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Tooner,

You kindly say: “I don’t think you understood that Kent has said that he will not be replying to you anymore.”...

Thanks but yes, I understood what Kent wrote at the end of his last post, nevertheless Kent may change his mind and, rather than privately, we may discuss publicly about Chas, that is what I would prefere.

...“You have not shown a mutually exclusive relationship between zero beating tuning and equal beating tuning.”...

On the theoretical ground, I have not shown a “ mutually exclusive relationship between zero beating tuning and equal beating tuning” merely because I can prove the opposite, i.e. Chas ET EB theory can also comprehend any zero beating ratio (section 3.3 - THE S VARIABLE). Let’s see together one example:

From Chas algorithm (3 – (∆*s1))^1/19 = (4 + (∆*s))^1/24

If s1 = 1 and s = 0

we can find a delta value that makes our equation true:
Δ = 0.0033858462466

In fact:(3–(0.0033858462466*1))^(1/19) = (4+(0.0033858462466*0))^(1/24) =
= 1.059463094359 = 2^(1/12) = 12th root of 2 zero beating ratio.

As for 12th root of 2 octave’s zero beating ratio, Chas model does not exclude any theoretical ratio, zero beating or what ever. This is why, at the end of section 2.0 you can read: “In conceptual terms, the model is trans-cultural; it also responds to a new requirement on the contemporary music scene, by providing an algorithm which can give form to all kinds of microtonal sound structures.”

On the practical ground, in my experience, 12th root of 2 zero beating ratio is useless, 7th root of 3/2 and 19th root of 3 are unnecessarily extreme as a final tuning form, although not “sharp” enough when tuning mid-strings in the treble. In fact in my practice, when tuning centre strings in the treble I go beyond the 7th root of 3/2 pure fifths ratio and I stretch progressive wide fifths. As I’ve said, this is done to compensate sound-board, bridge and strings elasticity, so considering my tuning settling-down.

You say:...“So, the failure to prove the validity of one does not prove a validity of the other.”...

True, then the question may be: what makes Chas beating-whole’s ratio better than any zero beating ratio? Leave all previous reasonings aside, you find an answer reading Chas article’s section 4.5 - Sequence of quotients..., and Table 6: Comparison between quotients deriving from ratios 3:2, 5:4, 3:1 and 5:1. There you’ll be able to evaluate Chas validity only with the help of numbers.

...“But here is something that you have not answered: How does the Chas theory predict that fifths “invert”? The answer is that it does not predict this. It is an effect of inharmonicity when choosing a particular stretch style.”...

I'd put it in a slightly different way: when choosing a particular stretch style, we need to invert fifths. In other words, inverting fifths in the temperament is the technique I use for achieving Chas form.

12th root of 2 predicts zero beating octaves, 7th root of 3/2 and 19th root of 3 predict respectively zero beating fifths and zero beating 12ths, Chas model’s basic form predicts opposite and constant 12ths and 15ths equal beating. How to get to Chas basic form in practice is a different matter, and Chas algorithm, with its s variable, allows you to make use of all ratios you need.

...“Although you think that your theory is coherent, it is not because it is incomplete. It does not predict what actually happens.”

I would not fuse coherence with completeness and we can still discuss both concepts. Chas theory describes a beating-whole and it gains the scale’s frequency values with a delta (∆) beat-factor. I can not immagine anything more coherent, can you?

About prediction, fifths inversion and Chas theory’s completeness, maybe this can help. In section 3.4 - CHAS SET…and section 3.5 - EFFECT OF ±DELTA…, you can read that the Chas model opens up a module of 49 sound elements, in a semitonal order, from 0 to 48, whose scale ratio is (4 + Δ)^2. If 12th root of two zero beating symmetry encompasses 8 scale’s degrees (one octave), Chas model’s beating symmetry encompasses 29 degrees of our semitonal scale. Now, in section 4.3 (graph 5, 6) you can compare 12th root of two and Chas difference curves for ratios 4:3 and 3:2, and in section 4.8 (graph 10) you can see what happens: within Chas compass, the beat curve for ratio 3:2 inverts its progression. Using your words I ask you: is this an effect of inharmonicity? I do not think so but, as you'd say, I think it is the effect of a particular stretch style, in this case the (predicted ?) effect of Chas ET EB.

