Alfredo:

I have taken a fresh look at graphs 5 and 6 now that I understand what you mean by degrees. What they say to me is intuitively understood. The twelfth root of two predicts beat rates that double every octave. The twelfth root of a number greater than two (such as Chas) predicts beat rates of wide intervals to double more often than every octave and narrow intervals less often than every octave.

But there are a few problems. First is the way you present the information. You seem to have no misgivings from stating “The differences, divided by themselves…” This is a mathematical show-stopper (and there are others.) You expect too much from others to bend their vocabulary to match what you mean instead of what you say.

Likewise there is the form of graphs 5 and 6. The x axis is labeled 1,2,3,4 etc, but actually are values 1,4,5,8 etc for the degrees of a scale! And the y axis are logarithmic values, but indicate 0 and negative values! Logarithmic values do not reach zero let alone become negative! Again it is up to the reader to try to figure out what you mean instead of what you say.

The biggest problem is that all this is only pertinent to harmonic instruments and not pianos. This is obvious to you, and yet you still put forth your paper as something important to piano tuning. You should not be surprised that there is little interest in your theory and paper. It is not useful for pianos, and puts a great burden on the reader to understand what it is that you are saying.

Tooner,

you say:...“I have taken a fresh look at graphs 5 and 6 now that I understand what you mean by degrees. What they say to me is intuitively understood. The twelfth root of two predicts beat rates that double every octave.”...

Good, I’m glad you could intuitively understand. Consequently you may also understand why 12th root of two, based on the 2:1 ratio, is an unreal and unusable model (section 4.5).


...“The twelfth root of a number greater than two (such as Chas) predicts beat rates of wide intervals to double more often than every octave and narrow intervals less often than every octave.”...

Correct. The next question may be: why Chas? Then you may have to look at Chas system from two different perspectives:

1 – Chas as a dynamic and comprehensive theory, capable of gaining infinite logarithmic ratios, including 12th root of two and any other pure-interval based ratio;

2 – Chas as the model that describes a perfectly stable beating-whole, an ideal logarithmic scale where all intervals are non-pure (impure) and where beats give rise to a synchoronic event, i.e. the 12ths (narrow) and 15ths (wide) equal beating.

You say:...“But there are a few problems. First is the way you present the information. You seem to have no misgivings from stating “The differences, divided by themselves…” This is a mathematical show-stopper (and there are others.) You expect too much from others to bend their vocabulary to match what you mean instead of what you say.”...

I’m sorry, luckily though you could understand. Would you kindly suggest a better way?

...“Likewise there is the form of graphs 5 and 6. The x axis is labeled 1,2,3,4 etc, but actually are values 1,4,5,8 etc for the degrees of a scale!”...

This is written inside and below each graph.

...“And the y axis are logarithmic values, but indicate 0 and negative values! Logarithmic values do not reach zero let alone become negative! Again it is up to the reader to try to figure out what you mean instead of what you say.”...

Tooner, me too I’m trying to figure out what you mean with what you are saying. I can immagine how you feel, people move bits while you shovel mountains. Please, look at table 9 (section 4.8 – graph 10). Like those ones, all the values in the graphs are differences. For example, from table 9:

1.4985392354 – 1.5 = - 0.0014607646

You write:...”The biggest problem is that all this is only pertinent to harmonic instruments and not pianos.”...

What you are stating has a lot of implications. Are you aware of it?

You then say:...“This is obvious to you, and yet...”...

Please Tooner be carefull, never force your mind in someone else’s head. Overconfidence and conjectures may take you ill, this is why I ask you to always argue your positions.

...“you still put forth your paper as something important to piano tuning.”...

So doing you may be misleading and off-putting for your readers. Talking about temperament and piano tuning, Chas model simply derives from a new approach to the logarithmic scale and from the combination of partials 2, 3 and 5. I ask you: to which model are we refering nowadays? Is’nt 12th root of two a logarithmic model? And Cordier’s 7th root of 3/2?

...“You should not be surprised that there is little interest in your theory and paper.”...

Are you talking about yourself? Was it the rattling of your pc’s keeboard that interested you during this last five months? No, I guess this may have been you in a bad day, nothing to do with my sharp devil's advocate. Anyway, about general interest in Chas theory and practice, for the time being I should be quite happy and there is no point in me saying why.

...“It is not useful for pianos, and puts a great burden on the reader to understand what it is that you are saying.”

Why and how Chas model can be useful, in its theory and in our practice, has extensively been written in this thread. Actually, with Chas Topic I’m trying out an alternative way for sharing the theoretical and practical results of my professional experience. Like any reader can do, I’ve freely chosen my burden and, if needed, I’m still willing to add more explainations.

