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Dear Alfredo,

I am a musician, but not a piano tuner or theoretician. Nevertheless, please allow me a question:

Imagine an ET tuning that
a) has the temperament octave size between 4:2 and 6:3,
b) is stretched so that 12ths and 15ths are equal-beating.

Is there any difference between this and CHAS? If there are differences, could you please elaborate on them?

I can answer this. The CHAS tuning has progressive beat rates in the 5ths over the whole of the keyboard. The narrowest 5th is A3-E4, approximately 1/2bps. The 5ths get wider from here, approaching pure in the bass, and eventually becoming wide in the top treble. Where this happens depends on the inharmonicity of the piano - keeping the equal beating octaves and 12ths causes the 5ths to widen (become less narrow then pure then wide) at a rate dependent on the inharmonicity.

As to how different this is to ET, I don't know. I suspect it is not very different at all. But one thing about CHAS theory is that it is not intended as a new theory. It is an updated mathematical theory that more correctly fits what we do when we tune pianos. The way we stretch the octaves, the checks that we do, these move our tunings away from the math-theoretical model of ET derived from the 12th root of two. What the CHAS theory does is provides a theoretical framework to look at what our tunings are actually like, which enables greater insight into the method of tuning. This has certainly enabled Alfredo to get a very solid handle on the way ALL the intervals should progress.

I disagree. Equal beating 12ths and 15ths do not produce 5ths that beat wide of just intonation. However, they do produce 12ths that are wide and 15ths that are narrow in the high treble. There is a point where they are both beatless.

Well, Alfredo, you are sort of underlining my thoughts. You resort into splitting my sentences up one after one, answering them, though you still didn't provide the 2nd part of your answer to my earlier post - which I eagerly await. One of the questions is exactly what Mark asked shortly after that:

Initial octave between 4:2 and 6:3 + equal beating 12ths/15ths = CHAS... yes or no?

Still, this implies only time and amount of experiences. You do not take talent, and - possibly the most important part - reflection into the equation at all.

A parallel from practicing an instrument: If you practice something wrong, for a long time, you get good at wrong stuff.

This NOT necessarily to question your own work, but definitely to challenge the sententious equation 'time=experience'.

Hello.

Thank you all for your feedbacks. It is pretty late here + 500 km. and seven hours work on a poor upright.

Kees, you say:...”How on earth can you tune an unequal temperament to Chas ET?”...

Sorry, I meant and understood “experienced” as “heard”.

..."Is what you lament (cadence) true for all ET tunings you have actually heard?
...Yes. And for all ET's that I haven't heard.“...

In my opinion, cadence enhancement may be a cliché. And when it really could be heard/noticed, they preferred a (theoretical yet intentional) juster tuning, ET. Otherwise, “enhancement” may well be a personal preference, though (in my logics) far from being universal.

..."What you refer to theory can be referred to practice?
...Sorry, I don't understand the question. If you mean can you hear the different size thirds in an unequal temperament in the cadences the answer is "yes". (Unless the difference is too small to be noticeable of course.)”...

The question was: could not you ear different size thirds too, in many Equal temperaments you've heard? Bill confirms about the large number of failed ET's he noticed. So I wonder what we are talking about. Is it fear for something that theory may cause, but we have not really experienced?

...“I don't want to be patronizing, but I really think you should educate yourself about temperaments if you make grandiose claims about having invented the ultimate temperament.”...

Be patronizing and relax, so to avoid being contorted. Ultimate temperament? No, Chas is only a Temperamental Theory that describes dynamic, ever developing, tuning forms. One of these forms can do for 12 semitones ET.

As for education and temperaments do not worry, I'm OK.

a.c.
.

Hello Jake, I'm trying to write fast, please excuse any mistake or approximation.

You write:...”I hope you know how much I love the sound...”...

You are so nice, thank you.

...”I feel as though I must say something here, however:

The word "gimmick" is very insulting in English--it means a cheap trick or deception. (The term doesn't mean "a device," or "a method of questionable value.") Any temperament could be called deceptive, in the broad sense of the word, which may be what you mean, since each temperament by definition involves a tempering of some intervals, but "gimmick" in English, is very strong.”...

Thank you, I could not guess that. In Italian we say “espediente” meaning one of many possible, convenient solutions, with no particular negative attribute.

