Chas temperament model

Hi,

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 actually found it more confusing than convincing.

Anyway, more than the paper's "technicalities", I would like to share the theoretical approach, some conceptual issues and this model' algorithm. And if you may suggest any other more correct representation or visualization I would be very grateful.

G.R.I.M. - Università di Palermo (2009)
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

Thank you, a.c.

PS: Sorry for my English.

Here you find five recordings on real pianos
http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en

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-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...> wrote:
>
> Hi,
>
> 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 actually found it more confusing than convincing.
>
> Anyway, more than the paper's "technicalities", I would like to share the theoretical approach, some conceptual issues and this model' algorithm. And if you may suggest any other more correct representation or visualization I would be very grateful.
>
> G.R.I.M. - Università di Palermo (2009)
> http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf
>
> Thank you, a.c.
>
> PS: Sorry for my English.
>
> Here you find five recordings on real pianos
> http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en
>

--- In tuning-math@yahoogroups.com, "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.

Are these axioms you are assuming or empirical claims?

12-ET may be described as the 12th root of 2 equal-step tuning.

Is there a similar description for Chas?

-Carl

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
>

Hi Carl,

I'm not sure, perhaps 12 ET may be described as one (of infinite) particular case "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-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?
>
> -Carl
>
> 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
> >
>

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.

-> ..."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-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "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.
>
> Are these axioms you are assuming or empirical claims?
>

> we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.

> > >From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s
> > >= 0 we gain 12TET.

> Chas enables also the modeling of pure-ratio based scales.

I found in your paper a limited number of formulas, those on page 67 (3.5). On page 68 is a Chas Scale list with Chas values in irrational numbers (4.0).

I'm interested in a complete scale ratios list, with the variable s included. Something similar to the one below, which could be used to create any scale just like you say: 12-ET when s=0 and also pure-ratio based scales.

1,00000000000000 ?
1,05948654435010 ?
1,12251173765892 ?
1,18928608192467 ?
1,26003260118204 ?
1,33498758639483 (4+∆)/(3-∆)
1,41440138465974 ?
1,49853923535714 ?
1,58768215604158 ?
1,68212788103081 ?
1,78219185582829 (4+∆)^2/(3-∆)^2
1,88820829070040 ?
2,00053127692738 ?
2,11953596945608 ?
2,24561983990476 (3-∆)^2/(4+∆)
2,37920400410472 ?
2,52073462861283 ?
2,67068442089264 ?
2,82955420814119 ?
2,99787461003480 (3-∆)
3,17620781098067 ?
3,36514943779371 ?
3,56533054906974 ?
3,77741974289974 ?
4,00212538996469 (4+∆)
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
5,34278771479711 (4+∆)^2/(3-∆)
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
8,98725217749490 (3-∆)^2
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
?,?????????????? ?
16,01700763699930 (4+∆)^2

with ∆=0,0021253899646

Hi Bogdan,

Check please if this can help: I can provide a list that includes a certain number of Chas algorithms + "s" values + delta values + relative scale incremental ratios. You could then use Excel and calculate yourself the scales frequency values relative to each s-scale ratio? Say yes, which other pure-ratio-based scale would you like to find in a list? For instance 3:2, and 3:1 and...?

Alfredo

--- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@...> wrote:
>
>
>
> > we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.
>
> > > >From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s
> > > >= 0 we gain 12TET.
>
> > Chas enables also the modeling of pure-ratio based scales.
>
> I found in your paper a limited number of formulas, those on page 67 (3.5). On page 68 is a Chas Scale list with Chas values in irrational numbers (4.0).
>
> I'm interested in a complete scale ratios list, with the variable s included. Something similar to the one below, which could be used to create any scale just like you say: 12-ET when s=0 and also pure-ratio based scales.
>
> 1,00000000000000 ?
> 1,05948654435010 ?
> 1,12251173765892 ?
> 1,18928608192467 ?
> 1,26003260118204 ?
> 1,33498758639483 (4+∆)/(3-∆)
> 1,41440138465974 ?
> 1,49853923535714 ?
> 1,58768215604158 ?
> 1,68212788103081 ?
> 1,78219185582829 (4+∆)^2/(3-∆)^2
> 1,88820829070040 ?
> 2,00053127692738 ?
> 2,11953596945608 ?
> 2,24561983990476 (3-∆)^2/(4+∆)
> 2,37920400410472 ?
> 2,52073462861283 ?
> 2,67068442089264 ?
> 2,82955420814119 ?
> 2,99787461003480 (3-∆)
> 3,17620781098067 ?
> 3,36514943779371 ?
> 3,56533054906974 ?
> 3,77741974289974 ?
> 4,00212538996469 (4+∆)
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> 5,34278771479711 (4+∆)^2/(3-∆)
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> 8,98725217749490 (3-∆)^2
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> ?,?????????????? ?
> 16,01700763699930 (4+∆)^2
>
> with ∆=0,0021253899646
>

Yes, if I have the list then the frequencies will come out in Excel. This is also useful if you want a concert pitch other that 440 Hz.

