Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Yet another SBI vs RBI thread

As some of you know, I am doing research on temperament sequences for my class in Denver in July at the PTG convention.

My premise is, we should be able to use math to determine how accurate a temperament sequence is.

Basing my results on the ability of a "super" tuner to tune exact beat ratios and beat speeds, allows me to determine how accurate a temperament would be.

If the sequence doesn't produce reasonably accurate results when used by the "super" tuner, then it will be even less accurate when tuned by us mere mortals.

I defined some inharmonicity curves that produce different sized beatless octaves.

Then I used those B curves and Fletcher's formula for partial frequencies to calculate beat speeds.

The graphs below are for scales where beatless F and A octaves are tuned as a pure 4:2/pure 6:3.

The first two graphs shows the RBI and SBI beat speeds after using a RBI tuning sequence where each interval is tuned between two other intervals. CM3 are one example of that type of sequence.

The second two graphs show the RBI and SBI beat speeds after using a SBI tuning sequence where P4 are tuned at 1bps wide, and P5 are tuned at 0.5bps narrow. The sequence used was the cycle of fifths.

I know that these temperaments could be revised by going over and over each note and interval multiple times to get the speeds and ratios better, but I believe that if we can develop a sequence that, used by our super tuner, results in beat speeds and ratios closer to ideal after only one pass, then that temperament sequence tuned by us mortals will produce a temperament closer to ET than others that do not score so well, resulting in less refinement needed and faster temperaments.

Mark C:

What if a super tuner could tune every 4th at 1 bps, and every 5th at 0.5 bps?

The cycle of fifths only hits a few of them, but tunes every note once. The RBI sequence does the same; each note is tuned only once, (except for the reiteration of F3 and F4 in the skeleton) and the result is a cleaner RBI progression and a tighter SBI range.

How could we tune every note only once, yet set each P4 and P5. After hours and hours of effort trying to precipitate out some answers here, I have an uneasy feeling that progressive (or static at 1 and 0.5) SBI and RBI are not possible. You have alluded to this before and that's what inspired me to look at both SBI and RBI.

My feeling is that if it were possible to get all the P4 = 1bps and all the P5 = 0.5bps, the RBI would be even more wonky.

Do you have a sequence of P4/P5 that I could enter into my spreadsheet that would set each P4 and P5?

Another important element is the scale. I have identified three inharmonicity ranges, each identifiable with aural checks, that require different beat speeds for the RBI. For example:

For pianos with a scale that produces beatless F and A octaves when tuned as pure 6:2/4:2, F3A3 = 7.22bps

For pianos with a scale that produces beatless F and A octaves when tuned as wide 4:2, narrow 6:3, F3A3 = 6.82bps

For pianos with a scale that produces beatless F and A octaves when tuned as pure 4:2, very narrow 6:3, F3A3 = 6.74bps

Have I missed something? Why would you deliberately want to set each 4th and each 5th to the same beat speed through the temperament?

Correct me if I'm wrong, but the beat speeds of 4ths and 5ths are supposed to progress just like the other intervals. Granted, it's very subtle, but when the temperament is done really well, the progression in beat speeds in 4ths and 5ths is definitely there and audible.

That being said, if one were to set all the 4ths and 5ths to exactly the same beat speeds through the temperament, of course there's going to be something wrong with the resulting 3rds and 6ths.

Personally, I tune the Sanderson/Baldassin sequence, where after tuning the CM3s four fourths are tuned and refined to fit in the given starting octave size. I've always tuned these four fourths beating the same, that is non progressive. I tune the rest of the fourths and fifths in the temperament, from F3 up to A4, just in the same way.

It has been demonstrated here that no tuner is able to tune an even progression of RBIs, no to speak about tuning an even progression of SBIs.

I think that SBI progression is just an illusion, kind of a "tuner's dream".

It is the only realistic goal we have at the moment. I do not have a sequence that produces progressive SBI without refinement.

I offer to you that you may think P4 and P5 can advance smoothly and also M3 and M6 can advance smoothly, but take one of your tunings and measure the beat rates using a band pass filter like I do in http://howtotunepianos.com/awesome-new-tool-to-help-students-tune-pianos-and-its-free/.

You will be surprised. I don't think it's possible. I was depressed after measuring one of my own.

One of David Andersen's defining parameters in his tunings is, within a very minor bandwidth, all the 4ths beat the same. Not only in the temperament, but the stretches compass wide also are set by consistently beating 4ths.

Following them can be somewhat tricky though.

