Hi Ralph.

..."Bare in mind I am assumming mostly beatless octaves.

Assumming mostly beatless octaves, are you referring to theory or tuning practice?

Regards, a.c.

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

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

If octaves aren't beatless then, at least in my mind, they aren't octaves anymore. How many beats are allowed until it's no longer a 2:1 ratio? I guess all pure perfect fifths will fit perfectly in 7 octaves if the octaves are stretched, but that's just my point. If you keep truely beatless octaves then they won't fit.

...“If octaves aren't beatless then, at least in my mind, they aren't octaves anymore.”...

Yes, this is what we have learned, that theoretical octaves can/should be beatless.

Before that, during my musical studies, I related the octave to the western scale degrees and did not know about this theoretical axiom. Later, when I started tuning, I also learned that we were expected to copy the 12 semitones module, so to extend the temperament up and down the keyboard. Soon after I could conclude they were both wrong assumptions, wrong references. The octave – meaning the eighth degree of the scale – can be described and justified as a beating interval.

...“How many beats are allowed until it's no longer a 2:1 ratio?”...

Well, the theoretical 2:1 ratio leaves no room for beats. But we know that three contiguous M3ds reach (encompass) the octave (the eighth degree), so the question could be: how can we have simultaneous wide-beating 3ds and 2:1 octaves? It seemed possible in theory (12th root of two ET), but it is impossible in practice.

...“I guess all pure perfect fifths will fit perfectly in 7 octaves if the octaves are stretched, but that's just my point. If you keep truely beatless octaves then they won't fit.”...

I agree. In fact the “comma” needed to be resolved by stretching fifths and octaves. Stretch narrow fifths by subtracting a quantity (negative delta), and stretch wide octaves by adding the same quantity (positive delta). By doing this onto the 12th degree (octave+fifth) and the 15th degree (double octave - 24th semitone) we manage to inter-modulate our 12 semitones compass. Chas ET model is as simple as that, it is a golden geometry that fits our semitonal scale in theory and in practice.

Regards, a.c.
.

That's very interesting. I've always been very frustrated with tuning. I find it to be like life itself, always a compromise. You have to steal a little bit from here and give a littler bit back there. In my mind a tuning sequence should work perfectly, but none of them do. I've always felt that we are missing something in our understanding of the scale as if somewhere there is a fundamental flaw that we have inherited for hundreds of years. It's like space is curved but we're trying to tune in a straight line. Hard to explain really.

...“I've always been very frustrated with tuning.”...

So did I, at least for the first years of my practice. Nobody had told me that octaves should be progressive and S shaped and that fifths must invert their beat rate progression, the fifths beat-curve must be spoon-shaped.

...“I find it to be like life itself, always a compromise.”...

I am not sure I get your meaning, but if you mean “adjustment” then yes, I wanted to go for the just solution. Historically, for 20 centuries, the octave had always been 2:1, so the first ET (12th root of two) was founded on that axiom. It represented a compromise between 3ds and fifths but it left out the octave. Actually, the theoretical beatless octave did not combine partial 2 with partials 3 and we know that the sound of a single string is the combination of all its vibrating, partial lengths.

...“You have to steal a little bit from here and give a littler bit back there.”...

Sure, you can look at it in that way. My way is that we needed to interpret beats as an engine, without setting consonance (beatless intervals) against dissonance (beating intervals), but aiming at the most natural and euphonious source of emotions, a beating whole.

...“In my mind a tuning sequence should work perfectly, but none of them do.”...

The execution of a tuning sequence implies a tuner and a piano...for sure it should be satisfactory and not frustrating.

...“I've always felt that we are missing something in our understanding of the scale as if somewhere there is a fundamental flaw that we have inherited for hundreds of years.”...

Me too, I think we have an heritage, passed along by matter, sounds and vibrational frequencies. And I like the idea of a fundamental flaw, a vibrational order intrinsic to nature.

...“It's like space is curved but we're trying to tune in a straight line.”...

Yes, it is not easy to move away from one, single dimension. This may be why temperaments have been elaborated on the basis of one “pure” interval. Even more difficult is thinking in dynamic terms and relying on beats.

