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#1201488 - 05/18/09 12:58 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bill Bremmer RPT]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1062
Loc: Sicily - Italy
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.
_________________________
alfredo

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#1201493 - 05/18/09 01:15 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Erus Offline
Full Member

Registered: 12/26/07
Posts: 386
Loc: Mexico
Check Bill's website, you can find the cents offset and aural sequences there:

http://billbremmer.com/ebvt/

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#1202028 - 05/19/09 09:26 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Erus]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3226
Loc: Madison, WI USA
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.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1202587 - 05/20/09 06:26 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Bernhard Stopper Offline
Full Member

Registered: 09/22/08
Posts: 211
Loc: Germany
Originally Posted By: alfredo capurso

Stopper:

I’m sorry, I was not interested in your “cello scrotum”.
I understand you are commercializing an ETD device, at a cost of $ 600. Could I know on wich basis? Does reading about scrotum help?


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


Edited by Bernhard Stopper (05/20/09 06:28 AM)
_________________________
Bernhard Stopper
www.piano-stopper.de

Salieri: "Mediocrities everywhere, now and to come: I absolve you all! Amen! Amen! Amen!"
(Amadeus, the movie)

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#1202602 - 05/20/09 07:31 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bernhard Stopper]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1062
Loc: Sicily - Italy
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

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#1202609 - 05/20/09 07:54 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4940
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1202952 - 05/20/09 05:58 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1062
Loc: Sicily - Italy
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

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#1203263 - 05/21/09 08:13 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
UnrightTooner Offline
4000 Post Club Member

Registered: 11/13/08
Posts: 4940
Loc: Bradford County, PA
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.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1204676 - 05/23/09 02:54 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso Offline
1000 Post Club Member

Registered: 07/10/07
Posts: 1062
Loc: Sicily - Italy
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)
------------------------------------------------------------------------------------------------
Step 2 – A4-A3 - tiny little narrow, just on the beating threshold
------------------------------------------------------------------------------------------------
Step 3 – A3-D4-(A4) - wide, close to 1 beat/sec. – D4-(A4) faintly beating
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
Step 5 – E4-B3 – narrow - tiny little faster beat than A3-D4, sensibly slower beat than E4-(A4)
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
So far, apart from A3-D4, we have stretched narrow - now we’ll stretch wide
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
--------------------------------------------------------------------------------------------
Beats curves are meant to be tuned temporarly. While you are tuning, bear all (few) doubts in mind.
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
Step 15 – B3-B4 - wide - increase octaves beats speed very, very slowly - 5ths towards pure
------------------------------------------------------------------------------------------------
Step 16 – C4-C5 - wide - increase octaves beats speed very, slowly - 5ths towards pure
------------------------------------------------------------------------------------------------
Step 17 – C#4-C#5 - wide - increase octaves beats speed very slowly – 5ths start transiting pure - evaluate one M10
------------------------------------------------------------------------------------------------
Step 18 – D4-D5 - wide - increase octaves beats speed very slowly – 5ths are transiting pure - evaluate M10’s progression

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

------------------------------------------------------------------------------------------------
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
------------------------------------------------------------------------------------------------
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 –
------------------------------------------------------------------------------------------------
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
---------------------------------------------------------------------------------------------
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

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#1204806 - 05/23/09 07:30 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Bill Bremmer RPT Offline
3000 Post Club Member

Registered: 08/21/02
Posts: 3226
Loc: Madison, WI USA
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.
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1205359 - 05/25/09 05:54 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bill Bremmer RPT]
UnrightTooner Offline
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Alfredo:

I am a bit confused. Parts of your sequence seem to indicate non-ET, like when saying a fourth should be narrow. And other parts seem to indicate ET, like when checking M3s and M6s for progression. It may be the use of the terms wide and narrow, and hopefully we can clear this up.

You wrote: “wide or narrow is referred to the note we are ment to tune” When I think of an interval being wide or narrow I think of it being wider or narrower than just intonation (beatless), so it would not matter which note is meant. But perhaps you mean wide and narrow to mean faster beating or slower beating, like a doorway being wide or narrow? Or perhaps by wide or narrow you meant sharp or flat (or even flat or sharp)?