Mark C:

Well, you are the one talking about a "super" tuner. Are you also going to define his kryptonite, after the fact. Let's call him Supertuner. If he can do what you cannot, tune your sequence exactly as intended, why can't he tune the sequence I propose but cannot, exactly as intended.

Really, it shouldn't be too hard to simulate on your spreadsheet. Kees has done this sort of thing. For a F3-F4 temperament octave, you cannot start with A. Instead start with C4 at 261.625. Tune F3 to beat at -0.5, G3 to beat at 1.0, D4 to beat at -0.5, A3 to beat at 1.0, etc. Finish with F4. See what you get.

With a logarithmic iH curve where F3 = 0.14 and F4 = 0.38 Here are the beatrates I calculate, simulating to the closest 0.10 cent, using Mr. Scott's tabulated values (Tunelab):

M3s: 6.85, 7.40, 8.10, 8.13, 8.78, 8.86, 9.56, 10.39, 11.20
M6s: 7.84, 8.42, 9.10, 9.70
The F3-F4 octave 2:1 +0.34bps, 4:2 +0.43bps, 6:3 +0.04bps.

Then again, I guess you could start with A. Just go one direction until you hit F3 and then the other until you hit F4. It should have the same results.

With all respect, Mark, you will make yourself crazy trying to achieve perfection in temperament tuning.

Let's consider the P.T.G. tuning exam as a "standard" against which to judge a good quality temperament. How many musicians do you know of who can hear the difference between a score of 90% and 100%? O.K. there are a few out there - I'd guess maybe one or two out of a thousand professional musicians could tell - but, only if they were specifically listening for it. They would never notice if simply playing as they normally do - Bach, Debussy, Gershwin, etc. How many would hear the difference between say 95% and 100%? Hmmm...- not many. At least in my 45 years as a professional musician and 38 years as a piano technician I've never met such a person.

Theoretical discussions about what a temperament is supposed to sound like are fine and can be enlightening. But, when you go into the customers home to tune the piano it might be best to leave your engineering brain in the car.



[i][/i]

Hi Jeff,

I will try. There are two problems with it though.

1) It means I have to check every fourth and fifth in the temperament. That's 24 checks instead of only 12. We probably do more than that in a normal temperament setting, but Supertuner is a lazy guy.

2) The iH curve. I have defined what I call the octave spread. It is the difference in cents between the 6th and 3rd partials of a pure 4:2, which equals the difference between the 2nd and 4th partials of a pure 6:3, given the same scaling, and the sum of the differences between 2 and 4, plus 3 and 6 for every size in between.

What's more, I have confirmed that there are three ranges of octave spread that require different beatless octave sizes. And these thresholds can be measured aurally! Imagine. Measuring B with your ear! Very cool. So arbitrarily choosing an iH curve and doing these calculations does not confirm that the resulting F3F4 will be beatless.

I will do what you suggest and post the results.

Hi Gerry,

Thanks for the post.

What used to make me crazy was having to keep refining my temperament after having set what I thought were accurate pitches.

The graph you see of the small sequence with small octave is not arbitrary math, it is a mathematical representation of how I actually tune. If my final tuning doesn't look like that, it just means my ear couldn't hear the small differences in beat rates which I think we should be able to. My research shows most tuners can easily hear a 3% difference in RBI. Chromatic intervals have a difference of 6%.

Also, my search for a superior sequence means I have less refining. Right now, my time is longer, 40 minutes for the temperament, but I expect it to get much faster. And that's using DSU!

Also, smooth RBI and consistent SBI are a requirement for being able to choose and set precise and accurate treble temperaments. I.e. Pure 12ths, pure 22nds, or pure 19ths. They all sound great but different. Without smooth RBI and consistent SBI, these treble temperaments wash into each other and it is a job just to get the treble to sound good, forget about trying for a specific colour.

The challenge of finding and evaluating a mathematical model that is built upon aural assumptions that are used to actually tune a real piano, means I can analyze the errors produced by those assumptions and thereby give my sequence more credibility; evidence that it is a very powerful way to target and execute ET without guessing.

Uh, check your math. There's only 12 4ths and 5ths in an octave.



OK, tune F4 to F3 as any octave you choose. It probably won't make a difference in the RBI progression.



Uh, check your math again, that's three problems.

I'll tell you what this reminds me of. Imagine a City Manager writes out a bid request for 1,000 cubic yards of #2 gravel, but it has to be blue-green in color. It is going to be used in french drains and so no one will ever see it. The color just doesn't matter. BUT, it just so happens that HE is the only one that owns a gravel pit with blue-green gravel. Guess who wins the bid? It would be called corruption, (especially when someone checks and finds out the gravel isn't green-blue after all.)

