Yahoo Tuning Groups Ultimate Backup tuning I just built my first JI scale

I am copying this e-mail to Claudio Di Veroli who
might evaluate the JI scale that I just worked out.

Mike mentioned the precise perform of (5/4) = 1.25
for note E, so by following Mike´s recommendation,
this tone frequency works in the new JI scale.
Classical JI frequencies in the scale are:
Eb= (32/27), E= (5/4)= 1.25, F= (4/3)= 1.3333....,
G= (3/2)= 1.5, Bb= (16/9) = 1.7777....,
B= (15/8)= 1.875.

Three narrow fifths with equal values of
1.49323976284 let (1.5/1.49323976284)=1.0045272282
which is one third of the Pythagorean comma:
(1.0045272282)^3 = 1.01364326477.
Additionally, the self cancelling pair are
1.5050913678 and 1.49492585512.

The remaining tone frequencies are given
in decimal numbers which will be transformed
to common fractions in a matter of days.

Tone frequencies of the JI octave follows:

C = 1
C# = 1.05826736797
D = 1.11992982213
Eb = 1.185185185= (32/27),
E = 1.25 = (5/4)
F = 1.33333......= (4/3)
G = 1.5 = (3/2)
Ab = 1.58740105196
A = 1.67232374199
Bb = 1.7777777.....=(16/9)
B = 1.875 = (15/8)
2C = 2

Below, I detail the twelve perfect
and imperfect fifths. Seven pure fifths:

G/C -- 1.5
2D/G -- 1.49323976284
A/D -- 1.49323976284
2E/A -- 1.49492585512*
B/E -- 1.5
2F#/B -- 1.5050917638*
2C#/F# -- 1.5
Ab/C# -- 1.5
2Eb /Ab -- 1.49323976284
Bb/Eb -- 1.5
2F/Bb -- 1.5
2C/F -- 1.5

*Cancelling figures

Pending: Major and Minor thirds.

Hope you reach to a positive conclusion.

Thanks

Mario Pizarro

Lima, March 05, 2009

piagui@...

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>
> Eb= (32/27), E= (5/4)= 1.25, F= (4/3)= 1.3333....,
> G= (3/2)= 1.5, Bb= (16/9) = 1.7777....,
> B= (15/8)= 1.875.

Hola Mario,

> Three narrow fifths with equal values of

using the:
http://superspace.epfl.ch/approximator/
yields for yours decimal numbers the following ratios:

1.49323976284 let (1.5/1.49323976284)=1.0045272282 = ~ 222/221

> which is one third of the Pythagorean comma:
(1.0045272282)^3 = 1.01364326477
= ~74/73 or better ~223/220 .... precisely 3^12/2^19 exact

> Additionally, the self cancelling pair are
> 1.5050913678
= ~444/295

> and 1.49492585512
= ~589/394

> C = 1
C# = 1.05826736797 = ~18/17
D = 1.11992982213 = ~~9/8 better ~28/25
Eb = 1.185185185= (32/27),
E = 1.25 = (5/4)
F = 1.33333......= (4/3)
????????? appearently here lacks F# pitch ?????, hence 45/32 inserted
G = 1.5 = (3/2)
Ab = 1.58740105196 = ~~19/12 better ~27/17
A = 1.67232374199 = ~~5/3 better ~97/58
Bb = 1.7777777.....=(16/9)
B = 1.875 = (15/8)
2C = 2
>

or when expressed in
http://www.xs4all.nl/~huygensf/scala/scl_format.html
terms as file:

! PizarroJIapprox.scl
!
Rational approximation of Mario Pizarro's JI, compiled by A.Sparschuh
12
!
18/17
28/25 ! 10/9 < 9.333333.../8.333333... < 9/8
32/27
5/4
4/3
45/32 ! lacking F# amended due to JI 5th "2C#/F# -- 1.5" specification
3/2
27/17 ! superparticular = 2.7/1.7 or more coarse ~19/12 ?
97/58 ! near ~5/3
16/9
15/8
2/1
!
!