Regards, a.c.
.

Ron Koval, hello.

In the EBVT thread you wrote (June 08, 2010):..."I've got a stretch really similar to the CHAS tuning which gets me really close for the tempered octaves. My temperament has a maximum offset of 2.1 cents, so it doesn't take much tweaking to get from the calculation to a nice aural finish."...

May I ask you to expand?

By using your ETD, would you be able to tune Chas ET?

Have you tried Ernest Unrau ETD settings?

When you design your own temperament, what do you refer your offset to? In other words, what do you "deviate" from?

Regards, a.c.
.

Any of the modern temperaments tunings are really based on the ideas and work of Owen Jorgensen. Right or wrong, we've used offsets from ET to replicate alternate temperaments. So we deviate from whatever passes for ET as set by that particular machine. As I understand them, they calculate based on user input THEN apply + and - offsets from that calculation for every note.

I have tried Ernest's settings for the Verituner and like it very much for the hybrid tuning that I've been doing - machine set above A3 and then Bill Bremmer's mindless octaves going down. For smaller pianos, it may even be C4 and above and then aural going down...

What I've noticed while working with many different styles is that on nicer pianos they tend to come up with very similar tunings. There are huge variences on pianos that have challenging scale designs.

Ron Koval
chicagoland

Thank you, Ron, for your reply.

...Any of the modern temperaments tunings are really based on the ideas and work of Owen Jorgensen."...

I'm not sure I understand what you mean. What are you referring to?

..."Right or wrong, we've used offsets from ET to replicate alternate temperaments. So we deviate from whatever passes for ET as set by that particular machine."...

Ok, but how can you/we check "whatever passes for ET as set by that particular machine"?

..."As I understand them, they calculate based on user input THEN apply + and - offsets from that calculation for every note."...

So, alternate temperaments would depend on the user's input and the one, particular machine?

..."I have tried Ernest's settings for the Verituner and like it very much for the hybrid tuning that I've been doing - machine set above A3 and then Bill Bremmer's mindless octaves going down. For smaller pianos, it may even be C4 and above and then aural going down...

You say "hybrid tuning", what are you referring to? Then, I was talking about Ernest Unrau's recent simulation of Chas and I would appreciate your opinion.

..."What I've noticed while working with many different styles is that on nicer pianos they tend to come up with very similar tunings."...

Yes, different styles may sound similar, but in my experience that depends on the user's "ear".

..."There are huge variences on pianos that have challenging scale designs.".

Somehow, this could represent ETD's limits?

Regards, a.c.

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

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

I guess it would be best to take a look at these one at a time!



In the states, the interest and resurgence of tuning historical temperaments can most likely be traced to the work of a piano technician from Michigan - Owen Jorgensen. I don't have a copy of any of his books, but the "big red" book covers most of the temperaments used by the tuning machines (ETDs) As far as I know, he is responsible for the work that allowed techs to set an alternate temperament using an ETD, making tuning any temperament as easy as pushing a few buttons. I do have some questions about the authenticity of these tunings...

1. how many pianos did he use to verify that offsets from ET provide an accurate path to tuning alternate temperaments?

2. which ET was used? Standard machine FAC tuning, or an aurally verified one?

3. There are questions about his sources for research - were they representative of all of Europe during those time periods, or more local to England?

But - his writings allowed a new generation of tuners to experiment with different tunings quickly and easily without having to go through the learning curve that would be required for aural methods.

Ron Koval
chicagoland

Actually, the common use is to use the machine to determine the targets and then tune the unisons aurally. So yes, you will find everything from people just using the default setting for the piano, to setting up a custom setting, to some tuners aurally verifying the entire tuning as ET before letting the machine apply the offsets and tuning the whole piano again...

Ron Koval
chicagoland

A hybrid tuning style is one that is neither entirely machine calculated nor aural, but one that uses both. Since I have been tuning the Chas emulation using the Verituner with non-equal temperaments, there are adjustments needed to reconcile the double octave and the octave fifth balance going down into the bass break. I haven't been tuning ET recently, so I can't speak to how well the verituner style does for ET.