...”EBVT is a well temperament, originally a development from meantone, isn't it,...”...

I cannot be sure. Actually I have my opinion and I'll post some June 2009 writings from Bill, together we may understand more.

...”with an attempt to avoid the wolf while at the same time being more precise in definition, and more regular, than an ordinaire?”...

To me, it seems that it is Victorian just out of “not ET” reasons, and Bill seems to confirm that the actual difference from modern ET is not noticed.

...”Well temperaments predate ET.”...

I do not get this.

...”If the desire to give the impression of consonant intervals while not creating actual consonance equates to deception, isn't ET really the largest gimmick?”...

Yes, good analysis. Chas modifies the concept of consonance. Pure, traditionally consonant intervals are dismissed so that now the sound-whole can be said pure.

Thanks a lot, a.c.

"...In Italian we say “espediente” meaning one of many possible, convenient solutions, with no particular negative attribute."

Ah! In English, the term would be the "expedient." Very different from a gimmick.

Mark, thank you for your question.

...”Imagine an ET tuning that
a) has the temperament octave size between 4:2 and 6:3,
b) is stretched so that 12ths and 15ths are equal-beating.

Is there any difference between this and CHAS? If there are differences, could you please elaborate on them?”...

Patrick made the same question and I'm about to post my reply.

One of my dearest friends is from South Africa. It must be amazing there...

Regards, a.c.

Thank you Jake and to all, I'm sorry, I shall say expedient.

Bill Bremmer, June 15, 2009.

...”Between what I do and what Bernhard Stopper does, although the two are aurally perceptive as different, there isn't really that much difference numerically or aurally as perceived by both casual and very educated listener alike. The difference amounts to a few cents here and there of manipulation of the temperament octave and midrange. If we take the "ET with pure 5ths" idea which I consider to be too extreme, there still is not all that much difference between a PTG Master Tuning and what would be required to produce those results in the midrange; a couple of cents worth at most. 

If there were very much more of a difference in any of these, the results would inevitably be perceived as unacceptable by at least some and that "some" would be far too many to try to convince. There is simply a limit on how far one can go before it is too far and that limit has a very narrow range.

Therefore, it begs the question whenever any one of these new and improved ETs come out which claim to have the ultimate answer to universal beauty, "Just where is that sweet spot?" If we don't hit it this time but get it the next, will anyone really ever notice or care?”...

What do you understand?

I understand that EBVT is a modern quasi-ET. If it was called FCDT (Few Cents Differing Temperament) it would be valid as much and it could open to other FCDT's, with custom-spotted color here and there.

Wouldn't this let Queen Victoria rest in peace?

a.c.

Well, most of my exposure is from CD's and concerts, where the temperament is usually OK I think.

A while ago I demo'd a recorder sonata with piano at my kids elementary school and had just a few minutes to touch up the (terrible) unisons on the old clunker. As it happened the temperament had drifted to have an almost pure CE M3. I actually liked it.

Another story is that in my music student days I bought a very cheap ($20) electronic keyboard just to have something at home to check my counterpoint assignments etc. Despite the horrible stops I grew quite fond of it. Later I discovered it is tuned in a bizarre quasi-randomized ET, undoubtedly to minimize electronic circuitry. It has perfect fifths FC F#C# and some wide fifths. I still like it. Bizarre but colorful.

See below for the temperament.



Kees

Patrick, I wrote: "...fund of field experience and elaborations, the fund...".

I meant to say Knowledge + practice + elaborations in time.

When you say...good at wrong stuff, I think you are right, so I tend to go for quality elaborations. This is why I'd talk about quality tuning. Sorry if I'm late with my second half reply, quite tight days.

...“Meanwhile here's a couple of really plain questions, that you shouldn't have any trouble answering. I think I'm not the only one who wonders about this, so please be as clear as you can:

1) ET can be tuned with wider and narrower temperament octaves, and with different stretches, but there is one thing that remains the same: ascending chromatic M3's and M6's have a progressive beat rate.”...

You may be back to theory now. Ask Bill about ET actual tuning, how many times could he find progressive M3's and M6's, when the tuning was meant to be ET. This could give you a measure of the gap between the first ET model and actual tuning practice. 

Anyway what does, in theory, remain the same? RBI's coherence, consistent and reliable progressiveness?

