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Sorry - I had a serious side-track...

Let's go with the U1 - looks like that's the easiest for all of us to work with.

I'll be back again to post some numbers. We still need come up with which partials would be most appropriate to measure/post.

Might not be till next week

Ron Koval
chicagoland

I wonder if this might be a case where the digital piano might actually be an advantage, rather than competition.

As far as I know digital pianos mimic an acoustic one to the best of their ability, including the inharmonicities of one tuned to some version of ET.

DPs now outsell acoustics 2 to 1 or so, right? So there's a pretty good user base.

Seems to me it would be possible to adjust the DP to mimic some of the ET variants now in use, and if people liked this enough they might start demanding their acoustics be tuned the same way.

Piano techs might even take on retuning digitals as an extra income generator.

Ron,

I do not know what a side-track is but I hope you are well.

For me Ya U1 is fine. Let's get down to the most appropriate partial and you may decide, I do not use ETDs normally...and let's choose a precise ETD, if you agree.

Best regards, a.c.

Side track is like a train taking a track that leads off to the side - instead of staying continuing on the journey. (kindof a detour) I'm fine, just was busy with other things...

Any ETD that can measure and give a cents value of a specific partial should work - I think Jeff has Tunelab, I'll look into the partials this week. I want to make sure that the measured partials are strong, clear and make sense for that particular piano.



Ron Koval
chicagoland

Ron:

I do not have an ETD at all. I have a simulator with iH curves derived from Verituner files. A U1 is one of the files.

The simulator provides beatrates when given the iH and either hertz or cent offsets of the fundementals. It also produces curves for iH, cents and beat rates. By adjusting each note to simulate a tuning sequence, the result of the sequence can be evaluted.

Although the simulator is set up for entering the fundamental frequency or cent deviation, this is not a problem. I wrote the program myself and can modify it to accept the cent deviations of partials. And if the iH curve of the U1 in the files is much different than what others are using, a new file can be made and used.

Jeff, what makes a difference is that when the string is stroked by the hammer , the spectra is different, all iH comes in play and modify the audible pitch - I see it that way.

Jake, you wrote:

"Thanks for posting these recordings. Were tuning sequences and checks ever written out for these two temperaments? (I remember seeing some lists of steps, and guesses, but not precise sequences.)"...

Chas ET tuning sequence has been published in PW, here is a recent version, written for another forum:

..."Here is the sequence I routinely use for Chas Preparatory Tuning:

sharp or flat is referred to the note (centre string) I'm ment to tune. The already-tuned note is in bracket -