In your case, by stipulating parameters that only you feel are necessary, and only your sequence can do, it becomes a self-fulfilling prophecy, doesn't it, (especially when someone checks and the RBIs aren't progressive after all)?



Look forward to seeing what you come up with.

First of all, a tuning like this guarants good sounding P4s and P5s but nothing can be said about the quality of octaves which may sound dirty depending on the piano's scale.

Second, a real piano has not a logarithmic iH curve and can have more or less random if not wild partials. If you tune the temperament based on P4s and P5s you are ignoring patials 5 and 6 of each note and thus M3s 5:4, M6s 5:3 and m3s 6:5 will have unknown random beat rates. And FWIK that can not be considered ET.

Rafael:

As the saying goes, "What is sauce for the goose is sauce for the gander."

If Normaltuner tried to tune 1bps 4ths and -0.5bps on Normalpiano all sorts of unpredictable errors would crop up. Some would make things better, some worse. But this is a simulation where Supertuner is tuning Superpiano.

But what does the simulation show us mere mortals? 4ths and 5ths need not be progressive, and making them beat 1bps/-0.5bps is going to be very close to ideal.

[Edit:] I don't think you can have bad octaves with good sounding, correctly tempered 4ths and 5ths. You might decide to make them "different", but if it makes the 4th and 5ths sound bad, it is a bad choice.

F3A#3
F3C4
F#3B3
F#3C#4
G3C4
G3D4
G#3C#4
G#3D#4
A3D4
A3E4
A#3D#4
A#3F4
B3E4

We're both wrong. Please try to understand that I am not trying to be some kind of bubble headed introverted mathematical bumble head. I am looking for mathematical models that confirm why my temperaments are so good after one pass so I can teach it to others. See?

Don't tell me I'm wrong about my own definition of octave spread. Obviously you don't have a clue what I'm talking about. (I will take some responsibility for that becuase I haven't defined it very well in my posts. But my book will be much clearer.)

Try it yourself. Tune a beatless octave.
Record your interpretation of the check intervals.
Record the check intervals and measure them.
You will find one of the following cases (example beatless A3A4)

Case 1:
F3A3 sounds equal to F3A4
A3C4 sounds equal to C4A4
This means the difference between F3A3 and F3A4 is less than 3.6% (From my research) I call this a pure 6:3 even though it is not mathematically, but aurally it is.
Calculate the octave spread. It will be less than 0.5 cents.

This creates a certain B curve.

Case 2:
F3A3 sounds slower than F3A4
A3C4 sounds faster than C4A4
AND this is a beatless octave as tuned.
This I call a median octave. Wide 4:2, narrow 6:3.
Calculate the octave spread. It will be between 0.5 and 1.1 cents.


Another B curve.

Case 3:
F3A3 sounds equal to F3A4
A3C4 sounds much faster than C4A4
Remember, this is a beatless octave, tuned by yourself directly.
This I call a pure 4:2 which accompanies a very narrow 6:3.
Calculate the octave spread. It will be more than 1.1 cents.

Also a different B curve.

Each case also creates a different m3/M3 equality. If anyone is using any m3/M3 equality when tuning their ET temperament, they need to know what the octave spread is, or they could be tuning incorrectly. No problem though, just more refining needed.

If you are close to one of these cases but you can hear differences, retune using the RBI and see if you don't get a more beatless octave. I wouldn't tell you to do that if I hadn't already found this to be true.

Try to tell me I'm wrong now. But first do all the recordings, calculations, and analysis that I've done.

I have no idea why you think I did so. Call your octaves what you want and define them as you care to. In the end they are just preferences.

Not preferences. Unless you prefer beating octaves. (You may be responding before I have finished editing my posts)

Isn't that telling me I'm wrong?

No, it was a joke about how many "problems" you saw. You started by saying "There are two problems with it though...", then numerated them (1, 2) and then continuing with "What's more..." which really makes three problems.

I have not looked at any math having to do with what you are doing with the octaves.

Sorry Jeff. My apologies. I didn't get the joke.

Here are some more graphs that may show you what research I've been doing. Please try to do the octave test I described. I hope you find it interesting.


There's an error here. Negative slope indicates a typical octave that can be tuned as a wide 4:2, narrow 6:3. Positive slope indicates a reversed octave that can NOT be tuned as a wide 4:2, narrow 6:3. It only works out as a wide 6:3, narrow 4:2. They're rare.


Same error here.


Note: TP refers to Tremaine Parsons' graph limits