Hello Kees,

I was on the point of posting my reply to you and Mark, when I could read you post.

..."If you're right it's up to Alfredo to explain why he needs a 26 page incomprehensible paper to explain this simple concept and how the prime number 5 fits in here.

I don't know why Alfredo keeps dodging the question. Perhaps the whole Chas thing is a practical joke?"...

You may wish to comprehend Chas reseach report more, you may think that I'm dodging a question and/or that Chas is a joke. Though, what you should comprehend first, even for tuning, surely for posting in this thread, say in general, is how to respect people.

You'll apologize, then you'll work out how you may get to know more.

a.c.
.

Kees:

In other words, "Pay no attention to the man behind the curtain!"

Hello Mark.

Patrick's question was:...”3) Now, If Bill sets up ET, he recommends the initial octave to be between 4:2 and 6:3. Then he extends the midrange utilizing equal-beating 12ths/15ths. Shouldn't Bills ET method give a high-quality CHAS?”...

I had answered: ”I cannot say how high, and quality may depend on the execution. I guess that method, like others, can get close to Chas ET. Why not? If you then want to improve from ordinary quasi-standards you need to know what to aim at, then even the initial octave, for instance, may need to be corrected, and you'd want to manage all 4ths, 5ths and octaves + RBI's. For practical tuning I find fundamental referring also to more accentuated beat curves.”

Kees wrote:...“Suppose the execution was perfect, could you then answer the question? You should be able to, since you know what Chas is precisely and Bill's method is crystal clear.
If it is close but not exact, can you explain the difference?”...

Mark, maybe you would like to know more about the “difference” in terms of cents and/or about perceivable differences in practice. If this is the case, unfortunately I do not know.

For this kind of conclusions, for stating how actual stretched version of ET (aural or ETD's) can be far from or close to Chas ET, I'd like to be able to refer to some precise, attentive analysis.

Nevertheless, I can tell you more about my tuning in practice, also in relation to what Bill could write in this thread last June, you may then suppose and/or deduce for the best, so to orientate your thoughts.

I (hopefully) get the Chas form by tuning a Pre-Form. In other words, I do not tune Chas form directly, I tune a form that can (hopefully) evolve into Chas (see: Chas Preparatory Tuning). In fact, I need to anticipate the final form, so I go for an “envelope” that can develop into Chas.

As far as the resulting beat curves (progressions), I give the same importance to SBI's and RBI's, though RBI's are in function of 4ths, 5ths, octaves, 12ths and 15ths. In fact, 4ths and 5ths represent the tuning form's skeleton.

In Chas form, 5ths do not get wide, although I tune also wide 5ths in the Pre-Form.
4ths must invert their beat rate progression.
5ths must invert their beat rate progression.
Octaves are progressive.
12ths and 15ths never invert their beat rate progressions.
12ths (narrow) and 15ths (wide) will be equal beating all along the keyboard, although I tune more often pure 12ths on center strings (see: Chas Preparatory Tuning).

Chas form is not the expansion of a mid-range temperament, beat wise it is a compact whole that (in principles and in practice) needs the accurate interweaving of all intervals.

Mark, you say:...”all semitones are the 12th root of an octave between 4:2 and 6:3”.

I've never measured Chas octave with that test. And the octave is not a constant in Chas, be it theory or practice.

In fact, on top of the octave size, the resulting tuning will depend also on 3rds, 4ths, 5ths and 6ths. I mean to say the the octave alone cannot stand for all the intervals within or outside the octave. All intervals, all together will make the tuning, that is why they cannot be copied.

And I do not know what the theoretical (or actual) leeway there might be when you measure the octave using that test. Do you?

Then I think that 12ths and 15ths equal beating can, to a certain extent, improve order in a temperament, and Ron Koval seems to agree. Perhaps Ron has made some progress with Chas and its ETD application, he may be willing to share some results.

Generally speaking, I get the idea that quite a few colleagues (from Europe, Japan, Canada) tune close to Chas, some of their feedbacks and my aural perceptions make me think so. Only when I read “There is a constant struggle with a piano tuned in ET” I strongly doubt Bill's ET can be anywhere close.

