After learning the 4th A4 Number, its easy to make a guess as to the location of A3.  If the 4th A4 number is a high number – above 3.0 or so, start with a template that results in a pure A3/A4 4:2 and using that template, tune A4, A3, D4, and E4 and measure the prime 5ths.

The prime 5ths’ combined width lets me know if the prime octave needs to be expanded or contracted and by how much.

At this point, look for a combined width of the prime 5ths to be -3.0 c.  At this point, it doesn’t matter if one of the 5ths is -2.0 and the other -1.0.  Simply add the 5ths’ widths together and if it is -3.0 c., the prime octave width is about right.   (Even though this combined 5ths’ width of -3.0 c. works on most pianos, of course there are exceptions.)

If one of them is -1.5 and the other is -2.3, that total is -3.8.  This combined width indicates the prime octave is .8 c. too narrow, (-3.8  vs. -3.0).   Find another template in the SAT with an A3 setting .8 lower and use that template to re-tune A4, A3, D4, and E4 and re-measure.

While working with the prime 5ths, keep an ear and eye on the width of the prime octave’s 2:1.   As long as the prime octave’s 2:1 stays < 2.8 c. wide, and the combined width of the prime 5ths is -3.0 c, the prime octave width is about right.

Most pianos will allow for that relationship.   But of course, some don’t.  It just depends on the piano.   Hopefully your first pianos on which you try this, won’t too much out of the ‘normal’ range.

All that is needed here is for the combined width of prime 5ths, to be -3.0 c.   The prime 5ths, will be balance a little later so they will be the same width.   Balancing them is done by raising or lowering the mid point (D#4) of the prime octave.

If  the combined width of the prime 5ths is -3.0. but the prime octave is 3.6 c wide and beating too much, some compromises will need to be made.   The octave can’t be allowed to beat too much.   When this situation presents itself,  a combined 5th width of -3.4 which would put the 2:1 octave around 2,8 or so would probably be a good compromise..

{Hopefully, the first few times you try this, it won’t be on a piano that requires too much compromising.    This routine works great on all those pianos too, but it takes some experience (and patience) with everything involved to find the best compromise.  Those pianos are often noisy and false, and often have wound bichords well up into the 3rd octave.  But those ‘exception’ or more challenging pianos, are our teachers, and with enough patience and experience, they too confirm that this system will work on every piano and help do a great job, every time.

It can be very rewarding making one of these challenging pianos sound really pretty good.   Our customers hear it and appreciate it too.   ‘My piano has never sounded that good!”  is a fairly common response.   It’s their piano, and if they play it, they will hear it.   The tuning may take a little longer, but since our LC spreadsheet has a database, their piano’s custom tuning is now stored and ready for me to use the next time I tune their piano.}

Start with the 5ths while keeping an ear and eye on the prime 2:1 width.   As long as the prime 2:1 is less than about 3.0 c. wide, the octave should sound good, and after the prime 5ths are balanced, they will sound good too.

Before the prime 5ths are balanced, the easiest ‘out of balance’ tell will probably be heard in the resultant 4ths (A3/D4  & E4/A4).   Since the 4ths beat faster than the 5ths or the octave, the most noticeable beating will probably be in the upper or lower prime 4th.   Which 4th beats the most depends on the direction the mid point of the octave will need to be moved to balance the 5ths.

If I can get the prime octave’s 2:1 width < 2.8 c., and once the -1.5 c. prime 5ths are balanced, the resultant 4ths will sound good too.

#### Prime 5ths more narrow than -1.5:

On pianos with a high 4th A4 Number, the prime 5ths may need to be more narrow than the -1.5 width.

Sometimes the prime 5ths @ the  -1.5 c. width will result in a prime octave’s 2:1  4.0 c. wide!

Sometimes to clean up some of the beating in the 2:1, the prime 5ths will need to be narrower than -1.5 c. each or have a combined width of -3.5 or -4.0.

Since the 4th A4 number is the width of the prime octave’s 2:1 after the prime octave has been tuned as a pure 4:2,  on these pianos, the A3/A4 4:2 will actually be narrow.

****Every time I come across a piano like this, I remember how aural tuning is often taught.  Invariably ithey teach to start off by tuning A3/A4 as a wide 4:2, with the F3/A3 M3rd beating slower than the F3/A4 10th by about a 1/2 beat per second.

But on these pianos that require a narrow 4:2 instead of a wide 4:2, the correct beating is just the opposite:  The F3/A3 M3 should beat faster than the F3/A4 10th.    Sadly, most beginner tuners are practicing on a poorly scaled spinet or console – a piano likely needing a narrow 4:2 rather than a wide one.*****

It’s a balancing act when working with a piano whose prime octave requires more beating in the octave and more beating in the 5ths.   What needs to happen is a good sounding compromise between the beating in the prime octave and the beating in the prime 5ths.

Knowing the width of the prime octave’s 2:1 (the 4th A4 number) and the widths of the prime 5ths, is a real advantage to finding the best sounding compromise.

When I first started doing this I didn’t have any ‘guidance’ as to a good width for the prime 5ths or the prime octave’s 2:1.   My starting point of -1.5 c. for the 5ths was purely the result of tuning a lot of pianos, doing a lot of listening and noticing the widths of what I was preferring.   Empirical data, I guess it’s called.

I found myself ending up with a width of -1.5 for the prime 5ths.   I liked the way the 5ths sounded, and the prime octave worked out nicely too on most pianos.

