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Jeff,

This example may be able to help. I ran the following experiment using TuneLab with some artificially entered (but still possible) inharmonicity readings. The made-up IH numbers are: C1: 0.2 C3: 0.05 C4: 0.2 C6: 2.5 Four readings is the absolute minimum number to define an IH model in TuneLab. I then let TuneLab auto-adjust a tuning curve using 6:3 octaves in the low bass and 2:1 octaves in the high treble. This resulted in a stretch of +37 cents at C8. High, but still within reason. Then I temporarily switched the treble interval to the 3:2 fifth to see what the fifths would be like. It turns out that with these settings, the 3:2 fifths transition between narrow and wide at about G7, ending up at about 1.2 cents wide at C8 (F7-C8).

This behavior is not typical. I deliberately chose the IH numbers to make the model think that the IH was increasing rapidly as you move up the scale. This rate of increase and not the absolute IH, it turns out, is the critical factor in determining how octaves and fifths compare. Normally 3:2 fifths are narrower than 2:1 octaves. But with a sufficiently high rate of increase of IH, striving to make 2:1 octaves beatless can make fifths wide.

Mr. Scott:

Thank you so very much for giving an example and explanation. I guess I had worried about this because I was thinking that if my fifths didn’t become wide, I wasn’t tuning “correctly”. But since this happens only in the very high treble, due to a greater slope of the iH curve, then fifths becoming wide is an inherent anomaly of some pianos, not the result of a tuning style.

Bill:

Welcome back and am glad to hear that both you and your computer are doing better. I hope neither of you had the Swine Flu Virus.

Tooner:

In my opinion, it is very important to distinguish theory from practice. If you do not, thinks will not work. For example, this is way the numbers you have used to find pure fifths in Chas could not work.

Jeff S. wrote "I think most of us would be interested in knowing if your theories have changed the way you actually tune a piano...". So I answered on the practical side of the matter.

Theory can only be singular, ways to get to the one theory could be many. So, generally speacking, the practical way to tune Chas requires that fifths go the way I said, but only to counterbalance string lenthening, and the bridge and the harmonic board's adjustement, and only if the piano you are tuning were flat.

Going back to theory, I'll add more this afternoon.

Bill Bremmen:

Thanks for your post. When you say "The "mindless octaves" idea is nothing more than making an exact compromise between the double octave and the octave and 5th (12th).", it makes me think we have had the same experience and we are supporting the very same euphonic set of sounds. Have you made any progress from the practical ground to theory? We could compare figures and get a more precise idea. a.c.

Yes I know it could not work. That is why I pointed it out, as the "Devil's Advocate", so that you now have the opportunity to present a more complete theory. I am not your enemy.

Tooner:

Do play the devil’s advocate and I’ll thank you for that, only I’d kindly ask you to read carefully Chas article, so that we don’t go in circles. Thank you very much.

You asked for Chas basic concepts. In the article you’ll find Chas basic concepts but, little by little, we can look at them together – In the abstract you read about the goals: Is E.T. improvable? Can we find a rule to manage string inharmonicity? Can we theorize stretched octaves?

Chas being a theory refers to E.T., i.e. our current international theoretical system. In section 2.0 you can read: “Thus two questions arise. The first: is it correct to theorise that the octave interval must have a 2:1 ratio? The second: which temperament model today is reliable in theoretical terms and is commonly applied in the practice of tuning?”

Answering these questions, Chas asserts that E.T. 2:1 theoretical ratio for the octave is a cultural/historical teaching, maybe deriving from the debatable idea that pure intervals sound better. A reliable model should be free of cultural or historical heritages.

For centuries we have calculated scale frequencies values giving the 2:1 ratio for granted. Today we stretch octaves, so “which temperament is commonly applied in the practice of tuning?”. We are not applying E.T., since it theorizes pure octaves.