Also, when you submitted your paper to the University for publishing, what was the process for acceptance? Was it checked by the math department? Did you have to defend the paper to a board of professors? Did anyone at the University understand it? Did they agree with it?
_________________________
Jeff Deutschle
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Who taught the first chicken how to peck?

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#1205476 - 05/25/09 01:09 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso Offline
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Loc: Sicily - Italy
Bill, Tooner, all collegues,

I’m so sorry, I used “narrow” and “wide” instead of “flat” and “sharp”. I had highlighted in red those notes to be tuned flat, and blue notes to be tuned wide, but when copying the text into the topic colours desappeard. So I wrote “wide or narrow is referred to the note we are ment to tune” forgeting that you would have understood that as been referred to the interval. When you see Step 2 - A4-A3 – narrow, means that A3 is flat, wich makes this octave wide (like any Chas octave); when you see Step 7 - F#4-C#4 – narrow, means that C#4 is flat, wich makes this 4th wide.

Bill,

Chas is an ET model’s theory, maybe the first ET model’s theory since traditional ET formula. Actually traditional ET, differently than what you could call a well described theory, looks more like an algebraic technique to maintain the pure octave and to distribute the so-called commas equally across 12 semitones (section 1.5).

To me, ET’s algebraic instrument can result been perfect, since nature seems to speak an algebraic language, nevertheless I’m denouncing traditional ET’s assumptions regarding the one octave module and the 2:1 octave ratio. So you were not wrong.

When iH was discovered, we runned to the conclusion that we could never put traditional ET into practice, because of iH, so ET could only be thought as an abstract “theory”. Probably then we also decided that no temperament theory can help in tuning.

I’m trying to correct this thinking, when I say: we could not put traditional ET into practice not really because of iH, but because traditional ET is a lame theory from birth.

In fact, traditional ET theory was spoilt by the “theoretical one octave module” and by “mathematical ratio 2:1”. These theoretical and mathematical assumptions, both wrong, lamed traditional ET and made it unrealizable and consequently unpleasent.

Since traditional ET could never be put into practice, we do not really know what tuners and musicians have been talking about in the past, when referring to traditional ET. Today two things are clear: RBI, like 3ths, 6ths, 10ths, 17ths and so on should have a smooth progression, octaves should be stretched.

Now, stretched-octaves do not come from traditional ET. So I ask: do we know of a reliable ET stretched-octaves theory?

You say you enjoy equal beating tuning in your dayly work, so I ask: do we know of a reliable EB ET stretched-octaves theory?

I'm sure it could be of great meaning for you to read Chas article, even only the two sections about approach and description of Chas model (2 pages). Meanwhile I'll prepare a mathematical description symbol-free.

Thank you also for your indications regarding Professor Jorgensen. You have been so kind, I'll follow your advice.

Tooner,

You ask: "Was it checked by the math department?"

Well, what do you think?

"Did you have to defend the paper to a board of professors?"

I had to rewrite the article 3 times, to explain things that on the way had resulted obscure. It took me almost 2 years.

"Did anyone at the University understand it?"

Yes, Chas maths is not that difficult and I'll demonstrate that.

"Did they agree with it?"

They checked Chas maths without playing any other role. We'd better talk about how could anyone disagree, don't you think?

Thanks, a.c.
_________________________
alfredo

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#1205506 - 05/25/09 02:14 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
BDB Online   content
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This discussion about how you cannot tune equal or whatever temperament because of this, that or the other reminds me of the old saying: There is no problem so difficult that you cannot look at it in such a way to make it much more difficult!
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#1206268 - 05/26/09 05:25 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: BDB]
alfredo capurso Offline
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Bill,

You say:

"In my understanding, ET can exist with any conceivable amount of stretch or even within an octave which is deliberately narrowed."

So far, if you are talking about progressive RBI, I'm with you. Then I start having difficulties:

"Stretching or narrowing an octave does not change any temperament, either ET or non-ET,..."

What do you mean, saying: does not change any temperament?

"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."

I ask: what effect will the smallest change to the size of the initial octave have?

You end up saying:

"Any non-ET will also be affected by octave stretching or narrowing decisions."

When you started saying:

"Stretching or narrowing an octave does not change any temperament, either ET or non-ET,...".

