Alfredo:

I know you know what narrow and wide intervals are, but there was a problem with the vocabulary when you described your tuning sequence, so I wanted to makes things very clear. Thanks.

You wrote: “Sorry, I should have written: the incremental ratio of 4:2 octaves using your theoretical iH tables. What significance? I told you, you may soon or later discover that there is a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB. You need to discover that yourself though.”

Thank you, I think I know where to go from here. With inharmonic tones, the octave (and incremental) frequency ratio changes from note to note when the octave type is constant. This is why the Railsback curve is not a straight line. So there cannot be “…a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB.” For a given iH curve any octave ratio for a 4:2 octave can be determined, but is of very limited value. The twelfth root of this ratio cannot even be used to determine the temperment because the next octave chromatically will have a different ratio.

But a picture is worth a thousand words. Here is a link with some graphs that show beat rates, cents deviation (Railsback curve) and the iH curve for a simulated studio sized upright piano:

http://www.box.net/shared/rxb631v2yz

The tuning was done note by note to try to have all 12ths beating narrowly at the same speed as the 15ths beating widely. It could be refined some but cannot be made perfect. For the upper notes, the common note must be on top. For the lower notes, the common note must be on bottom. For the middle notes, there had to be compromises, because the intervals do not all beat at the same speed; for the given beat speed of a 15th, it does not work to have both the upper and lower 12ths beat the same.

The graph that shows the beat rate for the 12ths and 15ths predict that a piano tuned this way will have these intervals beat progressively faster, then slower, then beatless and finally faster again with the 12ths being wide and the 15ths being narrow.

The graph that shows the 4ths and fifths show the fifths beating progressively faster and always being narrow, while the fourth beat progressively faster, then progressively slower, and finally faster again but being narrow.

But that is not how someone may actually tune when trying to tune with equal beating 12ths and 15ths. Nor is it how a person may think a piano sounds when tuned so that the 12ths and 15ths actually do beat equally. It can be an indulgent mystery.

As far as discussing the CHAS algorithm or model, sorry, NO!

Tooner, you have done a very good job.

You say:...“With inharmonic tones, the octave (and incremental) frequency ratio changes from note to note when the octave type is constant. This is why the Railsback curve is not a straight line.”...

I would never say that the octave type should be constant. I was giving you an exercise.

...“So there cannot be “…a ratio (only one) that can straighten 12ths and 15ths in what is Chas ET-EB.”...

Really? Let’s see why.

...“For a given iH curve any octave ratio for a 4:2 octave can be determined, but is of very limited value. The twelfth root of this ratio cannot even be used to determine the temperment because the next octave chromatically will have a different ratio.”...

I agree. In fact I’m talking of a ratio that is not only the octave’s ratio. Chas ratio is a combined ratio, i.e. it is a “difference” ratio that stands for all intervals.

You say:...“But a picture is worth a thousand words. Here is a link with some graphs that show beat rates, cents deviation (Railsback curve) and the iH curve for a simulated studio sized upright piano:

http://www.box.net/shared/rxb631v2yz

To be sincere, I think that you (if it was you) could have called it “ET-EB 12ths and 15ths test on a simulated studio sized upright piano”. We will be able to call it “Chas tuning” when we will get the right chance with a real Chas tuning. Anyway I’m not disappointed, mainly because ET-EB is what Chas is describing. I also think the results are surprisingly favorable.

...“The tuning was done note by note to try to have all 12ths beating narrowly at the same speed as the 15ths beating widely. It could be refined some but cannot be made perfect.”...

Nevermind.

...“For the upper notes, the common note must be on top. For the lower notes, the common note must be on bottom. For the middle notes, there had to be compromises, because the intervals do not all beat at the same speed; for the given beat speed of a 15th, it does not work to have both the upper and lower 12ths beat the same.”...

Do not worry, let’s look at the results together.

...“The graph that shows the beat rate for the 12ths and 15ths predict that a piano tuned this way will have these intervals beat progressively faster, then slower, then beatless and finally faster again with the 12ths being wide and the 15ths being narrow.”...

What I can see is that 12ths (3-1) and 15ths (4-1) are straighten, although not perfectly straght.

...“The graph that shows the 4ths and fifths show the fifths beating progressively faster and always being narrow,”...

Also in my final tuning form, as I’ve said, I do not think 5ths get wide. And in the highest section I tune octaves and check 10ths, 12ths, 15ths and 17ths. Listening to plucked strings (so with different iH) 5ths are narrow.

...“while the fourth beat progressively faster, then progressively slower, and finally faster again but being narrow.”...

Yes, this is what I find in my tuning form, excluding the bass section. Also the octaves go in the way Chas describes them. BTW, what was the range in this test?

...“But that is not how someone may actually tune when trying to tune with equal beating 12ths and 15ths.”...

In fact, to gain Chas form I invert the 5ths beat rate progression (on mid-strings preparatory tuning).

...“Nor is it how a person may think a piano sounds when tuned so that the 12ths and 15ths actually do beat equally. It can be an indulgent mystery.”...

In fact, Chas 5ths and 4ths do not sound flat at all, yet if you pluck the strings in the highest section you can hear them flat, 4ths more than 5ths. And this mystery is disclosed every time I tune.

...“As far as discussing the CHAS algorithm or model, sorry, NO!”