Chas model theorizes stretched octaves and combines theoretical harmonic partials. We could look at it the other way around: Chas model combines the scale effects of theoretical harmonic partials 2, 3 and 5 in a new set. The combination of prime numbers in a scale of sounds has been an age-old theoretical problem, today its solution stretches octaves and finds a new beats function.

Chas is a time-rhythm based temperament model that finds the biunivocal relationship between frequencies and beat frequencies. Chas describes a set where beats play the fundamental role (section 2.0).

So, Chas scale frequencies values come out as the result of synchronic beats, i.e. today a theoretical system based on proportional beats can order a scale of proportional frequencies, the opposite of what has been done so far.

Is it because of inharmonicity that we can not apply E.T.? Maybe not only. In fact E.T. calculates 13 frequencies and enlarges the scale by cloning this 13 sounds module. In section 4.3 – graph 5 we see the effects on the beats.

Chas model, referring to the traditional semitonal scale, adopts a two-octave module. From section 3.0: “A two-octave module gives the scale set an intermodular quality. From the minor second degree to the nth degree, all intervals will now find their exclusive identity”.

What does “all intervals will now find their exclusive identity” mean? It means that intervals greater than an octave, in terms of beats, can all play and support a ratio for the entire set.

Any questions so far? a.c.

Alfredo:

I am trying to think of the best way for us to communicate on this subject. I think a broader discussion, rather than a paragraph-by-paragraph study of your paper is worth a try.

There are some terms that we each may use but we each may use differently. ET is one of them. I understand that by ET, you mean the frequencies based on the twelfth root of two. To me, it means a scale where all keys have the same color and the feature of this type of scale is that M3s and M6 beat progressively faster. Since we are talking about your paper, we will use your definition.

I would say that since all pianos have iH, that any piano that has ever been tuned aurally has not been tuned to ET. And any piano that was tuned aurally by using temperament tests outside the initial octave (and there are many sequences that use more than one octave to set the temperament) will produce an “intermodular quality”. As you mentioned “A two-octave module gives the scale set an intermodular quality. From the minor second degree to the nth degree, all intervals will now find their exclusive identity”.

I really want to understand how you are tuning. Can you explain your tuning sequence? This may help me understand how you are using your discovery and thereby understand your paper.

By the way, I’ve been to Sicily. Augusta Bay is one of my favorite ports, very relaxing.

Tooner:

As you like, we can discuss broaderly, as long as we do not go to far from the main goal of this Topic.

You say: "To me, it (ET) means a scale where all keys have the same color and the feature of this type of scale is that M3s and M6 beat progressively faster".

To me it means the same, Chas theory derives from an ET approach and produces ET frequencies values, but talking about the original theory it will mean octave ratio = 2:1.

Then you say: "sequences that use more than one octave to set the temperament will produce an “intermodular quality”.

I agree but, in my opinion, a sequence is a tuning routine, not to be confused with a theory. Anyway, i'll write down the sequence I use and post it.

I'm very glad you've had a good time in Sicily.

Bremmer:

May I ask you what is the difference betwin "Maindless" and EBVT? Are frequencies values in Hz available?

Robert Scott, thanks for your post. a.c.

Jeff S.:

Thanks for contributing, please stay in touch.

You say: “…old teachings stated the beat rates of fifths should increase as one tuned up the scale. That model was based largely on a decades-old mathematical model, worked out before inharmonicity was understood. Current understanding leans more toward the idea that the beat rates of fifths should either stay roughly the same as one moves up the scale or decrease…”.

What you are saying gives me the opportunity to underline, for some of our colleagues, the importance of a reliable theory. In my opinion, even when we can “lean toward an idea”, we are left doubtful. An idea can be interesting, fascinating, even brilliant but it is not quite like having a precise and correct mathematical model deriving from a reliable theory. Should fifths stay roughly the same as one moves up the scale? How roughly? Should fifths decrease? About IH, you say it has been understood, i'm not that sure.

You end up saying: “My point is that… the practical way fifths are understood and tuned… has already changed away from the old model”.

