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I have some ideas how I would go about it, but have never tried. I am in love with 4:2 octaves in the mid section.

If no one else posts how to tune it aurally, I will share my ideas in a day or two.

Oh, yes it does. The article claims that any such measurement can't be done in that amount of time, but it can--over and over. Your appeals to the sampling theorem also do not dispute my counter example. Counters running at 10 megahertz, 100 megahertz or whatever, can be used to measure the period of middle C. That sample rate greatly exceeds the Nyquist criterion, so the sampling theory agrees that my example method is valid.

Finally, and most importantly, the article used the energy of a photon to purportedly prove a claim about measuring the frequency of musical tones. Using the energy of a photon may well make a valid proof about measuring the frequency of light, but unless someone can explain what the energy of photons has to do with measuring the frequency of sound waves or vibrating strings, then my conclusion is once again that the article is pseudo-science.

The Stopper tuning can be done aurally, in fact Bernhard Stopper was doing that for quite a few years before he developed the ETD.

To tune it aurally, you need to play the fifth above the octave along with the tonic. Unless you have pretty big hands, :^) you need a bar that will reach this intervall and play those keys together.

The formulas in that article are not using the speed of a photon (quote):

"the formula can be transformed into the relation between time and frequency of waves".

Then, it is transformed and even marks:

"...Here, there is no 'h'... The position and momentum of a wave in space have been changed in form to the time and frequency of a wave..."

Plank's constant is out and the connection with the electromagnetic world is gone. The inequations he uses deal only with time and frequency, which can be used for sound. The speed of a photon is not being used there.

I mentioned Nyquist-Shannon because you said "I believe by precision, he means resolution". The Nyquist-Shannon theorem is about sampling (it is relevant to resolution), and it is not being discussed here. Sampling frequency is no substitute to longer measurements according to the article being discussed. That's why I mentioned it, because you are talking about something else, not about uncertainty.

I know counters that run at high frequencies, how they work and what they can do. They work great (as in "good enough" taken to amazingly high standards), and that is not being discussed. That article is not about electronics, but about a physical limitation that can't be overcome.

The point of the article is not that those counters don't exist or don't work. It's point is:

"If you want to get the exact Middle C, you need an infinite time and a continuous wave of C."

If time increases, you can get a closer approximation, with shorter measurements you can't be that approximate. It doesn't matter what you use to make those measurements.

*Exact* is the keyword here. You cannot get exact measurements, period. What you can get is approximations. We can currently have great approximations, "good enough" for A LOT of applications. There are devices that give AMAZING measurements, but that doesn't mean exactness.

We can make a lot of things to try to get as close as possible, but that doesn't mean being exact.

I am not saying we cannot measure a frequency. I'm saying we cannot measure it and be 100% sure we are 100% right. I am saying we can have very good estimations and things work great: my computer works, ETD's work, software based ETD's running in PDA's/phones/laptops and what not work great, but are not exact (they are much more precise than we, humans, can be).

I have no elements to prove that point discussed in the article is wrong, but am not saying it can't be proven wrong. It could indeed be utterly rubbish, but so far I have no elements to prove it is.

To prove it is wrong, you need to show the maths are wrong, or that some physical principle is wrong or incorrectly applied. This kind of thing goes beyond current technology, because the article deals with a physical limitation and you want to use technology that has ITS OWN limitations. You need to overcome the limitations of that technology to attempt to prove the physical limitation discussed in the article.

To prove it wrong by making measurements you need to aim for *exact* repeatable measurements with current technology. Is that possible?

http://en.wikipedia.org/wiki/Measurement_uncertainty

He eliminates 'h' by using the energy of a photon, which contains 'h'. In so doing, he ends up an 'h' on both sides of the equal sign, and he is thus able to divide it out. Without using the energy of a photon, the 'h' is NOT eliminated.

That is the whole point of what I've been saying. His derivation relies on sticking in the energy of a photon, and then, with no explanation or justification, claiming that the result is valid for any kind of wave. Unless you or anyone else can give us valid reasons why using the energy of a photon in this context is valid, his thesis is just so much bologna.