I must admit that I am still a bit confused so for now let's just make the list below complete. As I will understand more, we will take it on from there.

--- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...> wrote:
>
> Hi Bogdan,
>
> Check please if this can help: I can provide a list that includes a certain number of Chas algorithms + "s" values + delta values + relative scale incremental ratios. You could then use Excel and calculate yourself the scales frequency values relative to each s-scale ratio? Say yes, which other pure-ratio-based scale would you like to find in a list? For instance 3:2, and 3:1 and...?
>
> Alfredo
>
> --- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> >
> >
> >
> > > we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.
> >
> > > > >From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s
> > > > >= 0 we gain 12TET.
> >
> > > Chas enables also the modeling of pure-ratio based scales.
> >
> > I found in your paper a limited number of formulas, those on page 67 (3.5). On page 68 is a Chas Scale list with Chas values in irrational numbers (4.0).
> >
> > I'm interested in a complete scale ratios list, with the variable s included. Something similar to the one below, which could be used to create any scale just like you say: 12-ET when s=0 and also pure-ratio based scales.
> >
> > 1,00000000000000 ?
> > 1,05948654435010 ?
> > 1,12251173765892 ?
> > 1,18928608192467 ?
> > 1,26003260118204 ?
> > 1,33498758639483 (4+∆)/(3-∆)
> > 1,41440138465974 ?
> > 1,49853923535714 ?
> > 1,58768215604158 ?
> > 1,68212788103081 ?
> > 1,78219185582829 (4+∆)^2/(3-∆)^2
> > 1,88820829070040 ?
> > 2,00053127692738 ?
> > 2,11953596945608 ?
> > 2,24561983990476 (3-∆)^2/(4+∆)
> > 2,37920400410472 ?
> > 2,52073462861283 ?
> > 2,67068442089264 ?
> > 2,82955420814119 ?
> > 2,99787461003480 (3-∆)
> > 3,17620781098067 ?
> > 3,36514943779371 ?
> > 3,56533054906974 ?
> > 3,77741974289974 ?
> > 4,00212538996469 (4+∆)
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > 5,34278771479711 (4+∆)^2/(3-∆)
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > 8,98725217749490 (3-∆)^2
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > ?,?????????????? ?
> > 16,01700763699930 (4+∆)^2
> >
> > with ∆=0,0021253899646
> >
>

Hi Bogdan,

I hope to be able to put the list together during this weekend.

In the meantime, thanks to Chris Vaisvil, I thought I could take John O'Sullivan's tuning as a starting point. Looking at John's website (http://www.johnsmusic7.com/) I could do some calculations based on his cents list:

12 Tone Raven Temperament (12TRT)

0.0, 113.7177, 197.1342, 312.1318, 389.8232, 498.045, 577.4304,
701.955, 811.81, 889.5014, 967.284, 1081.4929, 1200.0.

In order to represent how ratios and deviation-wedges work in the "background", I considered John's first step 113.7177 cents = 1.0678912604314432, and extracted this ratio from Chas algorithm.

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

say that (John's arbitrary) s = -1,73286500650591

(3 - delta)^(1/19) = (4 + (- 1,73286500650591 * delta))^(1/24)

Delta = - 0,4834823415293

(3 - (- 0,4834823415293))^(1/19) = (4 + (-1,73286500650591 * - 0,4834823415293))^(1/24) = 1.0678912604314432…

And the case:

s = -1,73286500650591 in fraction s/s1 = 1/-0,577078997063

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

(3 - (- 0,577078997063 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)

Delta' = 0,83780963089979

(3 - (- 0,577078997063 * 0,83780963089979))^(1/19) = (4 + (1 * 0,83780963089979))^(1/24) = 1.0678912604314432...