> Below, I detail the twelve perfect
> and imperfect fifths. Seven pure fifths:
>
recheck again Mario's cicle of 5ths once more anew:

G/C -- 1.5
2D/G -- 1.49323976284 = ~(3/2)(221/222)
A/D -- 1.49323976284 = the same as 2D/G
2E/A -- 1.49492585512 = ~589/394 = (3/2)(294/295)
B/E -- 1.5
2F#/B -- 1.5050917638 = ~444/295 = (3/2)(295/294) widend 5th
!
2C#/F# -- 1.5 ! use that JI-5th in order to replenish the scale on F#
!
Ab/C# -- 1.5
2Eb /Ab -- 1.49323976284 = the same as in the above 2D/G case
Bb/Eb -- 1.5
2F/Bb -- 1.5
2C/F -- 1.5

bye
A.S.

🔗Claudio Di Veroli <dvc@...>

Thanks Mario for the scale and Andreas for translating it into fifth ratios!

What I utterly fail to understand is why both of you are so fond of working
with all those unending numbers, meaningless for the non-specialist, when
this temperament can be most simply described by the Circle of Fifths in
terms of fractions of the Pythagorean Comma, thus:

0 0 0 0 -1/3 -1/3 -1/4 0 +1/4 0 0 -1/3
Eb---Bb---F----C----G----D----A----E----B----F#---C#---G#---D#

The above is not only very simple, but also shows that this is NOT a JI
scale.
It is instead a circular temperament in the style of d'Alembert, with some
Neidhardtian touches.
Millions of similar temperaments can be computer generated.
The major thirds C-E and G-E are pure(*), but they deteriorate rapidly and
E-G# is useless, 27.4 Cents wide.

Perhaps Mario can explain us better what this system meant for: what is its
practical use in musical performance.

Kind regards,

Claudio

(*) It looks strange that a P.c-based temperament produces pure major
thirds, like a S.c. one. This is because -1/3-1/3-1/4=-11/12 P.c. which is,
by a mathematical coincidence, ALMOST identical to the S.c. (as explained in
my UT book , p.79, footnote)

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Andreas Sparschuh
Sent: 06 March 2009 16:52
To: tuning@yahoogroups.com
Subject: [tuning] Re: I just built my first JI scale

--- In tuning@yahoogroups. <mailto:tuning%40yahoogroups.com> com, "Mario
Pizarro" <piagui@...> wrote:
>
> Eb= (32/27), E= (5/4)= 1.25, F= (4/3)= 1.3333....,
> G= (3/2)= 1.5, Bb= (16/9) = 1.7777....,
> B= (15/8)= 1.875.

Hola Mario,

> Three narrow fifths with equal values of

using the:
http://superspace. <http://superspace.epfl.ch/approximator/>
epfl.ch/approximator/
yields for yours decimal numbers the following ratios:

1.49323976284 let (1.5/1.49323976284)=1.0045272282 = ~ 222/221

> which is one third of the Pythagorean comma:
(1.0045272282)^3 = 1.01364326477
= ~74/73 or better ~223/220 .... precisely 3^12/2^19 exact

> Additionally, the self cancelling pair are
> 1.5050913678
= ~444/295

> and 1.49492585512
= ~589/394

or when expressed in
http://www.xs4all. <http://www.xs4all.nl/~huygensf/scala/scl_format.html>
nl/~huygensf/scala/scl_format.html
terms as file:

> Below, I detail the twelve perfect
> and imperfect fifths. Seven pure fifths:
>
recheck again Mario's cicle of 5ths once more anew:

bye
A.S.

🔗Michael Sheiman <djtrancendance@...>

--The above is not only very simple, but also shows that this
is NOT a JI scale.

---It is instead a circular temperament in the style of
d'Alembert, with some Neidhardtian ---touches. Millions of similar temperaments can be computer
generated.

So, basically it is a circular mean-tone temperament with a few notes 1/4 to 1/3 below or above the result of a circle of perfect 5ths?

>>>>>
My on-going question is, with mean-tone scales generated by numbers very close to 1.5 (the way most mean-tone scales are generated) is what does the change give as a notice-able advantage?