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

Excuse me, but your wikipedia link is nonsense. That is NOT normal ranges. They are of course not set in stone upwards, but they'd be closer to something like

Soprano: A#3-G5
Alto: F3-D5
Tenor: A#2-G4
Bass: E2-D4

... with less trained voices having a slightly lower tessatura (the basses down to D2 and the sopranos maybe just a few half steps lower than G5).

When you score for SATB choir, your 'safe' range would be E2-E5.

Which part of "it coincides pretty well" did you not get?

I thought you were talking about artists, they seem to be able to go much higher than C6, so maybe the data I posted were not that wrong, maybe they are averaged.

We were talking about tuning two good-ET octaves, I thought two octaves were not enough for evaluating cadences, and I was wondering if the all issue was a cliché. You said you did not think so. Then I did not understand this:

“The fact that there might be variations around the midrange doesn't necessarily take anything away from the different temperaments established here.”

C'mon Alfredo, I can't believe that I have to make this any clearer than it already is. Given the n number of times I've tried to get this message through to you, I can't help but starting to question just how familiar you are with beat rates and ascending 3rds and 6ths that get progressively faster?

So without assuming or implying or showing disrespect, I will ask you straight out: are you familiar with the nature of ascending 3rds and 6ths speeding up - true in any given ET?

If so, and if you tried EBVT or even just looked at the charts of the M3 sizes in the temperament, you should realize that there is NO way that a tuner setting the temperament (with just center strings open in the midrange) would mix these temperaments (ET and any EBVT) up.

Jeff, I must have I missed that - quite a lot of posts here. However, I do take a great interest in this, I will go back in this thread and try to find them.

Here's my guess as to what Alphonso might mean.

Assume we have ET over two octaves (say F3-F5) with an octave size x (slightly larger than 2). Let now find x such that F3-F5 and A#3-F5 are equal beating (a mindless octave). In practice x is found aurally, here I will compute it.

Let's denote by p3 and p4 the third and fourth partial numbers (a tiny bit larger than 3 and 4, depending on inharmonicity).

The frequency (in units of F3) of F5 is x^2. (x^2 means x to the power 2.) The fourth partial of F3 has frequency p4 (again in units of F3), So the 4:1 beat frequency is x^2-p4.

The frequency of A#3 (in ET) is x^(5/12), its third partial p3*x^(5/12), so the beat frequency of the 3:1 interval is p3*x^(5/12)-x^2.

Setting these equal gives the equation

p3*x^(5/12)-x^2 = x^2-p4

which we can solve. Without inharmonicity (p3=3, p4=4) the solution is

x = 2.000630601755216

which gives an octave which is 0.55 cent wide. This is almost exactly the number in the Chas paper.

With moderate inharmonicity 0.25 of a typical Steinway D (tunelab data) we get a stretch of about 2 cents.

Am I close?

Kees

Sorry Alfredo, I guess I remembered you saying you like to be called Alphonso and the name stuck. My apologies.

Kees

Looks like the thing, Kees!

Is CHAS theory taking inharmonicity into account, or is it just leaving that to the practical on-fly adjustments of the tuner?
(Sorry, I just skimmed through the very many pages. Still, I have no trouble following your math here... )

Well, the tuning of the octaves may be similar, but since the midrange, the bearing for EBVT, will be tuned differently, won't the tunings be completely different?

Well, Alfredo, thats a neat way to back off, but as you might have noticed (at least by now), wikipedia is not the ultimate source of knowledge.

To put an equal sign between artists and extended range upwards is hilarious.

If your wikipedia reference refers to average, it's just plain wrong. Ask anybody who deals with vocal music. Or trust me on this one. I've been accompanying, composing, arranging and conducting vocal music since I was in my teens - the last 15+ years on a pro level, in many different genres. In fact, that has been my main field of work, to the extent that I got tired of it. And then I got interested in tuning, and here I am...



This is not hard to understand either, if you're a musician and/or a piano player.

The midrange is the singing backbone of harmony. Outside of that, in piano tuning, there can be all kinds of stretch, choices, and compromises. If they sound good, they are good.

On the other hand, if the midrange is bad, the choices outside don't really help.

Jake, yes absolutely. But Kees' formula could work just as well for extending EBVT, you'd just have to find the EBVT values instead of the standard ET ones (ie 5/12 and so on). Isn't that so?