About tuning practice, have you tried inverted 5ths? Have you tested constant equal beating 12ths (narrow) and 15ths (wide)?

Regards, a.c.

Alfredo:

Just got handed a time-critical project, so please excuse my brevity. I will not address all that you posted.

I do not think that I have assumed too much. I am the only one that has shown continued interest in your Topic. And you have admitted that Chas does not explain how you tune. Perhaps this will:

The effects of iH cause higher partials to be at higher frequencies than theoretical. This causes octaves to be wider than they are, and frequency ratios to be greater than the 12th root of 2, or even the Chas ratio. (I know this has been said many, many times, but needs to be restated as a review for the following which is not said.) It is easily assumed that this would cause wide tuning intervals to beat faster and beat more than twice as fast for each octave, and for narrow tuning intervals to do the opposite. Oddly, this is not true, because iH increases logarithmically. With one notable exception, the opposite is true. The exception is the 3:2 partial match of the 5th. [Edit] The beat rates of fifths progress differently than other narrow intervals.

So, if we start with an equal temperment octave tuned on an actual piano within a 4:2 octave width so that the lower fourth beats at the same rate as the upper fifth, and tune upward, always keeping the fourth and fifth beating at the same rate for each octave, the following will happen: For a while both the fourth and fifth will beat faster, but not twice as fast each octave. Then they will beat at the same speed from one octave to the next and then start beating slower. (This is the “inverting” that you mention.) If they could be heard, they would eventually both become beatless and then the fourths would become narrow while the fifths become wide!

I don’t think this is well known for a number of reasons. First, 4:2 octaves will cause audible beating if continued too far up. Second, the higher partials become harder to hear. Third, tuners listen to other things in the high treble. I can hear the fourths and fifths speed up and then slow down as I tune, but not become beatless. By then there are other, more important things, to listen to.

So what happens if octaves are tuned wider than 4:2? Well, the fourths will beat faster, the fifths beat slower and what you call an “inversion” will happen lower in the scale. But also, if the octaves are tuned much wider than 4:2, this will be too wide for equal beating 12ths and 15ths with the 12ths being narrow. However, it can cause 12ths and 15ths to beat at the same speed (or at least seems to) but with the 12ths being wide! It would seem impossible for this to happen, but the effects of iH, being logarithmic, do unexpected things. I think this is how you are actually tuning because of where in the scale your fifths are “inverting.”

Must go now. Btw, most people have good days and bad days; I have good moments and bad weeks.

I've been following with interest. I just haven't had time or energy to try to decipher the meaning.

Well, I wouldn't go as far as saying "full of sound and fury signifying nothing," but it sure feels that way.

But maybe I should clarify in advance, when I say "feels that way," I mean the personal, subjective, and emotional response coming from a human when they don't understand the rationale behind what they've experienced, LOL.

[That's an attempt at a joke, up there, is what that is...]

Tooner,

Thanks for contributing, you are helping me a lot, and thanks for your continued interest.

To be precise, I complain when you assume to little, in terms of responsibility. In other words, I do not like when you offer hasty judgements or conjectural evaluations because to me that sounds illogical and superficial, what may result in being misleading, that’s all.

Next, you say:…”you have admitted that Chas does not explain how you tune.”…

I’ve explained that Chas is a model. Do you find hard to simply take note of it? What is that you do not understand about past and present models?

Thanks for describing iH’s effects in piano tuning. I also hope you take note of another undeniable fact: the actual approach to iH (on pianos) is still referred to pure-octave tuning.

You kindly say (from -/ to +// = skip):

...“The effects of iH cause -/ higher partials to be at higher frequencies than theoretical. This causes octaves to be wider than they are, and frequency ratios to be greater than the 12th root of 2, or even the Chas ratio. (I know this has been said many, many times, but needs to be restated as a review for the following which is not said.) It is easily assumed that this would cause +// wide tuning intervals to beat faster and beat more than twice as fast for each octave, and for narrow tuning intervals to do the opposite.”...

In your previous post, about the Chas graphs, you wrote:...“The twelfth root of a number greater than two (such as Chas) predicts beat rates of wide intervals to double more often than every octave and narrow intervals less often than every octave.”...

To me this sounds the same, and yet you talk about iH’s effects, Chas graphs show you theoretical values.

You then say:...”Oddly, this is not true, because iH increases logarithmically. With one notable exception, the opposite is true. The exception is the 3:2 partial match of the 5th.”...

So, “the opposite is true” means that 5ths progress like narrow intervals. But then you say:

...[Edit]“The beat rates of fifths progress differently than other narrow intervals.”...