...“They do not have that at all in EBVT III. how could the EBVT III ever develop into being a modern ET, when it goes against the very nature of ET itself?”...

Develop? I cannot say. Perhaps you can check: compare EBVT I, II and III, then tell me. Bill says it is a mild WT, maybe very mild, perhaps very very mild. All this with ETD cents figures. You may comprehend why EBVT in practice may be closer to modern ET than many 12th root of two ET's versions. I guess it may develop even more or remain as it is, a modern quasi-ET. 

The very nature of ET? Do you mean conceptually? You may know, Chas ET refers to geometric proportions in nature. What is the point of going, conceptually, against the very nature of nature? Then we get one more unjustified conjecture, that our ear would be spoiled by ET....In my opinion, our ear was disturbed and oppressed because of the impossibility to tune 12th root of two and pure octaves.

...“2) From reading your instructions for CHAS, and reading other peoples comments on the initial octave size in CHAS, I think (and correct me if I'm wrong) it ends up right between a 4:2 and 6:3.”...

You say “between a 4:2 and 6:3”. That may be, I'm not sure. In any case, that “between” may not be precise enough, though practicable. Chas octave is part of a beat-whole, so it must cohere with all intervals. 

...“This octave is extended in a way that leaves only one constant: the equal-beating 12/15ths...”...

Yes, constant and even beating 12ths (delta-narrow) and 15ths (delta wide) all along. Though Chas form is not really the extension of one octave, it is the weaving of all intervals into a form. 

...”3) Now, If Bill sets up ET, he recommends the initial octave to be between 4:2 and 6:3. Then he extends the midrange utilizing equal-beating 12ths/15ths. Shouldn't Bills ET method give a high-quality CHAS?”...

I cannot say how high, and quality may depend on the execution. I guess that method, like others, can get close to Chas ET. Why not? If you then want to improve from ordinary quasi-standards you need to know what to aim at, then even the initial octave, for instance, may need to be corrected, and you'd want to manage all 4ths, 5ths and octaves + RBI's. For practical tuning I find fundamental referring also to more accentuated beat curves. 

...“If so, why do it the hard way - from theory to practice? I think you yourself wrote about that earlier, that CHAS stems out of your tuning development, not vice versa.”...

Should I explain the relevance of theory as a practical reference? I do not think so. The point is not how to get close to Chas, the actual point is how to share a dynamic approach and a formal optimum, how to dismiss unjustified assumptions and wrong teachings and how to reveal the correct tuning of all intervals.

It gets hard only if you move without references, though equal beating 12ths/15ths, as you may experience, is much of a progress. 

...“I do not put scientific research down, quite contrary - I think it is needed. But if the end result is similar to another way of setting the temperament (your own method), but they both end up in a CHAS tuning, is there a practical difference?”...

What you could do is not mixing temperament theory and tuning practice. As I say, Chas theory is not only about a method for setting the temperament, it implies a new approach to tuning, it is about a developing form that needs to be described (and understood) in dynamical terms.

If, how and when you get a similar tuning is a different matter. What may the practical difference be? You've said it, the degree of similarity. 

...“If my thinking sounds reasonable, and If a good pitch raise is done before the fine tuning:

4) could CHAS be tuned using bills very smart and down-to-earth tuning instructions?”...

If you mean equal beating tests either than Chas PRE-Form, I do not really know, inharmonicity may distort the tuning.

Smart and down-to-earth, both valid points, must not negate reliability nor contrast high standards exactitude. Any tuner may be able to obtain Chas ET through correct teachings. 

...“Could CHAS even be described as "an temperament octave size between 4:2 and 6:3, and an extension of the temperament utilizing equal-beating 12ths/15ths?”...

Back to question 3?

I hope I've met some of your needs. I'll be happy to deepen any other issue.

Regards, a.c. 

a.c.

Hello Not a Mongoose,

...”As to how different this is to ET, I don't know. I suspect it is not very different at all.”...

I think nobody could ever tune 12th root of two ET. But many stretched octaves tunings I have herd in Italy and around Europe were very close to Chas. Today, from France and Canada I'm getting the same feedbacks, and also Steve Fairchild and Bill Bremmer, perhaps many other colleagues have worked on a similar line. So, I can suspect too.

