Yahoo Tuning Groups Ultimate Backup tuning Chas temperament model

Hi,

Perhaps I can describe Chas model with few lines.

One premise is that pure ratios do not necessarely translate into increased euphony.

A second premise is that the scale frequency values must be "coherent" with differences-from-pure-ratios values.

The third premise is that the correct amount of stretch relative to one single interval, will modify correctly the entire logarithmic scale.

The tool is a double-delta-wedge ((delta:delta) = (1:1)) which can stretch the forth. In fact 4:3 is modified in (delta + 4):(3 - delta). Now the 4th ratio is (4 + delta)/(3 - delta).

4 is the partial value for double-octave (2*2) and 3 is the partial value for 5th + octave (3/2*2). It is evident that by stretching the 4th we are combining the stretch of the fifth, the octave, the 5th + octave (12th) and the double octave (15th) and so on. The determining ratios appearing in Chas algorithm are 1:1, 2:1, 3:1 and 4:1. Don't you think Pitagora would have liked that?

Chas addresses one principle that makes this theoretical model adherent to tuning in practice: The scale must be considered in dynamic terms and therefor always adjustable.

That is in fact the function of the variable "s", a parameter that can modify delta and distribute the load of "differences" in infinite ways, by using rational values.

From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s = 0 we gain 12TET. With (3 - delta)^(1/38) = (4 + s*delta)^(1/48) (s=1) we gain the equivalent of 24TET, though this time no interval is pure.

Scale incremental ratios deriving from Chas algorithm may as well define single scale steps, so allowing to an infinite number of scales.

Perhaps you are willing to offer your feedback?

Thank you,

Alfredo

--- In tuning@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...> wrote:
>
> Hi,
>
> This group was pointed out to me by (perhaps) a partecipant.
>
> I would like to share a temperament model that somehow is meant to translate my experience as an aural piano tuner.
>
> I wrote a paper in order to fix some points and, not being a qualified mathematician, I was able to extend my elaborations only to a relative degree. So, the paper is not exhaustive at all and many people have found it more confusing than convincing.
>
> Anyway, more than the paper's "technical" compilation I would like to share its content, the theoretical premises and some conceptual issues.
>
> G.R.I.M. - Università di Palermo (2009)
> http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
>
> Your thoughts and comments are welcome.
>
> Thank you, a.c.
>
> PS: Sorry for my English.
>
> Here you find a digital compareson with 12 root of two ET:
> http://www.chas.it/index.php?option=com_content&view=article&id=46&Itemid=17&lang=en
>
> And five samples on real pianos:
> http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en
>

🔗Kees van Prooijen <keesvp@...>

Hey guys, I directed Alfredo to the group, and promised him an "active and
responsive audience". Please don't make me look stupid :-)

On Fri, Jun 22, 2012 at 10:26 AM, alfredo.capurso <alfredo.capurso@...
> wrote:

> **
>
>
>
>
> Hi,
>
> Perhaps I can describe Chas model with few lines.
>
> One premise is that pure ratios do not necessarely translate into
> increased euphony.
>
> A second premise is that the scale frequency values must be "coherent"
> with differences-from-pure-ratios values.
>
> The third premise is that the correct amount of stretch relative to one
> single interval, will modify correctly the entire logarithmic scale.
>
> The tool is a double-delta-wedge ((delta:delta) = (1:1)) which can stretch
> the forth. In fact 4:3 is modified in (delta + 4):(3 - delta). Now the 4th
> ratio is (4 + delta)/(3 - delta).
>
> 4 is the partial value for double-octave (2*2) and 3 is the partial value
> for 5th + octave (3/2*2). It is evident that by stretching the 4th we are
> combining the stretch of the fifth, the octave, the 5th + octave (12th) and
> the double octave (15th) and so on. The determining ratios appearing in
> Chas algorithm are 1:1, 2:1, 3:1 and 4:1. Don't you think Pitagora would
> have liked that?
>
> Chas addresses one principle that makes this theoretical model adherent to
> tuning in practice: The scale must be considered in dynamic terms and
> therefor always adjustable.
>
> That is in fact the function of the variable "s", a parameter that can
> modify delta and distribute the load of "differences" in infinite ways, by
> using rational values.
>
> From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s = 0
> we gain 12TET. With (3 - delta)^(1/38) = (4 + s*delta)^(1/48) (s=1) we gain
> the equivalent of 24TET, though this time no interval is pure.
>
> Scale incremental ratios deriving from Chas algorithm may as well define
> single scale steps, so allowing to an infinite number of scales.
>
> Perhaps you are willing to offer your feedback?
>
> Thank you,
>
> Alfredo
>
> --- In tuning@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...>
> wrote:
> >
> > Hi,
> >
> > This group was pointed out to me by (perhaps) a partecipant.
> >
> > I would like to share a temperament model that somehow is meant to
> translate my experience as an aural piano tuner.
> >
> > I wrote a paper in order to fix some points and, not being a qualified
> mathematician, I was able to extend my elaborations only to a relative
> degree. So, the paper is not exhaustive at all and many people have found
> it more confusing than convincing.
> >
> > Anyway, more than the paper's "technical" compilation I would like to
> share its content, the theoretical premises and some conceptual issues.
> >
> > G.R.I.M. - Università di Palermo (2009)
> > http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
> >
> > Your thoughts and comments are welcome.
> >
> > Thank you, a.c.
> >
> > PS: Sorry for my English.
> >
> > Here you find a digital compareson with 12 root of two ET:
> >
> http://www.chas.it/index.php?option=com_content&view=article&id=46&Itemid=17&lang=en
> >
> > And five samples on real pianos:
> >
> http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en
> >
>
>
>

🔗Mike Battaglia <battaglia01@...>

🔗alfredo.capurso <alfredo.capurso@...>

1 -> One premise is that pure ratios do not necessarily translate into increased
euphony.

2 -> A second premise is that the scale frequency values must be "coherent" with
differences-from-pure-ratios values.

3 -> The third premise is that the correct amount of stretch relative to one
single
interval, will modify correctly the entire logarithmic scale.

In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

-> ..."Are these axioms you are assuming or empirical claims?"...

Those are meant to be non-arbitrary premises and describe the approach to modeling. Perhaps the opposite should be proved, before being considered as worth modeling premises.

Do slight deviations from pure ratios reduce euphony? We cannot be that "sure" but, in any case... Chas enables also the modeling of pure-ratio based scales.

Can we manage/compute different degrees of "coherence" between scale-frequency values and differences-from-pure-ratios values? The answer may well be yes, as it is proved.

The third, isn't that simply true for any logarithmic scale?

Alfredo

--- In tuning@yahoogroups.com, Kees van Prooijen <keesvp@...> wrote:

Hey guys, I directed Alfredo to the group, and promised him an "active and responsive audience". Please don't make me look stupid :-)
>
> On Fri, Jun 22, 2012 at 10:26 AM, alfredo.capurso <alfredo.capurso@... wrote:

> > Hi,
> >
> > Perhaps I can describe Chas model with few lines.
> >
> > One premise is that pure ratios do not necessarely translate into
> > increased euphony.
> >
> > A second premise is that the scale frequency values must be "coherent" with differences-from-pure-ratios values.
> >
> > The third premise is that the correct amount of stretch relative to one single interval, will modify correctly the entire logarithmic scale.
> >
> > The tool is a double-delta-wedge ((delta:delta) = (1:1)) which can stretch the forth. In fact 4:3 is modified in (delta + 4):(3 - delta). Now the 4th ratio is (4 + delta)/(3 - delta).
> >
> > 4 is the partial value for double-octave (2*2) and 3 is the partial value for 5th + octave (3/2*2). It is evident that by stretching the 4th we are combining the stretch of the fifth, the octave, the 5th + octave (12th) and the double octave (15th) and so on. The determining ratios appearing in Chas algorithm are 1:1, 2:1, 3:1 and 4:1.

Don't you think Pitagora would have liked that?

> > Chas addresses one principle that makes this theoretical model adherent to tuning in practice: The scale must be considered in dynamic terms and therefor always adjustable.
> >
> > That is in fact the function of the variable "s", a parameter that can modify delta and distribute the load of "differences" in infinite ways, by using rational values.
> >
> > From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s = 0 we gain 12TET. With (3 - delta)^(1/38) = (4 + s*delta)^(1/48) (s=1) we gain the equivalent of 24TET, though this time no interval is pure.
> >
> > Scale incremental ratios deriving from Chas algorithm may as well define single scale steps, so allowing to an infinite variety of types of scales.
> >
> > Perhaps you are willing to offer your feedback?
> >
> > Thank you,
> >
> > Alfredo
> >
> > --- In tuning@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@>
> > wrote:
> > >
> > > Hi,
> > >
> > > This group was pointed out to me by (perhaps) a partecipant.
> > >
> > > I would like to share a temperament model that somehow is meant to
> > translate my experience as an aural piano tuner.
> > >
> > > I wrote a paper in order to fix some points and, not being a qualified
> > mathematician, I was able to extend my elaborations only to a relative
> > degree. So, the paper is not exhaustive at all and many people have found
> > it more confusing than convincing.
> > >
> > > Anyway, more than the paper's "technical" compilation I would like to
> > share its content, the theoretical premises and some conceptual issues.
> > >
> > > G.R.I.M. - Università di Palermo (2009)
> > > http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
> > >
> > > Your thoughts and comments are welcome.
> > >
> > > Thank you, a.c.
> > >
> > > PS: Sorry for my English.
> > >
> > > Here you find a digital compareson with 12 root of two ET:
> > >
> > http://www.chas.it/index.php?option=com_content&view=article&id=46&Itemid=17&lang=en
> > >
> > > And five samples on real pianos:
> > >
> > http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en
> > >
> >
> >
> >
>

🔗alfredo.capurso <alfredo.capurso@...>

🔗Mike Battaglia <battaglia01@...>

By the way, Alfredo, you may want to check out Facebook's Xenharmonic
Alliance group - it's a bit more active than the tuning list these days.