Step 1 – A4 – from 440.0 Hz to 442.0 Hz (concert or studio) - from 442.0 to 443.0 (for flat pianos)
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Step 2 – (A4)-A3 - tiny little flat, just on the beating threshold - beat raises after 2 or 3 secs and rolls slowly
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Step 3 – (A3)-D4-((A4)) - sharp, close to 1 beat/sec. – D4-(A4) faintly beating
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Step 4 – (A3)-E4 - flat
check overlaping 5ths and adjacent 4ths to set up Chas octave:
A3-E4 about 1,5 beat/3s - sensibly faster than D4-A4
E4-A4 about 2 beats/1s - sensibly faster than A3-D4
------------------------------------------------------------------------------------------------
Step 5 – (E4)-B3 – flat - tiny little faster beat than A3-D4, sensibly slower beat than E4-A4
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Step 6 – (B3)-F#4 - flat - little slower beat than A3-E4 since 5ths have already inverted -
faster beat than D4-A4 - evaluate M6 A3-F#4
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Step 7 – (F#4)-C#4 – flat - faster beat than E4-B3, sensibly slower beat than E4-A4
evaluate two M3’s progression + one M6
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Step 8 – (C#4)-G#4 – flat - slower beat than B3-F#4, tiny little faster than D4-A4
evaluate three M3’s progression + two M6’s progression
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Step 9 – (G#4)-D#4 – flat - tiny little slower beat than E4-(A4), faster than F#4-C#4
evaluate four M3’s progression
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Step 10 – (D#4)-A#3 – flat - tiny little faster beat than A3-D4, tiny little slower than E4-B3
evaluate five M3’s progression
------------------------------------------------------------------------------------------------
Step 11 – (A#3)-F4 – flat - tiny little slower beat than A3-E4,
tiny little faster beat than B3-F#4
evaluate seven M3’s progression
------------------------------------------------------------------------------------------------
So far, apart from A3-D4, I have stretched "flat" - now I’ll stretch "sharp"
------------------------------------------------------------------------------------------------
Step 12 – (D4)-G4 – sharp - tiny little slower beat than G#4-D#4, faster beat than F#4-C#4
evaluate eight M3’s progression + three M6’s progression
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Step 13 – (G4)-C4 - sharp - tiny little slower beat than B3-F#4,
tiny little faster beat than C#4-G#4 evaluate nine M3’s progression + four M6’s progression
--------------------------------------------------------------------------------------------
Beats curves are meant to be tuned temporarly. While I'm tuning, I bear all (few) doubts in mind.
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Step 14 – (A#3)-A#4 – sharp - increase octaves beat’s speed very slowly – 5ths go very, very slowly towards pure – F4-A#4 tiny little faster beat than D4-A4, as for the next 4ths
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From the octave beat threshold, first signs of beating come to me in a shorter and shorter lapse of time, this helps to S-shape octaves stretch
------------------------------------------------------------------------------------------------
Step 15 – (B3)-B4 - sharp - increase octaves beats rate very, very slowly - 5ths towards pure
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Step 16 – (C4)-C5 - sharp - increase octaves beats rate very slowly - 5ths towards pure
------------------------------------------------------------------------------------------------
Step 17 – (C#4)-C#5 - sharp - increase octaves beats rate very slowly – 5ths start transiting pure - evaluate one M10
------------------------------------------------------------------------------------------------
Step 18 – (D4)-D5 - sharp - increase octaves beats rate very slowly – 5ths are transiting pure - evaluate M10’s progression
------------------------------------------------------------------------------------------------
Step 19 – (D#4)-D#5 - sharp - increase octaves beats speed very slowly – 5ths are transiting pure - evaluate M10’s progression
------------------------------------------------------------------------------------------------
Step 20 – (E4)-E5 - sharp - increase octaves beats speed very slowly –
5ths have transited pure, evaluate M10’s progression –
chromatic M12s, like A3-E5 must be constant and temporarly tuned pure (on normally out of tune pianos) -
Step 21 – (F4)-F5 – sharp
Step 22 – (F#4)-F#5 – sharp
Step 23 – (G4)-G5 – sharp
Step 24 – (G#4)-G#5 – sharp
------------------------------------------------------------------------------------------------
Step 25 – A4-A5 – chromatic double octaves like A3-A5 must be constant and temporarly beat with a rate of about 3b/2s, or 3/2 bps - check the wideness of A3-A4
increase octaves beats speed very slowly –
5ths are very slowly widening, evaluate M10’s progression –
------------------------------------------------------------------------------------------------
Step 26 – (A#4)-A#5 – sharp - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
Step 27 – (B4)-B5 – sharp - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
Step 28 – (C5)-C6 – sharp - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
---------------------------------------------------------------------------------------------
Go back down for G#3 to lower notes using SBI and RBI, improve A#4 with F#3, never lose control of beats progressions for all intervals. 5ths will get slower, so will 4ths. Unison all these registers from left hand moving right, except last muted string on C6, then go up to higher notes. Chas delta-wide 15ths and delta-narrow 12ths beat’s rate is about 1b/3s or 1/3 bps.

Tune middle string first, then unison previous note’s right string (C6), next left (C#6), tune next middle (D6), unison previous right (C#6), next left, tune next middle and so on, checking also M17ths progression. While tuning, do not stop evaluating strings and sound-board rigidity/elasticity, so to be able to conveniently tune centre strings."

- . - . - . -

About Cordier's and Stopper's sequencies unfortunatelly I can not help you, I'd hope they were available.

..."Strange about the Chas--I recently reread the long thread\argument in which the ideas behind it were first put forth. Over 30 pages. And it turned out to have a lovely sound. (That 53 meg file is what first convinced me.)"...

In my opinion, it is not so strange...it takes time for the tuning to settle, it takes time for people to acknowledge.

..."Seems as though we have a stunning set of choices--good methods of reaching ET, the EBVT, the beautiful Stopper temperament, the CHAS."...

Yes, we have modern ETs that should be acknowledged as soon as possible, since they allow us to really and easly tune progressive and coherent intervals.