Here, I need you to conferm that: at first 5ths progress like narrow intervals, but then 5ths will progress differently than other narrow intervals.

I’d stop here and wait for your answer (take your time). You may also check 2ds (section 4.6) and maybe notice how they progress.

JDelmore,

Thanks for your interest, let me know if I can help you to decipher Chas model’s meaning or if it is only a question of time and energy.

Jim Moy,

Thanks for joining, I’m getting emotional responses too. About Chas rational understanding, is there anything I can do? Do you think another way would be clearer? For istance:

Chas model describes infinite scales of proportional frequencies deriving from all partials proportional differences.

Does Chas model deny previous ET models? No, actually Chas model includes any conventional ET pure-ratio based model, like 12th root of two.

What makes Chas a new model? A new theoretical approach to the sound scale.

What’s new about Chas approach? The way Chas gets to the frequencies. Chas gains the scale’s frequencies with a “difference” factor.

Why a “difference” factor? Because a difference factor can determine differences on any partial’s ratio.

Would this way be any better?

Have a nice w.e.,regards, a.c.

Dear colleagues,

Some of you asked for a recording of Chas tuning. Here,

http://www.megaupload.com/?d=QAVZ7RLE

you find the first of a series of recordings in order to demonstrate only one fact: no matter the size of the piano and despite the usual iH’s degrees, we can find our favorite form again and again, in my case Chas basic ET form, together with its intervals progression (on demand) and its constants, opposite equal beating 12ths (narrow) and 15ths (wide). This is why I’m choosing this conditions: small pianos, non-professional recording, non-professional playing. Finally, this is one of the ET forms that Chas theoretical model can mathematically describe.

This was recorded at Alessandro Petrolati’s lab (many thanks). On this Steinway S (5’ 1”, 155 cm), last month he put new strings, new hammers (still to be voiced) and new pins. For recording he used a 250 Euro device that he positioned about two meters away. I then asked him to kindly play whatever he liked to.

Ah, I hope you do not mind mega-muscles. Actually, I asked one of my sons to help me put this recording in the web. If you can suggest another place, so much the better.

Regards, a.c.

I use www.box.net It's very good and there are never any problems with it.

I have not been following this thread, as it is very technical and beyond my expertise.

Mr. Carpurso, I just listened to your recording, and to my ear, it's very pleasing. I would not hesitate to use your tuning stretch in my piano.

grandpianoman,

nice to get an immediate help! Are you the pianist that can tune his own piano?

Yes.....I do try! But, I am not a pianist, my 2 player systems are the ones that 'play' my piano.

grandpianoman,

I liked your ETD tuning, thanks a lot for your sharing. I'll tell you more tomorrow.

Regards, a.c.

Granpianoman,

In my opinion your recordings have well demontrated that with an ETD you can tune your piano. In the case of professional tuning, pin’s control and tuning stability are also very important. In fact, a pro tuner should never spoil the pins block while ensuring a stable, long lasting tuning. In my experience, my aural skill and my wrist’s sensitivity developed together, although implying opposite conditions: deepest relaxation and best body tension.

About Chas, the all question is much easyer than pro tuning and more handy than what it may seem to be. Today we are still referring to 12th root of two, so theorizing a pure, beat-less 2:1 octave. When I’ve asked why today should the octave be theoretically pure, I did not get any answer. Yet today, all the maths calculations for piano scaling, for piano iH and beats are based on 12th root of two, a model that nobody has ever been able to put into practice.

Chas model describes an ET scale where neither the octave is theoretically pure, a scale that is not based on a pure-octave module, nor a single pure ratio. In fact Chas finally combines all ratios like 2:1, 3:1, 5:1 and can theoretically describe a euphonic beating-whole. In stead of adopting the 2:1 zero-beating octave constant, Chas ET adopts two opposite equal beating constants, 12ths (narrow) and 15ths (wide), the two constants I can relate to my practical tuning.

If you would like to know more about Chas, you’ll be welcome with any question.

Regards, a.c.

Chas tuning on a Steinway S:

http://www.megaupload.com/?d=QAVZ7RLE

Mr.Capurso, thank you for your thoughts on a consumer being able to tune their own piano with an ETD. I have proven that it is possible.

Am I the only one here that likes your CHAS tuning...that can't be. My ears do not lie, and I find your tuning on the Steinway to be very pleasing to the ear. The whole piano sounds good!

I am really not technically minded to understand your CHAS tuniing theory, but I do know what I like, and I like your tuning. Is there a way I can implement your tuning into my Reyburn Cyber Tuner? I would like to try it on my grand piano.