So, may I ask you for a wider explaination? a.c.
_________________________
alfredo

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#1206346 - 05/26/09 07:27 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Roy123 Offline
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I finally had a chance to stick my nose into Mr. Capurso's paper, and I must say that it would be difficult to write a less understandable explanation of his thesis than he has done--perhaps it suffered in the translation. I went only so far as to find a math error, and perhaps Mr. Capurso would be willing to address my confusion.

Equation 4 is fine, Equation 5 is fine, but Equation 6 does not follow from Equation 5. To demonstrate this, I did a simple example in MathCAD. I arbitrarily selected a value of .1 for delta,and a value of 2 for S1. MathCAD solved for S, whose value is -0.3244117.... Now, if Equation 6 is valid, then we should be able to state that (3-.1*2)^(1/19) = (4+(-.3244117))^(1/24). However, this equality is invalid, and would only be correct if S1 = 1, which would cause Equation 5 to degenerate back into Equation 4. The error in the equality did not change much even for tiny values of delta. Perhaps, Mr. Capurso meant to suggest an approximate equality in Equation 6, or perhaps I made a mistake in my analysis.


Edited by Roy123 (05/26/09 07:28 PM)
Edit Reason: typo

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#1206607 - 05/27/09 07:38 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Roy123]
UnrightTooner Offline
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Roy:

I noticed the same error and posted about it in the beginning of this Topic. I am trying to go beyond the mathematical explanation and pursue the concept by looking at the tuning sequence.
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#1206633 - 05/27/09 08:38 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
Roy123 Offline
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So you did--it was so long ago in the thread that I missed it. It seems to me that the thesis in question is based in, or at least presented as based in, mathematics, and therefore must be judged on that basis. Mr. Capurso makes many hyperbolic statements and claims throughout his article, and if they are not supported by the analysis, what is his basis for making them?

I hope that Mr. Capurso will address the issue we have both raised. If not, I will be forced to judge his words as hollow.


Edited by Roy123 (05/27/09 11:34 AM)
Edit Reason: typo

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#1206638 - 05/27/09 08:47 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Roy123]
UnrightTooner Offline
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Roy:

That is you prerogative, of course. I am looking for a gem in the rubble. And even if there is not one, there may be something else to discover. If not for me, perhaps for Alfredo. He surely spent a great deal of effort. I think he is in earnest.
_________________________
Jeff Deutschle
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Who taught the first chicken how to peck?

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#1206729 - 05/27/09 11:01 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
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Loc: Bradford County, PA
Alfredo:

Thanks for the clarification on wide meaning sharp and narrow meaning flat.

I now understand your tuning sequence. Fourths beat progressively faster, while fifths beat progressively slower, become beatless, and then beat progressively faster but on the wide side of just intonation. This causes octaves to beat progressively faster also.

The fixed Chas ratio cannot do this when applied to either the note’s frequencies, nor to the beat speeds of the intervals. However, perhaps it describes the “change in the rate of change” of the beat rate curve or perhaps the frequency curve, which is a real ski slope.

Does this sound like what you are trying to say in your paper?
_________________________
Jeff Deutschle
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Who taught the first chicken how to peck?

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#1206760 - 05/27/09 11:57 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
Roy123 Offline
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Registered: 09/20/04
Posts: 1715
Loc: Massachusetts
Originally Posted By: UnrightTooner
Roy:

That is you prerogative, of course. I am looking for a gem in the rubble. And even if there is not one, there may be something else to discover. If not for me, perhaps for Alfredo. He surely spent a great deal of effort. I think he is in earnest.


Well, I hope you succeed, but I have my doubts. Even as I just read beyond Equation 6, Mr. Capurso starts talking about different values of s, without saying what the value of s1 would be. This paper should not have been published in its present condition. The figures are not properly annotated or explained, the claims he makes in the text are not backed up in the math, the math has at least some errors, and the whole presentation is loose, rambling, with extraneous information included, and essential explanations left out.

As you say, there may be a gem lurking in there, but one would have to start from the very basic premise, and then attempt to derive the analysis on one's own, I think.

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#1206820 - 05/27/09 01:34 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Roy123]
alfredo capurso Offline
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Roy123,

Thanks for joining in. You say:

..."I arbitrarily selected a value of .1 for delta,and a value of 2 for S1."...