-Mike

On Jun 23, 2012, at 12:35 PM, "alfredo.capurso" <alfredo.capurso@...>
wrote:

1 -> One premise is that pure ratios do not necessarily translate into
increased
euphony.

2 -> A second premise is that the scale frequency values must be "coherent"
with
differences-from-pure-ratios values.

3 -> The third premise is that the correct amount of stretch relative to one
single
interval, will modify correctly the entire logarithmic scale.

In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

-> ..."Are these axioms you are assuming or empirical claims?"...

Those are meant to be non-arbitrary premises and describe the approach to
modeling. Perhaps the opposite should be proved, before being considered as
worth modeling premises.

Do slight deviations from pure ratios reduce euphony? We cannot be that
"sure" but, in any case... Chas enables also the modeling of pure-ratio
based scales.

Can we manage/compute different degrees of "coherence" between
scale-frequency values and differences-from-pure-ratios values? The answer
may well be yes, as it is proved.

The third, isn't that simply true for any logarithmic scale?

Alfredo

--- In tuning@yahoogroups.com, Kees van Prooijen <keesvp@...> wrote:

Hey guys, I directed Alfredo to the group, and promised him an "active and
responsive audience". Please don't make me look stupid :-)
>
> On Fri, Jun 22, 2012 at 10:26 AM, alfredo.capurso <alfredo.capurso@...
wrote:

> > Hi,
> >
> > Perhaps I can describe Chas model with few lines.
> >
> > One premise is that pure ratios do not necessarely translate into
> > increased euphony.
> >
> > A second premise is that the scale frequency values must be "coherent"
with differences-from-pure-ratios values.
> >
> > The third premise is that the correct amount of stretch relative to one
single interval, will modify correctly the entire logarithmic scale.
> >
> > The tool is a double-delta-wedge ((delta:delta) = (1:1)) which can
stretch the forth. In fact 4:3 is modified in (delta + 4):(3 - delta). Now
the 4th ratio is (4 + delta)/(3 - delta).
> >
> > 4 is the partial value for double-octave (2*2) and 3 is the partial
value for 5th + octave (3/2*2). It is evident that by stretching the 4th we
are combining the stretch of the fifth, the octave, the 5th + octave (12th)
and the double octave (15th) and so on. The determining ratios appearing in
Chas algorithm are 1:1, 2:1, 3:1 and 4:1.

Don't you think Pitagora would have liked that?

> > Chas addresses one principle that makes this theoretical model adherent
to tuning in practice: The scale must be considered in dynamic terms and
therefor always adjustable.
> >
> > That is in fact the function of the variable "s", a parameter that can
modify delta and distribute the load of "differences" in infinite ways, by
using rational values.
> >
> > From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s =
0 we gain 12TET. With (3 - delta)^(1/38) = (4 + s*delta)^(1/48) (s=1) we
gain the equivalent of 24TET, though this time no interval is pure.
> >
> > Scale incremental ratios deriving from Chas algorithm may as well
define single scale steps, so allowing to an infinite variety of types of
scales.
> >
> > Perhaps you are willing to offer your feedback?
> >
> > Thank you,
> >
> > Alfredo
> >
> > --- In tuning@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@>
> > wrote:
> > >
> > > Hi,
> > >
> > > This group was pointed out to me by (perhaps) a partecipant.
> > >
> > > I would like to share a temperament model that somehow is meant to
> > translate my experience as an aural piano tuner.
> > >
> > > I wrote a paper in order to fix some points and, not being a qualified
> > mathematician, I was able to extend my elaborations only to a relative
> > degree. So, the paper is not exhaustive at all and many people have
found
> > it more confusing than convincing.
> > >
> > > Anyway, more than the paper's "technical" compilation I would like to
> > share its content, the theoretical premises and some conceptual issues.
> > >
> > > G.R.I.M. - Università di Palermo (2009)
> > > http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
> > >
> > > Your thoughts and comments are welcome.
> > >
> > > Thank you, a.c.
> > >
> > > PS: Sorry for my English.
> > >
> > > Here you find a digital compareson with 12 root of two ET:
> > >
> >
http://www.chas.it/index.php?option=com_content&view=article&id=46&Itemid=17&lang=en
> > >
> > > And five samples on real pianos:
> > >
> >
http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en
> > >
> >
> >
> >
>

🔗Carl Lumma <carl@...>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

Yes, also in my view Chas is a model "that can make different tunings depending on a parameter". Also what you are saying is true, I do highlight one theoretical "solution", namely an s = 1 logarithmic scale. And I say (only) "theoretical" because delta-wedges in 1:1 proportion can represent (numerically) the maximum (absolute?) degree of coherence, although when we think of tuning "practice" we have to be able to deal with other dynamic factors. In fact, the latter have suggested an "elastic" (s) parameter.

You wrote:..."Any equal tuning can be approximately described by its step size in cents. With 12-ET, it is 100 cents. Sometimes the step size is also a root of a rational number. For 12-ET, it is the 12th root of 2."...

Yes, I understand what you are saying and I don't see how I could not agree, but I'm not understanding why you have mentioned that. 12th root of 2 was "designed" under the influence of some precise age-old believes, for instance that octaves should be 2:1, or that it was impossible to combine all ratios (primes) in one scale. I believe that to be the reason why we could describe 12th root of 2 only as a "compromise" (meaning " getting the best from a bad job").

Perhaps we would still be able to describe Chas as a "compromise" but... Do you think it is mathematically wrong highlighting an optimum whole? This question of mine is not rhetorical, I'm really wondering.

Alfredo

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Alfredo,
>
> I understand you have a model that can make different
> tunings depending on a parameter.
>
> But from your website, it seems you are also promoting
> one solution of that model? Is this correct? If so,
> what is that tuning?
>
> Any equal tuning can be approximately described by its
> step size in cents. With 12-ET, it is 100 cents.
> Sometimes the step size is also a root of a rational
> number. For 12-ET, it is the 12th root of 2.
>
> -Carl
>
> --- In tuning@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> > >
> > > 12-ET may be described as the 12th root of 2 equal-step tuning.
> > >
> > > Is there a similar description for Chas?
> >
> > Hi Carl,
> >
> > I'm not sure, perhaps 12 ET may be described as one (of infinite) particular case (s = 1) "outside" 1:1 wedges; in fact we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.
> >
> > Or... were you on something else?
> >
> > Alfredo
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > On Fri, Jun 22, 2012 at 1:33 PM, Kees van Prooijen <keesvp@> wrote:
> > > >
> > > > Hey guys, I directed Alfredo to the group, and promised him an "active and responsive audience". Please don't make me look stupid :-)
> > >
> > > The conversation's taking place over on tuning-math.
> > >
> > > -Mike
> > >
> >
>

🔗Carl Lumma <carl@...>

🔗alfredo.capurso <alfredo.capurso@...>

🔗Carl Lumma <carl@...>

🔗alfredo.capurso <alfredo.capurso@...>

🔗Carl Lumma <carl@...>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

Perhaps this sounds strange, but I quite agree with you.

I read: ..."... my conclusion is that pure ratios (and their approximations) do translate into increased euphony."...

This is also my view. Pure ratios ((giving 1:1 (s = 1) the highest (purest / theoretical / traditional) rank)) work in the background and can determine deviations from pure ratios; their approximations, say the outcome, define and qualify the foreground, i.e. the scale frequency (irrational) values. The former, pure ratios (in the background), meaning 1:1, 2:1, 3:1, and 4:1 are combined together. Their 'combination' results in... the ratio 4:3, as (4/delta) / (3/delta) is again 4:3.

All pure-ratio@approximations translate into the scale irrational values, every irrational being double connected (proportion wise) with both, the logarithmic scale and the (pure-ratios) background. Pure ratios can as well modulate, in force of an all-arbitrary (s/s1) variable, which stands for "control".

You wrote:..."Regarding the second point, if it pertains to frequency differences (and therefore beat rates), I think they are of much less significance in regular musical situations. At any
rate, one would be hard pressed to distinguish a piano with 0.5 cents/octave stretch from one tuned in perfect 12-ET. Variance in different tuning stretch curves, and even the
error of the tuning hammer, are greater than this."...

A piano, we know, cannot be tuned in "perfect 12-ET". And it seems that singers too are used to stretch intervals.

So, the close lines are: ..."pure ratios (and their approximations) do translate into increased euphony."

Yes, in Chas model they form the "background".

..."Regarding the second point, if it pertains to frequency differences (and therefore beat rates), I think they are of much less significance in regular musical situations."...

Yes, "less significance", one more reason for saying that both pure ratios and their (foreground) "s" approximations do translate into increased euphony.

Let me know about the algorithm, if something doesn't make sense.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Alfredo,
>
> > Yes..., and much less than square root 9/8 (if we consider s = 1).
> >
> > In fact, the theoretical approach to the whole question is
> > quite "different". Perhaps you could define the context of our
> > comparisons?
>
> Well, I'm still not sure the derivation of your formula,
> but I don't agree with two of the premises you posted on
> the tuning-math list:
>
> > One premise is that pure ratios do not necessarely translate
> > into increased euphony.
> >
> > A second premise is that the scale frequency values must be
> > "coherent" with differences-from-pure-ratios values.
>
> Such matters have been extensively discussed on these lists
> over the years, and my conclusion is that pure ratios (and
> their approximations) do translate into increased euphony.
> Regarding the second point, if it pertains to frequency
> differences (and therefore beat rates), I think they are of
> much less significance in regular musical situations. At any
> rate, one would be hard pressed to distinguish a piano with
> 0.5 cents/octave stretch from one tuned in perfect 12-ET.
> Variance in different tuning stretch curves, and even the
> error of the tuning hammer, are greater than this.
>
> -Carl
>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

Perhaps this sounds strange, but I quite agree with you.

