I’ve measured the 4th A4 Number on one piano to be as high as 4.2 c..  That means when A3 was tuned as a pure 4:2, the A3/A4 2:1 was 4.2 c. wide.   A 2:1 octave there, at that width, will be beating too much.

That 4:2 c. wide 2:1 was found on a Baldwin Hamilton S#. 193416.
I only give the brand and model here, because these pianos are so common and we’ve all been called to tune them at one time or another.

Here are the A4 numbers for that piano:
4th P of A4:  6.4
8th P of A4: 29.2
2nd P of A4:  .6
4th A4 Number:  4.2 c. (When A3 was tuned as a pure 4:2 and then measured it as a 2:1,
the width of the 2:1 was 4.2 c. wide.

All of these A4 numbers are relatively low compared to most other pianos – especially the 2nd P number.

On this Baldwin Hamilton the 4th P of A4 @ 6.4.

To put those numbers in perspective,  I have over 800 templates I used for mapping the midrange.  The lowest A4 number in all those templates is 4:2.  But it is really rare I ever use those because pianos with that low of an A4 number are really really rare.   But I have templates down in that are for just those rare occasions.  (More on templates later.)

And since those low A4 number pianos are so rare, I just don’t have that many in that ultra low range.  I have only 10 templates in the 4.0 – 4.9 range, only about 19 templates in the 5.0 – 5.9 range, and about 30 in the 6.0 – 6.9 range, with 20 of them being 6.5 or above.

Of my 800 templates, 740 or so of them are higher than this Baldwin Hamilton’s 4th P number of 6.4.

The next number to look at here is the 2nd P number of .6 c.
The iH here is so low that the 2nd P of A4 is only .6 c. sharp.

What makes this 2nd P.  number so important is that it is used to tune the pure A3/A4 4:2 octave.

Low iH is often considered a good thing.  But that’s not necessarily true.

There aren’t many absolutes in tuning.  One of the remaining absolutes for me is that both the A3/A4 2:1 and the A4/A5 2:1 octaves should always be tuned wide.

What is not absolute however is the width of those 2 – 2:1 octaves.   We  know both of them need to be wide, we just don’t know how much.

A good sounding A3/A4 2:1 octave needs to accommodate a good sounding pair of (balanced) prime 5ths which will also produce a pair of good sounding  ‘resultant’ prime 4ths (A3/D4 and E4/A4).

A 4th A4 number of 4.2 tells us immediately that a pure A3/A4 4:2 will be beating too much at the 2:1.   Contracting the 2:1 by raising A3 will cause the A3/A4 4:2 to be narrow.

Knowing the 4th A4 Number is 4.2 means we’ll need to contract the octave by about 1.2 c. to get it to the 3.0 c. width.  Tolerances to beating here (A3/A4) can vary, but the 3.0 width is a good starting point for a ‘no wider than’ guideline.

Though not absolute by any means, there is a useful ‘relationship’ between the prime octave width and the widths of the prime 5ths.

When raising A3 to contract the A3/A4 octave to reduce the beating in the 2:1, the widths of all the intervals contained within the A3/A4 octave will be contracted.

This is where the ‘relationship’ between the prime octave and the prime 5ths need to be considered as one.

Most pianos work nicely with a prime 2:1 octave less than 3.0 c. wide.   That octave ‘size’ can easily contain a pair of good sounding balanced prime 5ths,  both around -1.5 c. narrow.

But some pianos – like this Baldwin Hamilton – need something different.

I keep using the word ‘balanced’ when talking about the prime 5ths.  Balanced here simply means that both prime 5ths are the same width.   Balancing the Prime 5ths is particularly important when working with a scale such as this.

After contracting the prime octave so it is close to 3.0 wide, the prime 5ths must be considered.

On this piano after contracting the octave so the 2:1 was closer to the 3.0 c. wide mark, one of the 5ths was -3.0 c.  and the other only -1.0 c.    Definitely unbalanced.   And a 3.0 c. narrow 5th will be unacceptable for most of us.

When the prime octave width on this piano is 3.0, the combined width of the 2 prime 5ths is -4.0  c.

But if we can balance the prime 5ths so they are each -2.0 c. narrow, they will at least beat equally and may sound pretty much OK.

But, on some pianos, when contracting the octave to the 3.0 c. width, the 5ths will end up beating too fast (too narrow).  Even balanced, they are beating too fast.

In that situation, the 3.0 c. width for the prime 2:1 may need to be compromised.

I’ve tuned pianos like this that actually had 5ths that sounded pretty good – and whose individual widths were fairly close to -1.5 c.    So contracting the octave down to the -3.0 width, caused unacceptable problems with the 5ths.

In situations like this, the ‘relationship’ between the octave and the 5ths needs to be seen as one and compromised as one.

I remember the best sounding compromise on a piano which end up with a 3.4 or 3.5 prime 2:1 and a pair of -2.2 c. narrow 5ths.   I also remember being quite surprised at how acceptable the tuning sounded when I was finished.   The tuning starts with the temperament, and only when the temperament is accurately fit to the pianos’ scale, can we expect to end up with a nice sounding tuning for that piano.

When working with a piano like this, the final determination is of course, an aural decision.   It’s our job to find the best sounding compromise.   Or as Al Sanderson used to say, ‘the least bad sounding compromise’.

Being aware of the specific compromises being made, is a great help in making one of these pianos sound as good as it can.