I simply agree, we have left E.T. original theory behind and we are now lacking a comprehensive model, well described by a solid theory that takes inharmonicity in account, what we may call a “inharmonic theoretical model”. This is what Chas is meant to be. a.c.

Alfredo, the Equal Beating Victorian Temperament (EBVT) is a non-equal temperament. The "mindless octaves" concept is an octave stretching technique and therefore it has nothing to do with the initial temperament octave, it is only a way of expanding the temperament over the rest of the piano.

Contrary to what many technicians seem to believe, causing double octaves and 12ths to beat equally (which is the mindless octave concept) does not require the temperament to be equal. Obviously, I use it with the EBVT and any other non-equal temperament, 18th Century style to the present. It would not work with the far more unequal temperaments of the 17th Century and earlier. It does work for 1/7 Comma Meantone and any other mild meantone temperament but an exception must be made when tuning the double octave G#-G# and comparing it to the D#-G# 12th since in any meantone, the G#-D# 5th is a wide interval.

Bill, thanks for replying. So, with EBVT M3s and M6 do not beat progressively faster. Does one have to use an EDT to tune the EBVT? What could aural tuners use? Are there any figures?

Reading in your previous post about cristal sound, pipe organ effect, customer satisfaction, I was really thinking you could be tuning Chas.

Actually, Chas is an inharmonic ET, adopts 12ths and 15ths as the scale constants wich determine model's scale incremental ratio. Nothing to do with a non-equal temperament. I'd really like to try your temperament. a.c.

Check Bill's website, you can find the cents offset and aural sequences there:

http://billbremmer.com/ebvt/

Thanks Erus, the EBVT was developed aurally and is the preferred way to tune it. However, so many technicians wanted the numeric data that I asked Professor Owen Jorgensen RPT to calculate them for me. The principal reason for aural tuning preference is that an ETD does not stretch the octaves the way I do aurally and it does make a significant difference in the final results. If you are an aural tuner, you will find the instructions very easy to follow.

Alfredo,
I have noticed that you are not interested in the cello scrotum article, unfortunately you can´t understand my intention then. This article is of great value, as we can learn from it, that even if something has been published in a serious scientific medium, we have to be very careful about the content.

The tuning software i am marketing is based on the tuning method i published in 1988 in euro-piano (based on the 19th root of three in case of abscence of inharmonicity, i.e. the theoretical case) and my own discovery of perfect beat symmetry in the 19th root of three temperament, dating from 2004.

I am presenting the software and some theory at the italian piano technicians convention (7-11 July 2009) in Cavalese, Italy. It is planned that i am tuning a Fazioli grand piano with my tuning software. There is a second Fazioli grand piano present to be tuned from someone else in a different (standard or whatever) tuning. You are welcome to participate and to tune it with your chas method!

Bernhard Stopper

Bernhard,

you say: "even if something has been published in a serious scientific medium, we have to be very careful about the content."

I'm sorry, my english is poor. I thought you were trivially insinuating that the content of the article about Chas is probably more rubbish. Yeah, how could you immagine readers being that naive or stupid and how could you have got to any conclusion without reading the article. Instead you were suggesting to read the article with care! So I have to thank you.

"The tuning software i am marketing is based on the tuning method i published in 1988 in euro-piano (based on the 19th root of three in case of abscence of inharmonicity, i.e. the theoretical case)...".

I could more or less read about this in an other Topic, but, like some others, I could not really understand your discovery. Could you please tell me more or do I need to necessarly come to Cavalese? By the way, thank you for inviting me but, having listened to your two recorded tracks, I think that sending you a sample of Chas will be enough.

Is there any graph or official document of yours available? a,c,

Alfredo:

Could you put an audio example of your Chas tuning on the internet and provide us with a link?