I read: ..."... my conclusion is that pure ratios (and their approximations) do translate into increased euphony."...

A piano, we know, cannot be tuned in "perfect 12-ET". And it seems that singers too are used to stretch intervals.

So, the close lines are: ..."pure ratios (and their approximations) do translate into increased euphony."

Yes, in Chas model they form the "background".

..."Regarding the second point, if it pertains to frequency differences (and therefore beat rates), I think they are of much less significance in regular musical situations."...

Yes, "less significance", one more reason for saying that both pure ratios and their (foreground) "s" approximations do translate into increased euphony.

Let me know about the algorithm, if something doesn't make sense.

Best wishes,

Alfredo

🔗alfredo.capurso <alfredo.capurso@...>

Thanks Andy.

Hi Carl,

You wrote: ..."I'm still not sure the derivation of your formula"...

In order to represent how pure ratios and delta-deviation-wedges work in the "background", I could take John O'Sullivan's tuning as a starting point and consider John's scale first-step ratio 15/14 = 1,07142857142857... This step-ratio too can be extracted from Chas algorithm.

15/14 = 1,07142857142857

Chas = (3 - delta)^(1/19) = (4 + (s * delta))^(1/24) Eq. (1)

say that (John's arbitrary) s = -1,7443693397446

Eq. (1) becomes:

(3 - delta)^(1/19) = (4 + (-1,7443693397446 * delta))^(1/24)

Delta = -0,709379301

(3 - (-0,709379301))^(1/19) = (4 + (-1,7443693397446 * -0,709379301))^(1/24) = 1,07142857142857 = 15/14

And the case:

s = -1,7443693397446 in fraction s/s1 = 1/-0,5732730891420

Chas = (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)

(3 - (-0,5732730891420 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)

Delta' = 1,2374195025407

(3 - (-0,573273089142 * 1,2374195025407))^(1/19) = (4 + (1 * 1,2374195025407))^(1/24) = 1,07142857142857 = 15/14

If we used that incremental ratio (15/14) for drawing a logarithmic scale (instead of a single scale-step), we would obviously find delta in position 19 and delta' in position 24.

Comment:
That is John's arbitrary "s" value, and by using John's "s" value we see that 1:1 wedge-delta-ratio, together with 2:1 (2*2:1), 3:1 and 4:1 are still effective. If we compare this case with the Chas s = 1 (the case highlighted), we see that here we loose contact with 4:3 (2*2/3): in fact (4/delta')/(3/delta) does not equal 1.333...

This is to say that in a logarithmic scale, the scale values relative to the ratios 3:1 and 4:1 can always be represented in terms of relation with their delta-deviation-wedges; and we can see that only in the case "s = 1" (in Eq. (1)) the unity enables to preserve also the 4:3 ratio.

I hope you can understand why I believe that Chas model enables to represent a "pot" entity, meaning that all ratios relations seem to be contained there, and why, also referring to John's tuning, I still cannot find a "rational" justification for its 2:1 octave.

I know that my English is limited, please do not hesitate to point out your issues.

Best wishes,

Alfredo

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

It is hard to say but I believe that, in the history of temperaments, the "point" (read "deviations") was understood as the problem.

In other words, in the whole history of temperament we have tried to zero "deviations" from pure ratios by... managing pure ratio in the "foreground", i.e. in-between the scale values. Scale frequencies have then been "sliced" in all sort of ways, (and still are today?) in the idea that "pure" intervals would return "consonance", and in the idea that only consonance would equal euphony and/or harmony.

Referring to Chas model, I wrote: ..."One premise is that "pure ratios" do not necessarily translate into increased euphony. A second premise is that the scale frequency values must be "coherent" with differences-from-pure-ratios values."

You wrote:..."Regarding the second point, if it pertains to frequency differences (and therefore beat rates), I think they are of much less significance in regular musical situations. At any rate, one would be hard pressed to distinguish a piano with 0.5 cents/octave stretch from one tuned in perfect 12-ET. Variance in different tuning stretch curves, and even the error of the tuning hammer, are greater than this."

I think that your "empirical" observation, referred to "regular musical situations", is confirming both premises of mine. You also mentioned 7th root of 3/2 (S. Cordier pure fifths idea) and 19th root of 3 (B. Stopper pure 12ths idea) and perhaps it is worth underlining that actually these two ratios (3:2 and 3:1) do not "meet" in a logarithmic scale: in fact pure 3:2 pushes 12ths above 3:1; likewise 3:1 pushes fifths below 3:2. So, again, both solutions end up favoring one pure interval to the detriment of all other intervals!

In all sincerity, I would like to understand if, in your opinion, there was room for improving our current ET model, i.e. 12 root of 2.

Based on what you have written, I should believe that you too are open to theoretical "stretched" octaves, but I do not understand if you would still attempt to a "one-pure-ratio scale" or if you are still preferring the first ET model. Nor I understand if today you believe that tempering 12 notes (out of 88) is enough to achieve a "sound whole".

And I have one question: did my example below demonstrate the meaning of the "s" variable?

I'm interested in your comments and I look forward to knowing your thoughts.

Best regards,

Alfredo

--- In tuning@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...> wrote:
>
> Thanks Andy.
>
> Hi Carl,
>
> You wrote: ..."I'm still not sure the derivation of your formula"...
>
> In order to represent how pure ratios and delta-deviation-wedges work in the "background", I could take John O'Sullivan's tuning as a starting point and consider John's scale first-step ratio 15/14 = 1,07142857142857... This step-ratio too can be extracted from Chas algorithm.
>
> 15/14 = 1,07142857142857
>
> Chas = (3 - delta)^(1/19) = (4 + (s * delta))^(1/24) Eq. (1)
>
> say that (John's arbitrary) s = -1,7443693397446
>
> Eq. (1) becomes:
>
> (3 - delta)^(1/19) = (4 + (-1,7443693397446 * delta))^(1/24)
>
> Delta = -0,709379301
>
> (3 - (-0,709379301))^(1/19) = (4 + (-1,7443693397446 * -0,709379301))^(1/24) = 1,07142857142857 = 15/14
>
> And the case:
>
> s = -1,7443693397446 in fraction s/s1 = 1/-0,5732730891420
>
> Chas = (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)
>
> (3 - (-0,5732730891420 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)
>
> Delta' = 1,2374195025407
>
> (3 - (-0,573273089142 * 1,2374195025407))^(1/19) = (4 + (1 * 1,2374195025407))^(1/24) = 1,07142857142857 = 15/14
>
> If we used that incremental ratio (15/14) for drawing a logarithmic scale (instead of a single scale-step), we would obviously find delta in position 19 and delta' in position 24.
>
> Comment:
> That is John's arbitrary "s" value, and by using John's "s" value we see that 1:1 wedge-delta-ratio, together with 2:1 (2*2:1), 3:1 and 4:1 are still effective. If we compare this case with the Chas s = 1 (the case highlighted), we see that here we loose contact with 4:3 (2*2/3): in fact (4/delta')/(3/delta) does not equal 1.333...
>
> This is to say that in a logarithmic scale, the scale values relative to the ratios 3:1 and 4:1 can always be represented in terms of relation with their delta-deviation-wedges; and we can see that only in the case "s = 1" (in Eq. (1)) the unity enables to preserve also the 4:3 ratio.
>
> I hope you can understand why I believe that Chas model enables to represent a "pot" entity, meaning that all ratios relations seem to be contained there, and why, also referring to John's tuning, I still cannot find a "rational" justification for its 2:1 octave.
>
> I know that my English is limited, please do not hesitate to point out your issues.
>
> Best wishes,
>
> Alfredo
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Hi Alfredo,
> >
> > > Yes..., and much less than square root 9/8 (if we consider s = 1).
> > >
> > > In fact, the theoretical approach to the whole question is
> > > quite "different". Perhaps you could define the context of our
> > > comparisons?
> >
> > Well, I'm still not sure the derivation of your formula,
> > but I don't agree with two of the premises you posted on
> > the tuning-math list:
> >
> > > One premise is that pure ratios do not necessarely translate
> > > into increased euphony.
> > >
> > > A second premise is that the scale frequency values must be
> > > "coherent" with differences-from-pure-ratios values.
> >
> > Such matters have been extensively discussed on these lists
> > over the years, and my conclusion is that pure ratios (and
> > their approximations) do translate into increased euphony.
> > Regarding the second point, if it pertains to frequency
> > differences (and therefore beat rates), I think they are of
> > much less significance in regular musical situations. At any
> > rate, one would be hard pressed to distinguish a piano with
> > 0.5 cents/octave stretch from one tuned in perfect 12-ET.
> > Variance in different tuning stretch curves, and even the
> > error of the tuning hammer, are greater than this.
> >
> > -Carl
> >
>

🔗Carl Lumma <carl@...>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl, thank you for your reply and the information provided.

You wrote: "Yes. These two rational numbers involve two primes: 2 and 3. By the, there is no interval that divides them both."...

What you say is correct, and considering the fundamental theorem of arithmetic, I would then ask: is there a point in trying to calculate the interval R by ignoring one of the primes (2, 3 and 5) that stand for octaves, fifths and thirds?