Ok, but you will avoid saying nonsense then, like “it could be an equation”. So, live the test’s imperfections and compromises alone, what are your latest temporary conclusions? Oh, have you heard Chas recording? Would you complain for iH’s effects?

Thanks a lot and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Alfredo:

One step at a time.

You posted: “I agree. In fact I’m talking of a ratio that is not only the octave’s ratio. Chas ratio is a combined ratio, i.e. it is a “difference” ratio that stands for all intervals.”

Please explain this more. I do not know what you mean by a combined ratio or a difference ratio. Examples are probably needed for me to understand you.

Tooner,

I’m going to answer you questions, then it will be your turn. Careful not to leave to many question marks behind.

You wrote:...“For a given iH curve any octave ratio for a 4:2 octave can be determined, but is of very limited value. The twelfth root of this ratio cannot even be used to determine the temperment because the next octave chromatically will have a different ratio.”...

I replyed: “I agree. In fact I’m talking of a ratio that is not only the octave’s ratio. Chas ratio is a combined ratio, i.e. it is a “difference” ratio that stands for all intervals.”

Now you kindly ask:...“Please explain this more. I do not know what you mean by a combined ratio or a difference ratio. Examples are probably needed for me to understand you.”...

Ok. What does “combined ratio” mean? In the way nature combines oxygen and hydrogen to make water, Chas combines two intervals ratios to make one scale ratio. You well know, so far ET scale’s incremental ratio could only be referred to a single interval’s ratio, in the way 12th root of two is referred to the octaves ratio, 7th root of 3/2 refers to fifths ratio and 19th root of 3 refers to 12ths ratio. Chas ratio is now composed of two combined ratios, 3:1 and 4:1 ratios.

This may explain more. When I realized the antagonism amongst 3ds, octaves and 5ths I guessed I had to look for a trivalent scale ratio, i.e. for a ratio that could represent the game I was playing with those three intervals beats, trying to find the best beat rate progressions. After a while, I understood that 3ds and octaves were on the same beat line, so that octaves could well represent 3ds (and vice versa). Since then I knew that 5ths and octaves were the two original stretchers, and that the correct scale incremental ratio would have had to combine those two intervals ratios in a balanced “beat regulator”. Then it could only be a beat-ratio.

If you wanted to cut a straight furrow, two oxen would do much better than one.

If anythyng then, I would have had to look for a double constant, i.e. a constant for two beating-intervals, so to gain the difference ratio that could balance the two original stretchers. Once the beats progression of all intervals could restore the same euphonic form again and again, the scale double constant could be extracted, and how 12ths and 15ths delta-differencies could include all intervals ratios, was quite evident.

Thanks and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Bill,

You say: "Alfredo, take Kent up on what he offers. You will benefit from it greatly."

May I ask you to tell me precisely what you mean? Do you mean Kent's open invitation to visit Kansas City?

Regards, a.c.

Yes, I do.

Alfredo:

Ok, a combined ratio is one that is calculated from two different ratios. I do not know of any intrinsic value that the ancestry of a ratio could have. What is important is what the ratio will accomplish on its own.

Maybe we can jump ahead a bit and see what such a ratio will accomplish. Let us use the Beat Analyzer as a simulator to see the results of using such a combined ratio. You would need to give me the ratio and directions in how to use it. Also, you should predict what the results will be ahead of time, so as to have an objective evaluation. I can also provide a table showing all the numerical values, for verification of the calculations and construction of the graphs.

You asked: “BTW, what was the range in this test?” All the graphs are for the full range of the piano, 88 notes. This is shown by the scale at the bottom of the graphs, A0-A7. The Beat Analyzer also provides mid-range (C3-C5) graphs for RBIs and SBIs but I did not include those.

Sorry, I did not listen to your recording. When I clicked on the link a number of questionable pop-ups appeared. I did not want to take a risk by downloading the file. And I am not sure how my opinion of the tuning would effect our present discussion, anyway.

Bill,

thank you very much, I will treasure your suggestion. At the moment I’m quite confused.

Do you think it would sound less subversive and maybe closer to actual tuning if I presented Chas model as (thanks to Robert) a variant of ET where the octave ratio is something a little bigger than two?

Kent,

It's kind of you to invite me to Kansas City and, if this gave us all a better chance, I would consider your offer. But tell me please, is it sincere?

Tooner, thanks, I'll reply asap.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Of course it is. We have the headquarters of the Piano Technicians Guild here with its piano museum which you might find interesting. The headquarters has more than just museum pieces; there is at least one modern 7' grand available there.

At my university, there are many very fine pianos that would be available, and we have professional audio recording equipment available there as well.

Yes Alfredo, the larger than theoretical, larger than typically chosen temperament octave size and consequent octaves is always whet I have understood your basic idea to be. It is my idea as well, as it is Herr Stopper's but each person has their own unique concepts.

To me, when any of these ideas was presented with ET alone, it represented only a subtle difference yet that difference has always seemed very important to those who advocate it as it does with you. Recently, I was impressed by the difference I heard between the Reyburn default stretch and the Stopper tuning from posts by Grandpianoman. Being able to compare the two tunings easily with each other, the difference proved to be more than subtle.