Grandpianoman,

thank you for your feedback. I do not use a Reyburn Cyber Tuner so unfortunately I can not help you. Maybe some expert will read this, or you could start a Topic, or we could ask the developer, Mr. Reyburn. About others who may like or dislike Chas tuning, let’s wait. After all, I’m not urging my colleagues to say how they like it and why I’m doing all this is because I would like to honestly contribute with a new approach to the sound whole, what has helped me in my tuning practice.

Dear colleagues,

now you can find the first recording of Chas tuning on a Steinway S (5’ 1”, 155 cm) at MediaFire:

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

Yesterday I wrote: “When I’ve asked why today should the octave be theoretically pure, I did not get any answer.”

In my opinion, one of the reasons why 12th root of two ET theorized a pure 2:1 octave is because, at that time, the pure octave was a dogma, or maybe because nobody knew how to mathematically combine also the 2:1 ratio in an ET scale.

So again, I ask: how can you think 12th root of two as an "ideal" when no one can give a logical theoretical reason for a zero-beating octave?

And yes, I would also like a technical comment about iH limits on the recorded baby grand. Last but not list, does this kind of effort make sense? Should I try out my Japanese?

Regards, a.c.

Alfredo & GPM:

I will listen to your recordings and reply, within a week.

GPM:

Your Avatar makes me think of a slice of mincemeat pie.

Alfredo, You're welcome. I understand, and I also hope at some point, your Chas system can be repeated in an ETD.

Jeff, lol...mincemeat pie?!!...you know, I tried to make that picture of my grand piano larger...I do have a large appetite!

Alfredo:

I have decided that I will no longer discuss nor study your paper for personal mental health reasons. It does not seem that I have been helpful, anyway.

We can discuss any and all other aspects of tuning and tuning theory, though.

Our discussions have caused me to look deeper into tuning and tuning theory, and also how to talk about them. I did not make myself very clear on the beating of fourths and fifths. Let me try again.

If theoretical, harmonic tones are tuned to 12th root of 2 pitches, the beat rate of fourths and fifths will double every octave. Also, the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave.

If a typical piano with inharmonic tones is tuned so that the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave (this being a definition of a 4:2 octave) the beat speed of the fourths and fifths will less than double every octave. Specifically, in the fifth octave they will stop increasing in beat speed, and start decreasing in beat speed. In the sixth octave they will become beatless and then start beating again, but with the fourths being narrow of just and the fifths being wide of just.

Of course, if the piano is not typical or the octaves are not 4:2, the beat rates may do something else.

Sometimes I ask too many questions in one post, or I forget that I had already asked a question. Here are just a few that I don’t think I have asked before:

You said that your 12ths and double octaves all beat at the same speed. What speed is this?

You said that your 12ths are narrow and your double octaves are wide. How do you know this for sure?

PLEASE, lets talk about other things and not your paper.

GP, the Verituner should be able to replicate this type of tuning. I haven't kept up with this discussion, but are we talking about balancing between two or more intervals as opposed to just working with the octave? Someone give me a short version.... tuning up, the note to be tuned (call it C6 two octaves above middle C)is compared/balanced between which notes?

You mentioned 3:1, which would be the octave+fifth below (F4) 4:1 double octave (C4), single octave types (C5), octave+fifth (G4)? Fourths and fifths? (F5, G5)?

RCT can be "pushed" to kindof replicate this type of tuning approach - but it is limited by only having a few inharmonicity measurements to project across the range of the keyboard. It's a great project for you because you are only concerned with one instrument. You'd create multiple tuning files aiming for specific interval matches - using the custom equalizer. Then you can try averaging the files, or flipping back and forth in the graph mode to find a happy medium... I think I still have a working copy of RCT on an old laptop - if I get the how-to specifics, I might be able to come up with a work-around.

Ron Koval

Thanks Ron, that sounds like a great idea. Let me know if you get the how-to specifics and work them with your copy of RCT.

I don't know anything about the 3:1 ratios etc that you mention. Perhaps Mr. Capurso can chime in.

Tooner,

You say:...“I have decided that I will no longer discuss nor study your paper for personal mental health reasons.”...

What’s this, the beginning of a bad week? If you were joking I’d answer: do not worry, it’s to late.

...“It does not seem that I have been helpful, anyway.”...

You well know that most of the time you are helpful, in this Topic like in many others, so what is this about?

...“We can discuss any and all other aspects of tuning and tuning theory, though.”...

As you like. For me, it has been this way since my second post (05/08/09) when I wrote: “...then we'll talk about anything you like.” Has it been so? Should it be different?

...“Our discussions have caused me to look deeper into tuning and tuning theory, and also how to talk about them.”...

I’m glad you have not wasted your time. But you seem to be regretting something...