..."I've only been learning about temperaments for the past year--not sure I'll ever catch up to a full understanding."...

I'd take a chance by stopping doubts about understanding, the all issue is absolutelly at hand.

..."Is there at least a conference, here--recent temperaments?"...

You are preciselly on one issue I'm after. I think it is time to gather together and refresh the actual knowledge about modern ETs.

Regards, a.c.

.

Merci.

bout Cordier's and Stopper's sequencies unfortunatelly I can not help you, I'd hope they were available.

I can provide the original Cordier method ans sequence, all is derived from a "pure 5th" hence the ocatve have a 3.5 cts enlarging.

The double octave get very wide...

After some thinking and may criticisms received, in 1982 Serge Cordier proposed something smoother wher the octave is only enlarged by 1.5 cts (1/3 bps) and the 5th became pure again only at a6 level.

That one is near the Chas for the plain wire. (no sequence for that option, but I have the one used by a collegue that use more or less the same).

But the approach stay based on the "pure 5th" intention, the 12th double octave relation is not the basis (in fact I was said that the twelve enlarge around 2.5 bps in the high treble (which is not much, but for some reason I believe that resonance begin to lower as soon as the 12the enlarge).

All those sequences are in French for today. I'll try to post them on the Canadian forum to see if someone have the desire to put it in English. Because the notes names is different in the French terminology (a3 is A440Hz) translation is necessary.
May be I will do that myself, or my colleague could if time permit.

Same basic process to stick on RBI progressiveness, but beat synchronism on P5th is tested with classical tests)
That tuning was even told at an university for some years in France, so we have tuners that use it regularly, it is not so often but not rare (many are from the south of France, as the University was in Montpellier). Some church organs (I have to look for twhich ones) have been tuned in Cordier, so the enlarged octave is not only to be used on inharmonic instruments (Jean Guillou who is a well known organist and play at the Ste Eustache church on magnificent organs, in Paris, is very found of that tuning) .

Should be interesting. records would be a plus indeed.


Best regards.

I believe you are slightly off the timeline here.

Codier published his book initially in 1982. Although the tuner colleagues attacked him immediately for his "too wide" octaves, he listed about 30 references from well known pianists applauding to his new tuning at the end of his book and confirming him what he was doing. I doubt that he gave up his model and proposed a smaller octave already in the year where is book was released.

Public criticism of his model came up the first time to my knowledge with my publication of the temperament based on pure duodecimes in euro piano in 1988, (translated and published also for the french tuning community, Cordier probably got knowlegde of it), where i strived the philosophical incoherence of Cordier´s pure fifth model, as the model is based on a pure 3/2 interval, which itself consists of two other pure intervals (i.e. the duodecime "3" and the octave "2"), which are both not pure in Cordier´s pure fifth ET model.

I appreciate if you can provide some provable facts that Cordier revised his model already in the year his book was published.

I'll have to verify those historical points as I am only refering to what a colleague wrote me . He tunes the "Cordier" and that modified version (he slighly adapted to his liking he says).

I have an extract (in French) from the writing of Cordier abouit that othe way. Indeed if the book was in 82 that modified method could not be at the same time.:

Voici ce que Cordier écrivait en 1982 à propos d'un tempérament intermédiaire :
"parmi toutes les solutions intermédiaires entre la gamme bien tempérée et le tempérament à quintes justes , l'une d'entre elles retiendra davantage notre attention : celle où la nécessité d'aboutir à un tempérament égal ne s'exerce pas davantage au détriment de la justesse de la quinte que de celle de l'octave mais où le prix à payer au tempérament paraît également réparti entre les deux intervalles . C'est précisément le cas lorsqu'on ne raccourci la quinte que de 1 cent (au lieu de deux dans la gamme tempérée) , ce qui permet de n'agrandir l'octave que de 1,7 cents (au lieu de 3,4 dans le tempérament égal en quintes justes) . Une octave et une quinte ayant la même note de basse présentent alors la même rapidité soit , par exemple , 0,3 bat/secpour fa2- fa3 et fa2-do3 ce qui est pratiquement imperceptible . "
Plus loin :
"C'est pourquoi nous avons récemment défini et réalisé un accord où la quinte , raccourcie de 1 cent dans le médium retrouve progressivement sa justesse pour l'atteindre au niveau de la quinte la4-mi5 (notation française (a5e6 in english) ndlr ) et l'outrepasser ensuite ... "


Indeed all that is dated 1882. Not from a second writing; my mistake, due also to the fact that Cordier stated something slighly different from his first eaasy a few years after, having not studied the points closely, I'll let my colleague chime in on the subject.