To my understanding, you'd only need to find the frequencies of the fundamentals of the 15th and 12th, then you'd be set to go.

Note a subtlety tough. In my scheme, in order to tune F5 you need to have A#3 which depends on s, which you don't know yet. I think this is what Alfredo means when he talks about refinement process etc.

In practice you set F4 as 4:2 6:3 octave from F3, tune A#3, then tune F5. The math is a bit easier then and the stretch slightly different (and different for F3-F4 and F4-F5).

OK since I say it's so easy let's do it.

We now have F4 = (p4 + p6)/(p2+p3) * F3, so octave F3F4 ratio y is (p4 + p6)/(p2+p3).
So A#3 = y^(5/12)*F3 and its 3d partial is p3*y^(5/12)*F3. F5=x*F4 where x is the F4-F5 octave stretch, so F5 = x*y*F3.

The 4:1 bps (wrt F3) is x*y -p4.
The 3:1 bps is p3*y^(5/12) - x*y.

So we get if we set them equal

p3*y^(5/12)-x*y = x*y-p4

which we have to solve for x which is high school stuff.

So x = (p3*y^(5/12)+p4)/(2*y) which is for no inharmonicity 2.001129890627526, or about
1 cent. If you want to measure stretch over F3-F5 you get (x*y)^/12 which is 2.000564865545492, even closer to the mysterious Chas number.

For typical inharmonicity all these variants are practically identical and about 2 cents.


Cheers,
Kees

I see your point now, UT.

Alfredo, don't take it personally, but I think you combine ignorance with arrogance in your posts here. Complaining about disrespect in the face of criticism does not refute criticism. That being said, your thread made me think more about octaves and I learned something in the process and I thank you and respect you for that.

As I said before, the traditional way to debate tunings is to hurl insults at each other and question the sanity and ancestry of your opponent. So I think by historical standards we are still behaving quite civilized.

Kees

Kees:

Alfredo's math is based on the common note of the 12th and 15th being on the bottom. That is probably where the slight difference is.

Pat:

Here is a link to the part of this Topic I was talking about. The boxnet link does not include tables as I remembered, but I still have the data if you are really interested. The graphs show what is going on pretty clearly:

Earlier post in this Topic

...“I can't help but starting to question just how familiar you are with beat rates and ascending 3rds and 6ths that get progressively faster?”...

I'd say quite a...Beat.

Patrick, my point was not provocative. My point is that you can/would only talk about two octaves. And when you refer to “energy” or “enhancement”, and compare WT to practical ET tuning, you are assuming that practical ET will have perfectly progressive 3rds and 6ths. Out of my experience, I can say that very few “freshly” tuned ET's I have heard around, had strictly progressive 3rds and 6ths. About this issue, I understand what Bill says, the many times he finds revers well instead of "ascending 3rds and 6ths that get progressively faster".

All this to explain my opinion, why the “energy” and/or “enhancement” issue might be referred to theoretical ET that very rarely is actual ET. So, the whole issue may be (my guess), more luckily, a purely academical cliché. A Test?. You could post a recording of chromatic 3rds and 6ths, say C2-C6, for you to report the ET's (?) that are actually being played in your conservatory.

Also, by reading your explanations in the Pianist Corner, you may refer to energy as a “switch” opportunity, from more energy to less energy (correct?). I refer to energy as to reliable power, continuous harmonic and melodic euphony sustain, enhanceable by beats synchronism and synergy. And I talk about Chas ET, neither about an aleatory ET nor an ETD stretched version of 12th root of two. All together, only my points, my preferences.

...“The midrange is the singing backbone of harmony. Outside of that, in piano tuning, there can be all kinds of stretch, choices, and compromises. If they sound good, they are good. 

On the other hand, if the midrange is bad, the choices outside don't really help.”...

If you say “singing” I tend to think of melody. About the rest, I have a different opinion and we do not need to agree. Thanks for letting me know about your involvement in music.

Hello Jake,

You write:...”Well, the tuning of the octaves may be similar, but since the midrange, the bearing for EBVT, will be tuned differently, won't the tunings be completely different?”...

We were elaborating on Patrick's question, posted a couple of days ago, to which I replied again yesterday:

Patrick wrote:...”3) Now, If Bill sets up ET, he recommends the initial octave to be between 4:2 and 6:3. Then he extends the midrange utilizing equal-beating 12ths/15ths. Shouldn't Bills ET method give a high-quality CHAS?”...