You kindly say:

...“If theoretical, harmonic tones are tuned to 12th root of 2 pitches, the beat rate of fourths and fifths will double every octave. Also, the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave.
If a typical piano with inharmonic tones is tuned so that the beat speed of any fourth will be the same as the beat speed of the fifth above it, sharing a common note, and spanning an octave (this being a definition of a 4:2 octave) the beat speed of the fourths and fifths will less than double every octave. Specifically, in the fifth octave they will stop increasing in beat speed, and start decreasing in beat speed. In the sixth octave they will become beatless and then start beating again, but with the fourths being narrow of just and the fifths being wide of just.”...

I’ve gone back in this Topic to see if I had said the same and yes, have a look at what I posted on 05/20/09.

You say...“Of course, if the piano is not typical or the octaves are not 4:2, the beat rates may do something else.”...

In my tuning, A4-A3 (A3 flat) goes together with A3-D4 (4th) beating at about 1 bps, D4-A4 (5th) beating slower at about 1/3.5 bps, A3-E4 (5th) about 2/3 bps, E4-A4 (4th) about 2 bps. As you notice, 4ths beat progressively faster but they sort of collapse at G4-C5, i.e. 4ths beating is not discernable anymore.

...“You said that your 12ths and double octaves all beat at the same speed. What speed is this?”...

On 05/23/09 I posted my sequence (please mind: wide = sharp, narrow = flat, but for any tuner this is obvious). At the bottom you also read: Chas delta-wide 15ths and delta-narrow 12ths beat’s rate is about 1b/3s.

...“You said that your 12ths are narrow and your double octaves are wide. How do you know this for sure?”...

Because to eliminate the 12ths beat I need to turn my tuning hammer clockwise. The opposite for double octaves.

About tuning and tuning theory, I ask: why should the octave be theoretically pure? Why should not we theoretically combine also 2:1 ratio in our ET scale, like we have done with 3:2 and 5:4 ratios?

RonTuner, GPM,

I need to read your post again and try to understand what the question is, maybe RonTuner you can kindly word it in a different way. Thanks a lot.

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:

No regrets, just pruning.

Your 5/20/09 post does mention what you call the “inverting” of fifths, but not fourths. (I have to be careful with your use of that term, it means something different to me.) And then mentions that the answer is in the Chas model. Sorry, I choose not to go there.

You mention a fault in tuning using a 2:1 ratio. I agree that there is a fault, but not with using the ratio of 2:1 instead of another number. The fault is in using a ratio at all. Tuning is done by matching partials so that they are either beatless, or beat at a specific (usually progressive) rate. Before inharmonicity was understood, it was believed that if an octave was beatless the frequency ratio was 2:1. After frequency measuring devices were available it was discovered that a 2:1 partial match does not mean a 2:1 frequency ratio.

Now if what you mean is that an octave should not be tuned to a 2:1 partial match, apparently they are not tuned that way, anyway. They are tuned closer to a 4:2 partial match, regardless of what people thought. This is shown to be true when considering the P4-P5 test for a 4:2 octave, which was a standard octave check before inharmonicity was well understood.

So what happens when 4:2 octaves are tuned? I wasn’t quite sure when listening to my tunings. But when calculating the beat rates, taking into account inharmonicity, it turns out that the double octave beats wide and about the same speed as the 12ths beating narrow in the mid section! I did not expect this. Equal beating 12ths and 15ths, at least in the mid section, seem to have been the norm all along.

But you are saying that your octaves are audibly beating, and your P4-P5 test confirms that you are tuning wider than 4:2 octaves. But the speed of your 12ths and 15ths are in range for 4:2 octaves in the mid section. This is just an objective analysis. What the discrepancy may be, I do not know.

Another discrepancy is that your 12ths and 15ths all beat at the same speed. Well, they may seem to. Mathematical analysis does not agree; they are progressive. I have proved this to myself when tuning this way and using a “drone tone” to make sure that the 12ths were narrow and the 15ths wide.

There are those that say all fourths should beat at the same speed and the beat rate of fifths is barely discernable according to modern tuning theory. I hear something different and mathematical analysis supports what I hear. I have no explanation for this discrepancy either.

Tooner very good, you have written a very well pruned post. May I seriously invite you to come and help me with my olive trees? Anytime, from now 'till next March.

You say:...“Your 5/20/09 post does mention what you call the “inverting” of fifths, but not fourths. (I have to be careful with your use of that term, it means something different to me.)”...