We all know about the "compromise" achieved with 12 root of two: the "pure" octave is the one postulate at the base of that model, and commas deriving from 3:2 and 5:4 are "spread" onto 12 semitones. Now, we might also have to acknowledge that 2:1 ratio, outside the octave-span, causes the progressive doubling of the scale differences for 2nds and fifths! Using your words, "in regular musical situations" the pure 2:1 ratio causes a dramatic "divarication" of all intervals deviations, for fifths, 12ths, 17ths, 19ths and so on, and this may explain why, in my view (but not only), 12 root of two was a lame-born unfeasible model. All this... only because we were not able to depart from the idea of one pure interval, nor could we think that the octave single-module needed to be inter-modulated with all the other octaves, into a "whole".

Now, perhaps we may agree on this point: focusing onto the scale values (and ratios), nobody could ever manage to "fuse" those "prime" ratios in One single incremental ratio, in other words nobody could obtain a scale ratio able to represent those primes "combined" action.

In my view, that happened simply because nobody could elaborate on a scale "deviation" ratio, which is merely what Chas model adopts and describes. I hope it is now clear how the Chas "deviation" scale incremental ratio is the result of pure ratios (1:1, 2:1, 3:1 and 4:1) combined together and "adjusted" by the (s = 1) unity. In fact, in this way both the scale values (the foreground) and the "deviation" values (the background) are most strictly interrelated.

> In all sincerity, I would like to understand if, in your
> opinion, there was room for improving our current ET model,
> i.e. 12 root of 2.

You wrote: ..."It depends if we are allowed to use more than 12 notes per octave! With 31 notes we can do very well indeed! If we are restricted to 12 notes, there is not much better than 12-ET."...

Well, I respect your outlook but, if it isn't for now, I hope to be able to better support my theses, that obviously is not the same as yours. In fact, also with Huygens 31-notes model we would suffer from the doubled-deviation effect of 2:1, and that obviously applies also to 24-TET, what now Chas represents with: (3 - delta)^(1/38) = (4 + (s * delta))^(1/48).

..."For music that only visits a few keys, like Baroque music, meantone and Baroque temperaments can do better. But if we want to play in all keys, 12-ET is very good."...

Hmm... Perhaps I can point out that Chas model is not based on 12 notes anymore, in fact it is not limited to a certain number of notes; intervals within the octave are not anymore "copied" outside the octave; what happens is that every single interval has its own meaning and character, in terms of "deviation" from pure ratios. Chas sound "whole" is expandable to infinite, although its period is 19*24 = 456.

..."There is a theory that says we can do better by *shrinking* the octave, down to 1198 cents. For music with many dominant 7th chords (e.g. blues music), the octave would shrink even more, to about 1196 cents."...

Hmm... if that's a logarithmic scale, we then get narrower fifths and octaves... Numbers and minimized errors may seem to get better, but I still wonder what would happen beyond a single octave, on chromatic fifths, 12ths and so on. This theory seem to move from the same old axiom, namely reduced-deviations, but in my view we needed to "harmonize" deviations. Do you know what I mean?

..."This theory is called "Tenney-weighted optimal tuning". It is the tuning which minimizes the weighted error among of primes we are interested in. For triads, that is primes 2, 3, and 5. If we also want dominant 7th chords, it is primes 2, 3, 5, and 7. The "weighted" error allows greater deviation on big primes like 7 than on small ones like 2. Long ago, Werckmeister noted that the ear tolerates more deviation from major thirds (prime 5) than from octaves or fifths (primes 2 and 3)."...

Hmm... perhaps I'm wrong but, as mentioned, a "shrunk" octave, in a logarithmic scale, would not help fifths nor intervals outside the octave.

Chas model, in its basic form (s = 1), does not consider any individual interest. Actually, I would appreciate if you could tell me about one arbitrary element in Chas basic algorithm: (3 - delta)^(1/19) = (4 + delta)^(1/24).

..."The weighting used, suggested by James Tenney, is the base 2 logarithm of the prime under consideration. Thus 1 cent deviation on the octave (1/log2(2) = 1) is judged equal to 1.58 cents deviation on the twelfth (1.58/log2(3) = 1). Finding the optimal step size for any list of primes is fairly easy using this method. I've done some experiments with these tunings on an upright piano, and will be trying on a grand piano soon. But to be honest, any improvement over 12-ET is likely to be small."...

Yes, perhaps it all depends on what we consider to be "optimal". For me optimal, referred to theory and modeling, means:

0 - a theory that is not based on arbitrary premises;
1 - a model that can "manage" all arbitrary deviations, starting from a Non-Arbitrary form;
2 - that is able to strictly relate (proportionally) scale values, all intervals and deviations;
3 - that is able to combine, shrink and/or stretch all ratios, no matter individual preferences;
4 - that is actually feasible in practice;
5 - that is able to represent our reality in dynamic terms.

How about you, what does "optimal" mean for you?

Best regards,

Alfredo

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Alfredo,
>
> > I think that your "empirical" observation, referred to "regular
> > musical situations", is confirming both premises of mine. You
> > also mentioned 7th root of 3/2 (S. Cordier pure fifths idea) and
> > 19th root of 3 (B. Stopper pure 12ths idea) and perhaps it is
> > worth underlining that actually these two ratios (3:2 and 3:1)
> > do not "meet" in a logarithmic scale: in fact pure 3:2 pushes
> > 12ths above 3:1; likewise 3:1 pushes fifths below 3:2. So,
> > again, both solutions end up favoring one pure interval to the
> > detriment of all other intervals!
>
> Yes. These two rational numbers involve two primes: 2 and 3.
> By the fundamental theorem of arithmetic, there is no interval
> that divides them both.
>
> > In all sincerity, I would like to understand if, in your
> > opinion, there was room for improving our current ET model,
> > i.e. 12 root of 2.
>
> It depends if we are allowed to use more than 12 notes
> per octave! With 31 notes we can do very well indeed!
>
> If we are restricted to 12 notes, there is not much better
> than 12-ET.
>
> For music that only visits a few keys, like Baroque music,
> meantone and Baroque temperaments can do better. But if we
> want to play in all keys, 12-ET is very good.
>
> There is a theory that says we can do better by *shrinking*
> the octave, down to 1198 cents. For music with many
> dominant 7th chords (e.g. blues music), the octave would
> shrink even more, to about 1196 cents.
>
> This theory is called "Tenney-weighted optimal tuning".
> It is the tuning which minimizes the weighted error among
> of primes we are interested in. For triads, that is primes
> 2, 3, and 5. If we also want dominant 7th chords, it is
> primes 2, 3, 5, and 7. The "weighted" error allows greater
> deviation on big primes like 7 than on small ones like 2.
> Long ago, Werckmeister noted that the ear tolerates more
> deviation from major thirds (prime 5) than from octaves or
> fifths (primes 2 and 3).
>
> The weighting used, suggested by James Tenney, is the base 2
> logarithm of the prime under consideration. Thus 1 cent
> deviation on the octave (1/log2(2) = 1) is judged equal to
> 1.58 cents deviation on the twelfth (1.58/log2(3) = 1).
> Finding the optimal step size for any list of primes is
> fairly easy using this method.
>
> I've done some experiments with these tunings on an upright
> piano, and will be trying on a grand piano soon. But to be
> honest, any improvement over 12-ET is likely to be small.
>
> -Carl
>

🔗Bogdan <baros_ilogic@...>

> In order to represent how pure ratios and delta-deviation-wedges work in the "background", I could take John O'Sullivan's tuning as a starting point and consider John's scale first-step ratio 15/14 = 1,07142857142857... This step-ratio too can be extracted from Chas algorithm.
>
> 15/14 = 1,07142857142857
>
> Chas = (3 - delta)^(1/19) = (4 + (s * delta))^(1/24) Eq. (1)
>
> say that (John's arbitrary) s = -1,7443693397446

Where did that s came from? "John's arbitrary" means that:

a) this is a value that John chose for this step (even though his tuning was not intended as a CHAS tuning, having in mind that CHAS can be applied to any interval, the s value is the only acceptabale value for John's first-step ratio)

b) it is an arbitrary value chosen by you to demonstrate the flexibility of CHAS

c) none of the 2 above.

If I understood the logic behind the algorithm right, CHAS is used to generate one elastic step in both directions (relative to the base note), for any given interval. In this example, you used the interval between 3/1 and 4/1 and calculated John's first-step ratio (which now has become the only interval between any given note and the next one in our scale) based on s.

I feel like I am close to figuring something out but right now I might be looking straight at it, without seeing it. I'm reading over and over your explanation from below, I understand the words but I am missing the point.

> Eq. (1) becomes:
>
> (3 - delta)^(1/19) = (4 + (-1,7443693397446 * delta))^(1/24)
>
> Delta = -0,709379301
>
> (3 - (-0,709379301))^(1/19) = (4 + (-1,7443693397446 * -0,709379301))^(1/24) = 1,07142857142857 = 15/14
>
> And the case:
>
> s = -1,7443693397446 in fraction s/s1 = 1/-0,5732730891420
>
> Chas = (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)
>
> (3 - (-0,5732730891420 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)
>
> Delta' = 1,2374195025407
>
> (3 - (-0,573273089142 * 1,2374195025407))^(1/19) = (4 + (1 * 1,2374195025407))^(1/24) = 1,07142857142857 = 15/14
>
> If we used that incremental ratio (15/14) for drawing a logarithmic scale (instead of a single scale-step), we would obviously find delta in position 19 and delta' in position 24.
>
> Comment:
> That is John's arbitrary "s" value, and by using John's "s" value we see that 1:1 wedge-delta-ratio, together with 2:1 (2*2:1), 3:1 and 4:1 are still effective. If we compare this case with the Chas s = 1 (the case highlighted), we see that here we loose contact with 4:3 (2*2/3): in fact (4/delta')/(3/delta) does not equal 1.333...
>
> This is to say that in a logarithmic scale, the scale values relative to the ratios 3:1 and 4:1 can always be represented in terms of relation with their delta-deviation-wedges; and we can see that only in the case "s = 1" (in Eq. (1)) the unity enables to preserve also the 4:3 ratio.
>
> I hope you can understand why I believe that Chas model enables to represent a "pot" entity, meaning that all ratios relations seem to be contained there, and why, also referring to John's tuning, I still cannot find a "rational" justification for its 2:1 octave.
>
> I know that my English is limited, please do not hesitate to point out your issues.
>
> Best wishes,
>
> Alfredo

🔗Carl Lumma <carl@...>

Hi Alfredo,

> I would then ask: is there a point in trying to calculate the
> interval R by ignoring one of the primes (2, 3 and 5) that
> stand for octaves, fifths and thirds?

