Mapping A3: Procedures
Procedure 1. For most pianos.
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. This indicates a reasonable and workable mapping of A3 and the prime octave.
Templates are used for mapping the midrange (A3/A4).
Using a template that results in a pure 4:2 prime octave can be a good place to start.
For example, if the 4th P of A4 is 10, and the 2nd P of A4 is 1.5, a good template for starters would be the one with A4 @ 10 and A3 @ 1.5.
Using that template, tune A4, A3, D4 and E4 to the template.
Now measure the lower (A3/E4) and upper (D4/A4) 5ths. Add their widths together and see how close their sum comes to -3.0. If it is -3.0 the prime octave width is probably about right.
If the sum is -2.6 c., the octave is probably too wide ( -2.6 c. is not as narrow as -3.0 c.).
In this case choose a template with a .4 c. higher A3, re-tune A3, D4, and E4, and then re-measure the prime 5ths. (If the first template started with an A3 of 1.5, look for a template with a setting of 1.9 c.)
The sum of the prime 5ths widths, should now be closer to -3.0.
If the sum of the widths of the prime 5ths is -3.0 c., and the prime octave 2:1 is less than 2.8 c. wide, A3 is probably properly mapped.
If the sum of the prime 5ths, is -3.6 the prime octave is probably too narrow. (-3.6 c. is more narrow than -3.0 c.). In that case widen the octave by selecting a different template with an A3 setting .6 c. lower. (If the first template started with an A3 of 1.5, look for a template with a setting of .9 c.).
Again, if the sum of the widths of the prime 5ths is -3.0 c., and the prime octave 2:1 is less than 2.8 c. wide, A3 is probably properly mapped.
The idea here is to use a few different templates for trying different locations for A3. The goal is to find a template that locates A3 so that the prime octave is less than 2.8 c. wide, and the sum of the prime 5ths, is -3.0.
Balancing the prime 5ths:
Now that the above two conditions are met, the next step is to ‘balance’ the prime 5ths. Balancing the 5ths, is done by raising or lowering the templates locations of both D4 and E4.
If the lower 5th’s width is -1.9 c. and the upper 5ths width is -1.1 c. , the lower 5th is too narrow by .4 c. (-1.9 c. vs. -1.5 c.) and the upper 5th’s width is not narrow enough by .4 c.. (-1.1 c. vs. -1.5 c.) Adding .4 to the setting of both D4 and E4 will balance the 5ths.
To balance these prime 5ths, look at the setting in the template for D4 and E4. Add .4 to the setting for D4 and re-tune D4 to that new setting. Do the same for E4 – just add .4 to the template’s setting for E4 and then re-tune E4.
Now you should have a good sounding prime octave less than 2.8 c. wide, a pair of good sounding balanced prime 5ths at -1.5 c. each, and a good sounding pair of ‘resultant’ prime 4ths.
In order to incorporate these settings for D3 and E4 into the final tuning, write them down as part of the mapping notes for the prime octave.
This balancing of the 5ths, assures the shape of the tuning curve will match the shape of the piano’s scaling in this area.
Procedure 2: The Exceptions.
The exceptions include pianos with a 4th A4 Number > 3.0 c.
The 4th A4 number flags an exception right away. A high 4th A4 Number >3.0, indicates the pure 4:2 results in a very wide 2:1.
The first priority is the octave. The prime octave can’t be allowed to beat too much. There is a relationship between the prime octave and the prime 5ths. If necessary, the prime 5ths may need to be more narrow than the -1.5 c. width. And that means that at this point, the sum of the prime 5ths widths, may need to be more narrow than -3.0. And after they are balanced sometimes each of the prime 5ths, may end up being -1.7 or -1.9 or even -2.0, depending on the piano.
A 4th A4 Number over 3.0 c. indicates the best sounding compromise for this piano may result in the prime 4:2 octave being narrow. It also may result in prime 5ths more narrow than -1.5. These exception pianos may push the ‘no wider than 2.8 c. wide’ guideline for the width of the prime 2:1 octave.
Double Octave Beat (DOB) adjustments can come in handy for these as well. DOB can be used to easily tweak the width of any template during the mapping of the prime octave, and it can also be used when there is no template with a low enough A3 number.
The procedure for working with these exception pianos is just the same as it is for the more ‘normal’ ones.
A high 4th A4 Number indicates a possible ‘exception’ piano.
My prime 2:1 octave tolerance is 2.8 c. wide. So, if the 4th A4 Number is much greater than 3.0, the piano falls into this category.
