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Folks:

I decided to find something out for myself. Yesterday on a customer’s piano with a friendly pinblock, I tuned Dr. White's sequence (circle of fifths) without any RBI checks or retuning previous intervals. The F-F octave ended up as a pure 4:2. The only note that needed adjusting for proper RBI progression was D#4. It must have been poor pin setting towards the end of the sequence because neither G#3 nor A#3 needed adjusting.

So now what? I guess I have been using the RBI checks as a crutch.

If you think I am lying, that’s OK. I would think the same thing.

Realizing that I can do this bothers me in ways that would be complicated to explain. I slept poorly last night. I think I will take a break from posting for a few days and consider the implications to Pride and Compassion. I am not sure the best way for me to post about tuning now.

The tuner teaching me, does exactly that and does it well, based on 35 years of practical experience, which makes it so difficult to learn from!

With all due regards Gadzar, his statement was a mere observation of human nature in some people, not a proclimation of his intentions. I've read many of Jeff's postings and don't really think he's trying to control anyone into becoming a Patty Hurst-like desciple of Braide White. Why all the paranoia?

Kees,

When I began to tune the CM3, I used to follow the instructions given by Randy Potter in his course of piano technology. But they weren't clear enough to me at how to tune F3. A sort of try and error procedure.

Then I've found the article of Bill Bremmer about tuning the midrange. I read it carefully and I followed his instructions step by step. And I discovered the correct way of setting the CM3s.

First of all, he explains exactly how to tune F3, which was not obvious in other documents I've read before. And secondly, the tuning of C#4 is a key step for the success of the CM3 setting.

You can start puting F3 where you hear a nice tempered M3. And it is the piano who tells you if F3 is sharp or flat. The way you tune C#4 is key in order to know how F3 must be retuned.

Before reading the Bremmer's article, I used to tune F3 as a 7 bps M3, then F4, and then I tuned C#4 by puting it in the middle between A3 and F4, striving to temper equally both M3s A3-C#4 and C#4-F4. That is an error!

If you do this, things smooth out, and you have a hard time to verify the even progression of M3s, as you already have smoothed the progression between A3-C#4 and C#4-F4.

So it is hard to say if F3 should be corrected and in what direction. F4-A4 is so fast that it helps only a little. And you finish with an acuraty of about 1 to 2 cents.

The point is to tune C#4 hearing the beat rate of A3-C#4 and tuning a 5:4 ratio from F3-A3. Do not hear at C#4-F4 when you are tuning C#4. Only tune a 5:4 ratio between F3-A3 and A3-C#4. Do not cheat, tune the 4:5 ratio!

How?

The speed of F3-A3 is at the limit of what I can reproduce with an alternating movement of my tonge between my teeth. And I can switch from a 4 ticks to the beat to 5 ticks to the beat. So it is easy to find the correct ratio. You can start practicing at a lower speed first, switching from 4 to 5 beats, and then at tempo (at about 7 bps). If your tonge is not fast enough you can play it mentally. Anyway there will always be a way for you to find the correct ratio. Everyone has his own means. Claping hands every 4 beats and every 5 beats, etc...

The trick is to find the correct rythm 4 to 5.

Once you have tuned C#4 in this way you can now check what the beat rate is for C#4-F4. If it is slow you have to raise F3 and if it is fast then you must lower F3.

When you have it right you check the whole set from F3 to A4, and make any needed adjustments, but chances are you have it already right.

The key here is to not put C#4 in the middle of A3 and F4, but try to tune the 5:4 ratio between F3-A3 and A3-C#4.

When I understood this, then my accuraty went to half a cent or even better.

And yes of course, there are pianos where F4-A4 is hard to hear.

My Petrof, for example, has a weakness in F3-A3 which is hard to hear, while F4-A4 is clearly discerned.

Even voicing can affect what you hear from M3s as the relative strength of partials is changed.

One more point.

In piano tuning there are no magic numbers!

The 5:4 ratio is not a magic number (of 1.25), but an approximation. It will change from piano to piano depending on it's scaling.

Once you have tuned C#4 to have a 4:5 ratio between the beat rates of F3-A3 and A3-C#4 you check C#4-F4, and you retune F3, F4 and C#4 until it is too in the 4:5 ratio.

Now, you finally have to check F4-A4. Why? Because the correct ratio of the CM3s for this piano can be other than 4:5. You can find that F4-A4 is faster or slower than you have expected, and then you have to do some adjustments in the tuning of F3.

