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I have read the threads about CHAS, and I must admit I don't get it, even if I have a solid mathematical background.

Can someone explain it to me in plain terms? Or at least in mathematically unambiguous terms?

Would it be possible to tune CHAS with TuneLab?

Chas is based on equal beating 12ths and 15ths.

The Chas paper (it's on the web) gives the equation; the semitone ratio comes out at 1.059486544. You can work out parameters for Tunelab from that.

If it looks like a duck, walks like a duck, quacks like a duck, then it is a duck.

CHAS looks, walks and quacks like self promotion wrapped in hodge-podge mathematic camouflage. Surely you have better things to do.

Pinkfloyd, you're not alone with the "I don't get it" sentiment.

Withindale, here I was thinking I was just too stupid to get it. Thanks for a simple and direct answer. Can you explain the bow and arrow part?

My arrows have a nasty habit of missing the target.

The aural tuning sequence is described here in an English translation.

I imagine the finer points might be lost when using Tunelab (and its inharmonicity model) to tune Chas by numbers, just as they might be if the target were pure octaves or pure twelfths.

TuneLab 15may give an approximation, or may help to gauge the even beating later.

With a temperament based one one octave it is yet something but only one octave can be tuned, anyway that is what I get when I tried.

The chas principle is to draw a curve not based on 2:1 3:1 or whatever, but with a balance between 3:1 et 4:1.

CHAS is crackpottery, that paper is nonsensical gibberish. Those tunelab "offsets" are nonsense, as they are too small to have any effect or be tunable.

Kees

Nevertheless, Kees, the Chas paper gives the equation and the result. Some, if not all, versions of Tunelab allow for custom offsets. You are right that the differences will be small.

Well, TuneLab only takes offsets for the temperament octave and then only one type of octave tuning (e.g. 6:3) for the entire range below the temperament octave and one type of octave tuning (e.g. 4:1) for the entire range above the temperament octave. There is no way to make TuneLab use the same ratio between semitones for the entire range (if that is the point of CHAS?) and there is no way to choose compromises between octave types (for instance, halfway between 6:3 and 4:2 or equal beating 12ths and 15ths). Probably TuneLab could be used in a more manual fashion to do this.

Not just small, these "offsets" are completely overwhelmed by effects caused by inharmonicity, hence irrelevant to piano tuning.

Equal beating 12/15ths in piano tuning is explained properly here:
http://www.billbremmer.com/articles/aural_octave_tuning.pdf

Kees

Reading the manual will show you how to do that.

Kees

Does it follow that equal beating 12/15ths cannot be tuned with an ETD like Tunelab without adjustments made aurally?

It is not an equation, Withindale. It is an equality.

Everyone has to bear in mind that the CHAS paper was not written by a mathematician, but a piano tuner who just wished to express his ideas more precisely. In this way, it is almost innocent and well-meaning if one does not take it too seriously.

Alfredo doesn't understand the need for precision in mathematics, as evidenced by his paper. He was loosely fluffing out an artistic concept. The paper was meant to explore the mystery of his ideas, rather than offer a scientific one with clarity.

When I made a mathematical differentiation in a thread months ago about the equality, Alfredo could not say which solution was the relevant one he intended to describe. This shows a disconnect between what he is hearing, and what he understands of the math behind it.

One valid concept that I think Alfredo has offered that I agree with, is the concept of nodal tuning (or the idea that tuning is not a continuous function). Hitting alignment points is important and stretch is dependent upon this.

From an earlier conversation with Alfredo:



I want to add one thing on the end here. After reading what Kees wrote above, I realized something that I had forgotten.



This is an extremely important point that most tuners (and Kees) are missing. These aren't offsets.

What Alfredo was describing (despite the rough math) is an entire concept of thinking about and understanding tuning that is not in the common lexicon. To put CHAS offsets into a tuning machine is to not understand what he was saying.

Machines are part of the problem in the way they approach tuning - as a series of point solutions from an algorithm, rather than an interrelated whole.

I know everyone's bullshit alarms will go off when I use these terms, but there's an important idea here:

You can't put CHAS offsets into a machine and have it work, because the way machines work, these offsets will be overwhelmed by the larger effects of inharmonicity. However, if you tune with the ideas represented by CHAS (not exclusive to CHAS, by the way), they can be incorporated into the inharmonicity. In this way, the tuning can be completed to a higher level, where small changes are maintained because of aural cross-referencing, and not a single-variable output.