How would you call the 5ths fenomenon, so that we can relax? About fourths, I thought it may have been to much although, asking Bill Bremmer about what he had heard, I wanted to introduce the question. This is what Bill wrote (06/02/09):

“5ths become wide on PTG Tuning Exam Master Tunings in the 6th octave.”…”It must be close to 20 years ago that I saw Steve Fairchild demonstrate that 5ths do become wide. He also said that 4ths become narrowed.”

I answered: “So we agree in saying that, somewere, 5ths do invert, now the questions are: can or should 5ths be progressive or can 5ths have casual beats/rate? And what about 4ths?”

Bill had also written: “I have met and had discussions with Bernhard Stopper and have also heard his tuning. It has a remarkably clear character to it. While I still do not fully understand it, I did gather from what he has said that the 12ths also become wide at some point in his tunings as well.”
And: “I have now long taken to the practice of tuning pure double octaves and 5ths from F6 to the top.”

I could not understand how he can have pure double octaves and 5ths, together with wide 12ths and progressive RBI. I would have talked about it but, as a result, since then I have not heard from Bill. This may be how tuning can become an indulgent mystery.

You now say:...“You mention a fault in tuning using a 2:1 ratio. I agree that there is a fault, but not with using the ratio of 2:1 instead of another number. The fault is in using a ratio at all.”...

This was your conclusion also five months ago, but now there may be some good reasons to update it.

...“Tuning is done by matching partials so that they are either beatless, or beat at a specific (usually progressive) rate.”...

Yes.

...“Before inharmonicity was understood, it was believed that if an octave was beatless the frequency ratio was 2:1.”...

Yes. Using your words, it was also believed that the octave could and should be beatless. It was also believed that there was no need to combine 2:1 ratio with 3:1 and 5:1 ratios, and that we could get by with 12th root of two model. It was believed that the traditional octave module was correct, that the temperament could be referred to 13 notes and that the octave module could then be copyed. All these being unjustified theoretical premises, all wrong beliefs.

You say:...“After frequency measuring devices were available it was discovered that a 2:1 partial match does not mean a 2:1 frequency ratio.”...

Yes, and a heavy curtain was drawn over those unjustified premises. In other words, not only a 2:1 partial match does not mean a 2:1 frequency ratio, a 2:1 partial match is a wrong target in that it comes (with Kent’s permission) from a lame model.

...“Now if what you mean is that an octave should not be tuned to a 2:1 partial match, apparently they are not tuned that way, anyway. They are tuned closer to a 4:2 partial match, regardless of what people thought.”...

Exactly. You say that nobody goes for a 2:1 partial match. You conferm that we are not referring to 12th root of two model, so we are lacking for a reliable model. You say that we are going for a greater ratio and you refer this to iH. Actually I do not. In fact, any ratio greater than 2:1 combines – perhaps you’d say compromises - partial 2 with partials 3 and 5. If this combining is ideal or not, this is another question and Chas is the answer.

..."So what happens when 4:2 octaves are tuned? I wasn’t quite sure when listening to my tunings. But when calculating the beat rates, taking into account inharmonicity, it turns out that the double octave beats wide and about the same speed as the 12ths beating narrow in the mid section!"...

Very good indeed. We may be on the right track.

..."I did not expect this. Equal beating 12ths and 15ths, at least in the mid section, seem to have been the norm all along.”...

You say the norm, I’d say the “about” tendency. In my opinion, all tuners have been and are seeking the model and the practical way that combines all partials and all relative ratios.

..."But you are saying that your octaves are audibly beating,”...

Yes, and progressive.

...“and your P4-P5 test confirms that you are tuning wider than 4:2 octaves.”...

Wider than 2:1 for sure, how much wider I’ve never measured. Can you calculate what the theoretical ET scale’s incremental ratio is with 4:2 octaves?

...“But the speed of your 12ths and 15ths are in range for 4:2 octaves in the mid section. This is just an objective analysis. What the discrepancy may be, I do not know.”...

Would you be more precise on this? I can not get the point.

...”Another discrepancy is that your 12ths and 15ths all beat at the same speed. Well, they may seem to. Mathematical analysis does not agree; they are progressive. I have proved this to myself when tuning this way and using a “drone tone” to make sure that the 12ths were narrow and the 15ths wide.”...

From your analysis, would 12ths and 15ths be convergent or divergent?

...”There are those that say all fourths should beat at the same speed and the beat rate of fifths is barely discernable according to modern tuning theory. I hear something different and mathematical analysis supports what I hear. I have no explanation for this discrepancy either.”...

In my tuning fourths are progressive. Around C5 they seem to collapse, so I can not say if they invert in the high section. I will have a go plucking the strings. The beat rate of fifths is very well discernable, like RBI, and fifths too in Chas tuning are progressive, first going narrower, then toward pure.

