#### The “4th A4 number” is the width of the A3/A4 octave’s 2:1after it’s been tuned as a pure 4:2.

First, tune A3/A4 as a pure 4:2.
To tune A3 as a pure 4:2, tune A3 to the 2nd Partial location of A4.

Once A3 has been tuned as a pure 4:2, to measure the A3/A4 (Prime) octave as a 2:1:
1.)  set the SAT to A4 (from A5)
2.)  play A3
3.)  stop the lights using with the cents buttons.  The number in the display (always negative) is the ‘4th A4 Number’.

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#### The “4th A4 Numbers” vary quite a bit from piano to piano, so knowing right away, at the beginning of the tuning, what the “4th A4 number” is will be immediately helpful for finding the best sounding setup for the Prime Octave.

*** The ‘minus’ sign when measuring the “4th A4 Numbers”:
Because the “4th A4 numbers” are measured by playing A3 with the SAT @ A4, when the width of the A3/A4 2:1 is wide, the SAT display will display a negative number.  A measurement of -1.0 c. means the width of the 2:1 is 1 c. wide.  A3 being tuned 1 c. ‘lower’ than pure, thus a negative number.  A measurement of -2.0 would be a 2:1, 2 c. wide, and so on.***

The “4th A4 number” is important because it lets us know a little about the inharmonicity of the piano we’re going to be tuning.

Everything has to work together for the best sounding tuning, and it all starts here in the prime octave – just like setting a temperament with aural only tuning.

We don’t want excessive beating in the 2:1. Since we start with by tuning a pure 4:2, there won’t be any beating there, but some pianos pure 4:2s will result in a fairly wide 2:1.   Anything over about 2.8 c. wide for the resultant 2:1 (4th A4 Number) the 2:1 may start to get a bit noisy. (For me, anyway.)

This is important when trying to find the best sounding setup for the prime octave.  If, when measuring the A4 numbers, the 2nd P of A4 was @ 2.0, tuning A3 to that setting of 2, there will be zero beating in the 4:2 since it’s pure.   If the resultant 2:1 (the 4th A4 Number) is 1.4 c. wide, if we lower A3 down to by that 1.0 c. to get a 1/2 bps in the 4:2.  The 2:1 will be fine, since lowering it down by 1.c.   will mean the 2:1 is only 2.4 c. wide – well within the ‘not wider than 2.8 c. wide’ rule.   That will sound pretty good.

But if the 4th A4 number on that setup gives us a 2:1 of 2.4 c. wide when the 4:2 is pure, lowering A3 down another 1. c. will result in a 2:1 of 3.4 c. wide.   That might be starting to become a little too ‘beaty’ for the prime octave.

Many pianos whose 4th A4 number falls in the 1 – 2 c. width range can handle the 1 c. lowering to get the 4:2 beating at 1/2 bps.   Those situations keep the 2:1 below that 3 c. wide threshold, while still getting the 1/2 bps in the 4:2.

But that’s just for the A3/A4 octave width.   That just gets us started.

The determining factor for me on all this is how the prime 5ths (A3/E4 & D4/A4) figure into the equation.  The widths of the prime 5ths are instrumental for finding the best sounding setup for the prime octave’s width that includes both the 4:2, and the 2:1 and a pair of prime 5ths.

#### The goal for every prime octave setup is the A3/A4 2:1 to be less than 2.8 c. wide, and a pair of balanced prime 5ths, each -1.5 c. narrow.   That setup will sound pretty good and is achievable on most pianos.   Believe it or not, the 4:2 doesn’t really matter all that much.  It’ll work with that setup. But everything depends on the pianos!!  Everything!! Read on. . . .

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#### The range of A4 numbers is from about 0.0 to about -4.0 c.

Low 4th A4 numbers (0.0 to -1.0):
Pianos whose pure 4:2’s result in a pure 2:1 are rare, but they do exist.  I didn’t know it was possible until I found one.    (I believe Dan Levitan refers to the situation in his book.   I think he says it means it’s a ‘well scaled piano’, but I’m not so sure.   I’d disagree with him on that point if I ever had the chance).

My first pure prime 4:2  = pure prime 2:1, wasn’t some weird brand either!  It was a Yamaha  P660S.   I triple checked it, and it kept coming up the same, so I went with it as a pure 2:1 and a pure 4:2.    Again, I didn’t know it was possible, but some pianos pure 4:2 prime octave yields a pure 2:1.

One might think that pianos with these lower 4th A4 numbers will be easy, because adding that 1.0 c. to the width of the prime octave (1/2 bps.) will be perfect.  But sometimes not.   Sometimes that 1.0 c. will be too much, and other times it won’t be enough.  Again, this is where the prime 5ths can be used to find the best sounding compromise to the prime octave setup.   The prime 5ths widths are not necessarily the same on every piano either.   One might think that on a piano with a low 4th A4 number, the 2:1 can be stretched to result in wider (less beating) 5ths, and that that would be the best sounding setup for the prime octave.   But that is just not correct.   The prime 5ths are a great help for finding the best sounding compromise for the prime octave.

But one thing fairly certain about pianos with these Low 4th A4 numbers, the prime 4:2 will most often be wide.  How much is uncertain, but for the tuning, the A3/A4 4:2 will probably be wide.