I have always placed fundamental importance on just how I stretch my octaves when I tune the EBVT. If I use my ETD's calculated program and the stretch it provides, even if I adjust it here and there a bit and use the device's "DOB" function to boost the high treble, the tuning just does not quite have the magic it has if I tune by direct interval. I don't get the signature "pipe organ effect" I am looking for. It doesn't sound bad and it has much of what I want but it is not the same.

I think everyone has given you the benefit of the doubt that you have discovered something that really appeals to you and you are very eager to let everyone know about it. The problem has always been in communication. Many of us, myself included, do not understand or use advanced mathematics in our thinking and reasoning. That does not mean that such a way of defining your concept is not valid. Perhaps it will ultimately be essential.

The developers of all of the ETD programs certainly did use advanced mathematics and have kept those secrets as protected information. Nevertheless, I know just from comments I have heard that they have had to struggle with each other over who had the right to use whatever findings they developed and how they came to know that information.

In your case, you have eagerly presented what you know for all to consider but very few if any could really grasp what it is that you are trying to say. That, I well understand is frustrating. I have encountered the same kind of frustration, for sure.

I have known Kent Swafford for many years, close to 25 in fact. We have always had respect for each other as piano tuners who seek excellence above and beyond the typical and ordinary. Kent is known to virtually every PTG member because he served the organization as its president and vice president and also a very long tenure as one of its examiners. That service had nothing directly to do with advancements in tuning concepts but because it was he who chose to serve PTG, the direction PTG took under his leadership has his own mark of distinction on it.

Kent is not the type of person to easily dismiss an idea. He listens. He will discuss and he will give his opinion but he also does not present himself as the final authority. He is also a very skilled pianist.

I could not imagine a better opportunity for you than what he has offered and I will make that offer even better. Kent has also expressed the desire to witness the way I tune in its best light. Presently, you have the advantage in currency exchange. The Euro is worth about $1.50. During winter, airplane flights from Europe to the USA are typically at their lowest.

I will also offer to meet with you and Kent to show you what I can do and listen to your idea as well. At the university where Kent is employed, you could also have the benefit of having skilled pianists play pianos tuned the way you advocate, the way I advocate and compare those two with standard practice. You can listen, Kent can listen and the pianists can comment on the effects of each.

It can be both a "blind" study, where the pianists do not know which piano is which and then an informed trial where they do. The feedback (our often used term for results or reaction) from the pianists would be interesting and informative in each case.

I can go even further in extending a welcome hand to you: My aunt lives in Kansas City and is always receptive to hosting guests such as for a week. If you do not have the means to pay for a hotel, you can be assured to have a place to stay at no further cost to you. I can meet you at the airport, have a car and take you to the PTG Home Office and the university on each of the planned days of your visit.

I would suggest planning a trip in February for a week. Arrive on Sunday and leave again the following Saturday or Sunday. I would imagine you could find a flight schedule for only a few hundred Euros and as far as I know, you would not need a visa, only a passport.

I would be willing to take the time off from my business for this and I am sure that Kent would make every accommodation he could as well. The only thing that would make it even more interesting would be to have Herr Stopper come too but let's not go too far.

I would mention with respect to Herr Stopper that he also encountered much negative resistance to his ideas and also had a similar problem in communication due to the difference in our languages. Kent was singly responsible for inviting Herr Stopper to the PTG convention one year and I am glad he did. Previous to that, I had not been very impressed with Herr Stopper's concepts through writing and a recorded sample alone. It took actually hearing his tuning and meeting Herr Stopper to gain a respect for him that I will always have.

So, you see, Kent is a man of his word, he has made such an offer before and he has made it again to you. So, I suggest that you take the offer as your very best opportunity ever to present your concepts. The language in your papers can be worked out so that a more general audience can understand and appreciate what you offer.

PTG has people with unique areas of expertise. Not all of the finest contributions come from RPTs either. People with writing skills like me, mathematics and editing skills like Jerry Viviano who is an Associate member and mathematics skills like Robert Scott who is also and Associate member can all possibly work together to help you come up with an English language paper which is presentable. It could be published in the PTG Journal or even be made into a book. Nothing is beyond possibility.

It would indeed be an example of where a PTG non-member, RPTs and Associates all can and do work together cooperatively with mutual respect. It does happen and is not uncommon. Advanced concepts of all kinds are what I believe should be presented at PTG conventions and those are sometimes the focus of the event. Your concepts could be presented in the future and you could have substantial numbers of people interested in them if you take the right course. Accepting Kent's and my offer would be an important step towards that.

Bill provides one good, possible scenario. We have had "tuning summits" before of course, but they usually happen at the annual PTG convention, which is not the best environment for such things due to the crowded schedule.

I would love to put together something. Of course, it is hard to say how much subsidy would be available for participants. It would depend upon the specific negotiated arrangements, who we could interest in being a sponsoring body, and I suppose, whose aunts happen to live nearby. <grin>

Let's see what we can make happen.

Thanks, Kent, I, for one, do not need any subsidy and I don't need to stay at my aunt's house but I was going under the assumption that such a trip and a hotel stay for Alfredo might be more than he could afford. I can say this: my aunt would be as thrilled to host a visitor from Italy for a week as she would me. She had been used to hosting visitors in earlier years and used to rent her upstairs boarders who were students at Rockhurst University which is just steps from her home on the 51st block of Virginia avenue. A few years ago, she hosted one of our Chapter members who could not afford the hotel stay at the convention. She was happy to do so.

