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When filling in the 6 remaining notes for a given 5th, the beat speed for the three 4ths must be determined by seeing how the progressive the beat speeds are for both the resulting Major 3rds (M3s) and minor 3rds (m3s). Here is the sequence starting with C5:

C5 to pitch source
F3 to C5, pure 3:1 12th
C4 to F3 and C5, best 5th and 8ve

* * C4-C5 about 4:2 8ve for mid size, slightly narrower for small, slightly wider for large pianos. (remainder of sequence is from F3 through C4)

A# to F, ~ 1 bps wide
G to C, ~ 1 bps wide

* * check G-A# ~ 10 bps narrow

F#-B, ~ 1 bps wide

* * check F#-A# and G-B M3 progression. Adjust F#-B both either up or down, maintain the same bps as other 4ths, and make G-B slightly faster than F#-A#.

G# to F and B as contiguous m3s

* *G#-B beats "somewhat" faster - takes practice.

A to F# and C as contiguous m3s

* *A-C beats "somewhat" faster - takes practice.

The progressiveness of the resulting chromatic M3s will indicate if the beat speed of the three 4ths is correct and which way to adjust the them. For instance:

If the 4ths are tempered only 1 cent wide (instead of 2 cents) they will beat only 1/2 bps (instead of 1 bps) with the following errors:

F: 0c
F#: + 0.5c
G: + 1c
G#: - 0.25c
A: + 0.25c
A# - 1c
B: - 0.5c
C: 0c

Notice how as the sequence progresses, the beginning 1c error is halved as each pair of notes is tuned. Let's look at the resulting errors in the M3 and m3 intervals.For a pair of chromatic M3s to beat the same speed, the error of the lower one must be about 1c wider than the one above it. For m3s, about 1c narrower:

F-A: +0.25c
F#-A#: -1.5c
G-B: -1.5c
G#-C: +0.25c

F-G#: -0.25c
F#-A: -0.25c
G-A#: -2c
G#-B: -0.25c
A-C: -0.25c

Two things to note. First, with the M3s, the middle pair are progressive because that is the way they were tuned. The outer pair are not because it is the result of the error of the 4ths. Second, with the m3s, only the middle m3 is not progressive. IT IS VERY IMPORTANT to tune G# and A as contiguous m3s and not try to smooth out the progression of all the RBI intervals by listening to the resulting M3s. Doing so will make any error in the 4ths less apparent.

This was the first iteration and showed the results of the 4ths not being wide enough.When starting the second iteration with A# to F it should be obvious that A# should be tuned higher to improve the progression of F-A and F#-A#. Removing about half the error OF THE M3 PROGRESSION (not the error of the 4th, which you cannot really know aurally...) and tuning the rest of the sequence with this new beat rate for the 4ths will result in:

F: 0c
F#: + 0.1c
G: + 0.25c
G#: 0c
A: 0c
A# - 0.25c
B: - 0.1c
C: 0c

F-A: 0c
F#-A#: -0.4c
G-B: -0.4c
G#-C: 0c

F-G#: 0c
F#-A: -0.1c
G-A#: -0.5c
G#-B: -0.1c
A-C: 05c

The result is 4ths tempered slightly less than they should be but all RBIs being progressive. In reality it is an exceptional situation where 1/4 cent stability can be realized and so the temperament will be as equal as typically possible.

Anyhoo, that's what I strive to do. If it's not working, I know I have a scaling or stability problem. Two things to remember: make the 4ths all have the same beatrate and tune G# and A as cm3s.

Please forgive any typos.

That sounds difficult. I must try it at the weekend...

Let us know how it works out. Everything is difficult the first time it is tried.

Here is something that may help: When tuning the middle note for the two minor 3rds, remember that if you are flat, the upper m3 will beat too slow because m3s are narrow from just intonation. (Sorry if this is obvious, I don't know your level of expertise.) And if you first get the two intervals to beat equally, you need only to raise the middle note less than 2 cents, not quite as much as when tempering a 4th.

For instance: when tuning A to F# and C, tune A until the beatrate of F#-A and the beatrate of A-C are the same as each other. Then raise A "somewhat" so that A-C beats "somewhat" faster than F#-A.

Also, when setting F# and B it is helpful to "shim" these notes. For instance: if F#-A# beats faster than G-B, but F#-B beats the desired 1 bps wide, first raise F# while listening to F#-B, making this 4th less tempered. Then raise B while again listening to F#-B, tempering this 4th back to 1 bps wide of just intonation.

Hope this helps.

I'm guessing you are tuning with temperament strips? If so this changes the equation substantially.

This aural checking does not take as long as you might think. Basically I'm tuning octaves but quickly checking the 17th against the 6th to make sure I'm not over or under stretching. The beauty of fast beating intervals is that your brain processes them almost instantly.

SInce I use open unison tuning style, I'm often "shimming" the unison to make slight adjustments to the 17ths to keep them in line with the 6ths - or put them somewhere between the 3rd (the check for the 4:1 double octave) and the 6th (for the 12th). Since the 6ths beat faster than the 3rds (at least they should) you get slightly more stretch using them as a reference.

I've been relying on Tunelab more the past few years to lock in the high treble. Tunelab makes it easy to tune directly off 3rd or 4th partials of notes lower down to establish the placement of the high treble notes.

Hi, my name is Jeff. Hi Jeff!

I am standing here to confess that, like many of you, I am a temperament-strip-alholic. I am ashamed to say that I cannot go a single day without using one. It all started so innocently. My blind tuning instructor used one, so it seemed only natural that I do so, too. How could I have possibly foreseen the heartache and destructiveness of using something, which at first glance seemed like a ribbon for a Christmas bow, that would eventually make me desire to use it as a hangman's noose! But there is hope through the 12-semitone temperance program!

Thanks for sharing Jeff. Someday you will wake up in the gutter, and having reached your rock bottom, slowly climb your way back.

I certainly don't have a problem with others using the strip - but I think it does affect the approach one has to listening to the piano. I used them for many years.

Honestly the thing that made me switch over was simple: One day while tuning at a college, I discovered I had left my strips somewhere and had to tune all day without them! I HATED it, but also a light bulb went off, and I began to understand why some technicians move away from them. After going back and forth for a while I finally settled on just using individual mutes. That being said, there are still occasional situations where I strip up the middle of the piano. Sometimes I just do it for the sake of variety!

My point in asking is this: I believe people who use strips focus more on slow beating intervals: 4ths, 5ths, Octs, etc and those who tune open unisons tend to rely on fast beating interval checks. Don't get me wrong, I still listen to the slow beating intervals, but with open string tuning they are more "nebulous". I'm not saying one is better than the other, but that the different approaches lead to different listening preferences.

Glad you joined in on the humor, Ryan.

Wonder if everything involving tuning preferences eventually boils down to RBIs vs SBIs...