Look carefully at his equations and you'll see his slight of hand. I have no interest in discussing this any more. I think I have clearly stated the flagrant problem with the author's derivation. The math doesn't lie. This is not an argument about what precision or exactness means--what's at issue is ONLY the author's mathematical derivation. I'm done.

Besides arguments about the theory behind the tuning, I still am curious about the tuning software. It's hard to get any information. There is virtually no information on his website. When I contacted Mr. Stopper a few weeks ago, he emailed back and just said to read the mailing lists and forum. That was it. Well, the PTG archives have been down. You can find some cached pages. There must have been some high praise on the PTG lists that he is banking on. There are no trial versions of the software available, so you can't try it out.

The basic concept of what he is trying to do has potential. If he says that the program doesn't use memory, I think he just means that it doesn't stored calculated tunings. If the program has one set stretch standard, this simplifies calculations, doesn't it? It wouldn't have to take measurements and do calculations before you start. It could be doing calculations in the background as you tune and it gathers information. It obviously would use some memory to run the program. It must take some sort of reading, and maybe it reads inharmonicity as you tune. But, since there doesn't seem to be a way to get information on this, we are left to guess.

Since no one has posted on how to aurally tune beatless twelfths I will give it a try, even though I have never tuned it. I choose to refer to this tuning as beatless twelfths rather than the “Stopper” or “OnlyPure” method. I am not sure what those are and do not want to give out misinformation on Mr. Stopper’s work.

Since this is based on beatless twelfths, it seems that it should only be used with Equal Temperament. Otherwise, any key color will shift by a fifth for each twelfth of the piano's scale. A full keyboard arpeggio would progress through many tonal colors.

When tuning perfect twelfths there is a stretch added beyond what is required by inharmonicity for beatless 2:1 octaves. The additional stretch is 2 cents per 19 semi-tones, or 0.1 cent per semi-tone. So an octave will have 1.2 cents, a fifth 0.7 cents and a fourth 0.5 cents of additional stretch. The fifths will beat one third slower and the fourths one quarter faster than they would when tuning with beatless 2:1 octaves.

The A3-A4 octave should be wide by one beat in three seconds. When this is established, a check of octave type can be used as a reference to tune other octaves until sufficient notes are tuned to allow beatless twelfths to be tuned. The m6-M3 test for 8:4 octaves may be helpful.

An RBI test for the beatless 3:1 twelfth is the M6-M17 test. Example: F2-D3 beats the same speed as F2-A4

A SBI test for a beatless 3:1 twelfth would be the P5-P8 test. Example: D3-A3 beats the same as A3-A4.

Since the twelfth uses the low partials of 1 and 3, but RBIs use the higher partials from 3 thru 6, a sudden change in RBI beat rates can be expected across the tenor break on smaller pianos.

A tool that spans a twelfth on the keyboard is probably a good idea, especially at the extreme ends of the piano.

I skipped the math, but if someone asks, I will provide it.

RoyP:

I suspect, but do not know, that Stopper’s tuning device tunes to theoretical pitches based on the 19th root of 3 and does not measure inharmonicity.

This could be a melodic model of the human ear and the best compromise between pianos. In other words, this may be the pitches that people want to hear and also may be a great way to tune pianos with different scales so that they sound good together.

It is clear that what you say is correct. But it is not enough to tune the Streched ET of Mr. Stopper.

How do you set the 19 notes in the temperament 12th? A la Braid White? You tune from the A4 note by fifths, expanded fifths, until you reach the 12th to check if it closes the circle of 12ths?

Or can you proced with another scheme a la CM3rds, that is: tune A4 to 440 Hz, then D3 a pure 12th below and from that point subdivide this 12th in smaller intervalls in order to tune the 19 notes that build up your 12th?

The problem here is that 19 is a prime number, it is not divisible, so you can't construct the 12th adding smaller similar intervalls, you can't subdivide the 12th in smaller similar intervalls.

You only can estimate expanded octaves,fiths, fourths, etc. to try to build up your pure 12th. And this estimations may be wrong and uneven, ruining the "expanded ET".