I guess we can say that Cordier opened the way, he was looking for the "secret" of concert tuners if I recall well the way things are stated at the beginning of its writing.

I also recall hearing cordier tunings that adbsorb iH on small pianos and that lend to a too vivid major harmony on low iH pianos (while may be the way it is used only conducts to that result)

To me a "pure fifth" at the piano will have differnt size depending of the iH, is not it the case or does it depend of the way the pure inteval is checked ?

Your point on the fact that 5th based on 2 others intervals yet pure is clear. for sure the 12ths weights more than the 5ths, very strange that it have not been evoqued more before.

SO to build an historical temperament one would better use a cycle of 12ths ?

OK, this text is from the last chapter of his book, where he writes about infinite intermediate possibilities (page 260). His intention was not to revise his model, but to explain that infinite possibilites between pure octaves and pure fifths exist, and that the pure fifths tempermant is superior to such intermediate solutions. This appears clearly some sentences later (page 261), where he is discarding this intermediate solutions by favorising the pure fifths in these statements:

"...Dans le premier Cas (pure fifths) on trouvera le piano brilliant, á la rigueur un peu haut, dans le second (with the intermediate solutions), on le trouvera tout simplement faux..."
and
"...Il ne presente pas á un meme niveau que le TEQJ (Cordier temperament) dont les rapiditées sont isochrones ou quasi isochrones et favorise donc moin..."

So the text you have interpreted as a rethinking and refinement of his pure fifth model, was rather an argumentation against the intermediate solutions. This is a nice example how the sense of a text can be converted into it´s opposite by leaving out important parts.

For what I know Serge Cordier did not train as a concert technician, I have seen no mention of the way the temperament octave is (generally) opened , in his book (I have read very fast).

I thought he focused on beats synchronism within the 5th, as he did not mention differnces in beats depending of the level of the partials, I did not see where he wanted to go (at the piano)

What makes me state that "second version" was the way things have been preseneted to me by my colleague. In the end I understand that for every day's work he uses that modified version (that he arranage to his liking he said) . With octave then at 1.7 cts and 1/3 bps for the 5th as for the octave.

It is in any case different than taking the 12th as the ruler of the tuning.

And again beats are seen as "the enemy of resonance" .

We have as tuners little learning to the question of resonance and harmony it is normal that false or one sided ideas are going around. (for instance the tuned duplexes)

In my view that should be the role of the ones who can explain to make those concepts and principles availeable to us.

Most of the colleagues I tolk of Chas and of your tuning ask for a temperament sequance and method to try those tunings. They seem generally speaking open to have the pôssibility to try something different, to analyse it as possible, and then decide with a more complete knowledge what looks more suiteable for them to their dayly work.

Those new informations are highly welcome, be it to seasoned tuners (more to seasoned tuners probably) than to beginners.

Do you have the envy or possibility to add your part to this ?

I find that the affirmation that a new way to determine the pitches of ET is not so excessive, at last for the piano it can't be questionned.
It always will be a compromizing question, but being aware of the kind of compromizing we are using, being able to compare the effect of differnt compromizing approaches sure may help the tuner to work more confidently.

What amazes me is that all this for a good part is using what some are or have been yet doing since a long time.

Many things we do are not done in conscience (particularely when tuning)because we have to learn to work fast.

Talking of Cordier, did he wrote some addendum or only stated something different from his book "piano and orchestral justness ?

The Cordier tuners here are very sure they provide a "better" tuning, with more clarity in the treble. To me this is mostly because they are confident in the way they stretch the tuning, they have a method. I see most others are presenting tuning as an art 'which certainly it is'.
Some find a way or another to stick to some rule.

It is good news that we begin to really analyse which ratios we use, and what is the result of those ones in terms of departing from the natural behaviour of the piano string.(while I understand well it can be of no interest to some, of course)

Yes Isaac,

also in my opinion it is foundamental for tuners to know what and why having to look for. This is how I understand Bill Bremmer's satisfaction when he features EBVT. If a tuner can rely on a re-foundable form, even if it is a non-equal temperament he/she may eventually know that his/her job has been done. And as I say (opinion), any intentional tuning form is better than any unhinged quasi-ET attempt. If anything, the intentional tuning of a precise form may say something about that one tuner and his/her own preferencies.