And, furthermore (especially concerning this new scale)...how does JI really differ that much in sound vs. mean-tone tunings built to approximate JI in general? Such, JI will almost always be purer on the whole...but by how much? I don't think calling the above tuning "quasi-JI" would be that much of a stretch.
<<<<<

It also seems to me, in most cases, different types of mean-tone just make certain types of chords more consonant and others less so...rather than expanding the number of good chords available. LucyTuning seems to do a half decent job of that but most other mean-tone tunings, at least to me, just seem to trade consonances and dissonances in different places rather than expand overall sense of consonant possibilities within the tuning.

On the surface, at least, it seems to go back to that, historically,
certain cultures prefer purifying certain intervals more than others,
and slightly altering generators in mean-tone can apparently make such
results.

It also seems to me that different types of 5-limit JI have the same "make something sweeter, another thing becomes more sour" trade-off problem...although I admit 7-limit JI could open some doors to everyday musicians far as offering more expressive possibilities with relatively little "dissonance-penalty" vs. 12TET.

-Michael
--- On Fri, 3/6/09, Claudio Di Veroli <dvc@...> wrote:

From: Claudio Di Veroli <dvc@...>
Subject: RE: [tuning] Re: I just built my first JI scale
To: tuning@yahoogroups.com
Date: Friday, March 6, 2009, 12:13 PM

Thanks Mario for the scale and Andreas for translating it
into fifth ratios!

What I utterly fail to understand is why both of
you are so fond of working with all those unending numbers, meaningless for
the non-specialist, when this temperament can be most simply described by
the Circle of Fifths in terms of fractions of the Pythagorean Comma,
thus:

   0    0
0    0  -1/3 -1/3 -1/4   0
+1/4   0    0   -1/3

Eb---Bb---F- ---C----G- ---D----A- ---E----B- ---F#---C# ---G#---D#

The above is not only very simple, but also shows that this
is NOT a JI scale.
It is instead a circular temperament in the style of
d'Alembert, with some Neidhardtian touches.
Millions of similar temperaments can be computer
generated.
The major thirds C-E and G-E are pure(*), but they
deteriorate rapidly and E-G# is useless, 27.4 Cents wide.

Perhaps Mario can explain us better what this system
meant for: what is its practical use in musical performance.

Kind regards,

Claudio

(*) It looks strange that a P.c-based temperament produces
pure major thirds, like a S.c. one. This is because
-1/3-1/3-1/4= -11/12 P.c. which is, by a mathematical coincidence, ALMOST
identical to the S.c. (as explained in my UT book , p.79,
footnote)

From: tuning@yahoogroups. com
[mailto:tuning@ yahoogroups. com] On Behalf Of Andreas
Sparschuh
Sent: 06 March 2009 16:52
To:
tuning@yahoogroups. com
Subject: [tuning] Re: I just built my first
JI scale

--- In tuning@yahoogroups. com, "Mario
Pizarro" <piagui@...> wrote:
>
> Eb= (32/27), E= (5/4)=
1.25, F= (4/3)= 1.3333....,
> G= (3/2)= 1.5, Bb= (16/9) = 1.7777....,

> B= (15/8)= 1.875.

Hola Mario,

> Three narrow fifths
with equal values of

using the:
http://superspace. epfl.ch/approxim ator/
yields
for yours decimal numbers the following ratios:

1.49323976284 let
(1.5/1.49323976284) =1.0045272282 = ~ 222/221

> which is one
third of the Pythagorean comma:
(1.0045272282) ^3 = 1.01364326477

= ~74/73 or better ~223/220 .... precisely 3^12/2^19 exact

>
Additionally, the self cancelling pair are
> 1.5050913678
=
~444/295

> and 1.49492585512
= ~589/394

> C = 1
C# =
1.05826736797 = ~18/17
D = 1.11992982213 = ~~9/8 better ~28/25
Eb =
1.185185185= (32/27),
E = 1.25 = (5/4)
F = 1.33333..... .=
(4/3)
????????? appearently here lacks F# pitch ?????, hence 45/32
inserted
G = 1.5 = (3/2)
Ab = 1.58740105196 = ~~19/12 better ~27/17
A
= 1.67232374199 = ~~5/3 better ~97/58
Bb = 1.7777777... ..=(16/9)
B
= 1.875 = (15/8)
2C = 2
>

or when expressed in
http://www.xs4all. nl/~huygensf/ scala/scl_ format.html
terms
as file:

! PizarroJIapprox. scl
!
Rational approximation of
Mario Pizarro's JI, compiled by A.Sparschuh
12
!
18/17
28/25 !
10/9 < 9.333333.../ 8.333333. .. <
9/8
32/27
5/4
4/3
45/32 ! lacking F# amended due to JI 5th "2C#/F#
-- 1.5" specification
3/2
27/17 ! superparticular = 2.7/1.7 or more
coarse ~19/12 ?
97/58 ! near
~5/3
16/9
15/8
2/1
!
!

> Below, I detail the twelve
perfect
> and imperfect fifths. Seven pure fifths:
>
recheck
again Mario's cicle of 5ths once more anew:

G/C -- 1.5
2D/G --
1.49323976284 = ~(3/2)(221/222)
A/D -- 1.49323976284 = the same as
2D/G
2E/A -- 1.49492585512 = ~589/394 = (3/2)(294/295)
B/E --
1.5
2F#/B -- 1.5050917638 = ~444/295 = (3/2)(295/294) widend
5th
!
2C#/F# -- 1.5 ! use that JI-5th in order to replenish the scale on
F#
!
Ab/C# -- 1.5
2Eb /Ab -- 1.49323976284 = the same as in the
above 2D/G case
Bb/Eb -- 1.5
2F/Bb -- 1.5
2C/F -- 1.5

bye
A.S.

🔗monz <joemonz@...>

Hi Claudio, Mario, and Andreas,

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
>
> Thanks Mario for the scale and Andreas for translating
> it into fifth ratios!
>
> What I utterly fail to understand is why both of you
> are so fond of working with all those unending numbers,
> meaningless for the non-specialist, when this temperament
> can be most simply described by the Circle of Fifths in
> terms of fractions of the Pythagorean Comma, thus:
>
> 0 0 0 0 -1/3 -1/3 -1/4 0 +1/4 0 0 -1/3
> Eb---Bb---F----C----G----D----A----E----B----F#---C#---G#---D#
>
> The above is not only very simple, but also shows that
> this is NOT a JI scale.
> It is instead a circular temperament in the style of
> d'Alembert, with some Neidhardtian touches.
> Millions of similar temperaments can be computer generated.

I agree that this is the clearest way to describe.
However, i would also say that no matter how one
describe a scale like this, the description will
be meaningless for the non-specialist! ;-)

> The major thirds C-E and G-E are pure(*), but they
> deteriorate rapidly and E-G# is useless, 27.4 Cents wide.
>
> Perhaps Mario can explain us better what this system
> meant for: what is its practical use in musical performance.
>
> Kind regards,
>
> Claudio
>
> (*) It looks strange that a P.c-based temperament produces
> pure major thirds, like a S.c. one. This is because
> -1/3-1/3-1/4=-11/12 P.c. which is, by a mathematical
> coincidence, ALMOST identical to the S.c. (as explained
> in my UT book , p.79, footnote)

I also explain that coincidence here:

http://tonalsoft.com/enc/number/12edo.aspx

about halfway down the page.