If we used that incremental ratio and draw a logarithmic scale, we would obviously find delta in position 19 and delta' in position 24.
Comment:

Of course, that is John's "s" value, and I would not ask myself why John preferred that. 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. In fact, compared with the Chas s = 1 case, we would only loose contact with 4:3 (2*2/3), in fact (4/delta')/(3/delta) does not equal 1.333…

--- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@...> wrote:
>
> Yes, if I have the list then the frequencies will come out in Excel. This is also useful if you want a concert pitch other that 440 Hz.
>
> I must admit that I am still a bit confused so for now let's just make the list below complete. As I will understand more, we will take it on from there.
>
>
> --- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@> wrote:
> >
> > Hi Bogdan,
> >
> > Check please if this can help: I can provide a list that includes a certain number of Chas algorithms + "s" values + delta values + relative scale incremental ratios. You could then use Excel and calculate yourself the scales frequency values relative to each s-scale ratio? Say yes, which other pure-ratio-based scale would you like to find in a list? For instance 3:2, and 3:1 and...?
> >
> > Alfredo
> >
> > --- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> > >
> > >
> > >
> > > > we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.
> > >
> > > > > >From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s
> > > > > >= 0 we gain 12TET.
> > >
> > > > Chas enables also the modeling of pure-ratio based scales.
> > >
> > > I found in your paper a limited number of formulas, those on page 67 (3.5). On page 68 is a Chas Scale list with Chas values in irrational numbers (4.0).
> > >
> > > I'm interested in a complete scale ratios list, with the variable s included. Something similar to the one below, which could be used to create any scale just like you say: 12-ET when s=0 and also pure-ratio based scales.
> > >
> > > 1,00000000000000 ?
> > > 1,05948654435010 ?
> > > 1,12251173765892 ?
> > > 1,18928608192467 ?
> > > 1,26003260118204 ?
> > > 1,33498758639483 (4+∆)/(3-∆)
> > > 1,41440138465974 ?
> > > 1,49853923535714 ?
> > > 1,58768215604158 ?
> > > 1,68212788103081 ?
> > > 1,78219185582829 (4+∆)^2/(3-∆)^2
> > > 1,88820829070040 ?
> > > 2,00053127692738 ?
> > > 2,11953596945608 ?
> > > 2,24561983990476 (3-∆)^2/(4+∆)
> > > 2,37920400410472 ?
> > > 2,52073462861283 ?
> > > 2,67068442089264 ?
> > > 2,82955420814119 ?
> > > 2,99787461003480 (3-∆)
> > > 3,17620781098067 ?
> > > 3,36514943779371 ?
> > > 3,56533054906974 ?
> > > 3,77741974289974 ?
> > > 4,00212538996469 (4+∆)
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > 5,34278771479711 (4+∆)^2/(3-∆)
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > 8,98725217749490 (3-∆)^2
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > ?,?????????????? ?
> > > 16,01700763699930 (4+∆)^2
> > >
> > > with ∆=0,0021253899646
> > >
> >
>

Hi Bogdan,

Referring to Chas algorithm, we need two ratios (in the Chas basic algorithm these are 3/1 and 4/1), and their relative scale ordinal position, starting from 1° = zero.

The list below may describe this new approach and well represent how 1:1 delta-deviations may be referred to any couple of intervals, so producing a wide variety of scale (or scale-step) incremental ratios. In the cases below we can imagine the "s-stretch" arbitrary variable being a ghost-variable, that equals 1.

- . - . - . -

2nd and 5th delta deviations (d.d.):

(9/8 - delta)^(1/2) = (3/2 + delta)^(1/7)

Delta = 0.00179253067136

Scale incremental ratio = 1.05981482784902

Octave ratio = 2.00798237793787

- . - . - . -

2nd and 4th d.d.:

(9/8 - delta)^(1/2) = (4/3 + delta)^(1/5)

Delta = 0.00227837646206

Scale incremental ratio = 1.05958559047296

Octave ratio = 2.0027766677249

- . - . - . -

2nd and 8ve d.d.:

(9/8 - delta)^(1/2) = (2 + delta)^(1/12)

Delta = 0.00232095768128

Scale incremental ratio = 1.05956549694614

Octave ratio = 2.00232095768128

- . - . - . -

2nd and double-octave d.d.:

(9/8 - delta)^(1/2) = (4 + delta)^(1/24)

Delta = 0.00247997487864

Scale incremental ratio = 1.05949045541777

Octave ratio = 2.00061989765139

- . - . - . -

5th and 8ve d.d.:

(3/2 - delta)^(1/7) = (2 + delta)^(1/12)

Delta = 0.001178134272

Scale incremental ratio = 1.05951508823057

Octave ratio = 2.001178134272

- . - . - . -

Best wishes,

Alfredo

--- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...> wrote:
>
> Hi Bogdan,
>
> I hope to be able to put the list together during this weekend.
>
> In the meantime, thanks to Chris Vaisvil, I thought I could take John O'Sullivan's tuning as a starting point. Looking at John's website (http://www.johnsmusic7.com/) I could do some calculations based on his cents list:
>
> 12 Tone Raven Temperament (12TRT)
>
> 0.0, 113.7177, 197.1342, 312.1318, 389.8232, 498.045, 577.4304,
> 701.955, 811.81, 889.5014, 967.284, 1081.4929, 1200.0.
>
> In order to represent how ratios and deviation-wedges work in the "background", I considered John's first step 113.7177 cents = 1.0678912604314432, and extracted this ratio from Chas algorithm.
>
> Chas = (3 - delta)^(1/19) = (4 + (s * delta))^(1/24)
>
> say that (John's arbitrary) s = -1,73286500650591
>
> (3 - delta)^(1/19) = (4 + (- 1,73286500650591 * delta))^(1/24)
>
> Delta = - 0,4834823415293
>
> (3 - (- 0,4834823415293))^(1/19) = (4 + (-1,73286500650591 * - 0,4834823415293))^(1/24) = 1.0678912604314432…
>
> And the case:
>
> s = -1,73286500650591 in fraction s/s1 = 1/-0,577078997063
>
> (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)
>
> (3 - (- 0,577078997063 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)
>
> Delta' = 0,83780963089979
>
> (3 - (- 0,577078997063 * 0,83780963089979))^(1/19) = (4 + (1 * 0,83780963089979))^(1/24) = 1.0678912604314432...
>
> If we used that incremental ratio and draw a logarithmic scale, we would obviously find delta in position 19 and delta' in position 24.
> Comment:
>
> Of course, that is John's "s" value, and I would not ask myself why John preferred that. 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. In fact, compared with the Chas s = 1 case, we would only loose contact with 4:3 (2*2/3), in fact (4/delta')/(3/delta) does not equal 1.333…
>
> --- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> >
> > Yes, if I have the list then the frequencies will come out in Excel. This is also useful if you want a concert pitch other that 440 Hz.
> >
> > I must admit that I am still a bit confused so for now let's just make the list below complete. As I will understand more, we will take it on from there.
> >
> >
> > --- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@> wrote:
> > >
> > > Hi Bogdan,
> > >
> > > Check please if this can help: I can provide a list that includes a certain number of Chas algorithms + "s" values + delta values + relative scale incremental ratios. You could then use Excel and calculate yourself the scales frequency values relative to each s-scale ratio? Say yes, which other pure-ratio-based scale would you like to find in a list? For instance 3:2, and 3:1 and...?
> > >
> > > Alfredo
> > >
> > > --- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> > > >
> > > >
> > > >
> > > > > we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.
> > > >
> > > > > > >From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s
> > > > > > >= 0 we gain 12TET.
> > > >
> > > > > Chas enables also the modeling of pure-ratio based scales.
> > > >
> > > > I found in your paper a limited number of formulas, those on page 67 (3.5). On page 68 is a Chas Scale list with Chas values in irrational numbers (4.0).
> > > >
> > > > I'm interested in a complete scale ratios list, with the variable s included. Something similar to the one below, which could be used to create any scale just like you say: 12-ET when s=0 and also pure-ratio based scales.
> > > >
> > > > 1,00000000000000 ?
> > > > 1,05948654435010 ?
> > > > 1,12251173765892 ?
> > > > 1,18928608192467 ?
> > > > 1,26003260118204 ?
> > > > 1,33498758639483 (4+∆)/(3-∆)
> > > > 1,41440138465974 ?
> > > > 1,49853923535714 ?
> > > > 1,58768215604158 ?
> > > > 1,68212788103081 ?
> > > > 1,78219185582829 (4+∆)^2/(3-∆)^2
> > > > 1,88820829070040 ?
> > > > 2,00053127692738 ?
> > > > 2,11953596945608 ?
> > > > 2,24561983990476 (3-∆)^2/(4+∆)
> > > > 2,37920400410472 ?
> > > > 2,52073462861283 ?
> > > > 2,67068442089264 ?
> > > > 2,82955420814119 ?
> > > > 2,99787461003480 (3-∆)
> > > > 3,17620781098067 ?
> > > > 3,36514943779371 ?
> > > > 3,56533054906974 ?
> > > > 3,77741974289974 ?
> > > > 4,00212538996469 (4+∆)
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > 5,34278771479711 (4+∆)^2/(3-∆)
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > 8,98725217749490 (3-∆)^2
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > ?,?????????????? ?
> > > > 16,01700763699930 (4+∆)^2
> > > >
> > > > with ∆=0,0021253899646
> > > >
> > >
> >
>