No, I was wondering, hearing Jurgen's words and Stopper's claims about Onlypure tuning, if there was a procedure to tune a circle of "tonic-fith-octave" notes in which you obtain the cancelation of beats. I guess that it is this cancellation of beats that can help to be sure you are on the right way when setting the 19 notes of the temperament 12th.

Hasn't Mr. Stopper published a book or article about the Onlypure method?

I read somewhere that Mr. Stopper's software takes iH into account(I think it was in this very thread).

In order to achieve pure 12ths, the software must tune the third partial of one note to the first partial of the second note, if not the 12th won't be pure. That is: if it tunes the fundamentals based on the 19th root of three then the 12th won't be pure, because the third partial of the low note will not coincide with the fundamental of the high note.

Gadzar:

I think that any method of tuning ET could be used. As the twelfths are tuned, it would be corrected and refined by using the usual chromatic tests.

Unright,

I believe this is a different temperament than meantone, well or ET in that it is not based on octaves but on 12ths, thus the methods used on those temperaments are no more valid. Stopper's tuning asks for different techniques to set up this "cancel of beats" effect in the tonic-fifth-octave. I think it goes beyond the "tuning pure 12ths" concept.

Gadzar:

I respect your prerogative to believe that this is something different.

I think this is only a stretched ET. By ET I mean that there is no difference in key color.

If D3-A3-A4 is an example of the “tonic-fifth-octave” that is being mentioned, then it is the same as the P5-P8 SBI test for a beatless 3:1 twelfth. It does not matter where the middle note is, within reason. The fifth and the octave will have the same beat rate. This may be why I like 4:2 octaves so much. The fourth beats the same as the fifth in the P4-P5 test.

Ok. This is a stretched ET, but the amount of stretching must be exactly established in order to accomplish the cancelling effect. You can have beatless 12ths without it being a 19th root of 3 temperament. It is more than a beatless 12th that is involved in it.

In ET you can have a circle of nice tempered 5ths without it being ET, for ET you must also have an even increasing rate of beats in major thirds, minor thirds, etc.

In Stopper's temperament it is not enough to have your octave stretched and an even run of raising thirds, fourths, etc. You must establish exactly the amount of stretch in all intervalls in order to get the pure 12ths and the cancelling beats effect.

Gadzar:

Thanks for continuing this. I am really enjoying it.

Because of inharmonicity, the 19th root of 3 will not produce beatless twelfths any more than the 12th root of 2 will produce beatless octaves. Only by tuning so that the partials match can an interval be beatless.

Any note can be tuned so that there is a beatless 12th above or below it. The other notes do not matter. And as long as the test note for the P5-P8 test is reasonable there will be equal beating within the “tonic-fifth-octave”. The P5 could even be wide with a narrow P8! To also have progressive beat rates, then the fundamental frequencies must also increase at steady rate, something a little greater than the 19th root of 3 due to inharmonicity.

The above paragraph could be reworded and still be valid when using the 12th root of 2 and any chosen octave type, or even the 7th root of 1.5 and the M6-M10 test for beatless fifths.

That's the point! I am not talking about a beatless interval nor equal beating intervals. I am talking about a beat in an interval that cancels a beat in another interval when sounded together. Furthermore this two beats have not the same rate. For example in A3-E4-A4 the 2:1 octave A3-A4 beats at +0.31 BPS, the 3:2 fifth A3-E4 beats at -0.47 BPS (minus sign for narrow interval, positive numbers for wide)and the fourth E4-A4 beats at +1.88 BPS.

The amazing is that when the three notes are sounded together you hear no beats, they cancell each other. That's unique to this tuning, no other temperament does that.

If you calculate de deviation in cents from ET you'll find that Onlypure tuning has an increasing deviation from 0 cents for A4 to -4.94 cents for A0 and to +4.01 cents for C8. As it is an expanded ET the amount of deviation in cents increases constantly as you get far from A4. In that aspect it is also unique! No other temperament or tuning system has this characteristique. In all octave based temperaments the deviation of a given note remains constant no matter what octave it is in. For example in Bill Bremmer's EBVT the notes C0, C1, C2, C3, C4, C5, C6, C7 and C8 all have the same amount of deviation of +3.8 cents from ET.