Good if you manage to post Cordier's sequence. Have you tryed it? BTW, do you think is worth contacting that university and ask what is left with that model?

Also, if really Cordier's model has been applyed to pipe organs, you may argue about the efficiency of theoretical models for harmonic and inharmonic tones, what I consider a crucial, very relevant issue, as this argument (O) is being abused. ((O) is my opinion).

Stopper,

I join the others requesting your practical tuning sequence, it should not take long for you and all of us may get some benefits.

About Cordier's pure 5ths model you write:

..."Public criticism of his model came up the first time to my knowledge with my publication of the temperament based on pure duodecimes in euro piano in 1988, (translated and published also for the french tuning community, Cordier probably got knowlegde of it), where i strived the philosophical incoherence of Cordier´s pure fifth model, as the model is based on a pure 3/2 interval, which itself consists of two other pure intervals (i.e. the duodecime "3" and the octave "2"), which are both not pure in Cordier´s pure fifth ET model."...

It would be nice if you managed to deepen about that (is Cordier still alive?). Also you may explain the theoretical and philosophical aspects regarding pure 12ths theory.

For instance, you claim pure 12ths superiority on the basis of some (practical?) beat symmetries (wich you could post here too), and I would ask you how this is related with a single theoretical pure ratio. In other words, how do zero-beating (12ths) symmetries cohere with beating symmetries?

Also, would you say that zero-beating intervals can still be theorized? And if yes, would you say for what purpose?

I took seven months to realize one more bottom question: not acknowledging a substantial difference between 12th root of two and Modern ETs. So, I think it is up to us explaining how the first ET has evolved and how modern theory can now take tuners to a practicable and naturally-harmonious form.

Bill Bremmer was mentioning some help that may come from PTG's mathematicians, like Robert Scott. I still hope they'll join us in renewing the equal tempering horizon.

Regards, a.c.

.

Bernhard Stopper,

I join the others requesting your practical tuning sequence, it should not take long for you and all of us may get some benefits.

About Cordier's pure 5ths model you write:

..."Public criticism of his model came up the first time to my knowledge with my publication of the temperament based on pure duodecimes in euro piano in 1988, (translated and published also for the french tuning community, Cordier probably got knowlegde of it), where i strived the philosophical incoherence of Cordier´s pure fifth model, as the model is based on a pure 3/2 interval, which itself consists of two other pure intervals (i.e. the duodecime "3" and the octave "2"), which are both not pure in Cordier´s pure fifth ET model."...

It would be nice if you managed to deepen about that (is Cordier still alive?). Also you may explain the theoretical and philosophical aspects regarding pure 12ths theory.

For instance, you claim pure 12ths superiority on the basis of some (practical?) beat symmetries (wich you could post here too), and I would ask you how this is related with a single theoretical pure ratio. In other words, how do zero-beating (12ths) symmetries cohere with beating symmetries?

Also, would you say that zero-beating intervals can still be theorized? And if yes, would you say for what purpose?

I took seven months to realize one more bottom question: not acknowledging a substantial difference between 12th root of two and Modern ETs. So, I think it is up to us explaining how the first ET has evolved and how modern theory can now take tuners to a practicable and naturally-harmonious form.

Bill Bremmer was mentioning some help that may come from PTG's mathematicians, like Robert Scott. I still hope they'll join us in renewing the equal tempering horizon.

a.c.

.

Alfredo:

I would also like to hear more about aurally tuning the Stopper temperament. It seems to be based on a "sweet spot" between the 3:1 and 6:2 partial matches of the P12. It may be another one of those artistic things that cannot be explained, only experienced.

But let me warn you about looking at the theory of pure twelfths. It is not about the 19th root of 3 any more than ET is about the 12th root of 2. These ratios are meaningless when dealing with inharmonic tones. Until you accept this concept, you will not be able to understand nor contribute to any discussion of inharmonic tuning theory.

But then to accept this concept is also to leave Chas theory behind because Chas theory is about frequency ratios. It would be “a bitter pill for you to swallow”, Acquaintance.