I hope you will still post your tuning sequence. The numbers for any tuning scheme are fascinating to me, but to understand what is really happening, when used on different pianos, the sequence means more. For example, the width of the mindless octave’s twelfth and double octave are determined by the piano’s iH and the width of the fourth formed by the lower notes of the intervals. This is easier to understand when looking at the sequence of tuning the intervals to be equal beating. It would be less easy to understand with an equation, because if iH is not included the theory is incomplete, and if iH is included then the theory must show how it is affected by different values and slopes of iH.

Oh, and I think Mr. Stopper’s cello comment had many meanings. Take your pick.

Tooner, thanks for your feedback.

Your idea of a Chas tuning example will probably work much faster than words and numbers. Last year, with the help of a brilliant sound expert, we digitally compared ET and Chas frequencies, in answer to an italian collegue who thoght Chas tuning could not work for pipe organs. I'll start puting these evidencies on internet and I'll proceed with a recording from a real piano.

I've almost finished with my sequence. Meanwhile I can propose some reasoning about Chas tuning in practice.

Chas model, as you may have red in section 2.0, theorizes an ET sound set that, to deal with inharmonicity, is not based on a theoretical pure interval. So, at the end of our tuning we should’nt look for any pure interval. We’ll also see that, to translate Chas inharmonic theory into practice, we’ll need to temporarly raise all frequencies above average Chas inharmonicity theoretical values. Nothing to worry about because anyway this is more or less what we have empirically done so far, although only on the bases of an approximate calculation of inharmonicity.

So, in addition to precise theoretical inharmonicity’s Chas standard values, we’ll also consider the sound-board and the strings while-tuning settling. In fact, once we have tuned and stabilized middle strings, tuning the right and left string of each flat note, from middle-high register upwords, will cause an overall lowering of frequencies. No ETD can foresee or evaluate the fall in frequencies, consequent piano settling (by measuring crhomatic 12ths after your ETD tuning, you may confirm this statement).

So again, considering inharmonicity and depending on how flat our piano was, while tuning the middle strings we will temporarly have to go for a more accentuated stretch. Anyway, the final evidences I can find after tuning are the ones that Chas theory describes:

1)the well known ET progression of M3’s, M6’s, M10’s, M17’s
2)the Chas inharmonic progression of 4ths and 5ths including inversion of the latter
3)the Chas inharmonic “S-shaped progression” for the octaves
4)the constant, equal beating of Chas delta-wide 15ths and delta-narrow 12ths

Let’s have a look at these evidences. In point 1 we find nothing new: it is a well known fact that ET progression of RBI comes from the geometrical esponential increase of frequencies. So, to get stretched octaves in that kind of geometrical progression we only need to use any ratio higher than 2^1/12. Thus one question arise: how much higher does the most correct theoretical incremental ratio need to be?

In point 4, I mention equal beating. Well, I do not seem to be the first, having red Bremmer’s posts, and that makes me very happy. Bremmer well describes the extraordinary effects of 12ths and 15ths equal beating, something that he him self, with other collegues, are still experiencing. Thus a second question arises: how do you get to the most correct equal beating value and still enjoy an ET progression of RBI?

In point 3, I am talking about an S-shaped progression for octaves, a shape that should be familiar to us, since Railsback's measurements. Two more questions arise: is there a chance to find the most theoretically correct standard curve deviation from the 2:1 ratio? Will it ever be possible to adopt a natural and reliable standard curve of reference that deals with inharmonicity?

In point 2 you may find a fresh piece of news: the precise beats progression for 4ths and 5ths and the observation that the latter’s beat curve invert. Then one last question arises: when should 5ths ideally invert?

Simply answering to all these questions would take you straight to Chas model and there you may also find the relevance of any comprehensive theory.

Please, tell me about the many other cello meanings...I might like them more. a.c.