Ron Koval
chicagoland

There were other people who did research on old temperaments who both predated Owen Jorgensen and gave more practical information about them. Jorgensen reads like a laundry list of old temperaments.

I have a friend whose doctoral thesis is a similar list of early German literature. Today, he has the book mounted on his wall in tiny shreds arranged into an artful hanging.

Here it is, Patrick.

To All, regards, a.c.

Stopper, in the Appassionato Topic you wrote:

"Signore Capurso,

we may take up further discussions if you can negate (or find someone else doing for you) the three falsification points i provided ealier in the CHAS thread about your CHAS paper: (link below)

http://www.pianoworld.com/forum/ubbthreads.php/topics/1340096/Re:%20CIRCULAR%20HARMONIC%20SYSTEM%20-.html#Post1340096

- . - . - . -

I went to the link you provided and could not stop reading many other following posts. What to say?

As the Chas author, I would have nothing to add, precisely for you. I understand that you may like to improve Chas paper (thanks) but still now you would not be the reviser I would choose, and the reasons (sob) have been exhaustively explained.

I also understand that you do not like the idea of Chas ET model being more suitable than pure 12ths, when we need to fit the 12 semitones compass and fuse the basic partials primes together. Since I can not explain your difficulties (maths? business?), I've decided to rely on Time.

More, you do not seem to realize that Chas theory describes a thoroughly new approach to temperament, and that the resulting ET model needs to be understood in dynamic terms. Tell me/us Stopper, do you get the difference between a dynamic temperamental theory and a static one (read zero-beat intervals)? What you see in section 3.5 (Research report) represents a precise geometrical entity, but it looks “still” only because it is a diagram. In actual facts, that is a “temporary” form, it is the beating-whole that I hear passing by, after Chas pre-tuning.

So, all together, Stopper, if I think of you as a researcher I can not weight your skills, nor your intentions.

To you, I wrote: “Sure Cordier is still alive? Have you already posted your tuning sequence? How do you justify "beat-symmetry" and "zero-beat" 12ths? What is the meaning of a 19 semitones module? Let's tell each other, we are still alive, perhaps now it may be Time.”

Then, I was addressing your “alive” suggestion, I was not going for capricious games. If you want me (piano tuner) to understand your preference (pure 12ths) you can answer my questions. And if you really wanted to improve Chas paper you'd try a different, unconditional/unreserved way.

Regards, a.c.

Hello Patrick,

you wrote (Topic: Extending Unequal temperaments):...”What we don't need though, is to go like "Yeah, I can see that you're dedicated and all, but why really bother? Just look at my stuff over here, it's way beyond. Forget old school and step into my world!"...

I can confirm you that the only reason I'm in PW is because I would like to share Chas temperamental theory, Chas ET model and numbers.

It has happened that (on request) I've posted about my tuning sequence but, as I said, different tuning methods or techniques may work as well. Actually, you may need different beat curves, depending on how the piano may behave and react. What is important, in my opinion, is to know exactly what you can aim at, and why.

Here I say “what” and I mean which “form” we may be able to temper, and I say “why”, offering a new approach and the evidences of my research.

That's all.

To you (All), have a nice Summer.

Regards, a.c.

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

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

Hello.

I'm happy to be able to host this contribute by Robert Scott, posted today. It is about "errors in tuning", cents and offset in tuning design. Interesting for further elaborations.

Re: Possible EBVT III implementation on TuneLab [Re: Bill Bremmer RPT]

Originally Posted By: Bill Bremmer RPT

...What Robert Scott did, at least as I see it, was to find some "typical" inharmonicity characteristics and directly apply them to the published offsets for the EBVT. I discussed that with him and frankly don't understand why he did that...

Robert Scott:

"Actually, I did not create my EBVT offsets by applying correction to any published offsets for EBVT. Instead I attempted to derive offsets from scratch using Bill's aural tuning directions as the only basis.

