint of the curve within the prime octave curve.
(Moving the midpoint in the A2/A3 octave (subprime) is also possible with the LC spreadsheet. Being able to have control over the placement of D#3 by using D3 and E3, makes for a better fitting and better sounding tuning in the tenor.)
Knowing the width of the prime octave’s 2:1 after it’s been tuned as a pure 4:2 can be very helpful when mapping A3.
The prime octave’s 2:1 width seems to be very piano dependent, while the width of the prime 5ths seems to be surprisingly reliable.