I hope this can help.

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
.

(Sorry. Egg on face. Saw Bill's name and somehow thought that someone was saying EBVT and CHas could end up being almost the same...)

Well, that's a sly way of throwing down the gauntlet, as you know very well that any tuning at the conservatory would be mine. I especially like the question mark in parentheses.

But since I know you are a dissector by nature (just look at your replies here), I will take you up on it. I'll throw in a bonus, too - I'll post the same test for EBVT III… as useless as the test is for setting that temperament, it still gives a pretty good picture of how much it differs from ET.

To mix them up and arrive at the same kind of quasi-ET (as you like to speculate about) would take a really unexperienced tuner. I hope you will grasp that when you listen to my ascending 3rds and 6ths.


Yes, I've been thinking about that for a while (and about the limitations of english as a third language), and I'd like to rephrase that and make it much more simple. To me it is the different sizes of the thirds that lightens and darkens the cadences in an interesting and, very human, way.

You commented on my use of the word 'human' before, and I might have to clarify that. The differing thirds work as any community at its best - together they make the bigger picture clear (in this particular case, they solve the riddle of the phytagorean comma). Still they do that without sacrificing their uniqueness (personality).

Ok, Alfredo, here we go...

I tuned two pianos at the conservatory tonight. No, not just for this test... there are exams coming up next week, so it was a nice opportunity to get people (teachers and students) happier, and to make these recordings at the same time.

A few liner notes:

• The ET is set on a Yamaha C2, the EBVT III on a Yamaha G2. Both have a couple of false beats in the fifth octave, but they are all pretty slow ones, so you shouldn't have any problems distinguishing them from the actual beat rates.
  
  
• The tenor break is handled slightly different in the two instruments - the Yamaha C2 has its first wound string at C#2, the Yamaha G2 enters the wounded section at A#1.
  
  
• The recordings are made with only the center strings open.
  
  
• These are both what you refer to as 'practical' tunings. I do not consider any of them to be perfect.
  
  
• That said, there should be no doubt as to why EBVT III is no quasi-ET. Fact is, I think that implication is degrading towards Bill. It implies that one aims for ET, but does not really get there.

EBVT III is not something you arrive at out of pure luck, or lack of luck, or whatever. It is a deliberate choice. I hope these audio files will make that clear to you, and others who might be following this discussion.

----------------------

ET ascending 3rds C2-C6
ET ascending 6ths C2-C6
ET 3rds in temperament range
ET 6ths in temperament range
----
EBVT III ascending 3rds C2-C6
EBVT III ascending 6ths C2-C6
EBVT III 3rds in temperament range
EBVT III 6ths in temperament range

Great Jeff, I'll check it out! And yes, I'm truly interested in this. I've been thinking a lot about the proportions of 4:1, 4:2 and equal-beating 12/15ths.

...“Well, that's a sly way of throwing down the gauntlet, as you know very well that any tuning at the conservatory would be mine. I especially like the question mark in parentheses.”...

These comments, Patrick, do not add much to this discussion. Neither I know about what happens in your conservatory, how many tuners work there. So, why making assumptions? The question mark in parentheses means that we cannot certify which ET we are talking about, if aural, equal beating or one amongst many ETD's versions. The only common reference we have is a model, 12th root of two, which is not practicable. Then, in actual practice, I would not expect to find much the same ET (?) from two different tuners. Would you?

...“But since I know you are a dissector by nature (just look at your replies here), I will take you up on it. I'll throw in a bonus, too - I'll post the same test for EBVT III… as useless as the test is for setting that temperament, it still gives a pretty good picture of how much it differs from ET.”...

Dissector by nature? More simply, I try to reply with accuracy, despite anything else you may deduce. In any case, it is not here that I would expand on my nature as it has nothing to do with the whole issue.

Please note, I was not asking for a comparison, because for that to have sense, all possible parameters have to be the same (especially energy in playing and chromatic timings, sensibly faster in these ET rec). Also, I was not talking about fresh tunings, nor single strings, so that we could get an actual idea of how the lamented issue can be referred to ordinary realities.