...“But one thing about CHAS theory is that it is not intended as a new theory. It is an updated mathematical theory that more correctly fits what we do when we tune pianos. The way we stretch the octaves, the checks that we do, these move our tunings away from the math-theoretical model of ET derived from the 12th root of two. What the CHAS theory does is provides a theoretical framework to look at what our tunings are actually like, which enables greater insight into the method of tuning.”...

Thank you for your comment, N.a.M. It is 2.54 am, I must take myself to bed.

Regards, a.c.
.

Nobody can tune a perfect fifth either.

Kees

Alfredo, Owen Jorgensen told me that to label the EBVT or EBVT III as "quasi equal" was not proper. That designation belongs to attempts at tuning ET which were imperfect, indeed the ET via Marpurg idea that I have is an example. The results are very close to true ET but not quite; close enough to receive a perfect score on the PTG tuning exam but still not perfect.

It may depend upon what any one person's idea of what "almost equal" is. Generally, however, the temperaments which are known as Quasi Equal are those 19th Century style temperaments that have no distinct well tempered kind of key color; no alignment of the Major thirds with the cycle of 5ths. Most often, the inequality does not seem to have a purpose; is is more or less random.

The temperament most often called "Marpurg" was not what Marpurg himself actually did but is a modern interpretation of a composite of ideas by both Marpurg and Neidhardt with the even more modern 4:5 ratio of contiguous Major thirds thrown into the mix. It has properties which are considered superior to the sound offered by true ET by the technicians who tune it: pure 4ths & 5ths, equal beating tempered 4ths & 5ths, evenly proportionate beat synchrony, etc.

Both versions of the EBVT follow the rules and reasons for tuning a Well Temperament. While it is true that those in the late 19th Century (the "Victorian" era) thought in terms of having all 24 Major and minor keys equally useful and may have called what they did, ET, the fact is that because of the sequence they used to tune the notes of the scale and the trial chords they played, they still tuned in the Well Tempered style.

I'll point out again that Owen Jorgensen wrote to me to tell me that the EBVT III is very nearly identical to one of the near ET ideas that Neidhardt had late in his life but long before the late 19th Century.

We are lucky to have the participation of DoelKees. I will say this with all respect for all participants here: It is easy to understand why piano technicians consider the ultimate perfection of ET to be the most worthy goal, really the only goal. It is easy to understand why technicians would believe as a premise that the more perfected the temperament (and whichever style of octave stretch is advocated), the more beautiful the music from the piano would sound. Virtually all of the piano tuning books have that as a premise. The creators and designers of electronic tuning devices (ETD) and software all have that as a premise.

The problem is that not everyone agrees. From professional musicians, both pianists and non-pianists alike to technicians who have taken the trouble to explore beyond what is considered to be conventional wisdom, to ordinary piano owners, there is, at the very least, not universal agreement about what the very best way to tune the piano really is.

We are coming into a time where ET can be and has been truly perfected, at last; but now people are finding that it doesn't necessarily satisfy their musical desires and sensitivities. Piano tuning is an art. You can't simply say that you have found the one and only way to do it and expect everyone to embrace it and never look beyond that one idea, whatever it is and no matter how and how well you may justify it. You can only offer what you know how to do. The future market will surely demand choices.

Piano technology has many such parallels. There is not just one kind of hammer felt for replacement. There is not just one kind of action part, not just one way to voice, not just one way to regulate, not just one way to refinish or otherwise detail a piano. There are really just as many tuning choices as there are for any other aspect of piano technology. This does not mean that I dismiss or discourage either the CHAS idea or that of Bernhard Stopper; all that I have heard of either one sounded just fine to me. I believe there is room for it all and people have the right to choose which style they like best.

As a designer of a tuning style, I believe you have the right to market or otherwise promote your concepts but to try to tell people that your ideas are the only one and correct way to tune will ultimately work against you. You need to respect opinions that differ from yours, even if they go against everything you have always believed. You may believe you have the perfect mathematical model, for example and the others may seem chaotic and even nonsensical but if the others seem to appeal to people's sensitivities and desires, it must mean that there is something that these other ideas offer that you have not yet understood. That is your challenge. Understand why at least, some people, prefer what you consider to be imperfection.

