Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum CHAS for Dummies

-

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