Now I understand the limitations of using only 12 offsets to define an unequal temperament. Bill's aural EBVT tuning instructions cover more than just the one octave, so no set of 12 offsets can ever completely describe a temperament that is inherently a full-scale tuning. But one can at least hope to match the aural instructions in that one octave, and then further hope that the discrepancy outside of that one octave is not too bad.

If we take the aural tuning directions as the gold standard, then the goal of coming up with the 12 ETD offsets should be to try to duplicate the results of the aural tuning instructions as closely as possible. One could imagine doing this with a real piano. Suppose you had a very nice piano with a very well-designed scale and average inharmonicity profile. Imagine following the aural tuning directions for EBVT-III on that piano. Then imagine carefully measuring the offsets from that piano as compared to an ET tuning on the same piano. These could then be used as the 12 ETD offsets, at least on this same piano. And, subject to the limitations discussed in the previous paragraph, these offsets could be used to put a reasonable EBVT-III tuning on most any other piano.

The problem with doing this with a real piano is that every tuning has some error. Even on his best day, Bill's EBVT-III tuning is not going to follow his tuning instructions precisely. And even if they did, the process of measuring the pitches afterwards introduces some little error. So to avoid these two sources of error, I devised a software simulation of the "real good piano" imagined above. My software simulation is called "Temperament Designer", and it is a free download from my website. It is not an easy program to use, and to my knowledge, no one besides me has ever actually used it. Anyway, what this simulation does is calculate the consequences of any specific setting of pitches for about 2 octaves. Using the functions provided, you can "tune" any note to any other note based on beats difference or cents. It is slow and tedious, but not impossible to follow the EBVT-III tuning instructions step by step until the simulated piano is completely tuned. Then you can read out the offsets from ET for one octave and that is how I developed my 12 EBVT-III offsets. They are similar, but not identical, to the ones published by Jason Kanter.

But if I run that same simulation with inharmonicity set to zero, then I get the exact 12 offsets that Jason published. So I know that the simulation is basically correct, and I also know that Jason's 12 offsets were derived assuming a zero-inharmonicity instrument.

Now the argument has been made that it is inappropriate to take inharmonicity into account when doing this because inharmonicity has already been taken into account in constructing the underlying Equal Temperament tuning to which these offsets will apply. So it is sort of like applying them twice. But that is not what is happening. Taking inharmonicity into account when running the simulation is just being true to real piano that it models.

Let me give a very simple example. I wish to develop ETD offsets for an unequal temperament I want to call the One Perfect Third temperament. This temperament has a beatless major third, A-C#. I don't care what else it has. For my purposes I am going to focus only on this one fact. That is the aural tuning direction: Tune A-C# beatless. And just to be specific, let us say that this means the 5:4 major third, using the lowest possible coincident partials. Now if we follow the conventional wisdom of calculating the offsets based on zero inharmonicity, we would have an ETD offset for C# of -13.69 cents from ET (where A is offset 0 cents). Now suppose we were to apply this offset to a real ET piano tuning. I picked a sample Mason & Hamlin A from the TuneLab samples. In that ET, the 5th partial of A4 is 2217.135 Hz and the 4th partial of C#5 is 2234.107 Hz. After applying the -13.69 cent correction to C#5, the 4th partial now becomes 2216.510 Hz. This is close, but not exactly equal to the 2217.135 Hz from the 5th partial of A4, so this major third will not be beatless as promised.

So this illustrates that using correction figures that were based on zero-inharmonicity will not exactly achieve the aural goals that were stated. Taking inharmonicity into account during the simulated tuning reduces this error considerably. It does not eliminate the error entirely, because the assumed inharmonicity in the simulation may not match the real inharmonicity of the piano to which it is applied. But it will match better than if zero inharmonicity were assumed in the simulation."

Regards, a.c.

Chas tuning mp3 - live recording on Fazioli
http://myfreefilehosting.com/f/07c3ca3905_6.32MB

CHAS Tuning mp3 - Amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. - Department of Mathematics, University of Palermo - 2009, Italy:
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

Article by Prof. Nicola Chiriano - published by P.RI.ST.EM (Progetto Ricerche Storiche E Metodologiche) - University "Bocconi" - Milano, 2010 - (Italian):
http://matematica.unibocconi.it/articoli/relazioni-armoniche-un-pianoforte

Hello.