Jeff I am unsure you understand well the effect of inharmonicity on piano tone. Ih only give you room for enlarging the intervals from the basic relations, this does not mean that the relation can't be tuned.

I am still expecting links for records of "pure 5ths" based tuning on an organ. It have been done (I wonder about the speed of the thirds !).

To me that is the whole partial match approach ans listening that complicate our job. Most often, the good tuners tune "in the tone" without focusing on a particular partial level (which is a good method to learn, but can be left aside after-that, in my opinion).
Or at last listen to the beating at the most fundamental level availeable.

Isaac:

We seem to have more and more in common. Likewise, Isaac, I am unsure you understand well the effect of inharmonicity on piano tuning.

Consider the m3-M6 test for a 6:3 octave. If an octave in the middle of a piano is tuned to sound beatless, the 6:3 octave test will show that the octave is narrower than the 6:3 partial match; the m3 will beat faster than the M6. This means that the effects of iH on the beat rate of these intervals is opposite of what would be expected from merely “stretched” intervals. If the intervals were merely stretched due to iH, the m3 would beat slower than the M6! You see, iH affects the beat rates of some intervals in a way that is opposite to the stretching effect of iH.

But now I also agree with you about there being more important things to listen to than beat rates. Although the complex interaction of many beat rates due to difference tones are, I believe, what gives some intervals “color”, the ear can normally only separate a few of the strongest and obvious ones. It is a start but, as you say, tuning “in the tone” is a more advanced way.

Jeff, you wrote:

..."I would also like to hear more about aurally tuning the Stopper temperament."...

So would I. I find strange that Stopper has not released its pure 12ths aural tuning sequence, and that he is not replying on his approach's incongruencies.

..."It seems to be based on a "sweet spot" between the 3:1 and 6:2 partial matches of the P12. It may be another one of those artistic things that cannot be explained, only experienced."...

Well, he himself could say what it is, not you nor me.

..."But let me warn you about looking at the theory of pure twelfths. It is not about the 19th root of 3 any more than ET is about the 12th root of 2.

Sorry, but yours is a wrong reduction.
ET is about geometrical progressions and equally distant notes.
12th root of two is only the first ET, the ET based on a pure, zero beating octave. Modern ETs models have well improved that first attempt to use a geometry we can find in nature.

...”These ratios are meaningless when dealing with inharmonic tones."...

I've acknowledged your opinion about ET models and how meaningless you think they are. And more, I've acknowledged your approach to tuning, to beats, to tuning-checks and, to a certain extent, I've also acknowledged your taste in terms of piano performance.

...”Until you accept this concept, you will not be able to understand nor contribute to any discussion of inharmonic tuning theory."...

Talking about iH, (O) we have said enough in Chas Topic. About the existence of “inharmonic tuning theories”, this is the first time I hear of this category. Do you have an "inharmonic tuning theory"?

Also, you'd better realize that my contribute is meant to go towards Temperamental Theories and the sharing of the latest ET's evolution. In other words, the latest variant of the first ET. Would you like a realistic refresh? Have a look:

http://www.huygens-fokker.org/docs/bibliography.html

..."But then to accept this concept is also to leave Chas theory behind because Chas theory is about frequency ratios."...

Well, you too may take advantage by letting me say what Chas Theory is about.
Chas theory is about proportional differences. When you are ready, you can read (maybe re-read) sections 2.0 and 3.0 in the research report and simply acknowledge what is written in there. So, in Chas case, the frequencies values are for the first time derived from proportional differences.

...”It would be “a bitter pill for you to swallow”.

If only you knew how sweet the tuning of Chas form can be, you would not worry about bitterness any more.

a.c.

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

.
_________

ANd now it seem to be ready to be tested in the Pianoteq software (even on the demo version).

The first comments of the users are as usual, more harmony, increase of harmonic resonance, sweet sound, etc...

As what said the pianists when hearing their piano , they speak of "beauty". Even when making good tunings, you fairly know that is not the word that comes in mouth of pianists naturally...

If you knew Alfredo, you would understand that he could take a lot of bitter pills , he have a natural medecine that protects againts that bitter taste (if not he would have left that forum for long !!)..

Please Alfredo, have a look at the logic of the "scala " file provided for Pianoteq. (an incremental ratio non one octave, reproduced all along the keyboard) At first it sound as Chas.

Best regards to all !