For what I hear, actual ET's (?) differ one another and can not be perfect either. Although it is not very much, the differences and the imperfections can be enough for making your “energy” and “enhancement” issue a theoretical (?), philosophical (?), academical (?) argument. Your ET ascending 3rds and 6ths, at a first listening, being what they are, do “enhance” some progressions – where I hear both ET-type progression and (quite normal) WT-type inversion or sudden beat accelerations – surely less in the mid-bass, where to me they sound identical.

...“To mix them up and arrive at the same kind of quasi-ET (as you like to speculate about) would take a really unexperienced tuner. I hope you will grasp that when you listen to my ascending 3rds and 6ths.”...

In my opinion, you could precise also your target, is it a concert pianist, a pro tuner, an educated music amateur, a practicer? In your idea, should the tuner prefer EBVT? Aurally (intonation wise) and/or conceptually (ET is bad)? Or?

Some comparative tests have already been offered and commented, and all together it was not easy to distinguish ET (?) from EBVT, nor to express a preference.

I was not talking about “the same kind” of quasi-ET, but quasi ET and you may not take quasi-ET as an odd definition, nor as a reduction. I was and I'm saying that EBVT cents differences can, today, be described as deviations from ET, in fact they are. In other words, like crystals and quasi-crystals, ET can represent the reference geometry. My tunings too could be defined quasi-ET or FCDT (Few Cents Deviating Temperament). Yet I have good reasons for referring and aiming at Chas ET, as regards theory's reliability, intonation (in tune), harmoniousness, resonance and shareability.

Did you read when I wrote that EBVT may sound better than some ET's (?) ? I also explained why, and what could improve the overall tuning. This, just to reaffirm that I'm not discussing your preferences. I simply can not share any of the aural, aesthetical and conceptual reasons that would justify the EBVT supremacy and the relative against-ET campaign. This may also explain why, as a tuner, I'd be happier to talk about Chas ET actual updates and, in general, about a new approach to temperaments.

I thank you for your efforts, I will listen more carefully to those recordings and I'll post my comments. Perhaps, anytime, other readers may post theirs too.

Regards, a.c.
.

You can go into semantics all you want, Alfedo, but I know that you know that I tune the instruments at the conservatory. If you are in doubt, please trust me, or confirm it by sending a request to info@musikhuset.fi.



No, this is true. The A/B test might not be made by a machine, but the differences should be evident to the educated listener.


This is major BS, Alfredo. if you think the mid-bass sound identical in my ET/EBVTIII samples, you are really off target. Simple rhythmic excersices (ie counting beats) would be next for you - if you need references, I'd be glad to provide them. This goes for the rest of the discussion about temperament(s) as well. I really can't belive that youre debating this, it severely hurts the validity of this thread.

Alfredo - excuse me for being blunt once more... I DO NOT KNOW what the h**k you are trying to say.

ET, be it your 'invented' version, or any other enhancement of the 12th root of 2, would demand 3rds and 6ths that progressively speed up. That is, no 3rds or 6ths would be slower than the ones a half step below.

Somebody, please? This can all go into semantics, but please - anybody - confirm that you do hear the difference between the progressive 3rds and 6ths in ET, and the non-progressive ones in EBVT III.

*

Come on Patrick, this is utterly trivial; only a deaf person could not hear this.

More interesting perhaps is how well the beat speeds compare to theory. Below I list the beat rates computed by Scala first assuming no inharmonicity, and then for typical Steinway D inharmonicity values over the temperament range.

C4 is 0 in Scala notation.

No iH:

Base frequency : 261.6256 Hertz
Beat frequencies of 5/4
0: 0.000: 7.1800
1: 94.910: 13.4879
2: 197.060: 10.6530
3: 297.790: 13.7066
4: 395.790: 14.0834
5: 498.040: 11.9757
6: 595.910: 18.0361
7: 699.310: 11.9714
8: 796.870: 20.2319
9: 896.200: 15.7747
10: 999.060: 15.7783
11: 1096.170: 21.8811
12: 1200.000: 14.3599

Base frequency : 261.6256 Hertz
Beat frequencies of 5/3
0: 0.000: 8.9780
1: 94.910: 15.8877
2: 197.060: 12.5432
3: 297.790: 16.1030
4: 395.790: 14.0786
5: 498.040: 14.8335
6: 595.910: 18.7737
7: 699.310: 13.7655
8: 796.870: 20.2259
9: 896.200: 19.5517
10: 999.060: 21.4818
11: 1096.170: 23.3677
12: 1200.000: 17.9560