The theoretical value of the CM3s ratio is near 1.26 but the iH of the piano changes this ratio all along the scale. However it can be considered constant in the span of the temperament octave, with a little error.

So tuning with a 4:5 ratio will put you very near, and you'll have to do only some little adjustments to have it right.

Thanks Gadzar, indeed I never tried to set the 5:4 beat ratio accurately.
I can do the tongue movement up to 12bps (a left over from the days I was playing baroque concerto's on recorder). I'm going to work on it that way.

Kees

Please let me know if your accuraty gets any better.

Another interesting thing I noticed, when F3-A3 is tuned right and you tune A3-C#4 for the same beatrate as F3-A3, C#4-F4 and F4-A4 beat almost the same, with C#4-F4 beating 0,6 bps slower than F4-A4. The beatrates would be 6,9 6,9 13,25 and 13,85. If you tune F3-A3 less than 6,9 bps and make A3-C#4 the same, C#4-F4 will beat faster than F4-A4. For example if F3-A3 is 6,5, the others are 6,5 14,475 and 13, but if you tune F3-A3 7,5 bps, the others are 7,5 11,625 and 15.

So you can check the accuracy of F3 by tuning the first two contiguous thirds equal beating and if F3 is right the other two should beat roughly the same, with the first of the two beating 0,6 bps slower.

That´s very interesting, this is in fact a way of testing F3 for doing a first approximation of near 1 cent!

For F3 tuned at -0.5 cents we have:

F3-A3: 7.2 bps
A3-C#4: 7.2 bps
C#4-F4: 12.9 bps
F4-A4: 13.9 bps

For F3 tuned at + 0.5cents we have:

F3-A3: 6.7 bps
A3-C#4: 6.7 bps
C#4-F4: 13.6 bps
F4-A4: 13.9 bps

But for F3 tuned at + 1 cent we have inverted CM3s:

F3-A3: 7.4 bps
A3-C#4: 7.4 bps
C#4-F4: 13.88 bps
F4-A4: 13.86 bps

In order to have inverted CM3s the error in F3 must be greater than +1 cent.

I think you made an error somewhere, or took into account inharmonicity or some other variables, because the F4-A4 should beat twice as fast as F3-A3 if we forget inharmonicity.

Here are the numbers I get by inserting the beatrates for the first two thirds:

F3-A3: 7,2 bps
A3-C#4: 7,2 bps
C#4-F4: 12,5 bps
F4-A4: 14,4 bps

And

F3-A3: 6,7 bps
A3-C#4: 6,7 bps
C#4-F4: 13,9 bps
F4-A4: 13,4 bps

Beatrates when F3 is correct:

F3-A3: 6,9 bps
A3-C#4: 6,9 bps
C#4-F4: 13,3 bps
F4-A4: 13,9 bps

I haven't tried this in practice and cannot say how hard it is to determine whether one third is beating 0,5 bps faster or if they would sound the same at almost 14 bps, but it seems if the inversion of the CM3s beating that fast can be heard you could set the F3 with 0,5 cent accuracy using this without having to guess anything.

You are right, I made an error.

In fact it is as you say: For F3 tuned at + 0.5 cents CM3s have already inverted with C#4-F4 = 14.0 bps and F4-A4 = 13.4 bps.

So we can detect a deviation of less than + 0.5 cents in the tuning of F3.

I've not tried this technique neither but I can anticipate it's easy to discern if one of the CM3s is faster and if they are inverted or equal beating.

Thank you Rafael,

This had always been my sense of it although I was unable to prove it mathematically. If properly executed with all five notes from F3-A4, the greatest possible error would be within 1/2 cent.

Incidentally, the first piano I tuned on Monday was a Steinway model A. I tried my latest suggestion. I fist tune A4, then F4 to it. There is a very small margin between a beat that is easily heard (too slow) and one that is too fast to discern. I placed F4 at the spot where it can just barely be heard clearly.

I then tune F3 to F4 and filled in C#4. I don't really think so much in terms of 4:5 but merely what I call a "small difference" (not the smallest difference that can be discerned but a small difference). The sequence revealed a perfectly even progression on the very first attempt.

Chapeau!

I guess you already have the ear of how a M3 should sound on a S&S. I seldom have the opportunity to tune such a nice piano.

When I set the CM3s, usually it takes me from 2 to 3 tunings of F3: my initial estimation and then a second tuning after hearing the progression, sometimes I have to retouch F3 if F4-A4 doesn't fit the rest.

I've never tuned F4 to A4. I'll give it a try.