I'm not a big fan of CHAS because I feel it is overinflated and confused in its technical scope. However, there are some important ideas within it, which are represented in many other areas of the tuning community.

In tunelab you can set the tuning curve to have pure 12ths or pure 15ths. Then manually tweak the curve to be the average of those. That will get you close enough.

Kees

I was just thinking about the Title for this Topic:

CHAS for Dummies

Yep, that says it all!

Do what exactly? I _have_ read the manual, many times.

Yes, I realise that. I think you have answered the OP's question.

How can I miss that point when I bring it up myself???

I guess we are dealing with CHAS logic here?

Kees

Yeah, maybe it's better to talk about these concepts separate from CHAS, since CHAS is such a confused conglomeration of claims and concepts with inappropriate math to describe tuning ideas.

What I meant was that in bringing up the idea of offsets, you weren't understanding the point of the idea Alfredo was trying to express. CHAS is an attempt to explain a whole way of tuning that is antithetical to offsets. More than offsets, I saw the real point as describing a way of tuning.

But who am I to say? It's possible I am reading in some logical idea to CHAS that isn't even there.

Point taken, Withindale. I made the mistake too, in talking to Alfredo.

Anyway, I think the real point here is: CHAS is an unnecessarily complicated and confusing explanation about tuning that is otherwise already available in other forms from various sources.

pinkfloydhomer, don't worry about not getting CHAS theory, it is best to see it for what it is worth and then move on.

CHAS is in reality a normal 4th/5th based tuning with RBI checks and expansion that is essentially the same as that which tuners have been using for many decades, but re-branded. The difference is that it has been self-promoted in a way that dupes reader into thinking that there is a special mathematical basis that has roots in natural phenomena and the beauty of nature.

Trying to program offsets that represent CHAS theory into an EDT is ludicrous. Offsets are normally based on 12-root 2 semitone spacing, and then, starting from all zero's for ET, ETDs alter or stretch them to accommodate for inharmonicity for the individual piano. To suggest some pre-stretching by altering offsets at the start would defeat the stretch algorithms that the ETD is designed to compute.

CHAS is an equation that describes a well-stretched piano using the balance between a stretched 15th and a narrow 12th. This is what many of us habitually try to tune anyway. It has merit in the way the equation allows you to see what happens to the beatrates of other intervals if this stretch is maintained consistently - specifically the way the 5ths and 4ths behave. If I remember correctly, the fifths reach a minimum beatrate close to pure somewhere below the temperament region (depending on the scaling) and get progressively narrower as you move upwards and downwards from that minimum. This is the consequence of semitones being the size that they are supposed to be in this system. It also has progressive 4ths.

Alfredo's tuning technique attempts to incorporate these progressions into the tuning, but the accuracy required is very high, with diminishing returns.

There's no denying that this size stretch results in a beautiful tuning. But there's a massive disconnect between the theory (which is actually very simple, it just suffers from a lot of obfuscation and bad communication) and the practice, and unfortunately that gap has been filled with alfredo's own tuning philosophy. (I'm not saying your tuning technique is bad, alfredo, it's just separate from the model you have presented, yet you treat it as the same. Slow-pull technique and pin 'charging' is nothing to do with CHAS)

There's also space within his maths for other 'delta' values, amounting to different stretch patterns. I find it an adequate model for a theoretical 'good' tuning on a theoretical piano. I believe if it was programmed into a dedicated ETD then it would produce good results similar to StopperStimmung.

Not so. Are intervals remain the same size everywhere in chas as chas doesn't take inharmonicity into account at all, hence it has no relevance to piano tuning.

Kees

Hey Phil, how do you see this?

It's not in the original paper, Tunewerk, but it came up in the discussions, so it'll be buried somewhere in the many threads. The derivation would be tedious but you can see how it might be done.

Inharmonicity is not taken into account in the equation, Kees, but he compares his values to the railsback curve, which is an idealised inharmonicity model of a piano, and it compares favourably.

As I said, an idealised model for an idealised piano.

Thanks for the reply, Phil.

No, I don't see how this could be done. Reversal of 5ths and 4ths was an inharmonicity artifact first discovered by Fairchild. No math will predict or model this, unless you are talking about finite element analysis of soundboard loading and scales.

CHAS too easily becomes a Shopsmith talisman for answering everything piano. I think this is a problem that leads to its discredit.