I'm happy for the shareable knowledge I've found. It is about iH approximations, soundboard loading and static measurements, this time covered in good English.

#1652600 - April 01, 2011 06:45 PM Re: High treble Stretch - 4:1 and 3:1 (15ths and 12ths)
[Re: UnrightTooner]

Tunewerk:

..."I have to add that if the Fairchild emulation was done completely internally using static Ic measurements, the data will certainly not be accurate. Inharmonicity changes slightly with tension added to a string.

It is not talked about, but the inharmonicity constant is not really a constant. It simply is for all intents and purposes, enough of a constant. Also, soundboard loading effects affect partial fields non-linearly - the subject of which this entire thread is about. Those effects cannot be known by measuring one constant for each string."

Thank you, Tunewerk.

Regards, a.c.

Hi.

Ron Koval wrote:
#1783559 - 11/05/11 08:54 AM
Re: Unequal Temperament Recommendations [Re: DavidWB]

..."I now believe that altering the way the temperament is spread out to the rest of the piano is MORE important than the specific choice of temperament - and in fact may explain the beauty of EBVT more than any of the offsets that have been provided."...

Ron, I'm glad to read that. Now you are applying the concept of "whole". Next step will be realizing that the way the temperament is spread out to the rest of the piano is AS important as the specific choice of temperament. In fact, "the rest of the piano" happens to be the direct multidimensional*() reflection of your temperament. You will soon acknowledge that all intervals, with their tensions, may then contribute to the sound whole; all intervals can be correctly interrelated.

* (one dimension per interval)

Regards, a.c.

Amazing thread!

Post #1788167 - November 13, 2011 06:06 PM
Re: Does it make sense to ask for a UT? [Re: PianoStudent88]

Bill Bremmer:

..."I think that this is an area where both Alfredo and I would agree, as well as other fine technicians such as David Andersen of Los Angeles and Berhard Stopper of Germany. We all go for the "beat masking" concept. I think of it as "beat cancellation" and Alfredo may have another term for it.

Since there are two undefeatable problems in piano tuning, the requirement for temperament and inharmonicity, we all use those two foes against each other to essentially eliminate some of the unwanted "noise" that the piano produces.

The only argument there really is between myself and the others is that they seem to believe that only ET provides for the most possible beat masking. I know for sure that I can get more of it with the other temperaments I use. Even when I choose to tune ET, I use a very slight quasi ET version of it that has more beat masking potential than theoretically true ET affords."...

Hi.

Thank you, Bill, for pointing out these issues. I'll take this opportunity and clarify some concepts.

In all my tuning practice I've never aimed at "beat masking" nor "beat cancellation"; in fact, I approach beats as the source of color, tension and energy. Color, in that the outcome of partials and the overall resonance depend on the beats that two or more sounds generate; tension, as the well known phenomenon we perceive; energy, in that proportional beats reduce noise and distortions, so enhancing sounds richness, roundness, smoothness and duration. For these reasons I tune one single interval, one bi-chord at the time, and always check all beat curves. I would not play three notes together, simply because the "beat cancellation", in case, would not help me correct the previous two notes that were tuned.

Perhaps the idea of "beat masking" works for a fast tuning, perhaps it is again referred to the ancient idea that beats equal dissonance or that beats reduce resonance. That was not my approach, as I thought that beats only needed to be ruled. You can read about this in Chas research report, sections 2.0 and 3.0.

So I would have no reason to say "only ET provides for the most possible beat masking". Actually I'm saying: let's talk about quasi ET versions and "theoretically true ET", in that - as a matter of fact - ET has evolved.

Regards, a.c.

Hi Alfredo

I would have to spend more time to understand exactly how CHAS generates its numbers but I gathered that it depends on constant values in the foreground and background:

- In the foreground, in the values representing scale frequencies, we find the incremental logarithmic ratio: 1.0594865443501…

- In the background we find the difference values in 1:1 proportion: ± 0.002125389965…

So there is a steady progression from note to note like ET where there is of course only a foreground value 1.059463094... By monochromatic I meant such a progression.