Allright Alfredo, I got it! Perfectly clear, uploaded here under Files / CHAS:

http://f1.grp.yahoofs.com/v1/8CfyT2J2xhQS7tgBIbSd_vQG4woX79t3IYtxQTUE-SGr2qEx3TBbNctU7V4d8DrbPWG-dp3R1CaWhXSWeHowROQGKBvIzr1-9Zg/CHAS/CHAS%20formulas%2001.docx

Your last post was an eye-opener, thank you. You still didn't answered my question about completeing the list and we will get to that - if I don't figure it out myself.

All the best
Bogdan

--- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@...> wrote:
>
> Hi Bogdan,
>
> Referring to Chas algorithm, we need two ratios (in the Chas basic algorithm these are 3/1 and 4/1), and their relative scale ordinal position, starting from 1� = zero.
>
> The list below may describe this new approach and well represent how 1:1 delta-deviations may be referred to any couple of intervals, so producing a wide variety of scale (or scale-step) incremental ratios. In the cases below we can imagine the "s-stretch" arbitrary variable being a ghost-variable, that equals 1.
>
> - . - . - . -
>
> 2nd and 5th delta deviations (d.d.):
>
> (9/8 - delta)^(1/2) = (3/2 + delta)^(1/7)
>
> Delta = 0.00179253067136
>
> Scale incremental ratio = 1.05981482784902
>
> Octave ratio = 2.00798237793787
>
> - . - . - . -
>
> 2nd and 4th d.d.:
>
> (9/8 - delta)^(1/2) = (4/3 + delta)^(1/5)
>
> Delta = 0.00227837646206
>
> Scale incremental ratio = 1.05958559047296
>
> Octave ratio = 2.0027766677249
>
> - . - . - . -
>
> 2nd and 8ve d.d.:
>
> (9/8 - delta)^(1/2) = (2 + delta)^(1/12)
>
> Delta = 0.00232095768128
>
> Scale incremental ratio = 1.05956549694614
>
> Octave ratio = 2.00232095768128
>
> - . - . - . -
>
> 2nd and double-octave d.d.:
>
> (9/8 - delta)^(1/2) = (4 + delta)^(1/24)
>
> Delta = 0.00247997487864
>
> Scale incremental ratio = 1.05949045541777
>
> Octave ratio = 2.00061989765139
>
> - . - . - . -
>
> 5th and 8ve d.d.:
>
> (3/2 - delta)^(1/7) = (2 + delta)^(1/12)
>
> Delta = 0.001178134272
>
> Scale incremental ratio = 1.05951508823057
>
> Octave ratio = 2.001178134272
>
> - . - . - . -
>
> Best wishes,
>
> Alfredo
>
>
>
> --- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@> wrote:
> >
> > Hi Bogdan,
> >
> > I hope to be able to put the list together during this weekend.
> >
> > In the meantime, thanks to Chris Vaisvil, I thought I could take John O'Sullivan's tuning as a starting point. Looking at John's website (http://www.johnsmusic7.com/) I could do some calculations based on his cents list:
> >
> > 12 Tone Raven Temperament (12TRT)
> >
> > 0.0, 113.7177, 197.1342, 312.1318, 389.8232, 498.045, 577.4304,
> > 701.955, 811.81, 889.5014, 967.284, 1081.4929, 1200.0.
> >
> > In order to represent how ratios and deviation-wedges work in the "background", I considered John's first step 113.7177 cents = 1.0678912604314432, and extracted this ratio from Chas algorithm.
> >
> > Chas = (3 - delta)^(1/19) = (4 + (s * delta))^(1/24)
> >
> > say that (John's arbitrary) s = -1,73286500650591
> >
> > (3 - delta)^(1/19) = (4 + (- 1,73286500650591 * delta))^(1/24)
> >
> > Delta = - 0,4834823415293
> >
> > (3 - (- 0,4834823415293))^(1/19) = (4 + (-1,73286500650591 * - 0,4834823415293))^(1/24) = 1.0678912604314432�
> >
> > And the case:
> >
> > s = -1,73286500650591 in fraction s/s1 = 1/-0,577078997063
> >
> > (3 - (s1 * delta'))^(1/19) = (4 + (s * delta'))^(1/24)
> >
> > (3 - (- 0,577078997063 * delta'))^(1/19) = (4 + (1 * delta'))^(1/24)
> >
> > Delta' = 0,83780963089979
> >
