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#2405385 04/01/15 02:25 PM
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After some new insights on setting aurally the pitch of A4 have come up recently, I am considering writing a PTG Journal article about it. I plan to write about various tools, including the tuning fork and a tone generator. However, a few things are still not clear to me, so I need some help to make sure I am correct in what I say.

A tone generator will have a strong third partial with no inarmonicity. That partial is a natural harmonic, so it will read +2.0 cents (perhaps +1.99 on a device which can read hundredths) but the piano string will have a difference (on the average) of about 4.5 cents between the 1st and 3rd partial.

If the inharmonicity is 4.5, we would expect to hear a faint and moderately rapid beat of 3.46 beats per second. I have been taking inharmonicity readings of many pianos on a daily basis to find out what the range is. An amazing number of different brands and sizes are all 4.5.

Outside of the Kawai 501 that we were puzzled at during the Chapter meeting which I recall was only a 2.0 (but I would have to do that again to be sure), the lowest reading I have found was on an old Mason & Hamlin at 3.4 (2.62 bps) and a Shigeru Kawai at 3.5 (2.69 bps).

On the high end was an 100+ year old Mehlin (NYC) Upright at 6.5 (5.00 bps), an Asian made Story & Clark console at 6.1 (4.69 bps) and a Baldwin Acrosonic at 5.7 (4.38 bps).

The question is how well anyone can distinguish between the usual approximate beat of 3.5, 3.0 or 4.0 bps? At the fundamental level of A4, how man beats per second error does it take to create a difference of 0.1 cents? This is the most important question because the claim is being made that virtually anything between 3-4 beats per second would result in a 0.0 reading on the fundamental of A.4 on virtually any piano.

I can easily see how a person such as Lucas Brookins, even though he is a novice tuner, could get that "certain beat rate" in his head because he is a percussionist. Every Steinway and every Yamaha piano is 4.5 and many Kawai pianos are either 4.5 or very close. He had been practicing on the Steinway and every piano he tuned the day before his exam was a Yamaha and he was "nailing" a 0.0 every time!

Where this will be important, of course will be on the PTG Tuning exam and virtually nowhere else that I can imagine. However, there are and always will be people wanting to take that exam and do the best they can. The pianos used at the PTG Institute are almost always a Kawai or Yamaha but could be another brand like Young Chang or perhaps Hailun which I have not yet had a chance to measure.

The question I had about how many cents at E6 does it take to equal 1 bps was answered as 1.3. Does this mean that to make a 0.1 cents difference at A4, it only takes a 0.13 bps error or am I completely wrong? Please help!

Whoever can provide the right information will also be confirming or refuting the claim that all you have to do to "nail" the pitch is to find that faint little beat and it does not have to be very exact.

After all, they did change the exam after some people were saying that "all you have to do is pound" to get a 100% on Stability.


Bill Bremmer RPT
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Originally Posted by Bill Bremmer RPT
A tone generator will have a strong third partial with no inharmonicity. That partial is a natural harmonic, so it will read +2.0 cents (perhaps +1.99 on a device which can read hundredths)...
Not sure why it wouldn't be 0 cents instead (since an electronic sound) - but I could be wrong as I don't know which type of generator you use.

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I can easily see how a person such as Lucas Brookins, even though he is a novice tuner, could get that "certain beat rate" in his head because he is a percussionist.
That's true. I'm no pro tuner at all, but as a musician it's easy for me to hear a metronome at 60 in my head and from that create a 4-beat or 3-beat, so in other words pretend to hear 3 Hz/4 Hz.

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The question I had about how many cents at E6 does it take to equal 1 bps was answered as 1.3.
That's correct (frequency around 1323 Hz). Just tried it on my little custom Excel sheet thing. If you have MS Office, it's easy to program (can tell you via PM if you need).

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Does this mean that to make a 0.1 cents difference at A4, it only takes a 0.13 bps error or am I completely wrong? Please help!
A 0.1 cents difference at 440 Hz gives a 0.025 bps/Hz error. A 1 cent increase at 440 Hz gives a frequency of 440.254 Hz, so about 0.25 bps/Hz or 1 beat per 4 seconds. The formula is :

resulting freq = initial freq * (2^(number of cents/1200))

Then by subtracting both frequencies, you get the "beat" speed. Hopes this can help.