You can pick any primes you like. Your choice should reflect
the music you want to play. If you are playing triads, you
should choose three primes: 2, 3, 5. Some 20th century music
(e.g. Russian piano school) may prefer only primes 2 and 3,
whereas American blues music would prefer 2, 3, 5, 7.

> We all know about the "compromise" achieved with 12 root
> of two: the "pure" octave is the one postulate at the base
> of that model,

It is true that the idea of tempering the octave was not
seriously considered until recent decades. But 12-ET is
close to optimal anyway for primes, 2, 3 and 5, and as you
discovered, even closer for primes 2 and 3.

>> ..."There is a theory that says we can do better by *shrinking*
>> the octave, down to 1198 cents. For music with many dominant 7th
>> chords (e.g. blues music), the octave would shrink even more, to
>> about 1196 cents."...
>
> Hmm... if that's a logarithmic scale,

Around here we would say, it is an "equal division" or
"rank 1 temperament".

> we then get narrower fifths and octaves... Numbers and minimized
> errors may seem to get better, but I still wonder what would
> happen beyond a single octave,

Ratios of 5 continue to improve, e.g. 5/2 is better than 5/4
and 10/3 better than 5/3.

> Chas model, in its basic form (s = 1), does not consider any
> individual interest. Actually, I would appreciate if you could
> tell me about one arbitrary element in Chas basic algorithm:
> (3 - delta)^(1/19) = (4 + delta)^(1/24).

It is a special case of the algorithm I described, with no
weighting for the primes, and considering only 'primes'
3 & 4. To include weighting, we would get

((3 - (delta/log 3))^(1/19) = (4 + (delta/log 4))^(1/24)

This is also equivalent to

((3 - (delta/log 3))^(1/19) = (2 + (delta/log 2))^(1/12)

To add prime 5, we can no longer use a single error term
"delta". Rather, we minimize the total squared error of
three terms.

As you already know, to evaluate systems other than 12/oct,
we change the exponents. To evaluate 31/oct, we need

((3 - (delta/log 3))^(1/49) = (2 + (delta/log 2))^(1/31)

> Yes, perhaps it all depends on what we consider to be
> "optimal". For me optimal, referred to theory and modeling,
> means:
> 0 - a theory that is not based on arbitrary premises;
> 1 - a model that can "manage" all arbitrary deviations,
> starting from a Non-Arbitrary form;
> 2 - that is able to strictly relate (proportionally) scale
> values, all intervals and deviations;
> 3 - that is able to combine, shrink and/or stretch all
> ratios, no matter individual preferences;
> 4 - that is actually feasible in practice;
> 5 - that is able to represent our reality in dynamic terms.

I think you would like the theory I'm referring to.
Here is more information:
http://lumma.org/tuning/erlich/
http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning

-Carl

🔗alfredo.capurso <alfredo.capurso@...>

Thank you, Bogdan, for your feedback. By reading your post I believe you are well getting the point. The way I see it, in general, it is a very precise point that opens to infinite.

...a) this is a value that John chose for this step (even though his tuning was not
intended as a CHAS tuning, having in mind that CHAS can be applied to any
interval, the s value is the only acceptabale value for John's first-step ratio)..."...

Yes, Chas algorithm can target and produce any "chosen" incremental ratio, referable to either a single step or to the entire (logarithmic) scale. And yes, if we consider Chas basic algorithm and use 3:1 and 4:1, 19 and 24 as two scale ratios and their respective scale ordinal positions, then the only "acceptable" "s" value is precisely that one (without considering the ± signs).

Of course, if we considered only the algorithm' mere structure and say we wanted to use two other scale ratios and/or other ordinal positions, we could still calculate a (different) "s" value and gain John's 15/14 ratio... Let me know if you would like a numerical example, I'll be happy to provide it.

...b) it is an arbitrary value chosen by you to demonstrate the flexibility of CHAS..."...

Yes, in a way... It is a value that can represent John's "arbitrary" choice (15/14), and be translated into delta wedge-deviation (s*delta) value, and that I've used to demonstrate the complete flexibility of Chas.

...c) none of the 2 above."...

Well, it seems more like... d)... ;-) ... in fact you are quite there.

..."If I understood the logic behind the algorithm right, CHAS is used to generate one elastic step in both directions (relative to the base note), for any given interval. In this example, you used the interval between 3/1 and 4/1 and calculated John's first-step ratio (which now has become the only interval between any given note and the next one in our scale) based on s."...

Yes, Chas "s" variable enables to generate ratios either for single steps or for an entire (logarithmic) scale, say in "elastic" terms, for any given interval. And yes, I ..."used the interval between 3/1 and 4/1 and calculated John's first-step ratio".

If that step-ratio can (in any way) ..."become the only interval between any given note and the next one in our scale..." that depends only on the actor's choice.

..."I feel like I am close to figuring something out but right now I might be looking straight at it, without seeing it. I'm reading over and over your explanation from below, I understand the words but I am missing the point."...

I might be wrong, but I think that you only need to know about a couple of things. For instance, this is your question now: "Where did that s came from?"

The "s" value can be "translated" (just for demonstrative purpose) in this way (now going backwards):

- consider one ratio (I considered John's case 15/14 = 1,07142857142857 (periodic))
- calculate the log. scale values
- number the scale ordinal positions starting from zero (upwards)
- consider the scale values in position 19 and 24 (John's respectively 3,709379301 and 5,237419503)
- calculate the deviations from 3 and 4 (opting for the correct ± sign: -0,709379301 and 1,2374195025407)
- calculate the quotient: -0,709379301 / 1,2374195025407 = -0,5732730891420
- calculate the inverse of that quotient: 1/-0,5732730891420 = -1,7443693397446

That's all:

(3 - delta)^(1/19) = (4 + (-1,7443693397446 * delta))^(1/24)

Delta = -0,709379301

(3 - (-0,709379301))^(1/19) = (4 + (-1,7443693397446 * -0,709379301))^(1/24) = 1,07142857142857 = 15/14

And the case:

s = -1,7443693397446 in fraction s/s1 = 1/-0,5732730891420

Chas = (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)

(3 - ( -0,5732730891420 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)

Delta' = 1,2374195025407

(3 - ( -0,573273089142 * 1,2374195025407))^(1/19) = (4 + (1 * 1,2374195025407))^(1/24) = 1,07142857142857 = 15/14

Now you may be able to extract Chas ET ratio too, by using also other partial values (other ratios, not necessarily 3 and 4) and two relative ordinal position values (not necessarily 19 and 24).

So, when you are ready, and if you wish, I'll be happy to expand on: why the highlighted basic form, why partial 3 and 4.

Best wishes,

Alfredo

--- In tuning@yahoogroups.com, "Bogdan" <baros_ilogic@...> wrote:
>
> > In order to represent how pure ratios and delta-deviation-wedges work in the "background", I could take John O'Sullivan's tuning as a starting point and consider John's scale first-step ratio 15/14 = 1,07142857142857... This step-ratio too can be extracted from Chas algorithm.
> >
> > 15/14 = 1,07142857142857
> >
> > Chas = (3 - delta)^(1/19) = (4 + (s * delta))^(1/24) Eq. (1)
> >
> > say that (John's arbitrary) s = -1,7443693397446
>
> Where did that s came from? "John's arbitrary" means that:
>
> a) this is a value that John chose for this step (even though his tuning was not intended as a CHAS tuning, having in mind that CHAS can be applied to any interval, the s value is the only acceptabale value for John's first-step ratio)
>
> b) it is an arbitrary value chosen by you to demonstrate the flexibility of CHAS
>
> c) none of the 2 above.
>
> If I understood the logic behind the algorithm right, CHAS is used to generate one elastic step in both directions (relative to the base note), for any given interval. In this example, you used the interval between 3/1 and 4/1 and calculated John's first-step ratio (which now has become the only interval between any given note and the next one in our scale) based on s.
>
> I feel like I am close to figuring something out but right now I might be looking straight at it, without seeing it. I'm reading over and over your explanation from below, I understand the words but I am missing the point.
>
>
> > Eq. (1) becomes:
> >
> > (3 - delta)^(1/19) = (4 + (-1,7443693397446 * delta))^(1/24)
> >
> > Delta = -0,709379301
> >
> > (3 - (-0,709379301))^(1/19) = (4 + (-1,7443693397446 * -0,709379301))^(1/24) = 1,07142857142857 = 15/14
> >
> > And the case:
> >
> > s = -1,7443693397446 in fraction s/s1 = 1/-0,5732730891420
> >
> > Chas = (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)
> >
> > (3 - (-0,5732730891420 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)
> >
> > Delta' = 1,2374195025407
> >
> > (3 - (-0,573273089142 * 1,2374195025407))^(1/19) = (4 + (1 * 1,2374195025407))^(1/24) = 1,07142857142857 = 15/14
> >
> > If we used that incremental ratio (15/14) for drawing a logarithmic scale (instead of a single scale-step), we would obviously find delta in position 19 and delta' in position 24.
> >
> > Comment:
> > That is John's arbitrary "s" value, and by using John's "s" value we see that 1:1 wedge-delta-ratio, together with 2:1 (2*2:1), 3:1 and 4:1 are still effective. If we compare this case with the Chas s = 1 (the case highlighted), we see that here we loose contact with 4:3 (2*2/3): in fact (4/delta')/(3/delta) does not equal 1.333...
> >
> > This is to say that in a logarithmic scale, the scale values relative to the ratios 3:1 and 4:1 can always be represented in terms of relation with their delta-deviation-wedges; and we can see that only in the case "s = 1" (in Eq. (1)) the unity enables to preserve also the 4:3 ratio.
> >
> > I hope you can understand why I believe that Chas model enables to represent a "pot" entity, meaning that all ratios relations seem to be contained there, and why, also referring to John's tuning, I still cannot find a "rational" justification for its 2:1 octave.
> >
> > I know that my English is limited, please do not hesitate to point out your issues.
> >
> > Best wishes,
> >
> > Alfredo
>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

I'm so glad we can exchange our interpretation of temperament in general and Chas model in particular.