He just states that it compares favourably, without ever showing the CHAS and Railsback curves together. The CHAS "curve" which is supposed to be like the Railsback curve is actually just a straight line. It does not "compare favourably" at all.

Kees

I've never understood how 4ths and 5ths "reverse". When and where on the keyboard does this supposedly occur? Aren't all 5ths supposed to be narrow and all 4ths wide, no matter the register?

Well I just took his word for it really, so I'm happy to be corrected.

erichlof, it wasn't that the 5ths become wide, it was that the fifths become less narrow, and then more narrow. So the direction of the change of narrowness reverses.

Good question, Erichlof. This is something I'd really have to pull out the data for, which I don't have around at the moment.

To the best of my memory, increased soundboard loading causes negative inharmonicity on 3rd partial of most pianos, around the 5th-6th octave. This causes the 5th to be pure with less stretch and, at the same time, the 4th to be pure with more stretch (neg. iH acting on both sides of these complimentary intervals).

Very rarely is there a crossover. It has been documented and is possible with extreme negative iH, but it is rare.

The real world outcome of this effect is you'll notice it is easier to get both 4ths and 5ths pure as you rise out of the temperament region.

This is an excellent point to post here, because tuning models never represent variable partial fields on a real piano. When partial fields fluctuate, the rules of tuning fluctuate.

Phil - glad to hear, and to know that all together we can clear up misinformation!

Ahh... thank you Phil and Tunewerk for the explanations. I think I understand a little better now. I guess it's kind of like in physics when you can have negative acceleration. This concept always tripped me up in High School and here I am many years later encountering it again with a completely different system!

Thanks!
-Erich

Many intervals exhibit anomalous behaviour. All aural tuners who use all the test intervals in the whole piano must have noticed them, particularly how single octave size and double octave size change places on several notes in and around the top octave, even in the best pianos..
This is maybe the most readily apparent of them, at least it's the first one I ever noticed as an ardent student of all this. It's only about 30 years ago that I started to choose the single octave to tune to instead of the double octave when the alternatives were too noisy in the smaller intervals. That was the time I started working in more recording studios and spending more time listening to playbacks with other musicians than tuning. (1 hour tuning/3hours listening).

Bass strings exhibit even more random exchanges but that's to be expected even in the largest pianos. Far too random to try to marshal them into some sort of order.

ETD's never pick up on them or pick up on the wrong ones.

A wizened old tuner once told me, don't listen too hard, you'll go crazy. Maybe that's what's happnin'.

Hi,

Posted in the "Sample" thread:



rxd, when you mention "..the emotional content carried over from other threads..", do you mean this thread? Or is it the "How long should it take?", where you wrote:



What I am saying, rxd, is that your insinuations do not help, and in the Sample thread I was not concerned about me, but Isaac.

Sure, I cannot say I am happy to read that I would be a charlatan, or an amateur, or one that would try to take you hostage, or... crazy, as you last wrote in this thread.

Let me suggest: take it easy, be respectful and filter your thoughts and fantasies three times before you post.

About theoretical and practical/technical contents... later on.
.

Sorry, everyone, I appear to have picked up another self obsessed stalker in the above post who, not content with insulting us all and being censured for it, has chosen to mix up the content of two different threads for their own ends.

I apologise to you all for any inconvenience or confusion this has caused.

IMO, there is little to be sorry about, when we try to be clearer.

You confirm your attitude again, rxd. BTW, do you have a name and surname?

Hi pinkfloydhomer,

Unfortunately I do not know what the user can do with TL, and I am not familiar with recent ETD's; I guess Robert Scott may say, perhaps it is worth a PM?

As for the Chas maths, which is the point you do not get?

Regards, a.c.
.

No he does not have a real name. He is a robot from a Boy Scout project that went horribly wrong. rxd is just a model number.

Oh no, he found it..

This thread, even though already resolved, is about to become extremely confusing. And long.



This is great advice that you should follow yourself, Alfredo.



It always is later on, even if unsolicited. And when you do explain, you never do really explain.

Yes, I think it is clear to everyone that CHAS is more about your own personal show than presenting something of worth.

Xd is the original model designation but it all went horribly wrong.
Started to exhibit human traits and developed attitude problems so the R prefix stands for "rebuilt". Now the attitudes are just right.

Some in the piano profession in a few parts of the world know exactly who I am. Some even know who I used to be, still fewer know what I used to be.

Nobody knows why.