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Bill,

What device or method are you using to measure these numbers? And what is your protocol for measuring it? The reason why I ask is because your posting above is really ambiguous without explaining exactly what the cents readings are referencing. Cents is a ratio and doesn't mean anything unless you state unambiguously what the cents offset is in reference to.

When you say something like "The inharmonicity of the A4's I've been encountering ranges from 3.4 to 6.5", that means nothing unless you specifically state what is it you're measuring. It's better to say "On the A4 strings that I have encountered, there are a range of inharmonicities, such that when I measure these strings at the E6 pitch level (A4 being as best tuned to 440), the cents offset from perfectly harmonic has ranged from 3.4 cents to 6.5 cents sharp."

It seems like a convoluted way of arriving at a beat speed anyway...

You could just avoid this whole problem and measure everything in Hertz and be done with it. Giving the frequency in Hertz of a fundamental, a partial, or a theoretically perfect harmonic would be totally unambiguous. Then you wouldn't even have to bother converting cents offset at the E6 third partial into an estimate of beats per second. You could just arrive at beats per second by subtracting.

Oh, and I disagree with your sentence "A tone generator will have a strong third partial with no inharmonicity". Sorry, but you can't possibly state that all tone generators out there will meet one, the other, or both of these criteria. The KORG ones are pretty close, sure. But not all tone generators create a tone with a timbre like the KORG.

Hope this helps,

Last edited by Chris Storch; 04/01/15 08:36 PM.

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Using BPS exclusively would help aural tuners like me. Are you using cents to include the ETD crowd?


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I crunched some numbers, summarized by the table below.

Code
        bps tuned: 2.5   3.0   3.5   4.0   4.5   5.0   
bps correct: 2.5  +0.0C +0.7C +1.3C +2.0C +2.6C +3.3C 
bps correct: 3.0  -0.7C +0.0C +0.7C +1.3C +2.0C +2.6C 
bps correct: 3.5  -1.3C -0.7C +0.0C +0.7C +1.3C +2.0C 
bps correct: 4.0  -2.0C -1.3C -0.7C +0.0C +0.7C +1.3C 
bps correct: 4.5  -2.6C -2.0C -1.3C -0.7C +0.0C +0.7C 
bps correct: 5.0  -3.3C -2.6C -2.0C -1.3C -0.7C +0.0C 

I assume the user tunes A4 to beat at various rates at E6 with the tone. Then I consider pianos where the correct bps should be specific values from 2.5 to 5.0. The C entries are the resulting errors of the tuned A4 in cent, given what the bps should be and what it was actually tuned to.

BTW all electronic tone generators (at least the ones which have a musical purpose) will have E6 a perfect harmonic of A4. If not it would not be useful as a reference pitch for tuning, and they would have had to make a special effort to make it non-harmonic as a sawtooth, or any strictly periodic waveform, has perfect harmonics.

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The tone generator I believe Bill is referring to is one that I built. And Bill, it was the 5th harmonic that I was unable to suppress in that design. The 3rd harmonic is quite suppressed. You may still be able to hear it, but it is much weaker than it would be in a standard square-wave tone generator. Of course all the even harmonics are suppressed naturally because the waveform is symmetrical. I suppressed the 3rd harmonic by synthesizing a 5-level stepped waveform: 0,+A,+B,+A,0,-A,-B,-A,0. With the proper choice of levels for A and B, the 3rd harmonic content is zero. This design was cheaper to implement than an ideal sine-wave synthesizer that would have had no harmonics at all.

As for why a 3rd harmonic would "read +2.0 cents" on an ETD, I think Bill is referring to reading the tempered 12th, for example, if the tone generator produces 440 Hz and the ETD is set to read E6 (tempered, of course).

@Chris: You wrote:
Quote
"Oh, and I disagree with your sentence "A tone generator will have a strong third partial with no inharmonicity". Sorry, but you can't possibly state that all tone generators out there will meet one, the other, or both of these criteria. The KORG ones are pretty close, sure. But not all tone generators create a tone with a timbre like the KORG."

Bill is quite right. All tone generators, regardless of their timbre or voicing, will have perfect harmonics with no inharmonicity. It takes a great deal of complexity to synthesize inharmonicity. High-end electronic keyboards do it by sampling real pianos and playing back those recorded samples. But simple tone generators that produce continuous periodic waveforms must have zero inharmonicity.