Alfredo:

You wrote –

”1)the well known ET progression of M3’s, M6’s, M10’s, M17’s
2)the Chas inharmonic progression of 4ths and 5ths including inversion of the latter
3)the Chas inharmonic “S-shaped progression” for the octaves
4)the constant, equal beating of Chas delta-wide 15ths and delta-narrow 12ths”

I think I now comprehend what you believe the Chas system to be and we can communicate better now. These four points are a great way to focus the discussion.

”1)the well known ET progression of M3’s, M6’s, M10’s, M17’s”

The progressive beating of intervals made from non-iH tones will occur as long as each semi-tone interval is the same. The semi-tone could be 1 cent or 1000 cents. The 2:1 octave ratio does not affect this. Of course, the cent (being 2^1/1200) is an entity of a theoretical 2:1 octave scale. But the unit of measurement could be something else. As long as the semi-tone ratio is the same for all semi-tones and greater than 1, all intervals made from non-iH tones will beat progressively faster. And for iH tones an important question is: what is the ratio of? Is it the ratio of the theoretical fundamentals or the first partials? This is important to understand whether the ratio is 2^1/12, 3^1/19, or the Chas ratio. It may be better to go into fixed tuning ratios deeper in another post.

”2)the Chas inharmonic progression of 4ths and 5ths including inversion of the latter”

As Mr. Scott showed in his post, it is the value and slope of the iH curve that produces the wide beating of the fifths. It is inherent in the scaling of some pianos, although a wider octave in the high treble can make this happen at a lower note. But is this a characteristic that has value in itself? Do listeners prefer a high treble with wide fifths? Or is this a feature of something else that is important, but is not a goal in itself. But then I have also showed that the Chas ratio does not produce wide fifths without iH. But here we get into fixed tuning ratios again.

”3)the Chas inharmonic “S-shaped progression” for the octaves”

I almost always see the same Railsback curve. I wonder if the piano was not tuned very well, or if it had scaling problems, or if the frequencies were not measured accurately. I am sure various pianos and various tuning preferences would show an S curve also, but with differing values. A piano tuned with 2:1 octaves (which is greater than a 2:1 frequency ratio because of iH) will show an S curve. As will also a graph of frequencies of non-iH tones with a semi-tone ratio greater than 2^1/12. Since the Chas ratio is greater than 2^1/12 it will produce an S curve. But that does not mean it accounts for iH, just that the ratio is greater than 2^1/12.

”4)the constant, equal beating of Chas delta-wide 15ths and delta-narrow 12ths”

OK, I was correct in thinking that the mindless octave is the basis of Chas. I have to be a bit intuitive on what I am going to say here because I have not actually worked out the math. Since the beat speed of the 12th and 15th is dependant on the width of the fourth that is formed from the lower notes of the intervals, then neither mindless octaves nor Chas prescribe the overall stretch of a tuning, but only a final outcome from an initial stretch. Also, again being intuitive, unless 12ths become wide first, I don’t think fifths can become wide. After all, a 12th is an octave and a fifth. If these together are not wide (and in the mindless octave and Chas they are not) how could the fifth be wide separately?

Maybe we can get into fixed and variable semi-tone ratios another time. I don’t have them really figured out yet.

To all aural tuning collegues :

I decided to go further my tuning sequence mainly for two reasons: firstly, because hundreds of interesting pages have and are been written about the most original and convenient sequencies, none of them leading to a solid, reliable theory that could deal with inharmonicity, so leaving tuners in a state of uncertainty. A sequence will be debatable, a mathematical evidence won’t. Secondly, because I do not think the sequence I use is any special, nor time saving or more confortable for listening to or comparing beats. In my opinion, any sequence can eventually work, as long as you clearly know what you can be aiming at and why, how and where you’ll get it.

The only novelty may regard the overall approach and the interweaving of SBI, i.e. 8ves, 4ths and 5ths beat curves, the results of research that opened to Chas algorithm. Chromatic 4ths are not only similar, going up the scale they get tiny little wider, chromatic 5ths are not only similar, from low notes they first stretch down and get tiny little narrower, in between C3 and C4 they invert and stretch up toward there pure ratio, going tiny less and less narrow.