> > (3 - (- 0,577078997063 * 0,83780963089979))^(1/19) = (4 + (1 * 0,83780963089979))^(1/24) = 1.0678912604314432...
> >
> > If we used that incremental ratio and draw a logarithmic scale, we would obviously find delta in position 19 and delta' in position 24.
> > Comment:
> >
> > Of course, that is John's "s" value, and I would not ask myself why John preferred that. 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. In fact, compared with the Chas s = 1 case, we would only loose contact with 4:3 (2*2/3), in fact (4/delta')/(3/delta) does not equal 1.333�
> >
> > --- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> > >
> > > Yes, if I have the list then the frequencies will come out in Excel. This is also useful if you want a concert pitch other that 440 Hz.
> > >
> > > I must admit that I am still a bit confused so for now let's just make the list below complete. As I will understand more, we will take it on from there.
> > >
> > >
> > > --- In tuning-math@yahoogroups.com, "alfredo.capurso" <alfredo.capurso@> wrote:
> > > >
> > > > Hi Bogdan,
> > > >
> > > > Check please if this can help: I can provide a list that includes a certain number of Chas algorithms + "s" values + delta values + relative scale incremental ratios. You could then use Excel and calculate yourself the scales frequency values relative to each s-scale ratio? Say yes, which other pure-ratio-based scale would you like to find in a list? For instance 3:2, and 3:1 and...?
> > > >
> > > > Alfredo
> > > >
> > > > --- In tuning-math@yahoogroups.com, "Bogdan" <baros_ilogic@> wrote:
> > > > >
> > > > >
> > > > >
> > > > > > we gain 12-ET when the delta:delta (1:1) proportion is modified in 1:0 by s = 0.
> > > > >
> > > > > > > >From Chas algorithm, (3 - delta)^(1/19) = (4 + s*delta)^(1/24) with s
> > > > > > > >= 0 we gain 12TET.
> > > > >
> > > > > > Chas enables also the modeling of pure-ratio based scales.
> > > > >
> > > > > I found in your paper a limited number of formulas, those on page 67 (3.5). On page 68 is a Chas Scale list with Chas values in irrational numbers (4.0).
> > > > >
> > > > > I'm interested in a complete scale ratios list, with the variable s included. Something similar to the one below, which could be used to create any scale just like you say: 12-ET when s=0 and also pure-ratio based scales.
> > > > >
> > > > > 1,00000000000000 ?
> > > > > 1,05948654435010 ?
> > > > > 1,12251173765892 ?
> > > > > 1,18928608192467 ?
> > > > > 1,26003260118204 ?
> > > > > 1,33498758639483 (4+∆)/(3-∆)
> > > > > 1,41440138465974 ?
> > > > > 1,49853923535714 ?
> > > > > 1,58768215604158 ?
> > > > > 1,68212788103081 ?
> > > > > 1,78219185582829 (4+∆)^2/(3-∆)^2
> > > > > 1,88820829070040 ?
> > > > > 2,00053127692738 ?
> > > > > 2,11953596945608 ?
> > > > > 2,24561983990476 (3-∆)^2/(4+∆)
> > > > > 2,37920400410472 ?
> > > > > 2,52073462861283 ?
> > > > > 2,67068442089264 ?
> > > > > 2,82955420814119 ?
> > > > > 2,99787461003480 (3-∆)
> > > > > 3,17620781098067 ?
> > > > > 3,36514943779371 ?
> > > > > 3,56533054906974 ?
> > > > > 3,77741974289974 ?
> > > > > 4,00212538996469 (4+∆)
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > 5,34278771479711 (4+∆)^2/(3-∆)
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > 8,98725217749490 (3-∆)^2
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > ?,?????????????? ?
> > > > > 16,01700763699930 (4+∆)^2
> > > > >
> > > > > with ∆=0,0021253899646
> > > > >
> > > >
> > >
> >
>