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I will start over. This topic is specifically about the Pitch test on the PTG Tuning Exam. It is the only instance that I am aware of where someone must tune aurally but the goal is to have the note A4, as measured electronically on the fundamental (not on any other partial) read as close to 0.0 as possible.

The test has a tolerance of +/- 1.0 cents. Beyond that, each 0.1 cents is a point off. That means that one can be +/- 3.0 cents but beyond that, the Pitch score is failed. It is considered to be so difficult and challenging that the examinee is given a second chance if the first attempt fails. If upon the second attempt, the Pitch score is below 80, the entire exam is failed, no matter how well one may do otherwise. The examinee is allowed 5 full minutes to attempt to tune the note A4 as best as can be done.

In my entire 24 years as an examiner, I had never once seen anyone's Pitch measure exactly at 0.0. It was always off to some degree or another, even if it was within the 1.0 cents tolerance.

Recently however, my apprentice, Lucas Brookins had discovered a technique that allowed him to tune the A4 to exactly 0.0 easily and quickly, within a matter of a few seconds, actually. He had been able to replicate that many times. Therefore, when it came time to actually take the exam, he did it perfectly. He only took 44 seconds but as he said, the remaining moments he took were spent using other aural tests which merely confirmed that what he did in only a few seconds was exactly right without so much as 0.1 cents readable error.

The device I use to read and score the exam is a Sanderson Accu-Tuner IV. It does not read 100ths, only 10ths. It is well known that any ETD is somewhat less sensitive when reading the note A4 on the fundamental compared to reading on a high partial.

The tuning fork has its obvious limitations, so many people wanting to take the exam prefer to use something else. An electronically generated tone is permissible, as long as there is no visual display. Therefore, a device such as the Sanderson Accu-Fork is allowed. So would be the Hale Electro Fork. An electronic metronome with an A4 pitch would also be allowed but one must be sure its pitch is properly calibrated and often, they are not exactly right.

The problem with one of these electronic tone devices is that they have an audible 2nd and 3rd partial which have no inharmonicity. The 2nd partial will be an exact multiple of the fundamental and the 3rd partial with be an exact octave and 5th with no tempering but also no inharmonicity. Therefore, the Sanderson Accu-fork I use will generate a perfect 0.0 on A4 read at the fundamental and on the 2nd partial. If the Accu-tuner is set to read the 3rd partial of A4, the Accu-Fork will read 2.0 as expected.

Any piano's A4 tuned electronically to exactly 0.0 at the fundamental will have the 3rd partial read sharper than 2.0, quite often 6.5 cents. If 2.0 is subtracted from 6.5, it leaves a difference of 4.5. When the piano string tuned exactly at 0.0 on the fundamental is sounded together with the tone from the Accu-Fork, a faint beat of approximately 3.5 beats per second can be heard.

So, what my apprentice discovered on his own and what some other people, notably Jim Coleman, Sr. have been saying for some years now, is that all that is really necessary to get a perfect score on the Pitch portion of the tuning exam is to tune an accurate unison with the A4 of the piano and the electronic tone, and listen for that faint and moderately rapid beat. The test notes, F2 and/or B1 are quite unnecessary and only redundant.

In this instance, the inharmonicity that is important is the difference between the fundamental and 3rd partial of A4 and nothing else. I have found that many pianos measure at 4.5 but some as low as 3.4 and others as high as 6.5. Those would produce a different beat rate but what I have found in my own trials is that if I heard a good unison and a faint, rapid beat, regardless of exactly what rate it is, I can get the A4 string to measure exactly 0.0 virtually every time.

In the March, 2015 PTG Journal, Jim Coleman Sr. claims an accuracy to within +/- 0.3 cents, well within the tolerance for the exam. It's another one of those, "All you have to do to ace the [whatever] score is...", claims.

When I wrote about it on another topic, there were immediately people who jumped in saying the whole thing was hogwash but one person produced a video of a man who can do that very operation perfectly virtually every time within a matter of seconds on just about any piano. I have found it to actually be true.

The kind of piano used on the tuning exam must meet certain specifications. Nearly any one of them will have that exact same amount of inharmonicity of the 3rd partial of A4 or at least be close enough to it that it doesn't really matter.