So when aurally tuning this temperament you can't simply use the same techniques or tests you use in ET. This temperament behaves differently.

You know I am an ETD oriented tuner. I am not able to program Onlypure tuning in my ETD, because I can not set the 12 figures it asks to set a temperament. In Onlypure system there are no 12 figures, there are 88 figures.

Even if I try to make measured tuning (interval tuning) with my ETD, I can't because I don't know how wide the interval must be and which partials are involved in.

The same thing happens when I try to tune it by ear, I don't know what to listen to! Except for the initial pure 12th D3-A4.

I hope you can understand what I am trying to explain, my bad english poses me strong limitations.

Gadzar:

I think your English is good, better than Jerry’s. How about an English as Second Language Story to lighten things up? On one particular ship I had a helmsman that was from Honduras. (All the Hondurans I’ve met were fine people.) Anyway, his English was adequate. One day a messman came up to the bridge to blab a while with the helmsman, but the helmsman did not say much. Later I asked him why. He said “Mate, I know just little English. (holding his fingers up about an inch apart) I do not want to waste it”

Did you actually hear the intervals that you mention and hear the resulting effect? Or are these beat rates what you calculated and are assuming that they will sound a certain way from what others have described? Also, I thought that we were talking about the P5-P8 intervals not the P5-P4 intervals. We seem to have been discussing different things. Glad you made this clear.

The idea of equal beating intervals “cancelling each other out” always bothered me. I know now that is not what you are talking about, but I thought I would mention it. Since the beats are in phase, they would reinforce each other, not cancel each other out. But since they are equal beating, the sound would be more pure because there would not be the sum and difference of the two beat rates confusing the sound. I suppose it is just semantics.

I don’t know much about ETDs so I can’t say whether my suggestions will work for tuning perfect twelfths. But if one of the “octave types” that can be selected is 3:1, then this should produce beatless twelfths. Or you could tune a beatless twelfth, measure the frequencies, do the math, tune the other notes and then tune beatless twelfths by measuring the third partial (direct comparison). Another possibility is to calculate the frequency for A3 so that the A3-A4 2:1 octave beats +0.31 bps, determine the octave type, enter this octave type into the ETD, and then tune the notes in this octave. Direct comparison can than be used for tuning the twelfths.

Looking at this mathematically I see it as a cut and dried interpolation / extrapolation problem.

The beat rate numbers that you gave brings up a problem. The numbers you give may be for a particular piano scale, or may be theoretical, but they could not possibly be for all piano scales. So, if these beat rates do produce a certain effect, it would not be for all piano scales due to differing iH.

You mention that beatless twelfths are unique: “As it is an expanded ET the amount of deviation in cents increases constantly as you get far from A4.” Any octave tuning scheme that accounts for iH will have this effect. This is shown with the Railsback Curve.

I guess you didn’t find any math errors in my earlier post. I used some “rules of thumb” and did it in my head.

Too bad the archives for the PTG lists have been lost by the new server. Stoppard did reply. He said it is not 19th root of 3. Very close. Fred Sturm from the U of T Austin covered that. I don't know enough of what he said to relay it here.

Thanks Keith. I will continue to refer to beatless twelfths and not "Stopper's" or "OnlyPure".

The beat rates I gave are theoretical values calculed from the 19th root of three.

You are right, this figures can not be used in an actual tuning because of iH.

The problem to tune direct intervals with an ETD is precisely the fact that those beat rates are theoretical. I can not get an accurate tuning if I tune a 2:1 A3-A4 beating at 0.31 BPS, because to take iH into account I must tweak this figure of 0.31 BPS to another value. Doing it aurally or mathematically leads to the same problem: what is the right figure for this particular piano?

In tuning ET the CM3 sequence is a way to tune accurate temperament without knowing the actual size of M3rds, what matters is the ratio on contiguous intervals.

There must be a similar way to tune a 19th root of 3 temperament 12th, in which it is possible to tune the piano to itself regardless of the theoretical or actual figures. Once you have tuned this 12th, you can tune the rest of the piano by tuning pure 12ths.