There is, by the way, a good example of an American taking a work written in English by a German whose English was nearly incomprehensible. Del Fandrich saw the value in it and took it upon himself to edit and annotate Piano Tone Building.

Bill, Kent,

Thank you very much for your generous offer and the spirit of cooperation. I will be very happy to personally know you as also our colleagues, and one week in February would be the right time for me too. You'll receive my e-mail address, so to define all the details.

Also thanks for your warm welcome hand, Bill, it will be very nice to meet your haunt in any case...She is not a piano tuner, is she?

Have a nice Sunday, a.c.

Alfredo:

I suppose I should wait until you answer my last post, but I came up with something you ought to be interested in and is pertinent to our present discussion.

You seem to have derived your CHAS ratio (1.0594865443501…) from the ratios of 3^1/19 and 4^1/24 (which is the same ratio as 2^1/12, btw) by using the CHAS algorithm. But there is a simpler, more “elegant” way that does not pretend to use other ratios. This means that the CHAS ratio need not be considered a “combined ratio” at all.

The desire is to have an incremental (semi-tone) ratio that produces equal beating 15ths and 12ths.

If “x” is this ratio,

and the multiplier of the fundamental frequency to determine the beat rate of a 15th is:

(x^24)-4.

and the multiplier of the fundamental frequency to determine the beat rate of a 12th is:

3-(x^19)

then for equal beating 15ths and 12ths with the common note on the bottom:

(x^24)-4 = 3-(x^19)

or:

(x^24) + (x^19) = 7

and, by trial and error:

x = 1.0594865443501…

If the common note is on the top, it is a different equation and a different ratio.

This could be looked at as a “combined ratio” when your algorithm is used (which is not necessary for this purpose), but I cannot look at it that way.

I wonder if I will regret posting this. I want to avoid getting drawn into a “tar-baby” discussion about your paper.

Tooner sorry, I have been away.

You kindly wrote:...“Ok, a combined ratio is one that is calculated from two different ratios. I do not know of any intrinsic value that the ancestry of a ratio could have.”...

I’d say that the intrinsic value of a ratio is its ancestry or, a ratio can have an intrinsic value depending on its ancestry. Chas ratio returns the qualities of two algebraic magnitudes, 19 root of three and 24 root of four.

...“Let us use the Beat Analyzer as a simulator to see the results of using such a combined ratio. You would need to give me the ratio and directions in how to use it.”...

Yes, any direction you need, could you be more precise on this?

...“Also, you should predict what the results will be ahead of time, so as to have an objective evaluation.”...

Do you mean results in terms of beats?

...“I can also provide a table showing all the numerical values, for verification of the calculations and construction of the graphs.”...

This will be relevant too.

About me you say:...“You asked: “BTW, what was the range in this test?” All the graphs are for the full range of the piano, 88 notes. This is shown by the scale at the bottom of the graphs, A0-A7. The Beat Analyzer also provides mid-range (C3-C5) graphs for RBIs and SBIs but I did not include those.”...

Very good indeed. Are you using that iH constant, the one that doubles every 8 semitones? Do you think there is a way to reduce the approximations?

...“Sorry, I did not listen to your recording. When I clicked on the link a number of questionable pop-ups appeared. I did not want to take a risk by downloading the file. And I am not sure how my opinion of the tuning would effect our present discussion, anyway.”...

If you like I can always e-mail you that recording. I did it also because you suggested me to do so (05/20/09). You also wrote (06/23/09): ...“If everyone tuned the same and everyone liked the same tuning, this would indicate there is a universal optimum, and through empirical observation a model could be developed. But since this is not the case, this indicates that there is no mathematical model that will satisfy everyone.”

I prefere to think that a universal optimum does exist, although it may not be needed. You'll find a curious analogy in "Pareto (improvement)" linked below:

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

Then you said that ET is a region, and I think I understood what you meant, but in another reasonable way Chas ET EB is a precise location, and that recording is meant to be the first demonstration. I would also like to demonstrate that small-pianos iH does not impede the finding of Chas ET EB form.

Now you say:...“You seem to have derived your CHAS ratio (1.0594865443501…) from the ratios of 3^1/19 and 4^1/24 (which is the same ratio as 2^1/12, btw) by using the CHAS algorithm. But there is a simpler, more “elegant” way that does not pretend to use other ratios. This means that the CHAS ratio need not be considered a “combined ratio” at all.”...

No, it does not need to, but this is what it can be said wanting to use descriptive terms. Numbers themselves do not call for any kind of consideration, yet we can give attributes to them. You say “...that does not pretend to use other ratios”.... Then you would have to explain what 3, 19, 4, 24 and 7 are. Btw, about 7, this is were the gem sparkles.

You say:...“The desire is to have an incremental (semi-tone) ratio that produces equal beating 15ths and 12ths.