It can be called an exception basically because the prime octave 4:2 on this piano will probably end up being narrow. If the 4th A4 Number is 4.0 the octave will need to be contracted to quiet or slow down the beating at the 2:1. In order to get the prime octave width down to 2.8, the prime octave will need to be contracted by about 1.2 c. (4.0 – 2.8 = 1.2)
A template will need to be selected with an A3 about 1.2 c. higher than the pure 4:2 setting. I need to raise A3 accordingly so that the template I select has an A3 setting that gives me a prime 2:1 octave close to my 2.8 tolerance.
Here is an example with some numbers:
The 4th P of A4 is 9.5
The 2nd P of A4 is 1.0
The 4th A4 number is 3.6
That means when A3 is tuned as a pure 4:2, the A3/A4 2:1 is 3.6 c. wide.
Select a template with (A3 @ 1.0 c.) and (A4 @ 9.5 c.).
Using that template, tune (A4), A3, D4, and E4, measure the 5ths, and add their widths together.
The sum of those prime 5ths may be close to -3.0 c., which means that even though the prime 2:1 octave is too wide and beating a more than desirable, the 5ths when balanced will be @ -1.5 c. each.
But right now all we care about is the sum of the widths of the prime 5ths. It doesn’t matter now, if one is -2.0 c. and the other is -1.0 c.. Balancing them makes them equal and that will be done later. Again, all we are about now is the sum of their widths.
But, if the prime octave’s 2:1 is 3.6 c. wide, it is probably beating too much. So, the prime octave needs to be contracted by raising A3 by about .6 c. so that prime octave’s 2:1 is closer to 3.0 c. wide.
Select a template with an A3 of 1.6 c. (instead of 1.0 c.) and A4 @ 9.5 c.
Re-tune (A4), A3, D4, and E4 and re-measure the 5ths. The 5ths have been narrowed as well, and their sum should now be about -3.6 c. instead of -3.0 c. When those 5ths are balanced, they will each be about -1.8 instead of -1.5. The .6 c. difference in the octave only affects the 5ths by .3 c. (each).
This exception now has a prime octave 2:1 about 3.0 c wide and a pair of prime 5ths, that will end up being -1.8 c. instead of -1.5. This is really not a bad sounding relationship.
Here’s where I remember the -2.0 c. theoretical width of a 5th I got from Rick Baldassin.
Even if the octave is contracted another .4 c. which will put it @ -2,6, the 5ths will still probably sound OK since they will be close to that -2.0 c. theoretical width.
Since there are two 5ths and only one octave, the width of the change in the octave, whatever that is, is split up between the two 5ths. So a change of .8 c. in the octave effects each 5th by only .6 c.
Using Double Octave Beat:
DOB is a really slick way to make these small changes without having to look up templates! DOB can be used to widen or contract the tuning while leaving A4 alone, and gradually expanding or contracting the tuning outward in both directions.
You can find more on the Double Octave Beat feature in the Sanderson Accu-Tuner instruction manual. It is a feature found on both SAT IIIs and SAT IVs.
.1 DOB effects the location of A3 by .2 c. so to try a slightly higher by .2 setting for A3, use a -.1 DOB setting, re-tune A3, D4 and E4, and re-measured the 5ths.
Again, at this point all I’m looking for is the sum of the two 5ths. Even if one 5th is -3.0 c. and the other is -1.0 c., don’t worry about it now because we’ll be balancing the 5ths later – the balancing comes last.
There is a relationship between the octave and the prime 5ths that will sound better than any other on any particular piano. On a normal piano, finding that good sounding relationship can be fairly straightforward using the guidelines. But on a borderline or worse scale, finding the best relationship between the 5ths and the octave can be a balancing act between the two. When working with a more challenging piano, the 5ths will need to be balanced before finalizing a decision as to what the best compromise may be.
For example, if the lower prime 5th is -2.3 and the upper prime 5th is -1.5, the sum is 3.8 c. (2.3 + 1.5 = 3.8) Dividing 3.8 by 2 means that each 5th, when balanced, will be -1.9 c. (narrow).
To tune them that way. add .4 to the settings on the template used for both D4 and E4. Tune both D4 and E4 to those new ‘balanced’ settings and give everything a listen.
If done correctly, everything should sound petty good.
You will need to write down the settings for (A4), A3, D4 and E4 so they can be used in the LC spreadsheet for creating the tuning.
The mapping of A3 is complete on this exception piano.