An italian collegue pointed out that SBI are much harder to evaluate than RBI. True, I would also agree in saying RBI give you the general idea of what you are doing in the shortest lapse of time. Nevertheless in my opinion, if one truly wanted to achieve excellence in aural tuning, would have to master a maximum control of any interval’s beat. A matter of wrist, both in the figurative and the anathomic sense, and a matter of rhythmics. In my case, SBI control took me to the 7th decimal point (section 4.5).

So what happened was, first I empirically found the univocal SBI and RBI chromatic proportional order, finding an astonishing euphonic set that would prove how inharmonicity can be made tractable. Then I simply elaborated its essence, to finally construct an updated and comprehensive ET IH EB temperament model (lucky us with all those abbs.), reliable in both theoretical and practical terms. Since I know all this comes from practice, simplicity and utmost exactitude, I’m disclosing Chas model with a serene soul.

In tuning, as I have learned, each sound is only temporarly tuned, since every single added sound may indicate the need to correct previously tuned notes. At the end, it is the Chas form that releases me from all doubts and only then I am absolutely certain to have done my best. Anyway, here are a few suggestions introducing and commenting the sequence.

A - do not take this tuning sequence as a must -
B - octaves, 4ths and 5ths shape the skelethon of the entire set -
C - start tuning only middle string, mute from C6 down to strings crossing, dampers up -
D - tuning single strings and unisons, get always the same moderate sound intensity -
E - octaves have a low beat-threshold and a high beat-threshold, this helps me when tuning octaves in middle register -
F - possibly, stabilize middle string frequencies by playing a Forte sound -
G – do not tire your ears, by playing louder you will not hear better nor more -

wide or narrow is referred to the note we are ment to tune

Step 1 – A4 – (Hz) from 440.0 to 442.0 (concert or studio) - from 441.5 to 443.0 (for flat pianos)
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Step 2 – A4-A3 - tiny little narrow, just on the beating threshold
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Step 3 – A3-D4-(A4) - wide, close to 1 beat/sec. – D4-(A4) faintly beating
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Step 4 – A3-E4-(A4)
check overlaping 5ths and adjacent 4ths to set up Chas ET EB inharmonic octave:
A3-E4 about 1,5 beat/3s - sensibly faster than D4-(A4)
E4-(A4) about 2 beats/1s - sensibly faster than A3-D4
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Step 5 – E4-B3 – narrow - tiny little faster beat than A3-D4, sensibly slower beat than E4-(A4)
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Step 6 – B3-F#4 - narrow - 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 – narrow - 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 – narrow - 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 – narrow - 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 – narrow - tiny little faster beat than A3-D4, tiny little slower than E4-B3
evaluate five M3’s progression
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Step 11 – A#3-F4 – narrow - tiny little slower beat than A3-E4,
tiny little faster beat than B3-F#4
evaluate seven M3’s progression
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So far, apart from A3-D4, we have stretched narrow - now we’ll stretch wide
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Step 12 – D4-G4 – widw - 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 - wide - 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
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Beats curves are meant to be tuned temporarly. While you are tuning, bear all (few) doubts in mind.
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Step 14 – A#3-A#4 – wide - 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 us in a shorter and shorter lapse of time, this helps to S-shape octaves stretch
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Step 15 – B3-B4 - wide - increase octaves beats speed very, very slowly - 5ths towards pure
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Step 16 – C4-C5 - wide - increase octaves beats speed very, slowly - 5ths towards pure
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Step 17 – C#4-C#5 - wide - increase octaves beats speed very slowly – 5ths start transiting pure - evaluate one M10
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Step 18 – D4-D5 - wide - increase octaves beats speed very slowly – 5ths are transiting pure - evaluate M10’s progression

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Step 19 – D#4-D#5 - wide - increase octaves beats speed very slowly – 5ths are transiting pure - evaluate M10’s progression