Thank you very much for those links, I'll read that material and reply to your post asap.

..."It is a special case of the algorithm I described, with no
weighting for the primes, and considering only 'primes'
3 & 4."...

I think we are going in the right direction, in fact Chas model describes also one special case, together with infinite other "special" cases. And I hope you too like that.

Let me know about the algorithm you described, I look forward to acknowledging your numerical results and share our outlooks.

I still believe that a "shrunk" octave does not help chromatic fifths and octaves, 12ths and double octaves and so on, although it is obvious that "Ratios of 5" are improved. I'm sure we will have more opportunities to deepen on "favored" ratios and the tempering of a "whole".

I'm going to be traveling for some days, I hope to be back here soon.

Best wishes,

Alfredo

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Alfredo,
>
> > I would then ask: is there a point in trying to calculate the
> > interval R by ignoring one of the primes (2, 3 and 5) that
> > stand for octaves, fifths and thirds?
>
> You can pick any primes you like. Your choice should reflect
> the music you want to play. If you are playing triads, you
> should choose three primes: 2, 3, 5. Some 20th century music
> (e.g. Russian piano school) may prefer only primes 2 and 3,
> whereas American blues music would prefer 2, 3, 5, 7.
>
> > We all know about the "compromise" achieved with 12 root
> > of two: the "pure" octave is the one postulate at the base
> > of that model,
>
> It is true that the idea of tempering the octave was not
> seriously considered until recent decades. But 12-ET is
> close to optimal anyway for primes, 2, 3 and 5, and as you
> discovered, even closer for primes 2 and 3.
>
> >> ..."There is a theory that says we can do better by *shrinking*
> >> the octave, down to 1198 cents. For music with many dominant 7th
> >> chords (e.g. blues music), the octave would shrink even more, to
> >> about 1196 cents."...
> >
> > Hmm... if that's a logarithmic scale,
>
> Around here we would say, it is an "equal division" or
> "rank 1 temperament".
>
> > we then get narrower fifths and octaves... Numbers and minimized
> > errors may seem to get better, but I still wonder what would
> > happen beyond a single octave,
>
> Ratios of 5 continue to improve, e.g. 5/2 is better than 5/4
> and 10/3 better than 5/3.
>
> > Chas model, in its basic form (s = 1), does not consider any
> > individual interest. Actually, I would appreciate if you could
> > tell me about one arbitrary element in Chas basic algorithm:
> > (3 - delta)^(1/19) = (4 + delta)^(1/24).
>
> It is a special case of the algorithm I described, with no
> weighting for the primes, and considering only 'primes'
> 3 & 4. To include weighting, we would get
>
> ((3 - (delta/log 3))^(1/19) = (4 + (delta/log 4))^(1/24)
>
> This is also equivalent to
>
> ((3 - (delta/log 3))^(1/19) = (2 + (delta/log 2))^(1/12)
>
> To add prime 5, we can no longer use a single error term
> "delta". Rather, we minimize the total squared error of
> three terms.
>
> As you already know, to evaluate systems other than 12/oct,
> we change the exponents. To evaluate 31/oct, we need
>
> ((3 - (delta/log 3))^(1/49) = (2 + (delta/log 2))^(1/31)
>
> > Yes, perhaps it all depends on what we consider to be
> > "optimal". For me optimal, referred to theory and modeling,
> > means:
> > 0 - a theory that is not based on arbitrary premises;
> > 1 - a model that can "manage" all arbitrary deviations,
> > starting from a Non-Arbitrary form;
> > 2 - that is able to strictly relate (proportionally) scale
> > values, all intervals and deviations;
> > 3 - that is able to combine, shrink and/or stretch all
> > ratios, no matter individual preferences;
> > 4 - that is actually feasible in practice;
> > 5 - that is able to represent our reality in dynamic terms.
>
> I think you would like the theory I'm referring to.
> Here is more information:
> http://lumma.org/tuning/erlich/
> http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning
>
> -Carl
>

🔗Carl Lumma <carl@...>

🔗Bogdan <baros_ilogic@...>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

Thank you for your reply.

I've had a look at the material you addressed me to and I found it interesting, those pages actually reassume the works of some other researchers I could read about in the web.

In my view, the fulcrum of those works is (again) "avoidable errors" and "reduced deviations", in the sense that "optimal" seems to be referred to "just" intervals and to those (numerical) solutions that may reduce deviations from pure ratios. The fundamental axiom then seems to be (again) the equality "pure = consonance = euphony", fairly in line with the centuries-old approach to the ordering of a scale. And I'm sure we all know about the large amount of solutions (and the wide variety of number of notes per octave) that have been proposed over the centuries, looking at "just" intervals as the main target and trying to manage primes and all commas.

Perhaps here I ought to expand on some concepts that may enable to focus on some precise points and help to frame Chas theory and model. Also, I would like to distinguish clearly one special (highlighted) case from the Chas general algorithm and infinite other "s" cases.

Point 1 - Scale and/or step-structure - As mentioned, Chas theory does not limit "per se" the number of steps that (in theory or practice) may be calculated and arbitrarily ordered within an octave, although the highlighted Chas solution (s = 1) calculates indeed one precise incremental ratio. In other words, this approach to the ordering of the scale opens to any scale geometry, which can be drawn either step by step - or - by using one single ratio for the whole (log) scale. In this sense, Chas theory conceives any possible (evolutive) step-or-scale variant.

Point 2 - The highlighted ratio - The highlighted Chas ratio derives from the smallest integers 1, 2, 3 and 4; if used for modeling the whole scale, that k-ratio enumerates (and fits) the actual number (13) of notes-per-octave. We know that having one-pure-ratio scale as the target would give reason to variate indefinitely (arbitrarily) even the amount of steps within an octave; beyond that, the Chas highlighted case, (s = 1), shows a (log) scale that is actually shaped by ratios 1:1, 2:1, 3:1, 4:1 and 4:3, when the "s" variable equals the unity. Am I wrong when I say that this was considered impossible?

Point 3 - Derivation - This is to say that the Chas ET scale derives only from small integers and pure ratios in non-arbitrary terms.

Point 4 - Variety of scales - Well beyond (and yet together with) infinite varieties of step/scale geometries, the Chas model then demonstrates that one and only one perfectly-coherent non-arbitrary scale exponential geometry exists. In this sense, this unique geometry can well be defined as a "special case", no more no less than how regular polyhedrons could be said "special" cases, and I really hope we can share this analogy.

Point 5 - The key - Chas theory considers "deviations" from just (pure) ratios as THE key-phenomenon, namely as the "dynamic" and variable event (here referring to beats) that gives "life" (light/brilliancy/energy) to any interval and to any combination of notes (played simultaneously).

Point 6 - Proportions - In this sense, "deviations-from-pure" are now regarded as quantities (magnitudes?) that can be "harmonized"; and here "harmony" is also referred to perception, i.e. to a sense (translatable also into numerical evidences) that calls for "proportion", "symmetry", "equilibrium" and "invertibility".

Point 7 - The phenomenon - By focusing on "deviations", Chas theory enables to describe "flows of beats" deriving from infinite (chaotic and/or arbitrary) partial combinations.

The above may explain how the historical fulcrum has been shifted and why I decided to carry on my empirical research having proportional beat-curves as the ultimate target.

Point 8 - Scale geometry - Having departed from the historical "pure" (debatable) axiom, all the work on actual deviations (i.e. on beats) and their optimal proportions could be carried out directly on intervals, empirically, in the idea that a one-ratio-sensitive (log) scale could only behave as a system of levers, say an interlaced web, whose exponential geometry could not ignore the interdependency of each and every interval, of each relatable (and to-be-related) partial-ratio.

As we know, 12th root of two favors partial 2, but I could find no logical justification for favoring one single partial and for ordering a (log) scale within such a restricted span. One observation helped to visualize one numerical relation amongst partials: in fact, in our semi-tonal scale, the partial 5's ordinal position is number 28, and this position shows in fact a sub-module of the octave (7*4), meaning that three contiguous thirds C-E, E-G#, G#-C make an octave. This means that by stretching the octave (as in a system of levers) we will also stretch all thirds, i.e. partial 5.

So, while I was empirically searching for chromatic proportional beat-curves, the fundamental question (for me) became: Is there a way to gain a "k-constant" (a scale incremental ratio) simply out of combined ratios? Is there an ideal proportional "stretch", i.e. one precise deviation-degree that can be said to be optimal?

