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And here lies one of my problems with the CM3 approach. I have very real problems discerning a beat that is this fast (at least on my piano). So I can never really say whether the F4-A4 M3 is "right" or "wrong", compared to the other three. Even C#4-F4 is sometimes difficult for me to discern.

On the other hand: to disregard beat rates and simply grade the CM3s in terms of "progressive sourness" isn't very effective or accurate for me either.

I haven't tuned many pianos, and I'm not trying to bash the CM3 approach either. I'm just saying that as a relative beginner, I can't place the uppermost M3 with great confidence, so unfortunately for me, it feels like a guessing game.

If you want to use thirds as a check, you should tune from E3 to E4. There is no reason to restrict yourself to one particular octave when setting the temperament. For that matter, there is no reason to restrict yourself to an octave.

Excellent point, BDB. One octave is a good starting point, but a two octave temperament will give you a lot more information to go on. I will very often make corrections in my starting temperament octave as I move up towards the treble. Ideally, in the end, the entire piano is a temperament.

I feel the same. I've had the opportunity to try it on a grand once and it works just as advertised. The F4A4 beats are fast but clearly audible. But on the less than perfect uprights I have the same problem. If I can hear the C#4F4 beats clearly I consider myself lucky. The crucial F4A4 check is not available it seems. On those pianos I do better by just timing F3A3 to 7bps with my watch, and make A3C#4 and C#4F4 progressive.

Not that I would ever tune equally except for the exam (some day I'll take it) of course.

Kees

To all of the above: you know you have it wrong when you can either hear the F4-A4 beat easily or it is so fast that you can't hear it at all. If you can hear it easily, it is too slow and that means F3 is too sharp. If it is so fast that you can't hear any beat, it means that F3 is too fast. It is as simple as that and very clearly and explicitly said in my ET via Marpurg article.

If you are in doubt, compare again C#4-F4 and then F4-A4.

Yes Bill, I understand but my dilemma on crummy uprights is that if everything is right I still can't hear the F4A4 beats. It's not that they are too fast, I just don't hear them.

Maybe I'll just have to focus my brain on them.

Kees

That doesn't seem very accurate, Bill.

F3-G4 is enough for me to refine the temperament. It gives some critical octave checks for the first SBIs and a few more RBI and inside/outside checks later on. It is pretty rare for me to need to adjust any of these notes later on, but sometimes I think I don't get things as stable as I'd like when getting used to a piano.

Very good point about continuing the temperament to the entire piano. 4ths, 5ths and 12ths are available if you know how to use them. But I wonder if those that set the temperament, or even just begin the temperament, with RBIs learn to use them to their full potential.

It's extremely accurate. There is only one place for each of the five notes to be for them all to be correct and that is why it is virtually infallible.

If the difference between "can either hear the F4-A4 beat easily" and "it is so fast that you can't hear it at all." is more than 1 cent, then I doubt if all the RBIs can be made to beat progressively.

Below some error analysis of the method.

Kees

You can move your CM3 ladder down to a slower beating area where the beat rates are easier to compare, to establish bearings. Then those are helpful when tuning up into the F4-A4 range.

Jim, you tune(d) C#3-F3-A3-C#4, if I remember correctly?

Yup. But recently I've been doing Eric Nikiforoff's 1-2-3, which switches immediately to complementary and opposing 4/5ths after the CM3 ladder. It seems to be more fun than how I was doing it before. I'm not sure why. Perhaps it's the symmetry of the sequence.

But back to the topic of this thread -- I'm not sure 1-2-3 is as good for the beginner, as it was touted in Eric's PTG Journal article, since you need a feel for 4/5ths. When you're just starting out, you mess with octaves and unisons, and then there's this chasm you have to leap to get to a temperament, which involves learning the sound and feel of all these intervals. When you're focused on M3rds and octaves, you can get a lot closer to ET, faster. Then work on refinement. Perhaps that's another contributor to the switch in emphasis by those doing the teaching. And of course the effect of ETDs.

On the other hand, perhaps that's why ET via Marpurg works, because you don't have to be as good at gauging the 4/5th beat rates, you go for equal beating. But I didn't come at it from that way, and once you "know," it's hard to go back and be in the frame of mind of "oh yeah, that would've been easier for me to learn."

Well, that is, when I'm not tuning EBVT, and I'm not using Tunelab to set temperament. I try to do a little bit of everything just to keep myself entertained

If the CM3s beat progressively as they should, there cannot be more than about 1/2 cent error at the most. The difference between being able to hear F4-A4 easily and for it to be too fast to be discernible is about 1 cent. Get it to where you can just barely hear it and you have it right.