Gadzar,

I while back, Patrick brought up an interesting point. While the beat rate of all Major thirds in ET is different, they are all of the same size and therefore the same quality sound. Perhaps the idea of tuning the F4-A4 M3 first would not work well for all people but it is worth giving it a try.

Some have said that they cannot hear the beats in that interval at all. Indeed, at some point, the coincident partials are too high and faint to be heard. In my writings, I have always said that the F4-A4 M3 is at or near the limit of discernibility. Normally, we would not listen to chromatic M3's this high in the scale.

However, for the A4 to be truly and exactly at pitch, we must work from that note. Therefore, the F4-A4 M3 must be correct. If one likes 4ths & 5ths, one can also tune D4 and E4 between A3 and A4 and make the resultant 4ths and 5ths be correctly proportionate.

To tune F4 as the first note after A4, sharpen or flatten F4 until you can hear a gentle beat, then flatten F4 until the sound becomes too fast, "sour" and indiscernible. Sharpen F4 again until you hear that moderate sound, not easily heard and barely discernible. Now, this may be no more accurate than guessing at the approximately 7 beats per second that the F3-A3 M3 should have but if the initial estimate is inaccurate, the rest of the sequence will show that.

Certainly, if one has F3-A3-C#4-F4-A4 with a reasonably even progression, no two exactly alike, no large differences, the two pairs of octaves, F3-A4 and A3-A4 are proven similar in width and D4 & E4 are tuned as proportionate 4ths & 5ths, one has fully half of the notes of the F3-F4 temperament octave already tuned plus one more outside note. How could anyone go very far astray with a framework like that?

Theoretically there can be an up to 3 cent error without reversing the beatrate progression of the contiguous thirds. But with a little practice I don't think I'd make an error of over 1 cent without noticing. With more practice it could become even more accurate.

The 0,5 cent error was for the other method of setting F3 by making F3-A3 and A3-C#4 beat the same and then the other two thirds should be almost the same, with C#4-F4 being 0,6 bps slower than F4-A4.

Ron:

I just skimmed through this entire Topic. It is obvious that you are not interested in CM3 sequences. Sorry that this Topic got hijacked and I apologize for my part in it.

I realize I have been “on the fence” about what 4th and 5th tuning is really all about. I had fallen for the trap of trying to beat someone at their own game. 4th and 5th tuning is not about making the RBIs progressive. It is about tuning properly tempered SBIs. RBIs can only go so far into helping this happen.

It would be like trying to be a good person by obeying all the rules rather than by developing good character. Obeying some rules shows poor character. And trying to explain what good character is, by putting forth a bunch of rules, is self-defeating. Likewise, if SBIs are properly tempered, the progression of RBIs is not an issue. It happens naturally. Just like a person with good character does not steal, unless it is to take something away that is meant to harm and probably many other reasons. (See what I mean about rules?) And RBIs should not be progressive when crossing a jump in scaling if the goal is to have SBIs properly tempered.

So putting aside RBIs, for now, how do you properly temper SBIs? It is pretty simple, really. In the temperament octave the 4ths should beat about 1 bps and the 5ths about 1/2bps. And by saying ”about” I do not mean that on some pianos it will be 1.2 bps for the 4ths and on others 2/5bps for the 5ths. I mean any 4th or 5th on any piano will be about these speeds in the temperament. Why? A number of reasons:

First, Inharmonicity is largely self-correcting when considering beatspeeds in the temperament, unless there is a jump in scaling. And a couple of tenths of bps on a SBI is very little change in pitch. It may not even be tunable depending on the pinblock, and won’t make a bit of difference in RBIs that are already properly progressive. RBIs are tempered 7 times as much as SBIs.

So what about RBIs and their checks? Let me quote Dr White again:

From page 85:

http://www.archive.org/stream/modernpianotunin00whit#page/84/mode/2up

”Experience shows that it is easiest to tune by Octaves, Fifths and Fourths; by Fifths and Fourths for the octave of tones, usually F2-F3 chosen for the “bearings” or foundation work and by Octaves up and down thereafter. The other intervals involved are best used for testing the correctness of the work as it proceeds.” (Bold added for emphasis.)

Of course we would now say F3-F4, the notation has changed. ”The other intervals involved are best used for testing the correctness of the work as it proceeds.” I now understand what this statement really means. Without getting into detail of the tests themselves, which Dr. White does later, let me explain the right and wrong way to use the RBIs as checks.