If “x” is this ratio,

and the multiplier of the fundamental frequency to determine the beat rate of a 15th is:

(x^24)-4.

and the multiplier of the fundamental frequency to determine the beat rate of a 12th is:

3-(x^19)

then for equal beating 15ths and 12ths with the common note on the bottom:

(x^24)-4 = 3-(x^19)

or:

(x^24) + (x^19) = 7

and, by trial and error:

x = 1.0594865443501…

If the common note is on the top, it is a different equation and a different ratio.
This could be looked at as a “combined ratio” when your algorithm is used (which is not necessary for this purpose), but I cannot look at it that way.”...

You can look at it the way you prefere, as long as we can share it as much as possible. As for simplicity, I was quite impressed by ROMagister when, commenting Chas model, he could explain it in (about) ten lines, five months ago.

ROMagister(05/29/09): “...it IS Equal Temperament, but with another ratio: not the classic one where 12 semitones = 1 octave of exactly 2:1 (Pythagorean octave still accepted as axiom in classical ET).
The basic version (s=1) makes an equal compromise between the 'justness' of 3rd and 4th harmonics (octave+fifth vs 2 octaves). "s" is just the compromise parameter which says how important is the error in the 3rd harmonic compared to the error in the 4th harmonic. It can be set "politically" as we want, and the Delta results as a solution of the (implied) equation, also the practical frequency ratio that results.”

You say:...“I wonder if I will regret posting this. I want to avoid getting drawn into a “tar-baby” discussion about your paper.”...

Do not worry, as I told you it’s up to you. You had written (10/28/09): ...“As far as discussing the CHAS algorithm or model, sorry, NO!”

You see, If you get steady, it’ll be easyer for me too. Now you talk about Chas algorithm and elegance, and yet you do not want to discuss it. Anyway, I prefered a delta-difference in Chas equation so that I am always able to recall beats and refer to my/our tuning practice.

Nevertheless I find (x^24) + (x^19) = 7 very much appealing. A "7th day" is when I was born.

Oh, in all this you are the oxygen, thanks.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Alfredo:

Interesting about Pareto efficiency. I believe in any human exchange, if it is not to the benefit of both, it is not to the benefit of either.

Let me continue to try to steer clear of your paper for the benefit of both of us.



The iH curve was shown on the last graph. It closely approximates the iH curve for a well scaled studio upright, and I suggest we use this for the simulation. Instead, I could use actual observed iH values, but since there are little deviations in the curve (perhaps due to instrumentation errors) for a simulation I think the computer generated curve would be better. But a curve for a smaller or larger piano could be chosen instead. The table that I will include will list the iH for each note. Yes, if you predict what the beat rates will do before seeing the graphs, this would be an objective comparison.

Yes, I need some direction from you, I guess to be your lab assistant. After all, it would do no good for you to predict the results for a different simulation than the one being conducted. Do you want me to use the CHAS ratio of 1.0594865443501…, or some other ratio? Do you want the same ratio used for all notes? (I ask this because if the semi-tone ratio is constant, the Railsback curve will be a straight line.) Is the same iH curve that was used for what I called “ChasTuning” satisfactory, or would you prefer a different iH curve, or no iH at all? What would you like to call the tuning results? May I make a suggestion? Since you are striving for “straightened” 12sth and 15ths how about a Straight we are both familiar with? “The Straight of Messina”

Oh, and the most important thing for any lab assistant to know, how do you take your coffee?

Bill, I wrote you and Kent an Email Post (I could not remember how to send a PM).

Did you get it?

Regards a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Tooner,

You kindly say:...“The iH curve was shown on the last graph. It closely approximates the iH curve for a well scaled studio upright, and I suggest we use this for the simulation."...

I agree, for a simulation the computer generated curve works fine.

...“But a curve for a smaller or larger piano could be chosen instead.”...

I suggest to try to refine one simulation at the time, but also the curve for a smaller or larger piano may be interesting eventually. Any idea about how to reduce approximations?

...“The table that I will include will list the iH for each note. Yes, if you predict what the beat rates will do before seeing the graphs, this would be an objective comparison.”...

Ok, I agree. The only thing is that I normally fix 4ths 5ths and octaves from C3 to C6 in my preparatory tuning (using RBIs too), and after unisons I check RBIs and 12ths and 15ths. Nevertheless I’ll give you the beat rates and I’ll try to be as precise as possible.

...“Do you want me to use the CHAS ratio of 1.0594865443501…, or some other ratio?”...

Let’s use Chas ratio first, then we’ll see.

...“Do you want the same ratio used for all notes?”...

Yes, please.

...“Is the same iH curve that was used for what I called “ChasTuning” satisfactory, or would you prefer a different iH curve, or no iH at all?”...

Yes, that curve was ok.

...“What would you like to call the tuning results?”...

Chas ET EB simulation of...on...with...using.... Thinking of Messina we should call it Bridge... all the administrations here talk about it when they want to poll more votes.

...“Oh, and the most important thing for any lab assistant to know, how do you take your coffee?”

Possibly, in good company.

Thanks Tooner and regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871
_________________________

Alfredo:

I have the graphs and values ready to publish when you predict the beatrates. The values are on an embedded worksheet. I hope you have Excel on your computer. If not, I can add some pages with the values printed out. Let me know.

Alfredo:

You should also predict the beatrate of the 12ths and 15ths.

Tooner,

I'm waiting to tune a reliable piano, so that the beat rates I'll talk about may be reliable too. You'll be given the beat rates relative to the preparatory tuning, i.e. from C3 to C6 on mid-strings only, separate from the after-unisons beat rates.