"The error in the equality did not change much even for tiny values of delta. Perhaps, Mr. Capurso meant to suggest an approximate equality in Equation 6, or perhaps I made a mistake in my analysis."

In this case we do not find any approximation. In section 3.3 you read: ..."When we add in the s variable, a rational number...". So, you can add an s value and you are not supposed to tuch delta, with or without s. Delta is not discretional. I'm sorry if my explaination was not as good you could have done.

Tooner,

I'll be back tomorrow and I'll answer you. thanks for your words and your attention, a.c.
_________________________
alfredo

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#1206860 - 05/27/09 02:22 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Roy123 Offline
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Registered: 09/20/04
Posts: 1715
Loc: Massachusetts
Thanks for your reply, Mr. Capurso, but your explanation does not make sense. You calculate delta from Equation 1 in your report. If we now use that value of delta in equation 5, we can see by inspection that the only possible value of s/s1 is 1. If we set s = s1 = 1, then all is fine, but Equations 5 and 6 are the same as Equation 1. If we take ANY other values for s and s1, such as s = s1 = 2, then Equation 5 still works, but Equation 6 doesn't, which is what I originally said. Basically, Equation 6 does not follow from Equation 5. The math is not correct.

Mr. Capurso, you have always been polite with your responses, and therefore it behooves me to behave similarly. However, both Tooner and I have addressed a serious question to you about your thesis--namely that Equation 6 is not a mathematically correct form of Equation 5. Therefore, unless you are willing to explain the veracity of your derivation, or to declare your mistake and correct your paper, it becomes difficult for me to take you seriously. Sorry to be blunt, but as the author of a paper that you present publically, you have an obligation to address any mistakes that may be in it, or withdraw it from the public until it can be corrected.

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#1207640 - 05/28/09 06:00 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Roy123]
alfredo capurso Offline
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Loc: Sicily - Italy
Roy123,

You say: "Mr. Capurso, you have always been polite with your responses, and therefore it behooves me to behave similarly."

I'm glad and I take that as a promise. I must admit that, reading about your conclusions in such final terms did not help my chemistry:

"The figures are not properly annotated or explained, the claims he makes in the text are not backed up in the math, the math has at least some errors, and the whole presentation is loose, rambling, with extraneous information included, and essential explanations left out."

I'm sure the text can be improved, but when you find extraneous information you can skip it, like we would do on any text, and when you are missing explanations you can ask me. First you claimed for maths basis, here you also talk about style, and you also say "the maths has at least some errors", when you should not be that sure.

You say: "Therefore, unless you are willing to explain the veracity of your derivation, or to declare your mistake and correct your paper, it becomes difficult for me to take you seriously."

Ok, let's declare our mistakes, by the way, is it clear why you cannot modify delta?

Then you say: ..."If we now use that value of delta in equation 5, we can see by inspection that the only possible value..."

You are not supposed to use delta value deriving from Equation 1. The value of delta will continuously change, depending on s.

What Equation 6 shows is that, if s is a fraction, the denominator will effect delta (i.e. differencies, i.e. beats) in the left expression (i.e. on partial 3). To check this, after having chosen a fractional s value, calculate the incremental factor (i.e. scale ratio), build up your scale values and you will be able to ascertain that the differencies on partial 3 and 4 will have the same proportions of your s fractional value.

"Sorry to be blunt, but as the author of a paper that you present publically, you have an obligation..."

I do not know what you are worried about, I think I'm aware of my obbligations, why would I be here?

Please, let me know if now Chas algorithm works better.

Tooner,

I have to postpone your question, hope you do not mind. a.c.
_________________________
alfredo

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#1207684 - 05/28/09 07:17 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Bill Bremmer RPT Offline
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Registered: 08/21/02
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Alfredo, while I can make no comment or judgment on the math, I can make a comment on the way the written sequence is described. I will have to take into account that you may not be familiar with the way temperament sequences are described in American (and probably any other variety of) English.

No interval, octave 3rd, 4th, 5th, etc. can be "sharp" or "flat" even though many people will describe them that way. An interval can only be beatless (also called "pure" or "just intonation"), wide or narrow (from the point where it does not beat).