Point 9 - Representation - Then, the idea of ordering deviations "harmonically" into an optimum "whole" system should have implied/involved two (necessary and sufficient) intervals (two ratios). That is, ruling the "degree of deviation" relative to two intervals, in order to obtain an intrinsic (in this sense "pure") ratio. I then waited for those two (deviated) intervals to show out, aiming at one ratio that could stand for deviation values as well as scale values, the latter (the foreground) being the direct representation of a bottom order, namely the (log-deviation) order that pure ratios (and the smallest integers) could themselves establish (in the background).

Of course, the first "equal division" (log) scale based only on partial 2 had its validity and could make sense, if only we set that solution against its historical background; and other more recent ET solutions (7th root of 3/2 and 19 root of two) indeed represent other "possible" (log) scales; and I am aware of the fact that Chas, on the one hand, is highlighting one "special" case that is one of many (admissible) variants. And we know that - not last - Guerino Mazzola too has produced a generic formula that includes all possible functions.

Point 10 - The Chas ET scale constant - On the other hand though, beyond any general representation (including Chas algorithm), I do not think I would have been able to extract (isolate/attain?) the Chas ET scale ratio (1.0594865443501) and that precise delta value (0.0021253899646...) had I not gone through the conceptual route I've partially described above.

Point 11 - The span - It should be clear that the theoretical tempering of 12 semitones is not enough for achieving a coherent "whole", in fact both partial 3 and 5 are positioned outside the octave compass (ordinal 19 and 28) and this, in general, might have suggested to "rule" a wider span. On top of that, as mentioned, the 2:1 octave produces an inconvenient effect onto the scale, in that all narrow interval deviations end up doubling their values every other octave. The Chas highlighted case is not based on one interval, no single ratio is favored, so the tempered span (now) is not limited. All partials are actually tempered across the entire scale.

Carl, you wrote "...there is not much better than 12-ET"... I think I would be able to understand your preference for 12 root of two, if you could kindly explain your reasons. In any case, seeing how familiar you are with maths representations, I am sure you understand both the Chas algorithm elementary structure and the one highlighted case, and I would be very grateful if you could help me represent this geometry with your favorite mathematical tool. And of course, in case you knew about someone that could already get numerically "there", I'd be very pleased to be informed.

Last, about cents, I do not use them mainly because they add approximations.

Hi Bogdan,

As you already know, the algorithm's basic structure is: two partial values, two scale ordinal position, two delta-wedges and one (s) variable.

The "special" highlighted case is: (3 - delta)^(1/19) = (4 + (s * delta))^(1/24) s = 1

The 4th is the first interval that, with its small circle (before the fifth's circle), enables to obtain all the other notes, before getting again to the generator. We also know that 4:3 is not really a partial sound, but the results of 2*2:3 and it is a fact that 2/3 = 0.666... At the numerator, all powers of two (and all integer numbers which are not power of 3) will produce a quotient (:3) with a periodic decimal value, either .333 or .666.

So, it is only by convention that 3/2 * 4/3 equals 2, and beyond convention the periodicity of 1/3, 2/3, 4/3 etc. can be managed and exploited. In fact 2/3 and 4/3 are the first n/n+1 and n/n-1 that produce an endless decimal value. For this reason, in my view, 2 and 3 together well represent the determining period-factor.

In general, not considering the "s" variable, delta values respond also to primes quotients; in particular, partial 4 squares partial 2 (the octave) so enabling to inter-modulate all octaves.

As mentioned, we could gain the Chas highlighted ratio (1.0594865443501...) also dealing with this whole issue in dynamic terms, i.e. calculating other delta values by using other partials and other "s" values; but the above "special" representation should (shall?) enable to show and help to acknowledge how this scale k-factor originates simply and precisely from those roots, so qualifying the Chas geometry as the correct (non-arbitrary) ET reference.

In conclusion, Chas ET (s = 1) 12ths and 15ths, beat-wise, are the two flow-factors that share the same delta-deviation value (degree). These two intervals shape this exponential "whole" form, meaning that they drive the linear continuous growth of the whole. Fourths, fifths and octaves, with their beat-curves, also affect the form and its torsion. Thus all intervals (all partials) have their precise role in determining this sound whole, and in this sense I would use the word "harmonic". In fact, Chas theory describes "s" as a dynamic and evolutive variable. In this precise (yet contradictory) sense, the highlighted s = 1 case is only meant to represent the direct product of the original primes.

Best wishes,

Alfredo

🔗Carl Lumma <carl@...>

Hi Alfredo,

> In my view, the fulcrum of those works is (again) "avoidable
> errors" and "reduced deviations", in the sense that "optimal"
> seems to be referred to "just" intervals and to those (numerical)
> solutions that may reduce deviations from pure ratios.
> The fundamental axiom then seems to be (again) the equality
> "pure = consonance = euphony", fairly in line with the
> centuries-old approach

Yes, correct. But so is Chas, where the intervals of
interest can be factored with primes 2 and 3.
The only other difference is that Chas considers constant
frequency deviations to give constant discord, whereas most
work in the field assumes constant log-frequency deviations
to do this.

> Point 4 - Variety of scales - Well beyond (and yet together
> with) infinite varieties of step/scale geometries, the Chas
> model then demonstrates that one and only one perfectly-coherent
> non-arbitrary scale exponential geometry exists.

No, I don't agree this is demonstrated, and don't understand
such a grandiose claim.

> Point 5 - The key - Chas theory considers "deviations" from
> just (pure) ratios as THE key-phenomenon, namely as the
> "dynamic" and variable event (here referring to beats) that
> gives "life" (light/brilliancy/energy) to any interval and to
> any combination of notes (played simultaneously).

The rate of beating is given by the frequency offset, but
in my experience, the rate matters less than the intensity
of the beating.

> Point 6 - Proportions - In this sense, "deviations-from-pure"
> are now regarded as quantities (magnitudes?) that can be
> "harmonized";

I don't agree that aligning or otherwise "harmonizing"
beat rates makes a noticeable improvement in concordance.
In fact, it may be detrimental, since aligning beat rates
increases the maximum intensity of the beating pattern.

> As we know, 12th root of two favors partial 2,

Again, the 12th root of two does not particularly favor
partial 2, but is an exceptional low-error tuning when a
broad range of targets (such as 2 3 and 5, or 2 3 5 and 7)
are considered.

> Carl, you wrote "...there is not much better than 12-ET"...
> I think I would be able to understand your preference for
> 12 root of two, if you could kindly explain your reasons.
> In any case, seeing how familiar you are with maths
> representations, I am sure you understand both the Chas
> algorithm elementary structure and the one highlighted case,
> and I would be very grateful if you could help me represent
> this geometry with your favorite mathematical tool. And of
> course, in case you knew about someone that could already get
> numerically "there", I'd be very pleased to be informed.

Here is one way to look at it

http://lumma.org/music/theory/TOPDamageOfETs.txt

The table shows TOP damage (which is the Tenney-weighted
error I explained last time) for different ETs at different
"prime limits". A prime limit, for example the 11-limit,
is the set of all rational numbers that can be factored with
primes no greater than 11.
http://en.wikipedia.org/wiki/Prime_limit

The ETs listed in my table are only those that improve upon
the 17-limit TOP damage of all smaller ETs. You can see that
12 is listed, because its 17-limit damage is lower than all
smaller ETs. But also, we can see something of how well 12
performs at other limits. For primes 2 and 3 only, we must
go to 29/octave. For the other prime limits, we really do not
get much better until 19/octave. So 12 is an outstanding
system no matter which chords we want to use. One must nearly
double the number of notes/octave to improve matters
significantly. And we assume musicians would like, all else
being equal, fewer tones rather than more.

You might object that all these systems favor the octave.
But in fact, when the tuning of the octave is allowed to
vary to minimized the total weighted error, most of the time
the results are fairly close to the exact division of the
octave.

-Carl

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

I appreciate your reply, thank you.

..."The only other difference is that Chas considers constant frequency deviations to give constant discord, whereas most work in the field assumes constant log-frequency deviations to do this."...

Hmm..., you say "...Chas considers constant frequency deviations...", I'd say "one constant frequency-deviation ratio" (4:3).

..."No, I don't agree this is demonstrated, and don't understand
such a grandiose claim."...

I agree, it sounds like a grandiose claim, perhaps due to the grandiosity of the smallest primes. In fact, that is the demonstration. ;-)

..."The rate of beating is given by the frequency offset, but in my experience, the rate matters less than the intensity of the beating."...

Not (in my experience). Actually, the rate... makes for both the intensity and the overall quality. In my mind it is intensity-within-quality.

> Point 6 - Proportions - In this sense, "deviations-from-pure"
> are now regarded as quantities (magnitudes?) that can be
> "harmonized";

..."I don't agree that aligning or otherwise "harmonizing" beat rates makes a noticeable improvement in concordance."...

The point is not "improvement in concordance", but "improvement".

..."In fact, it may be detrimental, since aligning beat rates increases the maximum intensity of the beating pattern."...

Yes, in a way. It is not really "aligning" but "proportioning" beat rates, so that the beating pattern can produce/release maximum intensity.

> As we know, 12th root of two favors partial 2,

..."Again, the 12th root of two does not particularly favor partial 2, but is an exceptional low-error tuning when a broad range of targets (such as 2 3 and 5, or 2 3 5 and 7) are considered."...

Well, I respect your own point.

Tomorrow... the other points.