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Step 20 – E4-E5 - wide - increase octaves beats speed very slowly –
5ths have transit 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 – wide
Step 22 – F#4-F#5 – wide
Step 23 – G4-G5 – wide
Step 24 – G#4-G#5 – wide
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Step 25 – A4-A5 – double octaves like A3-A5 must be constant and temporarly beat with a rate of almost 1b/s
increase octaves beats speed very slowly –
5ths are very slowly widening, evaluate M10’s progression –
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Step 26 – A#4-A#5 – wide - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
Step 27 – B4-B5 – wide - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
Step 28 – C5-C6 – wide - check 10ths, pure 12ths, wide 15ths, let 5ths go slowly wider
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Go back down for G#3 to lower notes using SBI, RBI and EB, never lose control of beats proportions. 5ths will get slower, so will 4ths. Unison all these registers from your 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 about1b/3s.
Tune as you know, middle string first, then unison previous note’s right string (C6), next left (C#6), next middle (D6), previous right (C#6), next left, next middle and so on, checking also M17ths progression. While tuning, do not stop evaluating strings and sound table rigidity/elasticity, so you’ll be able to conveniently set up middle strings. In fact, on pianos you have recently tuned, more often grand’s, to get to final Chas delta-wide 15ths and delta-narrow 12ths, could be enough to temporarly set up a milder 12thsV15ths proportion. The exact opposite, in case of badly flat pianos. This may produce a difference.

Tooner:

Thanks. You say:

“And for iH tones an important question is: what is the ratio of? Is it the ratio of the theoretical fundamentals or the first partials? This is important to understand whether the ratio is 2^1/12, 3^1/19, or the Chas ratio.”

Well, you judge. The intermodular combining of partials 3 and 4 in the way Chas algorithm does, includes partial 5, which in the scale is semi-tone 28 (adjacent 3th number 7). In fact partial 4, resulting from 6 adjacent M3 and 8 adjacent m3, can intermodularly mediate, together with partial 3, all partials. Pure theoretical ratio 3^1/19 = 1.0595260647382…, like any other ratio higher than Chas 1.0594865443501, increases differences, and therefor beats, on sounds relative to partial 5 and 10.

After all, I’m not talking about personal taste, one may prefere pure 12ths or pure 19ths (6^1/31), some others pure 5ths or pure 3ths, as we have seen. Chas model explains the reasons for aiming at a purly proportional and synchronic frequenciesVbeats set ratio. Fairly proportioned beats open to a proportional set of sounds, a pure set.

As you will have red, I use higher ratios than Chas only to compensate while-tuning strings and sound-board settling, so to finally get to the ET EB Chas form. Chas theory is meant to describe a new way to interpret beats, why and how profiting from beats, and to show the beauty of Chas form its self. Frequencies, throgh beats, can stir up (or awaken) all scale sounds partials and so lead to an extraordinary resonant set.

Then you ask:

“Do listeners prefer a high treble with wide fifths?”

We have no reason to talk about preferences.

And then:

“But then I have also showed that the Chas ratio does not produce wide fifths without iH.”

I do apply Chas in iH cases, so I do not get your point.

“Since the Chas ratio is greater than 2^1/12 it will produce an S curve. But that does not mean it accounts for iH, just that the ratio is greater than 2^1/12.”

To me Chas S-shaped octave curve meant that we can deal with iH. So far we have related the necessity to stretch octaves only to iH. Chas model suggests we have to recalculate iH’s effect, since up to now we’ve calculated iH giving for granted two unjustiefied ET assumptions (section 3.0). Moreover, Chas octave quotients are closest to pure n/n+1 quotients (section 4.5).

You say:

“OK, I was correct in thinking that the mindless octave is the basis of Chas”.

Well, I’d rather say that Chas theory is the ET height of a base EB idea, call it mindless or whatever. Bill Bremmer says: "The "mindless octaves" concept is an octave stretching technique and therefore it has nothing to do with the initial temperament octave, it is only a way of expanding the temperament over the rest of the piano."