So, here is the question: How much error in interpreting the inharmonic beat rate between the 3rd partial of the piano string and the tone generator must there be for the A4, read on the fundamental, to show an error of 0.1 cents?

The best that anyone has been able to say is that you listen for a good unison and that faint beat somewhere between 3 and 4 beats per second and you will "nail" it. For some people, a full 1 beat per second error seems like a lot but it seems to me that it can be at least that much, if not a little more.

My Dad would have said, you know its right when it "has that little 'ring' to it".


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Originally Posted by Bill Bremmer RPT

So, here is the question: How much error in interpreting the inharmonic beat rate between the 3rd partial of the piano string and the tone generator must there be for the A4, read on the fundamental, to show an error of 0.1 cents?

The 3rd harmonic of 440 Hz is 1320 Hz. At this frequency, one beat per second (i.e. 1321 or 1319 Hz) is 1.31 cents off. So to get to 0.1 cent accuracy, you would have to set the 3rd harmonic to within 0.076 beats per second, or in other words, one beat in 13 seconds. Still, that is 3 times easier than trying to do the same thing at the fundamental (440). There it would take one beat in 39 seconds to achieve 0.1 cent accuracy. This does not mean that A4 must sustain for 39 seconds, or even 13 seconds, because it is possible to sense a fraction of a beat. If something is beating one beat in 13 seconds, you may be able to tell there is a beat after listening only 5 seconds.

But why stop at the 3rd partial? How about using the 5th partial (2200 Hz). At that frequency you would only need one beat in 8 seconds to detect a 0.1 cent error. Of course the effect of inharmonicity, which may be small at the 3rd partial of A4, is somewhat larger at the 5th partial.


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Today, I have 4 pianos at a senior home to tune. I will document each one. The first thing I will do is turn on the Accu-fork, tune A4 and then measure the results and then the 3rd partial inharmonicity plus the beats per second that is expected to be heard.

Chickering Studio (Aeolian). 0.0. Nailed it! About 15 seconds. Partial 3 Inharmonicity (P3I): 7.3 BPS: 5.6

This is the highest amount of inharmonicity I have encountered yet since I have been studying this. I could hear a solid unison and I could also hear that the rapid beat was faster than usual but it still sounded right and it was!



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Originally Posted by Robert Scott
Originally Posted by Bill Bremmer RPT

So, here is the question: How much error in interpreting the inharmonic beat rate between the 3rd partial of the piano string and the tone generator must there be for the A4, read on the fundamental, to show an error of 0.1 cents?

The 3rd harmonic of 440 Hz is 1320 Hz. At this frequency, one beat per second (i.e. 1321 or 1319 Hz) is 1.31 cents off. So to get to 0.1 cent accuracy, you would have to set the 3rd harmonic to within 0.076 beats per second, or in other words, one beat in 13 seconds. Still, that is 3 times easier than trying to do the same thing at the fundamental (440). There it would take one beat in 39 seconds to achieve 0.1 cent accuracy. This does not mean that A4 must sustain for 39 seconds, or even 13 seconds, because it is possible to sense a fraction of a beat. If something is beating one beat in 13 seconds, you may be able to tell there is a beat after listening only 5 seconds.

But why stop at the 3rd partial? How about using the 5th partial (2200 Hz). At that frequency you would only need one beat in 8 seconds to detect a 0.1 cent error. Of course the effect of inharmonicity, which may be small at the 3rd partial of A4, is somewhat larger at the 5th partial.

Thank you very much, Robert. This us the information I was looking for. Bye the way, I am using the Sanderson Accu-Fork right now because I loaned the pitch box you made me to another student. The tone from a metronome is too loud.


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Story & Clark 5'2" grand (made in China). P3I: 4.9 BPS: 3.8
I forgot to say, I got the pitch to almost 0.0. Maybe 0.1 or 0.2 off.

Last edited by Bill Bremmer RPT; 04/02/15 11:47 AM. Reason: additional comment

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Originally Posted by Robert Scott
Originally Posted by Bill Bremmer RPT

So, here is the question: How much error in interpreting the inharmonic beat rate between the 3rd partial of the piano string and the tone generator must there be for the A4, read on the fundamental, to show an error of 0.1 cents?