And NOBODY, oops, nobody knows what you will become! (Or are you becoming?... naaah)

I am becoming very becoming.

Thanks for a good belly laugh, Jeff

Oh, the original Xd couldn't laugh at itself.

-

Not meaning to be rude, Mr Stopper, but without any knowledge of your own ideas beyond scraps and vague desriptions and the odd video, your ideas are no better communicated than alfredo's, IMO, and both seem to be wrapped very much in self-promotion (although you do actually have a product to sell, which is fine). You just have the good sense not to try and promote it heavily to the members of this board.

Hi Alfredo, thanks for answering.

What I don't/didn't get about CHAS is/was ... everything

I understand the tonal system, I understand various unequal temperaments, I understand equal temperament, I understand the ditonic and the syntonic comma, I understand inharmonicity, I understand what a 6:3 octave is or what a 3:1 twelfth is, I understand what beats are, I understand what equal beating is, I understand what fast and slow beating intervals are, I understand a lot of music theory and I understand the idea behind most aural tuning schemes that I encounter. They all seem to take all of the above into account.

But I didn't understand CHAS since it wasn't precisely described in the threads about it on this forum.

I guess I understand now that CHAS is about equal beating 12ths and 15ths and about a slightly larger semitone ratio than the 12th root of 2? Or is there more to it?

This must be even before we take inharmonicity into account. Taking IH into account, semitone ratio is always greater than the 12th root of 2 on a real piano with positive IH, even if it is tuned in equal temperament. So I guess on a real piano tuned in CHAS, the semitone ratio becomes even greater than the theoretical CHAS semitone because of IH.

Hi pinkfloydhomer,

It is as you say, Chas semitone ratio is slightly larger than the 12th root of 2.

Also the rest is correct: "This must be even before we take inharmonicity into account. Taking IH into account, semitone ratio is always greater than the 12th root of 2 on a real piano with positive IH, even if it is tuned in equal temperament. So I guess on a real piano tuned in CHAS, the semitone ratio becomes even greater than the theoretical CHAS semitone because of IH."

In Chas equation (3-delta)^(1/19)= (4+s*delta)^(1/24)

12ths and 15ths deviate from 3:1 and 4:1 depending on the parameter "s";

for s=1 12ths and 15ths deviate by the same (delta) amount.

That's all.

Cheers, a.c.
.

Okay then, but can you explain to me in short, precise terms why the CHAS semitone ratio is desirable and better than other suggestions (most notable 12th root of 2), and also why equal beating 12ths and 15ths are better than other approaches?

Why isn't CHAS just one more random way to tune? Why is it special? What sets it apart? It must have some kind of fundamental idea binding it together. And idea that it must be possible to express in short and precise terms to an educated audience. An executive summary.

Going back to your original question, PFH, this thread and the one about Stopper's temperament have shown that it's not possible to tune CHAS with Tunelab.

CHAS is a heuristic method of tuning rather than a deterministic one with offsets you can feed into an ETD.

Alfredo has set out the method in some detail (see this English translation) and explained that he allows for "variable stretch" in his tunings (see his post in the Stopper thread yesterday).

As I see it, you will never know in advance what the values of his "s" variables will be. To achieve the beat rate progression curves he is looking for stretch may swing between pure octaves and pure twelfths; conceivably more I suppose.

No doubt you are familiar with those Railsback diagrams with smooth curves approximating inharmonicity and jagged lines showing actual tunings. A Railsback curve represents a mathematical model which is a figment of the imagination. The jagged line is reality.

Don't get me wrong, I am a great believer in mathematical models. They can do a lot, but I know their limitations.

You will never get CHAS from Alfredo's equations, nor will anyone else. It's the tunings that matter.

This it does not have.

It does not have this because CHAS is a retroactive arithmetical explanation for subjective, qualitative field experience that has led a lot of tuners to believe tuning near the 12th works best on the modern piano.

It works backwards, not forwards. It provides a gum-and-tape explanation for complex interactions. As a result of this, it cannot be used as a model to work forward or predict anything physical in terms of real pianos.

Not only that, but the mathematical model is insufficient and incorrect. It does not model what Alfredo claims it does. I've modelled this in MatLab some time ago.

The positive role this model could have is giving new tuners a more quantitative view for the goal of tuning. Used in general conceptual terms, it could be helpful.



+1

Hi pinkfloydhomer,

Yes, I can try, but at some point we will have to address theory and practice separately, so that we do not get confused.