Thanks a lot and regards, a.c.

Alfredo:

Can you explain this in more detail:

“Now, in section 4.3 (graph 5, 6) you can compare 12th root of two and Chas difference curves for ratios 4:3 and 3:2, and in section 4.8 (graph 10) you can see what happens: within Chas compass, the beat curve for ratio 3:2 inverts its progression.”

You mention “ratios 4:3 and 3:2” but do not explicitly say what the ratios are of. Perhaps you mean partial matches? That is where beats come from, not ratios. Are you saying that using a semi-tone ratio of 1.0594865443501 will produce fifths that “invert”, in other words, beat faster and then slower when progressing up the scale?

Tooner,

...“You mention “ratios 4:3 and 3:2” but do not explicitly say what the ratios are of. Perhaps you mean partial matches? That is where beats come from, not ratios.”...

True. In section 4.3, at the bottom of graphs 5 and 6 you find the scale degrees that can be compared; as well as in section 4.8, were you also find Table 9 listing Graph 10 values.

...“Are you saying that using a semi-tone ratio of 1.0594865443501 will produce fifths that “invert”, in other words, beat faster and then slower when progressing up the scale?”

No, as I’ve said inverting fifths in the temperament is the technique I use for achieving the ET EB tuning form that Chas model describes. a.c.

Alfredo:

Then, your paper is incomplete as I said before. It does not predict the change in the beat rates of fifths that you find necessary for equal beating 12ths and 15ths on a real piano.

Tooner thanks, more often you help my mood.

Before you were talking about Chas theory being incomplete, now you are saying that my paper is incomplete. What happened in the meantime?

Could you tell me what can be average deducible from Graphs 5, 6 and 10? Could you get the partials-matching involving beats?

Maybe you know this:

http://www.pykett.org.uk/temperament_-_a_study_of_anachronism.htm

I'd like to know your opinion sometime. a.c.

Alfredo:

Because your paper is the only place that Chas theory is explained, and the only topic in your paper is Chas theory, I see them as one thing: A problem with one is a problem with the other. But I understand that you would view them as separate things. So, where do you think the problem is between how your beat rates are when you tune and how your paper explains what the theory predicts beat rates should be?

I have pondered far too many times what graphs 5, 6 and 10 are indicating and still have no idea. I do not know what you mean by degrees. The only mention of degrees in music that I can think of is in the Psalms, which may have indicated ascending stair steps while singing. But your graphs supposedly indicate “differences”, which should mean that one value is subtracted from another, but I have no idea what the values are.

Sorry, I have very little interest in unequal temperaments. A disturbing reason I have read for tuning unequal temperaments is that hardly anyone can tune ET anyway, so it is far better to tune an UT, especially one that is designed to be easily tuned. It reminds me of excuses for “new morality” which is really just old sin. If you started a new Topic on the paper that you provided a link to, there is sure to be a great deal of response.

Tooner, thanks for your reply, I came back yesterday and finally I can answer you.

You kindly say:...“Because your paper is the only place that Chas theory is explained, and the only topic in your paper is Chas theory, I see them as one thing: A problem with one is a problem with the other. But I understand that you would view them as separate things.”...

It is good that you understand me when I view Chas theory and the Chas article as separate things, yet I do not understand what benefit you get from viewing at them as one thing. While Chas theory and maths are solid, the Chas article can be improved for sure, so I’d rather keep them separate.

You ask:...“So, where do you think the problem is between how your beat rates are when you tune and how your paper explains what the theory predicts beat rates should be?”...

I can only think that the problem you are pointing out may derive from the way you look at Chas model, at its scale and at the scale’s values (to know more about scientific modelling and systems:
http://en.wikipedia.org/wiki/Model_(abstract)

Once you refere to general modelling, you will not expect Chas model to predict what beat rates should be on pianos.

Chas model’s aim is to represent an ET scale of frequencies related to partials 3 and 4 differences, i.e. an ET scale related to the Chas system’s constants, the 12th and the 15th intervals.