I had asked you: Are you using that iH coefficient that doubles every 8 semitones? Do you think there is a way to reduce the approximations?

About your own maths work, I think you are now very close to the Chas partials-combination key. May I supply one more hint?

You kindly wrote:

The desire is to have an incremental (semi-tone) ratio that produces equal beating 15ths and 12ths.

If “x” is this ratio,

and the multiplier of the fundamental frequency to determine the beat rate of a 15th is:

(x^24)-4.

and the multiplier of the fundamental frequency to determine the beat rate of a 12th is:

3-(x^19)

then for equal beating 15ths and 12ths with the common note on the bottom:

(x^24)-4 = 3-(x^19)

Let's stop here. Now that you can find a value for "x", you can substitute that value and see what you get. I'm asking you to do it yourself only because I think that this may help other colleagues too. Would you be willing to put your figures processing in words?

Regards, a.c.

Alfredo:

You wrote: “I'm waiting to tune a reliable piano, so that the beat rates I'll talk about may be reliable too. You'll be given the beat rates relative to the preparatory tuning, i.e. from C3 to C6 on mid-strings only, separate from the after-unisons beat rates.”

You may wait if you wish, but I do not see a real reason to. You have tuned enough pianos that I thought you would know about what the beatrates will be. What I am trying to do is what I mentioned on Oct 26, 2009. “We seem to be covering old ground (going in circles) in the discussion of frequency ratios, partial matches and inharmonicity.” I am hoping that comparing the simulation with your experience will shed light on this.

And you wrote: “I had asked you: Are you using that iH coefficient that doubles every 8 semitones? Do you think there is a way to reduce the approximations?”

No, the iH curve that I chose does not double every 8 semitones. The curve was smoothed by me from actual measurements of a well scaled studio upright and that was provided to me by another. The measurements given me were the actual frequencies of partials that I then calculated an iH curve from, which then was smoothed. For what I am hoping to accomplish, this seemed a good way. Since we are not dealing with the same piano in any case, reducing approximations could do more harm than good.

Here is a link to a website with many iH curves that you might find interesting: http://www.goptools.com/gallery.htm

These curves are from measured string data and not calculated from frequency measurements.

And you also wrote: “About your own maths work, I think you are now very close to the Chas partials-combination key. May I supply one more hint?

<SNIP>

(x^24)-4 = 3-(x^19)

Let's stop here. Now that you can find a value for "x", you can substitute that value and see what you get. I'm asking you to do it yourself only because I think that this may help other colleagues too. Would you be willing to put your figures processing in words?”

I would rather continue to the next step that I mentioned before and put all the terms containing x on one side and all the terms that do not on the other side. The equation is then simplified by combining like terms. Finally x is solved. That is how algebra is performed. I am not looking for things that do not really exist!

The minor point that I was trying to make is that the idea of this combined ratio is a mental fabrication.

The major point that I am trying to make is the relationship between frequency ratios, partial matches and inharmonicity. I have been trying to make this point for quite some time, and am hoping that a simulation will help.

But in an attempt to give you something philosophical to nibble on: Why is it preferred to have equal beating 12ths and 15ths with a common note on the bottom rather than the top? Or would it be better for these two sets of intervals to be non-equally beating by the same amount? But wouldn’t the non-equality need to be defined not linearly, but logarithmically? But the problem with all this is when inharmonicity comes into play, we find that the “earth is round.” A straight line on a nautical chart is rarely the shortest distance between two points on the earth. And a frequency ratio will not give the expected results when inharmonicity is included.

Tooner,

You write:...“You may wait if you wish, but I do not see a real reason to.”...

Strange, I think I’ve told you the reason. I normally check and fix the intervals beat rate progressions during my preparatory tuning, i.e. while I tune mid-strings only. These beat rates need to be different from the final tuning form’s beat rates, i.e. from the after-unisons intervals beat rates. More precisely, in my preparatory tuning, wide intervals need to be wider and 5ths need to invert their beat-rates progression.

So, I have already been able to tell you and all colleagues what my preparatory-tuning intervals beat rates are, and when it came to my practical final tuning form I could confirm exactly what Chas ET-EB model predicts, i.e. Chas theoretical ET EB constants, i.e. equal beating 12ths (narrow) and 15ths (wide) with all the other RBI check-intervals being progressive.

Now, if you are interested in my final tuning form’s beat rates in precise terms, you only need to wait for something I’ve never measured before. For me, tempering is done on mid-strings all along, while working on unisons results more into voicing. But is that what you are asking for? I may have misunderstood.

Still today, my impression is that you are not happy with Chas algorithm, a fairly simple equation that makes an equal compromise between the “justness” of the 3rd and the 4th partials (octave+fifth Vs double octaves – ROMagister), and still today I wonder how you could get by with 12th root of two.

You said:...“I am hoping that comparing the simulation with your experience will shed light on this.”...

In my opinion our goals may diverge, but in saying this I’m not even sure about your goals. What I understand is that the two of us move with quite different approaches, different experiences and different beliefs.