Telling you about my self, I first established in practice a congruent and coherent assumptions-free ET, then I elaborated the observable constants 12thsV15ths EB producing a comprehensive theory that could correct and apdate the approach to ET and iH.

“Since the beat speed of the 12th and 15th is dependant on the width of the fourth that is formed from the lower notes of the intervals, then neither mindless octaves nor Chas prescribe the overall stretch of a tuning, but only a final outcome from an initial stretch.”

This is not exactly correct. If mindless octave idea was intented for an ET scale, then mindless octave idea would be aiming at Chas model. But then he him self says that EBVT is a non-equal temperament. So, without “mindless” formula and with no precise frequencies values I can only say that mindless EB idea on his own, using your words, does not prescribe the overall stretch of a tuning, nor an initial 4th strecth in its sequence (since 4ths are meant to be similar). Chas model prescribes both. In fact, Chas tuning overall strecth and 4ths wideness are determined by Chas ET algorithm and its resulting incremental ratio. In the sequence I use, 4ths, 5ths + A3-A4 octave are the foundation of the whole. Nevertheless Chas is not featuring a strict form: with Chas algorithm, you could actually figure out other kind of EB, explore new ET’s and finally deal with iH, aware of what you are doing. For example:

(9/8 – Δ)^(1/2) = (4 + Δ*s)^(1/24)
s = 1
Δ = 0,00247997487864…
Semi-tone scale incremental ratio = 1,059490455417770…
1st partial’s ratio = 2,00061989765139…

Am I making any progress?

Bill Bremmer:

Do you think Professor Jorgensen could get to know about Chas?

Thank you. a.c.

Alfredo,

I am sure that Professor Jorgensen would be interested in your concepts. However, he is quite elderly and in frail health. He never has communicated via e-mail. I would suggest that you think about what kind of package of information you would like to present to him, print it out and send it to him by conventional mail in a large envelope.

He will reply to you in a hand written letter. Make sure you supply him with the properly written postal address as it is in your country. Professor Jorgensen understands the mathematics of tuning theory.

I have a very difficult time deciphering the math on these posts but I believe it is mostly because of the substitute symbols which are used. Since I am not familiar with the symbols you and other people may use, the mathematics I often see are beyond anything I can work with.

However, I can see from a cursory look at your sequence that what you describe is Equal Temperament (ET). Two points which confuse me are that I saw in earlier posts that you seemed to be denouncing ET and that attracted my attention. Yet, when I see your sequence, it looks like a very typical method of constructing ET using 4ths and 5ths. But that which has me most confused is that you apparently describe the first octave to be tuned as slightly narrow rather than slightly wide. Unless this was an error in transcription, it has me completely confused as to just what you are attempting to accomplish.

You mention lengthy scientific papers, none of which I have read and I am afraid they may prove to be unreadable by me. I have no education in higher mathematics. To me, aural tuning is a mechanical procedure which does have some foundation in mathematics but in the end is a physical job performed by a technician who listens and makes adjustments according to what is heard. Many technicians know nothing at all about tuning theory yet they manage to tune excellently. I have always said, "The essence of aural tuning is the perception and control of beats".

In my understanding, ET can exist with any conceivable amount of stretch or even within an octave which is deliberately narrowed. Stretching or narrowing an octave does not change any temperament, either ET or non-ET, it merely changes how the octaves sound but there is, of course an effect to be heard from even the smallest change to the size of the initial octave. However, that kind of effect is relatively small compared to the kind of effect which can be heard by deliberately tuning a non-ET. Any non-ET will also be affected by octave stretching or narrowing decisions.

If you would like Professor Jorgensen's mailing address, please send me a private message either on this forum or to my e-mail: billbrpt@charter.net. I do not believe it would be proper to post that information for all to see even though it can be obtained easily.