Chas model describes a beating-whole where partials effects are finally combined. In other words, in Chas scale no interval is pure; all intervals have proportional differences from their pure partial value, in an intrinsic correlation between frequencies and differences arising from the infinite combinations of the scale’s elements (section 3.0). As a result, in theory as in practice Chas octaves are progressively stretched (section 4.2, graphs 3 and 4), and 12ths and 15ths have opposite equal beating (section 4.1, table 2, graph 2).

You write:...“I have pondered far too many times what graphs 5, 6 and 10 are indicating and still have no idea. I do not know what you mean by degrees. The only mention of degrees in music that I can think of is in the Psalms, which may have indicated ascending stair steps while singing.”...

Here you can get an idea:
http://en.wikipedia.org/wiki/Degree_(music)

You then say:...“But your graphs supposedly indicate “differences”, which should mean that one value is subtracted from another, but I have no idea what the values are.”...

You can find graph 6 differences values in table 4 (section 4.4, graph 7), in table 8 (section 4.7, graph 9) and in table 9 (section 4.8, graph 10). Sorry if it comes out a bit confusing.

About Professor Colin Pykett’s paper you write:...“Sorry, I have very little interest in unequal temperaments.”...

I should have been more precise, in part 4 – Impure octaves, these are the lines I found intriguing:

“...For example, the beat rate of any interval played depends on the octave in which the interval resides. In other words, a fifth played in the third octave will beat faster than if it is played in the second octave, but slower than if it were to be played in the fourth octave. With any temperament which uses pure octaves, the ratio of these beat frequencies has a simple numerical relationship to the octaves considered...”,

“...Why not ease this problem a little by making the octaves themselves adjustable as well?...”,

“...Currently I have yet to decide on a definite road map for the study though. Doing it with the degree of emphasis on arithmetic and theory which constitutes current work on temperament is almost certainly debarred. It is debarred because pure octaves underpin the entire concept of temperament as it is understood today, and removing them will also remove the relative arithmetical simplicity of the subject. If the octaves are no longer pure, the subject could easily become theoretically anarchic and entirely experiential. Any note on the keyboard could in principle take any frequency value, and the frequencies actually chosen would then arise solely through empiricism – trial and error.”...

Tooner, would you say that Chas model is anarchic? That Chas values arise from error? By the way, last saturday I could record Chas tuning on a small grand, now I only need to make an mp3 of it.

Regards, a.c.

About Colin Pykett’s paper - it seems to be heavily tilted to temperament and octaves in organs... There is only a passing explanation of inharmonicity's effect on piano tuning.

While "pure" octaves are the "norm", there have been plenty of piano tuners over the years adjusting the octave width to achieve a pleasing balance of sound across the range of the instrument. Even the pure fifth temperament as well as the pure 12th temperament stretches have been discussed. Bill Bremmer was one of the first I remember tempering his octaves to match the octave+5th in unequal temperaments.

Ron Koval
Chicagoland

Alfredo:

I am going to take a break from your Topic. The idea of using the degrees (or steps) of a scale (a major scale presumably, although since it is not specified, it could be any number of scales…) as a basis for analysis is just too foreign to me. And when I try to make sense of it by examining paragraph 4.3 I again read “The differences, divided by themselves,…”. ANYTHING EXCEPT ZERO DIVIDED BY ITSELF EQUALS ONE!!!!!

I am going to take a break….

RonTuner,

you kindly say:..."About Colin Pykett’s paper - it seems to be heavily tilted to temperament and octaves in organs..."

In one post (06/30/09) I wrote: "I’m still trying to explain why I can not agree with Mr. Deutschle when he says:….” The octave is tuned wider than theoretical due to iH.”

In fact what I’m saying (since I can prove it) is that, with or without iH, we need to stretch octaves. Why? Because also partial 2, with the other partials, through stretching can practically contribute to hold up a resonant beating-whole system. Negating the beat’s value (or relevance), we would never get to the Chas concept of a beating-whole."

One thing I find interesting is that Professor Colin Pykett is a pipe organ tuner and, as you can read, he admits octaves stretch.

Tooner, in one post I had already explained what "The differences, divided by themselves,…” means. Anyway, in section 4.5 you can always see what that means.

Thanks and regards, a.c.

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.