For istance, you say that any piano tuning theory’s usefulness is limited in aural tuning, I do not. In my opinion, a correct and reliable temperament theory will address aural tuners towards a practicable and euphonic model, and I must say that 12 root of two ET needed to be improved. Today Chas ET-EB model can well represent an improved variant of ET (thanks Robert), a stretched-octave ET-EB variant that is in line with what (experienced) aural tuners may already be doing in their practice or be looking for. Not to mention that a temperament theory is not only directed towards pianos, but towards all keyboard instruments and more generally towards all musical instruments.

You say that we need to stretch octaves because of iH, I do not. I think that in aural tuning - i.e. when it comes to beats - iH is neutral, and I think we need to stretch octaves simply because the fairest compromise is to be found between two roots, 19 root of 3 and 24 root of 4; as a consequence, ET theoretical octaves should not and will not be in 2:1 ratio.

You want iH to come into play, I do not. Chas theoretical model, being a temperament model, does not consider whether you are tempering iH tones or what. Actually, I’m trying to share Chas theory because this model can describe our actual tuning, and because its constants agree with my practical results.

You kindly answer:...“No, the iH curve that I chose does not double every 8 semitones. The curve was smoothed by me from actual measurements of a well scaled studio upright and that was provided to me by another. The measurements given me were the actual frequencies of partials that I then calculated an iH curve from, which then was smoothed. For what I am hoping to accomplish, this seemed a good way. Since we are not dealing with the same piano in any case, reducing approximations could do more harm than good.”...

Maybe I did not get what you are hoping to accomplish. Could you also tell me more about that well scaled studio upright? Was it a real piano? Which temperament was used and how? What standard did you smooth the curve by? Do you know the approximation degrees, just to have an idea? In my tuning experience, reducing approximations has been and still is the real challenge.

About your maths work, you say:...“The minor point that I was trying to make is that the idea of this combined ratio is a mental fabrication.”...

For me this is disappointing, firstly because I’ve said that “combined ratio” is used as a descriptive mean, secondly because I was expecting you to understand the difference between using a one-ratio formula, like 12 root of two, and using a two-ratios equation. Differently also from the latest ratios, Chas ET incremental ratio is a new logarithmic average between two ET ratios.

You say:...“The major point that I am trying to make is the relationship between frequency ratios, partial matches and inharmonicity. I have been trying to make this point for quite some time, and am hoping that a simulation will help.”...

Yes, I understand that your interest is on the relationship between frequency ratios, partial matches and inharmonicity, then not only our goals may diverge, but also our methods. In fact, if I seriously wanted to make a comparison I would not use a somehow-approx-simulation, I would go for the real thing.

You say:...“But the problem with all this is when inharmonicity comes into play, we find that the “earth is round.”...

I can not understand how you understand logarithms. With 12 root of two ET the earth is flattened on every zero-beating octave. With Chas ET-EB the earth is rounded octave after octave.

Should I understand that you would like to find a way to improve ETDs? You see, I’m too keen on aural tuning and I do not see the point unless we try to do that with the least possible approximations.

About “something philosophical to nibble on”, I thank you and I’ll reply soon.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Alfredo:

You wrote: “So, I have already been able to tell you and all colleagues what my preparatory-tuning intervals beat rates are, and when it came to my practical final tuning form I could confirm exactly what Chas ET-EB model predicts, i.e. Chas theoretical ET EB constants, i.e. equal beating 12ths (narrow) and 15ths (wide) with all the other RBI check-intervals being progressive.”

Okay, I guess I really just wanted to confirm this. Here is a link to the data. I included an embedded spreadsheet in addition to hard copy tables and graphs. I will let you have first say about the results of using the CHAS incremental ratio on a piano with iH. I am not trying to evade your other questions. These can be dealt with. But I am hoping that you will see what a moot point much of what we try to discuss is when including iH into tuning theory.

BridgeTuning

Thanks very much Tooner for your elaborations.

You kindly say:...“I will let you have first say about the results of using the CHAS incremental ratio on a piano with iH.”...“But I am hoping that you will see what a moot point much of what we try to discuss is when including iH into tuning theory.”...

What can be said about your generous elaborations? I can say that Bridge tuning shows not only the results of using the CHAS incremental ratio on a piano with iH, but more generally the effects of iH on beat rates, if we were to use any set of theoretical frequencies values on a piano.

We may as well consider one evidence: 12th root of 2 predicts zero-beating octaves, 7th root of 3/2 predicts zero-beating 5ths, 19th root of 3 predicts zero-beating 12ths. Then I may ask you: when including iH, is it of value to theorize a zero-beating ET incremental ratio? In other words, taking your latest analisys to extremes, can an infinitesimal degree of iH agree with ET theoretical zero-beating choromatic intervals?

I do not think it is including iH into tuning theory that is making much of what we try to discuss a moot point, but a kind of deafness (what a nightmare), in my opinion that kind of deafness deriving from different ways of looking at the same issue, in our case being the way we look at piano tuning and temperament theory.

Four points, in my opinion, are mainly causing this phenomenon:

The relevance of iH in piano tuning
The relevance of tuning theory in piano tuning
The relevance of Chas theory as an improved ET-EB temperament variant
The relavence of zero beating octaves and more generally of zero beating intervals

It was June when we long discussed about iH, when you kindly posted some calculations (06/11/09) that again could prove to all readers how iH effects the actual frequencies values of partials, and how the actual frequencies will be different from theoretical values. That calculation could well prove that theoretical frequencies values and the relative theoretical beat rates will not correspond in practice, due to iH. That is to say that, theoretical beat rates can be fixed in practice, but the actual frequencies will differ from theoretical frequencies values.