Now, having said that, in order to widen a beatless interval, one may flatten the bottom note or sharpen the top note. To narrow an interval, one may sharpen the bottom note or flatten the top note.

In ET, 5ths are always slightly narrow and therefore some people say that they are flattened and we know what they mean but it is not the correct way to describe a tempered 5th. This is the most common example of misuse of the terms, "sharp" and "flat" when describing the tempering of intervals but it applies to all intervals.

So, I believe you need to review your written instructions for construction a temperament. The way you have described it is quite confusing. You have said that an octave should be slightly "narrow" when you really meant that the octave should be slightly wide. I believe there are some other examples of that where you say a 4th should be narrow when you meant it should be wide and the same possibly with other intervals where you have effectively said the opposite of what you mean.

I am sure that if you sent that material to Owen Jorgensen, he would write back the same as I have said and would provide corrections in red ink.

Writing temperament sequence instructions is very difficult and it is easy to make very bad errors and for the writer to not see them. I know this from experience and I am grateful for those who have helped me correct those kind of errors on many occasions.

Regards,
_________________________
Bill Bremmer RPT
Madison WI USA
www.billbremmer.com

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#1207904 - 05/29/09 07:22 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: Bill Bremmer RPT]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4940
Loc: Bradford County, PA
Bill:

Yes, there is a language error in how the sequence is written. I was able to decipher the sequence when I understood the error.

I think an Equal Temperament can be constructed with wide fifths. It certainly can be constructed with just fifths, so why not wide?

What Alfredo seems to be doing is having ever increasing octave widths. Looking at it with non-iH tones, I would say that the temperament octave would be about 1202 cents wide, and each octave higher being an additional 2 cents wider in order for the fifths to become wide. I normally think of ET as having beat rates that increase for all intervals. Seems very odd to think of one as having an interval that beats slower and then faster on the wide side, but such a beat rate can still be considered to be progressive. I think sometimes my twelfths do this, so probably my nineteenths actually do. Of course, having the fifths do this so low in the piano’s range will make very busy octaves!
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1207908 - 05/29/09 07:36 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
Roy123 Offline
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Registered: 09/20/04
Posts: 1715
Loc: Massachusetts
Originally Posted By: alfredo capurso
Roy123,

You say: "Mr. Capurso, you have always been polite with your responses, and therefore it behooves me to behave similarly."

I'm glad and I take that as a promise. I must admit that, reading about your conclusions in such final terms did not help my chemistry:

"The figures are not properly annotated or explained, the claims he makes in the text are not backed up in the math, the math has at least some errors, and the whole presentation is loose, rambling, with extraneous information included, and essential explanations left out."

I'm sure the text can be improved, but when you find extraneous information you can skip it, like we would do on any text, and when you are missing explanations you can ask me. First you claimed for maths basis, here you also talk about style, and you also say "the maths has at least some errors", when you should not be that sure.

You say: "Therefore, unless you are willing to explain the veracity of your derivation, or to declare your mistake and correct your paper, it becomes difficult for me to take you seriously."

Ok, let's declare our mistakes, by the way, is it clear why you cannot modify delta?

Then you say: ..."If we now use that value of delta in equation 5, we can see by inspection that the only possible value..."

You are not supposed to use delta value deriving from Equation 1. The value of delta will continuously change, depending on s.

What Equation 6 shows is that, if s is a fraction, the denominator will effect delta (i.e. differencies, i.e. beats) in the left expression (i.e. on partial 3). To check this, after having chosen a fractional s value, calculate the incremental factor (i.e. scale ratio), build up your scale values and you will be able to ascertain that the differencies on partial 3 and 4 will have the same proportions of your s fractional value.

"Sorry to be blunt, but as the author of a paper that you present publically, you have an obligation..."

I do not know what you are worried about, I think I'm aware of my obbligations, why would I be here?

Please, let me know if now Chas algorithm works better.


Mr. Capurso, you continue to miss or evade the point. You claim that Equation 6 can be derived from Equation 5. It can't. There is no reason for me to expound further, the math speaks for itself.

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#1207912 - 05/29/09 07:48 AM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4940
Loc: Bradford County, PA
Alfredo:

Take your time. Remember what I said before, I am not your enemy.