I suggest for anyone who has trouble hearing the F4-A4 M3 to do what I said for novices who are trying to hear the F3-A3 M3. Tune A4 to the fork, then move F4 sharp and flat until you can hear the beats. Surely, you will be able to hear the beats when that interval beats slowly.

Now, flatten F4 until the beats increase to the point where they are too fast to hear. Sharpen F4 again until you can just barely perceive the very rapid beat. F4 should then be about right. Tune F3 from F4 as the same type of octave you have tuned for A3-A4, then fill in C#4. The entire sequence should come out either correctly or within a range that you can make small adjustments for it to be exactly right.

Yes, you can tune the CM3s lower down first, such as from C3-E4, C#3-F4, D3-F#4, etc., but the only way to be sure that A4 ends up exactly at 440 is to start with A4 at A-440 and work from there. The F4-A4 interval needs to end up correctly at some point. If you start somewhere else and don't even check it, chances are that it will be incorrect.

Thanks to Kees and Jim M. for your posts.

Unfortunately, Jeff, we have been through all of this before. Gadzar provided some excellent analysis in the past as well. It seems that no matter what anyone says about this, you still come up with the "I doubt..." and "I still don't think..." remarks. Have you ever really followed the directions even one time yet? Or did you read the directions and then make up your own and find they did not work and then proclaim that the CM3s are inaccurate?

Braide-White wrote nothing of this but the CM3s have been taught by PTG now for thirty years.

Excellent table! It shows clearly that one can easily detect a 1 cent error in any note of the CM3s by hearing at the progression of the beat rates in the CM3s set.

In every row of this table a significant jump is found in the beat rates of the CM3s that tells us exactly how to correct the tuning of which notes.

Hi!
I studied the chart of the ladder of M3's and one question came up. Since the thirds are sensitive to adjustment, and if you adjust the common tone of two contigous thirds you can find the exact spot for the notes very accurattely. My question is: If I tune F3-A3-C#4-F4-A4 correctly, what happens if i tune D4 pure to A4, and F#3 pure to C#4, what if i tune A#3 so that F#3-A#3 and A#3-D4 are equalbeating, would this be the right spot for A#3 ?

Lets see.

D4 pure to A4 = 440/1,5
F#3 pure to C#4 = 2^(-8/12)*440/1,5

Beatrate of F#3-A#3 is 4*A#3-5*F#3
Beatrate of A#3-D4 is 4*D4-5*A#3

The thirds beat equally, their beatrates are the same, so 4*A#3-5*F#3=4*D4-5*A#3

Which gets down to 9*A#3=4*D4+5*F#3
A#3=(4*D4+5*F#3)/9

Popping in D4 and F#3 from above gives us A#3=233,0306041
Real A##=233,0818808

Difference in cents=log(A#3/A#3real;2)*1200 which is -0,3809 cents.

A#3 would be 0.3909 cents flat, theoretically. I hope I didn't make a math error somewhere.


On tuning the contiguous thirds, assuming that A3 is spot on at 220hz, when F3-A3 beats at 6 bps I got that F3 is 1,8419 cents sharp. But then it's hard to get the others beat progressively. For example, when F3-A3 is 6bps, and you make A3-C#3 9 bps, C#4-F4 is 12,15 bps and F4A4 is 12 bps, but if you make A3C#3 10 bps, C#4-F4 is 10,9 bps and F4-A4 is 12 bps. In the first one, an upper third would be beating faster than a lower one, in the second the beatrates doesn't get progressively faster. I haven't tried it too much so I cant say how noticeable the flaw in the second beat rate progression would be, but I guess it would be.

Let's say you make F3-A3 6,5 bps, then A3-C#4 8,9 bps, which gives C#-4F4 of 11,475 bps and F4A-4 13 bps. Errors for F3 and F4 is 0,85 cents and for C#4 0,26 cents. Is it possible to notice the error in such a beat rate progression?

I put together a small excel file where you put in F3-A3 and A3-C#3 beatrates and it gives you the other beatrates and how many cents each note is off. It uses perfect octaves

http://www.speedyshare.com/files/28124849/Contiguous_thirds_beatrates_calc.xls

interesting, how would things change if i tuned D4 pure from A3 instead of A4? the A3-A4 octave is 4:2.
How would things change if F#3 is temporarly tuned pure from B3 if B3 is at its correct position.I'm looking for a way to find B3s correct position by tuning it between other interwalls that are equalbeating. I don't know how to calculate it myself... got any tips on links for theoretical calculations?

// martin