@ ...why the CHAS semitone ratio is desirable...

Here I can only talk subjectively, I was longing for a ratio (and tuning criteria) that my sense_of_intonation could justify, and that ratio was lacking. That “desire” made me start with my research... on the one hand there was no way to tune pure octaves, on the other hand there was no need to avoid faintly beating octaves... This made me believe that perhaps a better scale_geometry could be found, that it could depend on strict application.

@ ...and better than other suggestions (most notable 12th root of 2)...

There is no way we can put 12th root of two into practice; that ratio favors pure_octaves (2:1), and in doing so it doubles all other intervals deviation values, every other octave; the Chas semitone ratio is “better” in that it spreads deviations amongst all intervals, so that - octave after octave - all intervals can progress together, as part of a whole.

@ ...why equal beating 12ths and 15ths are better than other approaches?...

12ths and 15ths..., because in this way we actually stretch the fourth (4:3 - in between the 12th and the 15th) which is the interval that first closes a circle (we say so, but it is a spiral), enumerating the number (4*3) of semitones.

@ ...Why isn't CHAS just one more random way to tune?...

The Chas model has nothing against “random” tunings, i.e. tunings that may result from any other semitone ratio, that is the meaning of the “s” variable; in fact, this is a fundamental passage: we are expected (and enabled) to modify the ratio in order to set the “desired” semitone progression; on the other hand, s=1 defines the most coherent geometry.

@ ...Why is it special?...

The Chas ratio (with s=1) is special in that it is self-referential: by stretching the fourth (4:3), we determine the constraint for 12ths (3:1) and 15ths (4:1), and s=1 fixes that constraint in 1:1 proportion.

@ ...What sets it apart?

Maximum coherence.

@ ...It must have some kind of fundamental idea binding it together...

No interval needs to be pure;

Deviations define “color”, more than pure intervals;

Deviations need to be ordered in proportion;

We can order scale frequencies and deviations with one ratio;

What we need to consider, represent and aim at, it's a dynamic (beating) whole, then we can set the premises.

@ ...And idea that it must be possible to express in short and precise terms to an educated audience. An executive summary.

I do make use of a PowerPoint with some short phrases and graphs... are you asking for that?

Regards, a.c.
.

Hi,

Phil, it is a bit of a shame that my sharings on this board appears as self-promotion, as if I wanted to sell something... Honestly, I do not understand what is giving this impression, if not my mere enthusiasm when I think that, three hundred years ago, these results would have remained the preserve of a few. Anyway, thank you for posting and for pointing that out (and thanks PW for providing this opportunity).

Tunewerk, I hope you will be able to compare the Chas model with other models and soon realize that the tonal scale is now tailored correctly. About maths, try not to confuse notions, for instance what “equality” means, and about tuning in general, try to help other colleagues understand now why the octave needs to be stretched. Oh, it would be great if you could help also Jeff, Chris, Kees and Bill.. :-)

Here is a link to some literature:
http://www.huygens-fokker.org/docs/bibliography.html#C

Ian, thank you for your posts... seeing how other posters mix up notions, concepts and practical issues, I find your lines refreshing.

To All, have a nice Epiphany.
.

In keeping with this topic for dummies, I would like to present an imitation CHAS -- for dummies like me.

This is an equal temperament very close to that of CHAS, but one that I will describe in terms with which I am familiar. I am hoping others will see this as a worthy contribution to the topic.

First, mathematical models are useful, in that they identify the target beat rates from which you will depart when actually tuning in the real world. A piano tuned to a 12th root of 2 model will have beat rates as close as possible to that of the 12th root of 2 model.

Pianos are scaled in general so that their beat rates can approach that of the models of equal temperament; this is obviously less true for some pianos than it is for others, but it would have been easy to make pianos that could not be tuned with beat rates that remotely resemble the beat rates of any model.

Modern equal temperaments come in a variety of widths.

The temperament denoted by the 12th root of 2 is 12-tone to the just octave equal temperament. It is characterized by clean octaves, and with 4ths expanded by approximately 2 cents and 5ths contracted by the same approximate 2 cents. The C4-F4 fourth will beat the same as the F4-C5 5th. (If that 4th is faster than that 5th, then the octave is expanded, and the model for the tuning is not the 12th root of 2.) This is considered to be the most narrow equal temperament that is musically useful.