Something that this discussion is doing for me is making me think about how I think about tuning theory.

I can think of it purely mathematical, with or without iH. And I can think about it purely harmonically (beats) with or without iH. Or I can think about it musically, in how to provide the listener with what they want to hear, or fool them into accepting what they hear as “correct.”

I am guessing that you found a way to tune, and you also discovered some mathematical phenomena and think they are related. I don’t know the evolution of your thinking, so I am only guessing. I am realizing that connecting harmonic tuning theory to mathematical tuning theory is quite a challenge.

Not too long ago I realized how the effects of iH are largely self-correcting on the theoretical beat rates of intervals. But unless there is some reason to express a harmonic tuning style mathematically, why bother? I suppose it is necessary to construct an ETD program. Or in my case, out of the desire to understand what others have said on the subject. What is your motivation to express your tuning style mathematically?
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1208044 - 05/29/09 12:50 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
UnrightTooner Offline
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Registered: 11/13/08
Posts: 4940
Loc: Bradford County, PA
Roy:

I agree that how the equations are presented are in error. But I think I now understand what Alfredo was trying to show with these equations.

Although he gives the correct solution for equation 1, and it seems that delta is really a constant and not a variable, I think he means to show that delta can have other solutions dependant on including an “s” factor. But the important thing is that after the “s” factor is applied, that the 19th root of the one term equals the 24th root of the other term. There may be an “s” and also an “s1” because iH affects the third partial differently than the 4th partial. "s1" is shown breifly but not explained.
_________________________
Jeff Deutschle
Part-Time Tuner
Who taught the first chicken how to peck?

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#1208057 - 05/29/09 01:19 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: UnrightTooner]
alfredo capurso Offline
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Registered: 07/10/07
Posts: 1062
Loc: Sicily - Italy
Dear colleagues,

I’m obliged to treat a mathematical matter regarding Chas algorithm. I know most of you will not be interested in this but Ive got no choise.

If anything, Chas model describes the beautiful set I have found in my tuning practice, this makes me hope I can share it with all tuners, despite the following necessary-in-necessity figures.

Bill,

You said: “So, while I have absolutely no idea of what Alfredo is talking about at this point, I say, go for it, I may find something I like after all…”. If you can, please have a go, you may discover that Chas maths is not difficult.

Roy123, Tooner,

I’m going to address the issue you have raised.

In section 3.2 you read: “In the chas algorithm, the Δ variable proportions the differences of two intervals, 8th+5th (12th degree) and 8th+8th (15th degree)…”. So Δ is an unknown quantity.

In section 3.3 you read: “…infinite exponential curves related to oscillations of partial values, and identifiable through a second variable, expressing an “elastic” potential and enabling the system to evolve. When we add in the s variable, a rational number, (s from the concepts of stretching, swinging and spinning), equation (1) becomes:….”, so, Equation (1) becomes Equation (4). This is to say: to our Equation (1) we can add in a rational number, the so called s variable that will change delta value, enabling the system to evolve.

Then you read about the scale effects of s variable:

“The s variable can swing the logarithmic scale… The variable affects the distances and proportion of scale values…”.

Then you read: “If s is a fraction (s/s1)…”, Equation (4) becomes Equation (5). Then you are told about the effect of s/s1 fractional value on the equality: “…the denominator multiples delta in the left-hand expression so that...”, so that Equation (5) equals Equation (6). Let’s check this together:

We choose a fractional value for s/s1:
s = -9
s1 = 8 and use the Equation (5) type, so we have:

Equation (5) type:

(3–Δ)^(1/19) = (4 + (Δ*-9/8))^(1/24)

true for Δ = 0.01018036614 = first found delta from Equation (5) type

Substituting this Δ value:

(3–0.01018036614)^(1/19)=
=(4+(0.01018036614*-9/8))^(1/24)= 1.05933652544275 this is our scale incremental ratio.

You were told that Equation (5) equals Equation (6), so that:

(3–Δ)^(1/19) = (4+(Δ*-9/8))^(1/24) equals
(3–(Δ*8))^(1/19) = (4+(Δ*-9))^(1/24)

It should be that there exists a value of delta (the unknown quantity that s can alter) so that our latter equality produces our previously found scale incremental ratio. Can it be true? Can we find this delta value?