The 3rd harmonic of 440 Hz is 1320 Hz. At this frequency, one beat per second (i.e. 1321 or 1319 Hz) is 1.31 cents off. So to get to 0.1 cent accuracy, you would have to set the 3rd harmonic to within 0.076 beats per second, or in other words, one beat in 13 seconds. Still, that is 3 times easier than trying to do the same thing at the fundamental (440). There it would take one beat in 39 seconds to achieve 0.1 cent accuracy. This does not mean that A4 must sustain for 39 seconds, or even 13 seconds, because it is possible to sense a fraction of a beat. If something is beating one beat in 13 seconds, you may be able to tell there is a beat after listening only 5 seconds.

So to get an error no larger than 0.9cent (which will be rounded down to 0 at the exam, correct me if I'm wrong) you have to set the bps to the correct (unknown) value, to within 0.68 bps. Since the variations across different piano in "correct value" is apparently larger than that this method appears fundamentally flawed.

Maybe you could argue that the pianos used for the PTG tuning exam are all similar and have some specific known target bps, but then the whole method goes so much against the spirit of the whole exam. Do you really want to teach people to set A4 with a method that only works for PTG exam style grands?

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Well, Kees, this is what I am trying to figure out. Since yesterday, I "nailed" the pitch on a 1970's Everett console with a P3I of 6.5 and BPS of 5.0 and this morning, "nailed" a studio with the highest P3I that I have ever seen yet of 7.3 and BPS of 5.6 but was slightly flawed but still well within tolerance on a small grand with P3I of 4.9 and BPS of 3.8, the advice that if you hear a pure unison and a moderately rapid beat, you will score a 100% Pitch score on the exam is apparently correct.


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Cable-Nelson spinet. P3I:5.8 BPS:4.5. Nailed it! And now I think I know why it works despute differing amounts of inharmonicity and beat rates. You do not hear the rapid beat unless you are within a very small range of what would be 0.0. Now that, I think can be proven with numbers. Unless you are within 1 cent of target, there will be no Partial 3 rapid beat. A skilled tuner can hit the spot within a matter of seconds.


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"Teaching to the test" is one thing. Teaching to get far beyond a perfect score is another. And using a device and technique that no one would bother with outside the testing room to do so is wasting everyone's time.

Bill, are you mentoring piano tuners or professional test takers?

Yeah, this is a bit of a rant. I was thinking that there was something useful to be found in this somewhere, but there is not.


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So, Kees, Robert, anyone that could, please take the typucal 4.5 Partial 3 Inharmonicity with a beat rate of 3.5 and move away from it in increments of 0.5 cents. I already tried it at the tolerance limit of 1 cent on the moderately high inharmonicity spunet that I am tuning now. I can hear both the indistinct unison and a Partial 3 beat rate which is much too slow.


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Originally Posted by Bill Bremmer RPT
So, Kees, Robert, anyone that could, please take the typucal 4.5 Partial 3 Inharmonicity with a beat rate of 3.5 and move away from it in increments of 0.5 cents. I already tried it at the tolerance limit of 1 cent on the moderately high inharmonicity spunet that I am tuning now. I can hear both the indistinct unison and a Partial 3 beat rate which is much too slow.

I already posted a table which should speak for itself.

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Perhaps a more complex interaction is occurring that allows one to achieve the accuracy claimed, similar to the change in the apparent partial structure/reinforcement that occurs as one tunes unisons, allowing one to 'voice' the unison.

If I am correct in my assumptions, these 'tone generators' are fed through an amp/speaker device that adds some harmonic distortion to the signal. Perhaps that added partial structure in the generated tone interacts at other frequencies and the resulting mix has a 'quality' that can be perceived, but maybe not measured.

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Yes, thanks Kees, I will look at that table again when I am at home and can see it on a large screen. Meanwhile, a Piano Disc 5'2" Korean made grand. P3I: 5.7 BPS: 4.4 (almost the same as the spinet. Once again, I could tune the A4 pitch perfectly to O.0 using this technique.


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Bill, I hope this is not a silly question, but what is wrong with simply tuning A4 directly to the wheel of the Accu-tuner in Tune mode? I thought that a stationary wheel would mean at least 0.1 cent accuracy.


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