Most pianos will have “4th A4 numbers” between -1.1 c. – 2.4 c.:
Most pianos fall within this 1.1 c. – 2.4 c. range.   This is the comfort zone.   No flags here, and here too, most of them end up with their Prime octave 4:2’s being wide.   But some of these pianos may end up with a pure or even narrow prime 4:2.

It depends on the piano.  The best sounding compromise involves compromising in all these intervals, the prime 4:2, the prime 2:1 and the prime 5ths.   But how the compromise is tempered is up to the technician.  Most of the time these pianos do give us options or decision to make as to the widths of these intervals.

For instance one tech may want as little beating as possible.  Another tech may prefer a certain interval over another and temper the others to work with his favorite while still keeping things within reasonable good sounding parameters.   For instance one tech may want to stretch the octaves as much as possible so the 5ths are as clean as possible.   Where as another may want to save that more octave stretching for up or down an octave.   Another technician might believe that the 4:2 just must be wide, period.  As long as the 2:1 doesn’t get too noisy for him, and the prime 5ths (and prime 4ths) don’t cause any issues, that’s his tuning style.

This group of 4th A4 Number pianos really gives the technician some technical leeway to ‘temper’ the compromise to his or his customers’ desires.  It’s up to them.

But, even with technicians working the compromises as they see fit, sometimes the best sounding compromise results in a prime 4:2 that’s pure or even slightly narrow – as well as of course being wide.  These midrange “4th A4 Number” pianos allow for probably the most versatility in their tuning.

Pianos with “4th A4 Numbers” wider than 2.8 c.:
Pianos with high “4th A4 Numbers” are more challenging.   Their compromises become pretty narrow.   When the prime’s 2:1 is up around -3.0 c., the beating in the 2:1 octave can start to get too fast.   Flags go up when I find a piano with a “4th A4 Number” over this -2.8 c. width.   Anything over -3.0 is going to take longer to map a good sounding, as well as technical, compromise.

I’ve measured pianos with 4th A4 numbers as high as 3.7 c.. That’s a noisy beating A/A4 2:1  And since the A3/A4 octave was tuned as a pure 4:2, the offensive beating is all in the 2:1.

The A3/A4 octave must be contracted to reduce the beating in the 2:1.  So on these pianos the A3/A4 4:2 will be narrow.

On those pianos, the prime 5ths will come into play and be really be important when it comes to finding the best compromise.    More on the prime 5ths later, but I should say this for now:  the 3 widths for prime 5ths I’ve heard are:  -2.0, -1.5, and -1.2 c..  My work with all this led me to the -1.5 c. width.  So for now I think the range of the widths of the prime 5ths could fall between -2.0 and -1.2 c,   I didn’t know about the other 2 numbers until after I’d been doing this for a few years.

On a piano with a high 4th A4 number, one that the prime octave needs to be contracted to reduce the beating in the 4:2, when contracting the octave we are also contracting the Prime 5ths.  As a result, sometimes the prime 5ths start to beat too much.

At least the 4:2 won’t be giving us any beating problems!   It’s the 2:1 and the Prime 5ths that need balancing in this scenario. Both the octave and the 5ths will need to be compromised.

As an example, I once ended up with a 3.3 c. wide 2:1(!), a .5 c. narrow(!) 4:2 and a pair of -2.2 c.(!) prime 5ths.  Everything was beating more than it should – except of course for the 4:2.  Trick on that was it was narrow!   I know it sounds strange, but it was the best sounding compromise I could find.  it was a very ‘balanced’ compromise given what was in front of me.  When I was finished, to my surprise, the thing really sounded pretty good too!  I was smiling all the way home.   Go figure.  Hey, it’s a piano!

Working with the prime 5ths can be difficult.  But they ae so helpful with all this.  In fact, I’d say they are mandatory at least to some extent.  But all it takes is a want to, and practice, practice, and more practice to measure them reliably, which puts our tuning skills and wrench techniques to the test.  But the payoff is huge.

Working with the Prime 5ths on most pianos is not that hard, but when the challenging pianos show up, it’s great to know how to deal with them intelligently, technically, and aurally.   All of this just helps us know what we’re dealing with and with these skills, know how to deal with it.

Again, my goal is a setup consisting of a prime 2:1 less than 2.8 c. wide, and a pair of prime 5ths, each -1.5 c. narrow.  If that’s not possible, I work from there.   Either widening or contracting the prime octave and then balancing the 5ths to get the best possible sounding compromise.   Again, the prime 4:2 is the least important interval here.

A plug for this system is how the spreadsheet allows us to tweak the midpoint of the A3/A4 octave.  What this means is that if we know where both D4 and E4 need to be tuned to get a pair of prime 5ths of equal width, we can raise or lower the midpoint in the tuning to hit those mini targets!   This is truly unique but very useful and especially so for the more challenging scales.

I’m convinced one of the reasons the tuning mentioned here sounded as good as it did was because I was able to have a pair of ‘balanced’ prime 5ths.  Balanced means the same width.   Each of the prime 5ths was -2.2 c. narrow.   If one of them had been -1.5 and the other -2.9, the octave would have been the same width, but the tuning wouldn’t have sounded nearly as homogeneous.  That’s the beauty of being able to map the mid point of the 4th octave and build that into the tuning.   Again, no other system comes anywhere close to doing this.