Δ = 0.0012725457675, second found delta from Equation (6) type, so that

(3–(0.0012725457675*8))^(1/19)=
=(4+(0.0012725457675*-9))^(1/24) = 1.05933652544275

The first 24 scale values deriving from our s/s1 fractional value and consequent delta values will be:

Scale values
1,0
Scale ratio 1,059336525442750
1,122193874137110
1,188780959501540
1,259319091150850
1,334042710443460
1,413200169673400
1,497054557496920
1,585884573337010
1,679985453672080
1,779669953287340
1,885269384750260
1,997134719564920
2,115637754664980
2,241172348122290
2,374155728178230
2,515029879948310
2,664263014409130
2,822351124549780
(3-(delta*s1)) 2,989819633859990
3,167225142633750
3,355157277892540
3,554240653076620
3,765136944017540
(4+(delta*s)) 3,988547088091660


Difference on partial 4 (element 24) = 3.98854708809166 - 4 =
= -0.0114529119 wich is our first found (delta*-9/8) and our second found (delta*-9), in fact:

First delta from Equation (5) type = 0.01018036614
0.01018036614*(-9/8) = -0.0114529119

Second delta from Equation (6) type = 0.0012725457675
0.0012725457675*-9 = -0.0114529119

Difference on partial 3 (element 19) = 3 – 2.98981963385999 = 0.01018036614 this is our first found delta from Equation (5) type and our second found (delta*8) from Equation (6) type, in fact:

0.0012725457675*8 = 0.01018036614

Last ceck: divide the difference value on partial 4 by the difference value on partial 3:

-0.0114529119 : 0.01018036614 = -1.125 = -9/8 wich is our discretional s fractional value.

Roy123,

I would not like missing or evading any point.

Bill, Tooner,

I'll be with you asap, thank you. a.c.
_________________________
alfredo

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#1208171 - 05/29/09 04:09 PM Re: CIRCULAR HARMONIC SYSTEM - CHAS [Re: alfredo capurso]
ROMagister Offline
500 Post Club Member

Registered: 04/26/08
Posts: 518
Loc: Bucuresti, Romania
Good fundamental idea, quite confusing presentation.
If I understood this well, it IS Equal Temperament, but with another ratio: not the classic one where 12 semitones = 1 octave of exactly 2:1 (Pythagorean octave still accepted as axiom in classical ET).
The basic version (s=1) makes an equal compromise between the 'justness' of 3rd and 4th harmonics (octave+fifth vs 2 octaves). "s" is just the compromise parameter which says how important is the error in the 3rd harmonic compared to the error in the 4th harmonic. It can be set "politically" as we want, and the Delta results as a solution of the (implied) equation, also the practical frequency ratio that results.

The 'tweaking knob' of s/s1 may result in different deltas and frequency ratios.

Equation 6 is equivalent to eq.5 only if the Delta in eq.6 is a different Delta from the one in eq.5 (say, notate it Delta').

I just don't see where's the "circular" part of CHAS. The octave being wider than 2:1 they deviate more and more.

The "attractor of size 19*24" is pompously written, since 456 semitones way exceed the audible range (the most used in MIDI is 128 semitones).

I don't understand how this method incorporates the prime number 5. Of course, one can use "politically" the 5th harmonic as the 19th (2 octaves+M3), like it's used in organs with the 1 3/5' Tierce stop. But there it's no inharmonicity, and that stop is meant only to be used together with a fundamental (8') stop. But if used across the whole instrument it deviates way too much from the consonance of 2:1 octaves and 3:2 fifths.

One may use a similarly designed CHAS-like algorithm of equal (or stated-weight) compromise between 3:2 fifths and 5:4 thirds etc.

The difference from classical 2^(1/12) is smaller than the unknown inharmonicity of piano anyway - and that is an unknown depending on many practical details of building.

The suggestion to tune 'narrow' not wide in the central zone I understand it so: the intrinsic piano's C5 (that sounds consonant to C4 on that piano) is in unknown ratio to the C4, but > 2.00. One tunes C5 lower than what sounds consonant to C4, so that the result is closer to 'true' ET (or even Chas) than that piano's inharmonicity may suggest.

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