Then iH was not a moot point, though our goals were already diverging, you wanting to prove the limited usefulness of any tuning theory in aural tuning, me wanting to say that iH effects could be re-calculated, so to reduce approximations. None of us could doubt about iH’s effects on actual frequencies values. Both of us had good reasons for going back to iH calculations: (posted in 06/16/09) ” Certainly you will have read where our cello-expert writes: “…the outstanding symmetry of the 19th root of three ET can still be preserved with proper consideration of inharmonicity.” What do you think he meant, saying “…proper consideration of inharmonicity”?”
You answered: “I am not certain what Mr. Stopper means. It may be similar to what I call “the largely self-correcting effects of iH on beat rates”.”

Where to go then to oxygenize this discussion? This is my proposal:

a)Let’s distinguish the general meaning and relevance of a temperament theory, what Chas ET-EB is, from what piano tuning’s issues are

b)Let’s analyse what logic is behind each one considered theory

c)Let’s separate iH issues from beats-control and tuning-form issues

d)Let’s see if we can address aural piano tuners towards a more reliable and practicable model

e)Let’s try to clear up how SBI and RBI intervals should go, and get rid of all misteries

f)Let’s evaluate if today there is a way to reduce approximations relative to iH and piano scaling

In your previous elaboration,

http://www.box.net/shared/rxb631v2yz

when you added iH to Chas theoretical frequencies values, you/we all could see how 12ths (narrow) and 15ths (wide) beat rates can be straighten in terms of opposite equal beating. In my opinion, there you find Chas theory’s usefulness in aural tuning, without forgeting that yes - due to iH - actual frequencies are different from theoretical values.

Regards, a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Alfredo:

Let's first finish what is going on now.

Please answer YES or NO.

Is an incremental (semi-tone) ratio useful for predicting the beat rates of intervals of inharmonic tones?

I had just written:

In your previous elaboration,

http://www.box.net/shared/rxb631v2yz

when you added iH to Chas theoretical frequencies values, you/we all could see how 12ths (narrow) and 15ths (wide) beat rates can be straighten in terms of opposite equal beating.

Tooner, you ask: "Is an incremental (semi-tone) ratio useful for predicting the beat rates of intervals of inharmonic tones?

The answer is: Yes, if you add iH to the theoretical values. No, unless you add iH to the theoretical values. a.c.

First recording of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

Alfredo:

You posted: "when you added iH to Chas theoretical frequencies values, you/we all could see how 12ths (narrow) and 15ths (wide) beat rates can be straighten in terms of opposite equal beating."

No, I did not add (a better term may be apply) iH to "Chas theoretical frequencies" in the tuning that showed equal beating 12ths and 15ths. I determined through trial and error what frequencies are required on a simulated piano with iH so that 12ths and 15ths would beat equally.

However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally.

This is why I see no usefulness in frequency ratios for predicting beatrates. The desired beatrate is determined, and then the frequencies are calculated. Finally, if wanted, the frequency ratios can be ascertained. But they are a byproduct, not used in the calculations.

Perhaps you misspoke.

Tooner,

you say: "No, I did not add (a better term may be apply) iH to "Chas theoretical frequencies" in the tuning that showed equal beating 12ths and 15ths. I determined through trial and error what frequencies are required on a simulated piano with iH so that 12ths and 15ths would beat equally."...

Sorry, I misunderstood.

..."However, when I did apply iH to Chas theoretical frequencies, the 12ths and 15ths did not beat equally."...

And what happened? Could you tell me more?

...This is why I see no usefulness in frequency ratios for predicting beatrates....

When I wrote (06/04/09): ”There is a fenomenon that I do not really understand, how is it possible to take a lame theory inside and out, one minute referring to it and the minute after negating it. Now theoretical wrong value from traditional ET have a sense, the minute after they do not.”

Your answered:..."Because if we take the beat rates (or at least the ratio between beat rates, including equal beating) that are predicted from a frequency ratio (such as 2^1/12) that does not take into account iH, and then tune a piano with iH using the beat rates we end up with a decent tuning, but a different frequency ratio, one that is non-linear. So on the one hand, the frequency ratio is wrong, but on the other, the beat rates are correct. And since when tuning aurally, we listen to beat rates, the model works even though it is incorrect."...

So, you try to agree with yourself.

Now you say:..."The desired beatrate is determined, and then the frequencies are calculated. Finally, if wanted, the frequency ratios can be ascertained. But they are a byproduct, not used in the calculations."

This is what I would do in practice too, I'd tune Chas form and then I would ascertain the frequencies values and ratios.

Granpianoman,

many thanks for converting the .rar file in an MP3 file, for using your site and for thinking that other people may prefere this or want a quicker way to hear the file.

Regards, a.c.

First recording (.rar) of Chas tuning on a baby Steinway S (5’ 1”, 155 cm) at MediaFire:

http://www.mediafire.com/?sharekey=20194ca8898fecef1bee9a6e9edd9c76e04e75f6e8ebb871

CHAS Tuning MP3 http://www.box.net/shared/od0d7506cv