The  < 3.0 c wide width for the prime’s 2:1, is also a personal setting.   I just generally found that when a prime octave’s 2:1 started to get wider than 2.8 c., it starts to sound a little too beaty, or too noisy, or too edgy for me.

So, that’s where those two guidelines or guideline combinations came to be.   Now they are a good starting point, from which to work.

In fact, they worked so well, and so often, whenever a piano would present itself where they wouldn’t work, I was taken aback and really had to do some extra work trying to figure out why not.   Remeasuring and retuning everything was done first.    Sometimes an error was made in tuning or measuring or something, but more often than not, it was the piano that just couldn’t take the -1.5 c. prime 5ths, and a prime octave less than about 3.0 c. wide.    It was like trying to fit a square peg into a round hole.

Those pianos were really the exceptions, and I just didn’t come across them every day.   Eventually I figured it out and have learned how to deal with them.

### Dealing with the exception pianos:

A few years ago,  I asked Rick Baldassin what he thought the width of a 5th in this area of the piano should be.   He said -2.0 c..  He knew the theoretical answer straight away.   I then asked him if he used that in any of his tunings, and he said, “No.”    But he gave me a width of the prime 5th!

Theoretical can be different that what we do when tuning pianos for real – especially when tuning pianos!   My target width for the prime 5ths is about .5 wider than theoretical.   (-1.5 c. rather than -2.0 c.)   What that means is that ‘in practice’ I’m tuning a slightly wider than theoretical prime 5th.   I’m probably also tuning a slightly wider prime octave.

I believe that there is more ‘variation’ tolerable in the prime octave than there is in the prime 5ths.   Using the -1.5 c. width for the prime 5ths has resulted in a prime octave’s 2:1 being anywhere from 1.2 – 3.0 c. wide.   When needing to make big compromises on some wild pianos, I have had to leave the prime 2:1 over 3.4 c. wide.  On that piano, the prime 5ths, were around -2.2 c. narrow.   The most important thing tuning that piano was that the prime 5ths were balanced!  They were both at the edge of objectionable at that -2.2 c. width.   If one of them would have been even a couple tenths of a cent more narrow, it would have been the straw that broke the camel’s back.

Tuning that piano was actually a rewarding experience for me.   When finished I was shocked at how decent it sounded.   Made me smile.

The range of the widths of the prime 5ths, is from about -2.2 c.  to -1.2 c..

There is no given width for the octave or the prime 5th that is right for every piano.   There is no ‘perfect width’ for any interval that works every time on every piano.   There is so much talk about the perfect 12th these days, and how it is the best way to make every piano sound it’s best.   My data sure doesn’t confirm that at all.

The easiest P12ths to check are  D3/A4 and/or  A2/E4.   Those can be checked after mapping the midrange.   With all the talk about the P12th, I started checking my work against the talk.   The easiest one for me to check is the D3/A4 12th.  After checking it a lot, it was so rarely pure, I quit checking it.   Sure it was close, but most times I found my placement for D3 at least .5 c. away from the pure setting.   And it could be either sharp or flat of the pure setting.

It’s too bad the systems touting that P12th aren’t capable of checking their work.   Those devices can’t do Direct Interval tuning, so they can’t check to see if they are ending up with P12ths.  The pianos just have too much to say about it on their own.

My overall approach to tuning is minimal beating and that is true for me in the midrange.   I don’t want to stretch any more than necessary to get a pair of good sounding prime 5ths contained within the prime octave.  Nor do I want the prime octave’s 2:1 to be any wider than about 2.8 c. unless absolutely necessary.

There was a series of articles in the PTJ Journal a year or so ago.   In one of them some numbers were given which I found interesting, helpful, and useful.   In the article the width of ‘their’ 5th was given as -1.2 c.   I now had 3 numbers for the width of the 5th:
-1.2 c.from the article,
-2.0 c. theoretical number I got from Rick Baldassin
-1.5 c. my own personal number

Over the course of my work, I have ended up with Prime 5ths, being from -1.2  to -2.2 c.    So, I guess everyone is correct, it just depends on the piano!

Most computed tunings contain ‘consistently shaped’ curves in the prime octave.
Even though the starting and ending point – say A3 and A4  – may vary, the ‘shape’ of the curve between those notes will probably be fairly consistent in virtually every tuning that system or software creates.   The shape of the curve will  probably be the same for every combination of A3 and A4 settings, regardless of the partial being used.   In other words, the basic shape of the curve will be the same, only the starting and ending points will be different.

Tuning software is designed to be consistent so that it will work on most pianos, but it will really only work on ‘some’ pianos.

If you are curious about this, start measuring the prime 5ths in your own tunings.  See if they are balanced or not.   Careful listening and measuring will expose a mis-fitting template or tuning to the piano.

The piano’s scaling will often – even most of the time – not be a good fit for that computed tuning’s curve.    The pianos are just not that consistent.   This is particularly noticeable with the less than well scaled pianos.  But it can show itself on any piano.

It’s easy to check.  Just tune A3, D4, E4, and A4, and measure the widths of the prime 5ths.  Simply measuring the prime 5ths will tell you all you need to know.  If they are the same width, they are balanced.  Depending on how out of balance they are,  you may be able to hear a difference in the 5ths.  Hearing a difference in the resultant prime 4ths, can be easier than listening the the prime 5ths.

When the 5ths are balanced, the piano will sound even better!