Let me offer an analogy that I believe may be helpful. Every year in Madison, Wisconsin, there is an "Art Fair On The Square". It is an exhibition of art of all kinds that is offered for sale on the streets surrounding the Capitol. I have attended it now for more than 30 years and have purchased certain items that appealed to me. One aspect I have consistently noticed during this entire 30+ year period is that much of the photography looked like paintings and most of the paintings looked like photography.

Obviously, a camera is capable of capturing a true image. But does it really? Are there other factors? Can the image be manipulated somehow so that it has more appeal? Is it the painter's task on the other hand to create an image so perfect that it looks like a photo? Should the painter create instead a very appealing impression rather than a forensic looking image that is exact in every way?

Which is true art in piano tuning? Mathematical perfection of a perfectly equalized scale or something else which is far more difficult and complex to define?

I appreciate greatly the opportunity to present my views and discuss what I believe in this forum.

I am a sort of new at this forum, but I really appreciate the way you are dedicated to temperament research and participating in this topic. The last few sentences are very vise and made me think a lot. I am sorry I haven*t discover this forum, and your articles much earlier.

Bojan:

As someone that used to work for the circus, let me advise you to not be decieved by the smoke and mirrors. Notice how little things are made important and important things are made little to suit the situation.

..."As someone that used to work for the circus, let me advise you to not be decieved by the smoke and mirrors. Notice how little things are made important and important things are made little to suit the situation."...

Be it referred to Bill's or to me, I've got no words. I formally complain for insinuating in such a way, with no arguments what so ever.

And I'm sorry, again I wonder if there is a point in all this.

a.c.

Hello Jake, Mark, N.a.M., Kees, Patrick, Bojan Babic and All.

I'd really like to elaborate more in depth, maybe once for all, what we were talking about last night? What do you understand by reading what Bill posted last June 15, 2009?

Bill, now you write:...Alfredo, Owen Jorgensen told me that to label the EBVT or EBVT III as "quasi equal" was not proper. That designation belongs to attempts at tuning ET which were imperfect, indeed the ET via Marpurg idea that I have is an example.”...

I say “quasi equal” after considering How you get it and What the aural result is.

...“The results are very close to true ET but not quite; close enough to receive a perfect score on the PTG tuning exam but still not perfect.”...

True ET? To me this is playing with both words. What is true? PTG's? Which ET do you refer to? 12th root of two ETD's stretched octaves variants?

...“It may depend upon what any one person's idea of what "almost equal" is.”...

I got that idea from you, from your own writings.

...“Generally, however, the temperaments which are known as Quasi Equal are those 19th Century style temperaments that have no distinct well tempered kind of key color; no alignment of the Major thirds with the cycle of 5ths. Most often, the inequality does not seem to have a purpose; is is more or less random.”...

Perhaps generally, but I'm now trying to talk precisely, possibly with few words.

...“The temperament most often called "Marpurg" was not what Marpurg himself actually did but is a modern interpretation of a composite of ideas by both Marpurg and Neidhardt with the even more modern 4:5 ratio of contiguous Major thirds thrown into the mix. It has properties which are considered superior to the sound offered by true ET by the technicians who tune it: pure 4ths & 5ths, equal beating tempered 4ths & 5ths, evenly proportionate beat synchrony, etc.
Both versions of the EBVT follow the rules and reasons for tuning a Well Temperament.”...

Yes, they seem to be reasons for a crusade against ET, as long as the rule, few cents ETD deviations, can result similar to an aural modern ET.

...“While it is true that those in the late 19th Century (the "Victorian" era) thought in terms of having all 24 Major and minor keys equally useful and may have called what they did, ET, the fact is that because of the sequence they used to tune the notes of the scale and the trial chords they played, they still tuned in the Well Tempered style.

I'll point out again that Owen Jorgensen wrote to me to tell me that the EBVT III is very nearly identical to one of the near ET ideas that Neidhardt had late in his life but long before the late 19th Century.”...

Near ET ideas? I'm getting more confused. If only you could state one firm, steady concept...

...“We are lucky to have the participation of DoelKees. I will say this with all respect for all participants here: It is easy to understand why piano technicians consider the ultimate perfection of ET to be the most worthy goal, really the only goal.”...

With all respect, you should not talk for all piano technicians, it could be more than enough if you talked coherently and congruently about yourself.