As the circle of fifths is constructed more and more RBIs become available. Each one gives an indication of how equal the temperament is. The first one only gives beatrate information, which is actually reliable despite iH. And if there is a question, simply go back and listen to the SBIs. If the 4ths are about 1bps and the 5ths are about ½ bps, continue on! That is the right way. The wrong way would be to adjust the 4ths and 5ths to obtain a theoretical beat speed for an RBI. If we do that, then we are letting the tail wag the dog.

The next RBI will give us two different beatrates to compare. Then later some beatrates will be the same. Then some beatrates will fall between two others. As the circle is formed it may be noticed, for example, that two beatrates that should be the same are not quite the same. It is then an easy thing to check the SBIs that formed those RBIs. If any seem a bit slower or faster than others, and bringing them in line with the others will make the RBIs beat closer together, then do so. If not, it is probably a due to a jump in scaling. If it is not a jump in scaling then perhaps you do not have a good feel for what a properly tempered SBI sounds like. This takes practice. But it is easy to recognize jumps in scaling.

A jump in scaling can be expected from one bridge to another, when going from wound to unwound strings on the same bridge, and when the number of strings per unison changes on the same bridge. Usually, major thirds will beat faster than expected when straddling a break and slower than expected above a break. Also if a break is much lower than it should be for the size of the piano, the lowest M3s above the break will beat slower than expected.

But there are exceptions. On the Kimball console, the M3s that straddle the wound/unwound break beat slower, not faster than expected when the SBIs are properly tempered. Kohler Campbell consoles have some surprises, too.

So then how to determine the correct beatspeeds for intervals that straddle or are near a jump in scaling? By tuning properly tempered SBIs to construct them. Don’t let the tail wag the dog!

OK, more about actually tuning properly tempered SBIs. It is challenging to play an SBI and say if it is properly tempered, although it is easy to play a series of them and tell which ones are different. Here is my suggestion for actually tuning SBIs. Start a bit below pitch so that the beat is easily heard. If both faster and slower beats are heard, lock onto the slower one. The faster one is from a set of higher partials and will lead you astray. Bring the pitch up with a smooth pull until it beats the proper speed and get this sound in your head! It is not just a beat speed it is also a tonal color. We are tempering intervals not just setting beatrates. It is this color difference that makes incorrectly tempered SBIs stick out more than the beatspeed. Now you are going to have to do whatever is necessary to actually leave this interval where it belongs. Usually some overshoot and then easing the pitch back down. But let’s not get into hammer technique. That’s a whole other subject.

Don’t know if this will help you, Ron. I hope it does. It helped me. I am now off the fence. I won’t fall for trying to beat someone at their own game in regards to 4th and 5th tuning. I am off the fence now. (Still have a couple splinters in interesting places, though )

Jeff: I tried a bit of 4th and 5th tuning.

A3 from fork (ETD actually). Then D4, G3 C4 F3. Now I check F3A3. It was too slow. So I adjust F3 so F3A3 beats at 7bps and go back and fix G3 C4 and D4 to all beat proportional.

Now why not set F3 right away at 7bps from A3 and then tweak G3 C4 and D4 to fit?

Kees

I computed the error from blindly tuning 5ths 1/2 bps and 4ths 1 bps, using White's sequence. "Error" as compared to taking the F3F4 octave that would result from such a scheme and then dividing that (stretched) octave in 12 equal parts. All with zero inharmonicity. I get the following errors:

C 0.0
C# 0.9
D -0.2
D# 0.9
E -0.0
F 1.2
F# 2.0
G 0.8
G# 1.8
A 0.9
A# 2.1
B 1.3

Doing the same with a typical IH model doesn't change this much.

Not sure what to conclude. I guess what I'm after is some instruction on what to do if the RBI tests (e.g. steps 5 and 8 in his sequence) fail the test. Not that I mind figuring out what to do myself, but the RBI tests should come with some instruction on what to do if the test fails.

Kees

UnrightTooner, I wonder how big is the jump in scaling, does the IH double?. How much does the expected beatrate change and if you make the M3s beat correct, is there an obvious flaw in the fifths?

Because then you are not

Because then you are not creating a M3 by tuning 4ths and 5ths, you are fitting 4ths and 5ths into a M3. (The tail is wagging the dog.) But how do you know that the M3 was beating too slow? If it was way, way slow then your SBIs were way off. If it was only a little slow, that may be the correct speed for that M3.

The first M3 is no place to work backward. If the first couple of RBIs show a problem it is best to start over. Including the complimentary SBIs and listening to the resulting octave will help things get started right.