Tooner, thanks for your efforts 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

RonTuner,

The constants of Chas tuning’s form are 12ths (octave+fifth) and 15ths (double octave). In other words, 12ths (narrow) and 15ths (wide) have the same opposite beat-rate all along the keyboard.

How to get this final tuning form is a different matter and it may depend on the piano’s conditions and settling.

I first tune only middle strings from C3 to C6. Generally speacking, on normally-flat pianos, when I go up the temperament section, i.e. from A#4, I stabilize (on mid-string) a preparatory wider stretch for all notes (and all check intervals) so to obtain - at list - chromatic pure 12ths. This is to say at list 3:1 matching. So, from A3-E5 12th up, all chromatic 12ths will be - at list - beatless (on mid-strings). All double octaves will then beat (on mid-strings) about 3/2 bps, so that after unisoning left and right strings I get (hopefully) equal beating 12ths (narrow) and 15ths (wide). If the piano was very flat I may tune it twice, anyhow I would stretch all check intervals so to obtain chromatic little-wide 12ths, with maybe a wide ¼ bps.

Is this of any help?

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:

When I read something like “The fifths are …..” what I imply is “The [beat rate] of the fifths are …” So when I read something like “The fifths invert ….” I imply “The [beat rate] of the fifths invert [from narrow to wide] …” But I understand that you mean “The [progression of the beat rates] of the fifths invert [from beating faster to beating slower] ….” It is not a big problem. I understand what you mean. But I want to make it clear in my mind and in the mind of other readers. If you choose, you could include the phrase “beat rate progression" when appropriate, such as "The beat rate progression of the fifths invert from becoming faster to becoming slower."

I will not speak for other posters including Bill and Kent. I do love your term “indulgent mystery”, though. It is how I look at what you say about your tuning.

We seem to be covering old ground (going in circles) in the discussion of frequency ratios, partial matches and inharmonicity. Perhaps we can break free of this if I play the Devil’s Advocate again. You ask: “Can you calculate what the theoretical ET scale’s incremental ratio is with 4:2 octaves?” Yes I easily can and will, but you must first tell me what you will do with the answer.

Now also consider this: You say your fourths beat progressively faster. When a 12ths is tuned up from the upper note of a fourth so that the resulting 15th from the lower note of the fourth is wide and beats at the same speed as the narrow 12th, the beat speed of the 12th and 15th will be ½ the beat rate of the fourth. Yet you say that all your 12ths and 15ths beat at the same speed. I consider this to be an “indulgent mystery.”

Thanks - that's what I was looking for. That should be easy to program into the Verituner. (Only down to D2- then I lose the 3:1 as an option...)

Ron Koval
chicagoland

Ron,

excellent! I really hope you can manage. Only remember that, if you go straight for the Chas final form, the piano's settling may cause a lowering of the frequencies, especially from mid-high section (C5) up.

This is why I mention a preparatory wider stretch.

Tooner,

thanks, I shall 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

I'll try it out - in my experience, the Baldwin school upright pianos prove to be the death of many of these types of tuning schemes. The added width to the octave by referencing the octave+5th as a basis leaves the single octave with a problematic, obvious beat. I'll see if tempering the 12th with the 15th works with these types of pianos. I'm confident it will be fine with larger instruments.

Ron Koval
chicagoland

Tooner,

You kindly say:...“If you choose, you could include the phrase “beat rate progression" when appropriate, such as "The beat rate progression of the fifths invert from becoming faster to becoming slower."...

Thank you, I’ll include the phrase you are suggesting.

...“I will not speak for other posters including Bill and Kent. I do love your term “indulgent mystery”, though. It is how I look at what you say about your tuning.”...

Ok, I feel the blow.

...“We seem to be covering old ground (going in circles) in the discussion of frequency ratios, partial matches and inharmonicity.”...

I do not think so, and you yourself talk about some fresh understanding.

...“Perhaps we can break free of this if I play the Devil’s Advocate again. You ask: “Can you calculate what the theoretical ET scale’s incremental ratio is with 4:2 octaves?” Yes I easily can and will, but you must first tell me what you will do with the answer.”...

This sounds more like the Devil’s Ambassador. Anyway ok, with that answer I’ll turn hell into heaven. Only then you yourself may be able to approach the only one ratio greater than 12th root of two that can straighten the beat rate progression of the 12ths and 15ths, in the way Chas model describes its equal beating constants. It can not be now though.

You write:...“Now also consider this: You say your fourths beat progressively faster.”...

Not exactly. I say that Chas fourths beat progressively faster up to G4-C5. I’ve also said that, tuning up the scale, I use many other check intervals and that I do not use 4ths because they seem to collapse. To be more precise (it may be useful), I do not even exclude that the beat rate progression of the 4ths invert also going down the scale, after C3-F3, from becoming slower to becoming faster.