...“It is easy to understand why technicians would believe as a premise that the more perfected the temperament (and whichever style of octave stretch is advocated), the more beautiful the music from the piano would sound. Virtually all of the piano tuning books have that as a premise. The creators and designers of electronic tuning devices (ETD) and software all have that as a premise.”...

Excuse me Bill, who are you talking to, is it me?

...“The problem is that not everyone agrees. From professional musicians, both pianists and non-pianists alike to technicians who have taken the trouble to explore beyond what is considered to be conventional wisdom, to ordinary piano owners, there is, at the very least, not universal agreement about what the very best way to tune the piano really is.”...

I'm here for talking to colleagues and piano tuning practicers.

...“We are coming into a time where ET can be and has been truly perfected, at last; but now people are finding that it doesn't necessarily satisfy their musical desires and sensitivities.”...

More often, you'll have found not very satisfied people, due to poor ET tunings, and you seem to confirm. This may better explain your and your customers satisfaction for your modern quasi-ET tunings.

...“Piano tuning is an art. You can't simply say that you have found the one and only way to do it and expect everyone to embrace it and never look beyond that one idea, whatever it is and no matter how and how well you may justify it.”...

Are you talking about yourself? Why don't you want to understand in depth modern ET's?

...“You can only offer what you know how to do. The future market will surely demand choices.”...

More than future markets, I'm concerned with present tuners and tunings.

...“Piano technology has many such parallels. There is not just one kind of hammer felt for replacement. There is not just one kind of action part, not just one way to voice, not just one way to regulate, not just one way to refinish or otherwise detail a piano. There are really just as many tuning choices as there are for any other aspect of piano technology. This does not mean that I dismiss or discourage either the CHAS idea or that of Bernhard Stopper; all that I have heard of either one sounded just fine to me. I believe there is room for it all and people have the right to choose which style they like best.”...

OK, so?

...“As a designer of a tuning style, I believe you have the right to market or otherwise promote your concepts but to try to tell people that your ideas are the only one and correct way to tune will ultimately work against you.”...

So, what is your crusade about?

...“You need to respect opinions that differ from yours, even if they go against everything you have always believed.”...

How many colleagues in PW have told you about this problem of yours?

...“You may believe you have the perfect mathematical model, for example and the others may seem chaotic and even nonsensical but if the others seem to appeal to people's sensitivities and desires, it must mean that there is something that these other ideas offer that you have not yet understood.”...

You give a wrong example and a fairly shareable evidence to come to a conjecture, if not a wrong conclusion. Your tunings have sense more than your arguments. And here I'm not seeking people's appeal, my tunings go for that. Here I expect to talk with colleagues for making some technical analysis.

...“That is your challenge. Understand why at least, some people, prefer what you consider to be imperfection.”...

As you could read, I've almost made up my mind about EBVT, if you were by chance referring to it. Imperfection? I love it, I do not need to aim at it, as it is already part of myself.

...“Let me offer an analogy that I believe may be helpful. Every year in Madison, Wisconsin, there is an "Art Fair On The Square". It is an exhibition of art of all kinds that is offered for sale on the streets surrounding the Capitol. I have attended it now for more than 30 years and have purchased certain items that appealed to me. One aspect I have consistently noticed during this entire 30+ year period is that much of the photography looked like paintings and most of the paintings looked like photography.

Obviously, a camera is capable of capturing a true image. But does it really? Are there other factors? Can the image be manipulated somehow so that it has more appeal? Is it the painter's task on the other hand to create an image so perfect that it looks like a photo? Should the painter create instead a very appealing impression rather than a forensic looking image that is exact in every way?

Which is true art in piano tuning? Mathematical perfection of a perfectly equalized scale or something else which is far more difficult and complex to define?”...

I find the “art” argument too easy and exploitable. I also think that you make EBVT “far more difficult and complex to define” than what it could be, so that it can be sold as non-ET art.

I tend to leave “art” to not-self-named artists, I tend to believe that “art”, if anything, can result from deep self-consciousness and deep elaborations. I would never confuse art with smart expedients. I hope future art, in piano tuning, will not depend on what you yourself have stated, a bunch of ETD “cents here and there".

Your tunings Bill? I like them.

Regards, a.c.

CHAS Tuning MP3 - Amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf