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For example with the following Steinway temperament sequence:

A4-A3-E4-B3-F#4-C#4-G#4-D#4-A#3-F4-C4-G4-D4

when do you find out that your initial fifth A3-E4 was 2 cents narrow than it should be?

the octave is too small. (for instance)

It is a 4:2 octave.

I do not know what it really means, if too small I open it a little so the first 4ths and th 5fths fit well within the octave.

if using M 3 10th, the 10th must be a little faster, but I never use those tests now.

Ok.

But what if the octave is correct, but the fifth is 2 cents narrow.

At what step on the above sequence do you wake up that you made this error?

Stop thinking of cents. You can't measure cents aurally. The whole thing needs to mesh. All intervals agreeing with each other.

BTW, that sequence uses SBI exclusively. It is very, very old. It is RBI that challenge the ears to listen deeper to the inaccuracies of a tuning.

Too imprecise. The answer is never.

No, the RBI are not given there , not on that page

The first RBI is A3-F#4 but it is a 16 cents wide interval. Then there is A3-C#4. And then comes B3-G#4 and then comes B3-D#4. But I don't think any of these checks will reveal a 2 cent error on A3-E4.

If the initial A3-E4 5th is two cents narrower than it should be, then that means it is contracted by four cents. A 5th which is that far off should be evident immediately - even without any additional checks. By the time you have tuned F#4 you can check the 6th, A3-F#4. Any "cumulative" errors will be apparent at that point. Next note tuned is C#4 making the 3rd A3-C#4 available as a check - again looking for so called "cumulative" errors.

It should be pointed out that it is possible to use notes from outside the temperament octave as checks. For example, by comparing the beat rate of C3-A3 against C3-E4 you can check for correct tempering of the 5th immediately. If the 5th is tempered correctly the 6th (C3-A3) will be slightly faster (about 1 bps) than the 10th C3-E4.

Bill Bremmer has stated quite correctly that many technicians who are not aware of, or do not use RBI's as checks, will end up with a very poor temperament when using a 4ths and 5ths sequence. However, with proper understanding and use of RBI checks it is possible to tune a very accurate temperament with these sequences.

No, not on their own.

You need a more precise temperament sequence that does not allow errors, early on.

Example, lower skeleton sets C#4, high accuracy. Upper skeleton sets F4 and hence F3, high accuracy. (Well, as accurate as the tuner's aural sensitivity to beat speeds)

Now:

F3A3 < F3D4 < A3C#4

F3D4 = G3B3, so F3D4 must be EXACTLY between F3A3 and A3C#4.

Note, close is not good enough. F3D4 has to be EXACTLY between F3A3 and A#C#4. You will have to fit another beat speed between F3A4 and F3D4, and F3D4 and A3C#4, eventually.

Now, you have set those M3's with scarily accurate beat speeds AND the P4 is wide, by exactly the right amount, very early in the sequence.

How's that for accurate?

Ooops - I forgot to mention that test notes from outside the temperament octave do not need to be in tune. They just need to be close enough to produce useable beat rates.

Sorry to disagree with you Mark. I will not argue against a contiguous 3rds based sequence. But, a 4ths and 5ths sequence can be very accurate as well. I once had the opportunity to watch Bill Garlick do it and also saw/heard Christine Lovgren
produce a beautiful temperament this way. Been doing it myself for 35 years and scored 98% on the temperament section of the PTG exam. I am not saying this sequence is better in any way - just that it can work very well.

If a temperament is well done, by any method, I defy any technician to listen after the fact and identify the sequence used.

This is a good relationship to know about. The tolerance of D4 is about 2 cents either way to fall out of the relationship. At those tolerances, the P4 is either beating too fast, or too pure. The beat rate has to be reasonably in-between for the P4 to be acceptable. Of course, this requires an accurate contiguous M3 skeleton to begin with in order to have a good enough C#3, and there is also a tolerance for the CM3 skeleton to consider.

I might add that, in practice, I may tune a P4 using a M3M6 test to prove it is wide, and then use my sense of P4 beat speed to decide how wide is right.

Use ETD to check the width of fifth. If you want the mathematical accuracy, better to use ETD.

Easy. When you tune the E4!. It will beat far too fast. It should be obvious.

But if you tune each and every 4th and 5th, say, 1/2 cent too tempered you may not notice until you tune F4. At this point F4-A4 will scream.

It seems that regardless of the sequence, you cannot know if you are really, really right until the ninth "named" note. (Octaves don't count)

Hi Gerry,

I'm sure if we got together, we would find that we agree on a lot more than we thought we disagreed on.

My point was that a simple temperament sequence of 4/5 is not accurate or fast enough, for me anyway.

However, I certainly do not doubt that someone could tune a high scoring temperament, as you obviously did. But you probably used check notes (RBI) and the ladder of chromatic 3rds and 6ths (RBI) and tweaked a lot of the notes after your first or second pass through the temperament.

I am developing a temperament sequence that provides high accuracy early on. Think of the Contiguous Major 3rds (my technique uses the Upper and Lower Skeleton) but applied for the whole temperament!

High accuracy early on = less refining and hence faster tunings.

The steps I wrote in my previous post prove this. If you do not tune D4 so that F3D4 is EXACTLY between F3A3 and A3C#4, you will have trouble fitting F#3A#3 and G#3C4 where they belong.

When I tune the temperament with this approach, and then check my fourths, they all increase in colour from F3A#3 to C4F4.

Evenly increasing thirds, sixths, AND fourths are a requirement for ET.

BTW, no one has ever posted, as far as I remember, a reason why anyone would care to create such a high level of accuracy in the temperament, when it's been proven that most pianists don't hear much of a difference between temperaments that are more or less ET. Anyone care to post their reasons why (or why not) highly accurate temperaments are better than approximations?

Cheers,

Hi Mark -

You are right, I am sure we don't disagree all that much.
There is no question that a contiguous 3rds based sequence can produce excellent results. There was a period of time when I used one of these sequences myself, but found that, for me anyway, 4ths and 5ths are faster and easier while being just as accurate. Of course, this means paying attention to 3rds and 6ths as they become available. Yes, there is a certain amount of "tweaking" after the first pass, but I find that to be true of any sequence.

Truthfully, I get a little sensitive hearing comments about how "old fashioned" and "unreliable" a 4ths and 5ths sequence is. Franz Mohr always tuned this way. Gee, I wonder if anyone ever told Horowitz that his piano had been tuned using an "unreliable" method!!! (Just joking of course.)

I think the question is: How do you correct errors after you have gone through a sequence and then they have been identified? Lets say there is a too narrow fifth, then which of the notes is to be adjusted without disturbing other relationships?

Is there a systematic way of doing this?