The temperament denoted by the 7th root of 1.5 is 7-tone to the just fifth equal temperament. It is characterized by clean fifths -- and octaves expanded by approximately 3 1/3 cents. The C3-A3 sixth will beat the same as the C3-E4 10th. This is sometimes considered to be the widest equal temperament that is musically useful. However, my PTG Atlanta institute class will demonstrate some equal temperaments that are substantially wider than this.

The temperament denoted by the 19th root of 3 strikes a middle ground in the range of equal temperaments. It is characterized by clean 12ths — and by octaves expanded by approximately 1 1/4 cents and 5ths contracted by the same approximate 1 1/4 cents. The D3-A3 5th will beat the same as the A3-A4 octave. If this temperament is executed evenly across a piano scale, some, including myself, claim that a particularly coherent tuning will result.

The source of the coherence of a tuning with an evenly-executed stretch may be the temperament itself, or perhaps it is that it is unusual for a chosen specific level of stretch to be accurately executed across a whole scale. This appears to still be an open question, because it seems to be reasonable to think that executing a consistent amount of stretch across a scale might contribute to coherence in a tuning. My class will deal extensively with these and related issues.

There are other widths of equal temperament that can be readily identified, including one denoted by the 31st root of 6. It is characterized by clean double-octave 5ths. The C3-C4 octave will beat the same as the C4-G5 12th.

Of particular interest to this PianoWorld topic is the equal temperament denoted by the 43rd root of 12. It is characterized (at least theoretically, in its zero-inharmonicity mathematical form) by clean triple-octave 5ths, and double octaves that are expanded by the same amount in cents (about 1.09 cents) as the 12ths are contracted (again, about 1.09 cents). The C3-G4 12th will beat the same as the G4-G6 double-octave. (I believe equal-beating double octaves and 12ths are claimed in CHAS.)

It would be of interest to me to understand how the 43rd root of 12 ET differs substantially from CHAS. They appear to be so similar that it would be difficult to aurally distinguish one from the other. Perhaps someone could identify for me a specific pair of 12th and double-octave that would be equal-beating in CHAS.?

Thanks.

Mr. Swafford:

I rarely view posts having to do with CHAS, but I made an exception when seeing that you posted to this Topic.

The purpose of CHAS is for Alfredo to promote himself. The model is for 12ths and double octaves with a common note on top bottom to beat at the same speed. This is virtually identical to "mindless octaves". So D3-A4 A2-E3 would beat (3:1 narrow of just intonation) the same as A2-A4 (4:1 wide of just intonation).

I believe pure 12ths to be an exceptional stretch and use it exclusively. I don't want to post about it on this Topic, but here is a current one on the subject: P12 Tuning Sequence

If you respond, I may not get back to you until Monday. I rarely post on weekends.

Dear Kent,

You start of by writing:



You then proceed to present various degrees of stretches:



May I ask: which of these schemes (or widths) is it that you would like to "present as an imitation CHAS"?

In Chas temperament method there is an unique feature that I never find anywhere , that is the installation of future enlarging that is done in an initial not particularely expanded octave (minimally expanded, in my opinion, when comparing with some basic octaves A3 A4 I have seen describe.)

Without that trick you cannot tune with higher ratios than 2:1 without raising a lot the speed of FBI and even the octaves and doubles.

In my opinion

And there is the 55th root of 24, the 61st root of 34, the 67th root of 48, the 74th root of 72, the 79th root of 96, etc. etc. ad infinitum.

These theoretical numbers (relevant to a theoretical zero inharmonicity instrument) differ by such small amounts that these differences are completely washed away by effects caused by inharmonicity in a real piano.

Kees

Kees writes:

"These theoretical numbers (relevant to a theoretical zero inharmonicity instrument) differ by such small amounts that these differences are completely washed away by effects caused by inharmonicity in a real piano."

Of course.

But _my_ point remains that there are various widths of equal temperament that _are_ readily identifiable by their characteristic beat rate patterns, as I detailed. These various ET's can be aurally differentiated from each other.

The ET denoted by the 43rd root of 12 would, I believe, be difficult to aurally distinguish from CHAS.

Since the 43rd root of 12 ET is easily described and CHAS is somewhat difficult to describe, I suggest that the 43rd root of 12 ET might be a good simpler alternative to CHAS.

Kent, Just a few figures:

If A6 on a typical Yamaha U1 is tuned we get about 1776 or so Hz using a reasonable stretch.