In unequal temperaments by definition, the ratios between notes varies and there are many practical possibilities. As people have associated colours and colourful descriptions with the keys in well temperaments, I called them polychromatic. You might say each well temperament is a palette, e.g. http://www.youtube.com/watch?v=LMU6szv8BFc

My question is whether it's possible to introduce one or more periodic elements that would "modulate" the foreground or background numbers in an elegant way. By "modulate" I meand affecting the frequency as in FM radio transmissions, e.g. the annual BBC Proms broadcast from the Royal Albert Hall every year which carry a lot of musical information!

Will that do?

Hi Alfredo,
I would like to attempt your CHAS tuning on my piano. Could you kindly direct me to a temperament and initial octaves sequence that you use from the very beginning (A4-A3) all the way to expanding the temperament? Or maybe repost here, because this would be the appropriate place to do so?

I don't know if you witnessed Jeff's 'pure 12th' method and our discussion about the theory and practice of that tuning, along with my recording of his method a while back on my home piano. I would like to see how your CHAS system differs from the pure 12ths sound.

I really enjoyed listening to your tuning example recording on the Fazioli. I like Faziolis to begin with, and then adding your precise tuning was the icing on the cake!

Thanks in advance.
-Erich

Hi Ian,

Thanks for your reply. I hope to get your question right.

I'm going to address you to a description of Chas algorithm that you'll find in this thread. If you are logged in, it's on page 9.

alfredo capurso #1211679 - June 04, 2009 05:30 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]

There, I also meant to demonstrate the dynamic essence of "s", that is the (natural) parameter that enables you to "modulate" and modify the scale incremental ratio. The new obtainable ratio could well be a semitone ratio, i.e. it could variate and be referred to single notes too. Let me know if that can work enough.

The "s" parameter translates exactly what I do in my practical tuning, when I want to modify the tuning curves and anticipate the re-adjustments of the bridges and the soundboard.

12 root of two ET forced the tuning into the "pure octaves" concept. Chas ET conceives the tuner as an actor, the one that is able to determine the premises.

Regards, a.c.
.

Hi Erich,

I've replied in this thread: Chas Preparatory Tuning

Regards, a.c.
.

Alfredo

Thank you for your reply. Maybe I will have some time one day to look at the maths but, I suspect, coming up with the numbers is one thing and translating them into a practical tuning method is another.

I have now seen David Pinnegar put my question much better when he wrote, "Were you to apply your mathematical approach of achieving better harmonic structure as you have with ET but to the UT temperaments you would do the musical world a considerable service."

Although I am no tuner, your tuning sequence tells me you are exceptionally skillful. The Fazioli recordings sound perfect, even too perfect, but I like David and Ed's tunings too.

How can we persuade you to modulate some of your tunings to demonstrate the dynamic essence of "s" ?

That really would prove your point!

You sure get something better out of that little S than I've heard elsewhere on such a small piano. I've been following this discussion about CHAS with the difficulty of a rube but hadn't listened to your files till now. Listening makes your theory more compelling than words have. Listening to the S&S and the Fazioli, I certainly hear a "musical whole".

(That dog barking at 10:33 on the S is a few cents off, though.)

Alfredo,

I just read your paper "A NEW MODEL OF INTERPRETATION OF SOME ACOUSTIC PHENOMENA CIRCULAR HARMONIC SYSTEM – C.HA.S."

Having spent some 25 years editing some hundreds of chemical engineering technical papers, I'm well acquainted with such tables and graphs. It's interesting to see your elegant approach to the whole point of distributing those semitones to create coherence in *difference* of the partials. I really like the notion of self-ordering (and the analogy of the fractal) and applying that form of math to the geometry of sound to achieve coherence. If that requires scrapping the 2:1 octave, I'm convinced enough about the beauty of your thinking to want to give it a go in tuning my piano (9-footer, low inharmonicity). So do we have instructions on how to go about tuning from scratch here? (Forgive me if I've missed that here). I could sort it out for myself, since you've provided the math (and I might approach it from that perspective anyway first, as I think it'd be instructive). The concept of the "chorale" seems to me to be the whole point and your effort to describe that mathematically is fascinating.