You then say:...“When a 12ths is tuned up from the upper note of a fourth so that the resulting 15th from the lower note of the fourth is wide and beats at the same speed as the narrow 12th,”...

Is this to say: when you tune a 12th and a 15th opposite equal beating?

...“the beat speed of the 12th and 15th will be ½ the beat rate of the fourth. Yet you say that all your 12ths and 15ths beat at the same speed. I consider this to be an “indulgent mystery.”

To me this looks like a tuning experience gap. Will a recording of 4ths with relative 12ths and 15ths be enough?

I also asked you: From your analysis, would 12ths and 15ths be convergent or divergent? Also this may be useful.

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:

The reason I like your term “indulgent mystery” is that I consider it to be a term of acceptance, not of rejection. An acceptance that people do not hear things the same, and come to different conclusions. I did not mean it as a jab, but more as a prod. Indulge is a warm, not a cold, word. Again, I am not your enemy.

Since we do hear things differently, if we want to reach a consensus, we need an objective way of looking at tuning. The most objective way that I know of is to look at beat rates. Especially beat rates that are tests: tests where a third note is used. The third note is used to understand what is happening with another interval or intervals. This is my basis of bringing up the conflicting statements of progressive fourths and unprogressive equal beating narrow 12ths and wide 15ths. But maybe we should put this aside for now, and hopefully bring it up again later. Other tests have been mentioned and you have not responded in a way that shows understanding.

I enjoy a challenge, and it continues to be a challenge to communicate with you. There is some vocabulary that we should define. (Remember, I have travelled, and am used to those that English is not their primary language. And why should it be!)

When the terms “narrow” interval or “wide” interval are used, it refers to the relationship of the nearly coincident partials that cause beats. If the nearly coincident partial of the lower note is higher than the nearly coincident partial of the upper note the interval is ”narrow.” So, if the upper note of a “narrow” interval is raised in pitch (or the lower note is lowered in pitch) the beat rate decreases. And if the nearly coincident partials are at the same pitch, the interval is “just” and beatless.

You asked “From your analysis, would 12ths and 15ths be convergent or divergent?” I am not sure what you mean by 12ths and 15ths being convergent or divergent. I will certainly answer your question, but need to understand it first. Could you give examples?

The reason I asked what you would do with the incremental ratio value for theoretical ET 4:2 octaves is to know what your understanding of the subject is. I think much of the communication problem is that we each assume that the other already knows certain things, looks at things a certain way, or is asking a question for a certain reason. And when we assume incorrectly we end up talking about two different things.

This reminds me of a story. A young boy asked his Mother where he came from. The mother wanted to be truthful, but did not know where to start. So stalling for time, she asked her son why he wanted to know. The son said that Suzie next door said she came from New York but he didn’t know where he came from.

So, I want to try to talk with you about just one or two things at a time, and make sure we understand what each other are saying before moving on.

So what significance do you think the incremental ratio of theoretical ET 4:2 octaves could have?

Tooner,

I well know you are not my enemy, I was joking. The same word can result in having different meanings though, I do not need to tell you and vice-versa.

You talk about “conflicting statements of progressive fourths and unprogressive equal beating narrow 12ths and wide 15ths.”

The beat rare progression of fourths also invert, for sure in the mid-high section, maybe also in the bass section. Will this be any better? You did not like the word "interweaving" but this is what 4ths and 5ths do with their beat rate progressions.

About “narrow” and “wide” intervals, and how they refer to coincident partials, I think I’m ok, maybe you wrote about this for the latest yung reader.

...“I am not sure what you mean by 12ths and 15ths being convergent or divergent…Could you give examples?"...

Yes. When you wrote:...”Another discrepancy is that your 12ths and 15ths all beat at the same speed. Well, they may seem to. Mathematical analysis does not agree; they are progressive. I have proved this to myself when tuning this way and using a “drone tone” to make sure that the 12ths were narrow and the 15ths wide.”

I answered: From your analysis, would 12ths and 15ths be convergent or divergent?

So, you say they are progressive. If it is so, how do they progress? Overcrossing, like 4ths and 5ths? Getting further apart?

...“So, I want to try to talk with you about just one or two things at a time, and make sure we understand what each other are saying before moving on.”...

Good idea, me to, I would like to know what you have understood so far about Chas model and Chas algorithm. You have written somewhere that delta is superfluous and that Chas algorithm could be an equation, when in fact it is an equation.

...“So what significance do you think the incremental ratio of theoretical ET 4:2 octaves could have?”...

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.

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