Best wishes,

Alfredo

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Alfredo,
>
> > In my view, the fulcrum of those works is (again) "avoidable
> > errors" and "reduced deviations", in the sense that "optimal"
> > seems to be referred to "just" intervals and to those (numerical)
> > solutions that may reduce deviations from pure ratios.
> > The fundamental axiom then seems to be (again) the equality
> > "pure = consonance = euphony", fairly in line with the
> > centuries-old approach
>
> Yes, correct. But so is Chas, where the intervals of
> interest can be factored with primes 2 and 3.
> The only other difference is that Chas considers constant
> frequency deviations to give constant discord, whereas most
> work in the field assumes constant log-frequency deviations
> to do this.
>
> > Point 4 - Variety of scales - Well beyond (and yet together
> > with) infinite varieties of step/scale geometries, the Chas
> > model then demonstrates that one and only one perfectly-coherent
> > non-arbitrary scale exponential geometry exists.
>
> No, I don't agree this is demonstrated, and don't understand
> such a grandiose claim.
>
> > Point 5 - The key - Chas theory considers "deviations" from
> > just (pure) ratios as THE key-phenomenon, namely as the
> > "dynamic" and variable event (here referring to beats) that
> > gives "life" (light/brilliancy/energy) to any interval and to
> > any combination of notes (played simultaneously).
>
> The rate of beating is given by the frequency offset, but
> in my experience, the rate matters less than the intensity
> of the beating.
>
> > Point 6 - Proportions - In this sense, "deviations-from-pure"
> > are now regarded as quantities (magnitudes?) that can be
> > "harmonized";
>
> I don't agree that aligning or otherwise "harmonizing"
> beat rates makes a noticeable improvement in concordance.
> In fact, it may be detrimental, since aligning beat rates
> increases the maximum intensity of the beating pattern.
>
> > As we know, 12th root of two favors partial 2,
>
> Again, the 12th root of two does not particularly favor
> partial 2, but is an exceptional low-error tuning when a
> broad range of targets (such as 2 3 and 5, or 2 3 5 and 7)
> are considered.
>
> > Carl, you wrote "...there is not much better than 12-ET"...
> > I think I would be able to understand your preference for
> > 12 root of two, if you could kindly explain your reasons.
> > In any case, seeing how familiar you are with maths
> > representations, I am sure you understand both the Chas
> > algorithm elementary structure and the one highlighted case,
> > and I would be very grateful if you could help me represent
> > this geometry with your favorite mathematical tool. And of
> > course, in case you knew about someone that could already get
> > numerically "there", I'd be very pleased to be informed.
>
> Here is one way to look at it
>
> http://lumma.org/music/theory/TOPDamageOfETs.txt
>
> The table shows TOP damage (which is the Tenney-weighted
> error I explained last time) for different ETs at different
> "prime limits". A prime limit, for example the 11-limit,
> is the set of all rational numbers that can be factored with
> primes no greater than 11.
> http://en.wikipedia.org/wiki/Prime_limit
>
> The ETs listed in my table are only those that improve upon
> the 17-limit TOP damage of all smaller ETs. You can see that
> 12 is listed, because its 17-limit damage is lower than all
> smaller ETs. But also, we can see something of how well 12
> performs at other limits. For primes 2 and 3 only, we must
> go to 29/octave. For the other prime limits, we really do not
> get much better until 19/octave. So 12 is an outstanding
> system no matter which chords we want to use. One must nearly
> double the number of notes/octave to improve matters
> significantly. And we assume musicians would like, all else
> being equal, fewer tones rather than more.
>
> You might object that all these systems favor the octave.
> But in fact, when the tuning of the octave is allowed to
> vary to minimized the total weighted error, most of the time
> the results are fairly close to the exact division of the
> octave.
>
> -Carl
>

🔗Alfredo Capurso <alfredo.capurso@...>

Hi Carl,

I hope we can share some issues that may be considered relevant, although I also hope not to force you into something you may not be interested in.

You wrote: …"No, I don't agree this is demonstrated, and don't understand such a grandiose claim."…

Then, my reply… "I agree, it sounds like a grandiose claim, perhaps due to the grandiosity of the smallest primes? ;-)… was meant to reduce some tension that, sometime, some words may convey on their own. In any case, I am simply referring to the ET geometry (the Chas highlighted case) that is designed precisely by the unity (s = 1), by the first primes (2 and 3) and the 3:1, 4:1 and 4:3 ratios.

Now, I would be willing to demonstrate the perfect coherency of that (log) geometry, but really I do not know what should be "demonstrated", beyond those numbers.

Perhaps I should demonstrate:

- that 2:1 (theoretical) octaves is/was an arbitrary axiom?
- that also the octave-span is/was arbitrary, not justifiable and not "sufficient" as a temperament (log) module?
- that 2:1 octaves have the effect of doubling deviation values?
- that tempering 3:2 and 5:4 was forcing two prime-ratios into an (octave) arbitrary ET model?
- that, instead of only thirds and fifths, we needed to temper (log) octaves (and thirds) and fifths?
- that the first scale "incompatibility" is between partial 2 and 3?
- that 4:3 is the (periodic) first n/n-1 that - in a circle - enumerates the number (12) of notes?
- that even only one single theoretical zero-beating interval makes any theory (and relative model) arbitrary and only "partially" congruent? (here referring to the interval R).
- that all intervals, all across the scale, can have their unique proportional beat-period (please, read unique "timing", in terms of beats), and yet all notes be "all centers"?

Should I demonstrate that, also in music, pure ratios and scale deviations together can all contribute to determining both a non-arbitrary (static) scale ET geometry (s = 1) and an ever-evolving (s*delta) whole?

In other words, perhaps something is missing, but what is it? Can you help me understand that?

Thank you and best wishes,

Alfredo

________________________________
From: Carl Lumma <carl@...>
To: tuning@yahoogroups.com
Sent: Saturday, July 14, 2012 11:18 PM
Subject: [tuning] Re: Chas temperament model

Hi Alfredo,

No, I don't agree this is demonstrated, and don't understand
such a grandiose claim.

The rate of beating is given by the frequency offset, but
in my experience, the rate matters less than the intensity
of the beating.

> Point 6 - Proportions - In this sense, "deviations-from-pure"
> are now regarded as quantities (magnitudes?) that can be
> "harmonized";

> As we know, 12th root of two favors partial 2,

Again, the 12th root of two does not particularly favor
partial 2, but is an exceptional low-error tuning when a
broad range of targets (such as 2 3 and 5, or 2 3 5 and 7)
are considered.

Here is one way to look at it

http://lumma.org/music/theory/TOPDamageOfETs.txt

-Carl

🔗Carl Lumma <carl@...>

🔗alfredo.capurso <alfredo.capurso@...>

Hi Carl,

Thank you very much, 1 is a relevant point indeed.

Yuo wrote:

> What I think is missing is
>
> 1. A statement of which chords we're interested in.
>
> 2. A model for how to measure the discord caused by
> tuning deviations of the chords we name in #1.

The way I imagine Chas is like a self-tensioning web, were all intervals and all relative deviations are not determined on the basis of a specific interest (on chords), but on the strict proportional, most intrinsic relation amongst partials and deviations.

In this sense, the idea is that of a self-ordering (harmonious) whole, where partial ratios themselves "adjust" all chromatic intervals deviation-curves. Partials, in a conjoint action, establish themselves those sequences of values relative to the scale and to every chromatic-interval-deviation-curve and deliver the (tuning) form. This is to say that partials and deviations are interrelated, so that we can consider them as one.

In other words, I was looking for a self-ordering optimum, I was not "interested" in one specific type of chord/tonality (nor selected partial). I thought that a "pure" (meaning non-arbitrary and self-contained) tuning-form could exist, and it should have been the result of partials-And-(proportional)-deviations. Then empirically, I considered chromatic intervals and looked for proportional beat-rate progressions, trying to identify/visualize the beat-curves relative to every interval.

So the focus was/is not on one or some chords, but on a "whole", say a multi-plane/level geometry that appears to be still (and unique), but can (should) also be seen as "elastic", like a 4D spiderweb. The accent is not anymore on "discord" that might be avoided, but on discord as the proportional "agent", discord as the multi-partial (s-variable) factor that actually keeps this entity together. And we see that the unity itself (for s = 1) does not admit zero-deviations.

From the above premises, how do I/we get to point 2?

Best wishes,

Alfredo

>
> --- In tuning@yahoogroups.com, Alfredo Capurso <alfredo.capurso@> wrote:
>
> > Now, I would be willing to demonstrate the perfect coherency
> > of that (log) geometry, but really I do not know what should
> > be "demonstrated", beyond those numbers.
> >
> > Perhaps I should demonstrate:
> >
> > - that 2:1 (theoretical) octaves is/was an arbitrary axiom?
> > - that also the octave-span is/was arbitrary, not justifiable
> > and not "sufficient" as a temperament (log) module?
> > - that 2:1 octaves have the effect of doubling deviation values?
> > - that tempering 3:2 and 5:4 was forcing two prime-ratios into
> > an (octave) arbitrary ET model?
> > - that, instead of only thirds and fifths, we needed to temper
> > (log) octaves (and thirds) and fifths?Â
> > - that the first scale "incompatibility" is between partial
> > 2 and 3?
> > - that 4:3 is the (periodic) first n/n-1 that - in a circle -
> > enumerates the number (12) of notes?
> > - that even only one single theoretical zero-beating interval
> > makes any theory (and relative model) arbitrary and only
> > "partially" congruent? (here referring to the interval R).
> > - that all intervals, all across the scale, can have their
> > unique proportional beat-period (please, read unique "timing",
> > in terms of beats), and yet all notes be "all centers"?
> >
> > Should I demonstrate that, also in music, pure ratios and scale
> > deviations together can all contribute to determining both a
> > non-arbitrary (static) scale ET geometry (s = 1) and an
> > ever-evolving (s*delta) whole?
> >
> > In other words, perhaps something is missing, but what is it?
> > Can you help me understand that?
> >
> > Thank you and best wishes,
> >
> > Alfredo
> >
>