Rick

Alfredo,

I need a clarification here for the sentence "Stretch narrow fifths by subtracting a quantity (negative delta), and stretch wide octaves by adding the same quantity (positive delta)."

Please forgive the following, slow moving, set of questions. Are you saying that:

1. In your CHas, M5's should be less narrow than they are in theoretical ET?

2. That octaves should be wider than 2:1? (but not necessarily to 4:2)

3. That the M5's should be widened (from theoretical ET) to the same degree as the octaves are widened?

4. Which do you tune first, or consider as the basis for the tuning--the octave or the M5?

5. Assuming that the answer to (3) is yes, is this widening of the octave a widening that starts at 2:1? In other words, if the M5 is stretched to +1 cent above the theoretical ET value, should the octave be stretched to +1 cent above the 2:1 partial match? (Or are you starting from 1:1?)

6. To ask (5) above in a slightly different way: Are these pitches set without first tuning the strings with a regard for inharmonicity? In other words, are you setting the M5 by listening only to the pitches of the fundamentals and then adding a stretch, or are you setting the M5 by a conventional method and then stretching slightly more?

7. What determines the degree to which the M5, and thus the octave are stretched? In other words, what gives you the number for Delta? (Is Delta the variable that accounts for the ih of the M5? If so, what partials or set of partials do you listen to, to determine Delta?)

I'm sorry if I'm asking you to explain what you may have explained before, but the answers to these questions would help me to understand CHas. The discussion of CHas is spread over so many pages that finding many details is difficult.

(What confuses me, in part, is your speaking of a negative and a positive Delta, for both a positive and a negative Delta lead to a rise in pitch.)

Hi Ian,

Thank you for your involvement. When I read David's wish/request I got puzzled, still now I'm not sure to get that point. David talks about "harmonic structure" and that's ok, I can easily share the concept of "structure". It is when David says "...better harmonic structure...", there I loose the logical connection.

UT's, WT's, non-equal temperaments in general could elaborate on an "irregular" division of the scale steps, within one (pure) octave. 12 root of two ET gained a proportional and thus "regular" steps geometry, within one (pure) octave. Chas model describes a proportional "structure" that involves the weaving of all intervals. In other words, proportional beats now define and order the scale steps into a coherent whole geometry. This is how the first ET structure has been improved, that is why this new proportional structure can be reasonably said "harmonic" and explains why David would ask for a "better harmonic structure". Now my question is: how can I apply Chas approach to the UT temperaments and their limited (steps division and pure octaves) geometry? In other words, considering theory, I'm afraid David's request might be loaded with a contradiction in terms. But, as I say, I may be wrong.

On the other hand, Chas algorithm and the "s" variable allow the calculation of infinite ratios, this has been demonstrated and you only need some time to look at the maths I mentioned above.

I also read "...coming up with the numbers is one thing and translating them into a practical tuning method is another."...so maybe you are talking about practice. I'm willing to demonstrate anything I can, but I don't understand when you say "...modulate some of your tunings...". Would you mind telling me what is the point that I need to prove, and exactly what am I asked to do? Is David asking for a new UT model? Are you asking for a UT practical tuning example, or a practical method?

Thanks again.

Hi Rick,

I'm so glad you like those results. Please, find the instructions in the Preparatory Tuning thread and let me know if I can help further.

Regards, a.c.
.

Alfredo

To my mind your recordings are a perfect demonstration of your approach to ET. Some pianists will find your tuning to their taste and ideal for the way they want to play. Others, such as Kamin, would like to hear more contrast. That is why David, I think, and I are suggesting you might turn your attention to variations in temperament.

Although David might like to see a new model, I hope he would happy with a practical UT example inspired by CHAS; one that might appeal to Adolfo Barabino - http://www.youtube.com/watch?v=9zxNrQuxfNY. His homepage talks about a multitude of shades and tones, delicate details and subtle differentiations in expression.

Whether it proves a point or not does not really matter, but I am sure everyone would like to hear what you can do.