With 12 root 2 the frequency is 1760 Hz
With 19 root 3 the frequency is 1762 Hz
With 7 root 1.5 the frequency is 1767 Hz
With 42 root 12 the frequency is 1820 Hz

Although I have only shown one note, none of these models resembles what a real piano should be given inharmonicity. A particular model may be close for some parts of the scale and some pianos but not others. I think it is pointless trying to characterise CHAS with a model such as these unless it is modified with other coefficients such as what Alfredo does.

12 root 2 is a default model and is fundamental reference for the way we describe tuning systems. It is not used in practice for pianos unless we describe tuning frequencies in terms of cents offsets from 12 root 2. With the other models we will still have to describe actual tuning frequencies in terms of deviations from the theoretical values of the model.

The value I get for A6 using the 43rd root of 12 is more like 1761.11.

Models assume zero inharmonicity, so there is obviously no claim that they "resemble what a real piano should be given inharmonicity".

You use the phrase, "we describe tuning frequencies in terms of cents offsets from 12 root 2". Yes, that is the way models are used in electronic tuning devices.

However, there is another way to use models, as I pointed out in my post. As Daniel Levitan puts it in his text, The Craft of Piano Tuning: "...Piano tuners always prefer to approximate to some extent the theoretical beat rates of equal temperament, purposefully mistuning 1st partials... This is because piano tuners listen, not to 1st partials, but to coincident partials. Through the artful mistuning of 1st partials, tuners strive to create the illusion that there is no inharmonicity in the intervals of a piano."

So, as I said, models provide target beat rates, not frequencies.

Any reasonable model of piano tuning should incorporate a model of inharmonicity, as do all ETD's.

2^(1/12) model is the coarsest model and gives you ballpark figures for beatrates. Any refinement of this model should first deal with inharmonicity, then the various stretching methods can be discussed.

Fooling around with other semitone ratios without modeling inharmonicity seems pointless to me. You can propose any real number close to 2^(1/12) as the basis of a "new tuning", it does not matter if it can be written as the foo root of bar or not.

That's why the whole chas "theory" is pointless.

Kees

<grin>

There is a way of tuning in which one can pick a stretch level by choosing a width of equal temperament and using the beat rates of that width of ET to provide the target beat rates for a tuning, regardless of inharmonicity.

The chosen width and its associated beat rates are executed across a scale artfully by the piano tech to provide a best-fit, coherent tuning in spite of inharmonicity.

I don't see how that is possible. Let's take a semitone of 2^(1/12) and try to use those zero-ih beatrates as guides.

The octaves (and double, and triple, etc) have theoretical beatrates of zero. Problem is that on a real piano a single octave has at least 3 beat rates, 2:1, 4:2, and 6:3, which can not all be zero. What do you now do with your "target beat rate"?

Kees

I have asked twice what those different beat rates are supposed to sound like, and I have gotten no response.

Beats occur according to when the maxima and minima of two interfering waves coincide. When the maxima and minima of a piano tone occur is affected by the shape of the wave, which determines the partials, but it does not mean that the maxima and minima of the partials can interfere with each other to the extent that they differ from those of the fundamentals. It just means that the beats will not be nice, clean, regular sine waves themselves, just as the fundamentals are not a nice, clean, regular sine waves.

There are other things that affect how beats sound. There is the difference in volume between any two notes, which is why beats on a piano never approach silence at their minima, and there is the decay which overwhelms the sound of the beat as the intervals come closer to coinciding. These make "beatless" a relative term. I would like someone to demonstrate how octaves can be beatless on a piano. If it is such an important phenomenon, someone should be able to record it so we could listen to it and hear exactly what it is.

In addition, the iH is not consistent across the break, and, the lower partials in the bass are anomalous, due to bridge rocking or some other not well researched effects. Any tuning paradigm must include those factors - in the case of aural tuners , they use their ears, in the of ETDs, compromises must be made.

OK, go on. Lets suppose that you determine beat rates for a chosen model. Then how do you propose to extend this to actually tuning a piano?
Have you actually tuned a piano like this?

Unless I am completely wrong, it is not possible to use any size octave other than precisely 2:1 octaves for instruments that do not exhibit iH, such as a pipe organ. As a result, on a piano, the iH is 'a priori', after which a stretch can be imposed.

Well, no.

If I understand, Mr. Capurso designed CHAS to be valid in instruments without iH.