Steinway D:
Beat frequencies of 5/4
0: 0.000: 7.0204
1: 95.290: 14.5231
2: 196.850: 12.5801
3: 297.220: 15.6673
4: 395.580: 15.6914
5: 499.700: 10.8098
6: 597.960: 15.5911
7: 700.910: 11.0596
8: 798.340: 18.4712
9: 896.700: 15.6260
10: 998.820: 15.8167
11: 1096.960: 19.9382
12: 1200.000: 14.0408

Beat frequencies of 5/3
0: 0.000: 9.3584
1: 95.290: 15.3906
2: 196.850: 13.3958
3: 297.220: 16.6145
4: 395.580: 14.6420
5: 499.700: 12.9470
6: 597.960: 15.9732
7: 700.910: 11.7147
8: 798.340: 20.4730
9: 896.700: 21.5417
10: 998.820: 23.9785
11: 1096.960: 24.3559
12: 1200.000: 18.7169

Kees

Kees. I agree. And I'll be happy to move on. But what to do?

Math put aside... Bill's method is ingenious.

I have my own way of extending the temperament, but I love Bill's way, because it resides in musicality, and harmony.

EDIT: Kees: why do we accept Alfredos fancy maths as a fact? It really shouldn't be because of the length of the paper...

... "You can go into semantics all you want, Alfedo, but I know that you know that I tune the instruments at the conservatory."...

Wake up, Patrick. I know you tune pianos in your conservatory, I did not know you are the only tuner. In any case, the question mark was not referred to this, as I have explained. That was you making assumptions.

... "No, this is true. The A/B test might not be made by a machine, but the differences should be evident to the educated listener."...

The differences are quite evident, for what I can say out of one listening, and I was not comparing, as you seem to have understood. The point was not a compareson, the point was giving you an idea of what ETs sound like in ordinary situations. This may help you understand that your lament about enhancement can not be referred to actual ETs.

... "if you think the mid-bass sound identical in my ET/EBVTIII samples, you are really off target."...

It is a shame, you have misunderstood. This is what I wrote: "Your ET ascending 3rds and 6ths, at a first listening, being what they are, do “enhance” some progressions – where I hear both ET-type progression and (quite normal) WT-type inversion or sudden beat accelerations – surely less in the mid-bass, where to me they sound identical.".

I was talking about "your ET ascending 3rds and 6ths", as you can read at the begging of the sentence. So I meant in the ET sample only.

... "Simple rhythmic excersices (ie counting beats) would be next for you - if you need references, I'd be glad to provide them. This goes for the rest of the discussion about temperament(s) as well. I really can't belive that youre debating this, it severely hurts the validity of this thread.

Alfredo - excuse me for being blunt once more... I DO NOT KNOW what the h**k you are trying to say."...

As a result of your misreading you hurt the validity of what you write.

... "EDIT: Kees: why do we accept Alfredos fancy maths as a fact? It really shouldn't be because of the length of the paper..."...

Fancy math? From your misreading I can learn about fancy math.

Patrick, I suggest you to read what you read twice. And always check (my English), if you can. You seem to be taking the whole thing so personal that you loose control and respect. Is this like you?

a.c.

I don't take it as fact and I don't think it is fancy math. I sincerely think either this Chas thing is an elaborate joke, trying to see if tuners can be made to believe anything, or it is just crackpottery.

Fancy math on the topic I consider to be Bill Bremmer's articles. They are crystal clear and everything is precise and logical. It is mathematics with sound instead of symbols, but good math nevertheless.

Kees

Kees: Well put. Goes back to the fact that any given, *sound* theory should resonate - and affect - everyday life.

As of now, CHAS seems to be constructed, compulsive knowledge. Math for math's sake... setting the temperament by ear (and extending it utilizing common sense) far exceeds (and naturally arrives) at a more logically tuned piano. This is the core of Bill's methods.

I'd love to see a theory that puts down what we do in writing, but I doubt this is it. Even more so as the author (original poster) seems to lack the basic knowledge necessary for defending his case.