Fine aural tuning predates a complete understanding of inharmonicity. There was a time when piano techs didn't even know inharmonicity existed and yet they tuned pianos as best they could using the beat rates of the model, because that was all they had.

You say, 2:1, 4:2, and 6:3 can not all be zero, which is obviously true. However, what does that have to do with artfully tuning a clean octave aurally? The whole sound of an octave can have a clean sound. And the octave can be tuned clean while tuning 4ths that beat much the same as 5ths in the 4th-5th test of the 4:2 octave.

Well, that's a problem then.

Take linearly stretched octaves of 2.005 (as I believe is the CHAS model) of say 100 Hz-200.5-402.0025-806.015 on a organ. Because the partials are, in fact, harmonic, you now have the following beating sequence - 100-200-200.5-300-400-401-402.0025-500-600-601.5-700-800-801-802-804.05.

Now, if the octave were precisely 2:1, the sequence would be 100-200-300-400-500-600-700-800.

Which do you think will sound better?

Of course I have tuned a piano like this. And I have said how to go about it, by artfully tuning as closely as possible to the beat rates of the model and by doing so, to give the illusion of zero inharmonicity.

For my PTG Atlanta institute class, I have prepared tunings of the Pianoteq modeled piano to demonstrate equal temperaments at various recognizable widths.

As as been mentioned elsewhere on Piano World, Pianoteq supports the Scala tuning protocol, and it even supports 88 note tunings, and so I have used Scala to tune the Pianoteq piano note by note to a high degree of accuracy, and to many different tunings.

Obviously, it would be better to demonstrate these tunings with real pianos and that can be done, but with Pianoteq in a classroom situation, all the tunings will be utterly stable, and will be immediately available and comparable at the click of a trackpad button.

This works well for a decaying sound envelope. Not so easy with a constant amplitude sound.

ET forces the existence of a semitone ratio. Usually this ratio is supposed to be 12th root of 2, as the ratio of the octave is 2.

But iH in pianos creates multiple kinds of octaves: 2:1, 4:2, 6:3, 8:4, etc.

Ths the magic octave ratio of 2 is blown off.

In real pianos there are no pure octaves. They beat and they have multiple beat rates, one for each pair of quasi coincident partials.

The same happens for all the intervals, they all have multiple simultaneous beat rates. In P5s for example we have two distinctive beat rates namely 3:2 and 6:4
Theoretically there is an infinite number of quasi coincident partials for each kind of interval. Thus for the P4 we have 4:3, 8:6, 16:12, etc... In the real world only 2 or 3 pair of quasi coincident partials are audible at some specific locations of the piano's scale.

IMO there is no fixed ratio for the semitone that can be taken to tune a puano from A0 to C8.

The iH of a piano is not constant across the scale, it's curve has "hokey stick" shape.

Thus, I think the size of the semitone can not be constant but it must change from A0 to C8.

The fact here is that we can no more talk about an Equal Temperament.

Gadzar wrote:

"The fact here is that we can no more talk about an Equal Temperament."

Beat rates. Define equal temperament in terms of beat rates.

For me, the recently passed Bill Garlick provided the best definition of equal temperament for the modern world.

"In equal temperament on the modern piano there is not a single interval which is tuned just or perfect -- even including the octave. Due to the Comma of Pythagoras and inharmonicity all intervals must be contracted or expanded from perfect and will therefore beat... All the tempered intervals of equal temperament should gradually increase in beat speed evenly, ascending chromatically. It should be noted that such chromatically ascending progressions, particularly of M3rds and M6ths is a characteristic unique to equal temperament and distinguishes it from any other temperament."

-- Bill Garlick

Of course, but what does that have to do with exotic zero-ih models with weird roots?

Kees

"Of course, but what does that have to do with exotic zero-ih models with weird roots?"

Kees

I'm not so sure they are so weird.

They provide a good way of quantifying stretch, and executing their characteristic beat rate patterns evenly across the scale can promote coherent-sounding tunings.

It's impossible.

You can tune M3s in an even progression.
Or you can tune P5s in an even progression.

But in a real piano there are iH jumps in the scale and you can not tune an even progression of M3s and P5s.

You have a jump at the break, you have a jump when passing from wound strings to plain strings and you have a jump each time the diameter of plain strins changes.

So it is impossible to tune all intervals in even progressions of beat rates.

It sounds good in a vague sentence, but how do you propose to "execute their characteristic beat rate patterns evenly across the scale" exactly? Beat rates of which intervals?

Kees