It is indeed a very strange coincidence that
1/12 pythagorean-comma is so close to 1/11 syntonic-comma.
That is because the difference between the two commas
is so small, the skhisma with ratio 32805/32768 or
2,3,5-monzo [-15 8, 1> of ~2 cents -- which also happens
to be ~1/11 of 81:80 and ~1/12 of pythagorean-comma.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

🔗Mark Rankin <markrankin95511@...>

Mario,

I can see that you like to get your discoveries published on the internet tuning list, that you take pleasure in getting this information "out there" to the public, to any and all readers, as soon as possible, but I have a suggestion for you:

Instead of informing us, your readers, that:

A) You are "copying this email to Claudio di Veroli who might evaluate the JI scale that [you] just worked out", and that:

B)  "The remaining tone frequencies are given in decimal numbers which will be transformed to commom fractions in a matter of days", why not approach things in the following manner:

Why not 1) Send a "personal" email to Claudio di Veroli asking him if he would evaluate the JI scale that you just worked out, and, why not:

2) Hold off mentioning to your public that "The remaining tone frequencies are given in decimal numbers which will be transformed to common fractions in a matter of days"?

This way, X)  You can avoid the possibility that, for whatever reason, Mister Di Veroli might not be able to evaluate the JI scale you just worked out, or that he might not be able to evaluate it any time soon, and Y) You can avoid the possibility that the tone frequencies might not be able to be transformed to common fractions in "a matter of days".

This way if, by chance, neither the evaluation of the new JI scale, nor the transforming of the decimal numbers to fractions take place when you thought they would, you will not have "wasted your time and the time of your readers".

In other words Mario, why not hold off on publishing these things until they are complete and ready to be published on the internet?

Why "Jump the gun" and give us a partially finished story one day, and then an almost finished story the next day, and perhaps, if we're lucky, a finished story on the third day?

Why not "Get all your ducks in a row" first, and then present them to us as a
"fait accompli" on the third day? Don't you see, it makes for better reading that way.

Finally, Mario, I would like to tell you, and the tuning list, that once before, many months ago, I addressed an email to both you and the tuning list. I don't remember the details, but I do remember that I tried to write it in Spanish because I had spent nine months in Peru and know some Spanish, but you got very angry and exploded.

I want you and everyone tuning in now to know that I meant nothing small minded or angry toward you then, and I mean nothing small minded or angry toward you now.

-- Mark Rankin

--- On Thu, 3/5/09, Mario Pizarro <piagui@...> wrote:

From: Mario Pizarro <piagui@...>
Subject: [tuning] I just built my first JI scale
To: "tuning yahoogroups" <tuning@yahoogroups.com>
Cc: "Mike Battaglia" <battaglia01@...>, "Claudio Di Veroli" <dvc@braybaroque.ie>, "Norma Arias" <Norma.Arias@...>
Date: Thursday, March 5, 2009, 7:27 PM

A new scale whose temperament differs from scales
that were given out in the past cannot be produced
any more; all kind of scales were already built. The
probability to achieve a new one is practically null.
This information I received recently seems to exclude
the Just Intonation scales.

I am copying this e-mail to Claudio Di Veroli who
might evaluate the JI scale that I just worked out.

Mike mentioned the precise perform of (5/4) = 1.25
for note E, so by following Mike´s recommendation,
this tone frequency works in the new JI scale.
Classical JI frequencies in the scale are:
Eb= (32/27), E= (5/4)= 1.25, F= (4/3)= 1.3333....,
G= (3/2)= 1.5, Bb= (16/9) = 1.7777....,
B= (15/8)= 1.875.

Three narrow fifths with equal values of
1.49323976284 let (1.5/1.49323976284) =1.0045272282
which is one third of the Pythagorean comma:
(1.0045272282) ^3 = 1.01364326477.
Additionally, the self cancelling pair are
1.5050913678 and 1.49492585512.

The remaining tone frequencies are given
in decimal numbers which will be transformed
to common fractions in a matter of days.

Tone frequencies of the JI octave follows:

C = 1
C# = 1.05826736797
D = 1.11992982213
Eb = 1.185185185= (32/27),
E = 1.25 = (5/4)
F = 1.33333..... .= (4/3)
G = 1.5 = (3/2)
Ab = 1.58740105196
A = 1.67232374199
Bb = 1.7777777... ..=(16/9)
B = 1.875 = (15/8)
2C = 2

Below, I detail the twelve perfect
and imperfect fifths. Seven pure fifths:

G/C -- 1.5
2D/G -- 1.49323976284
A/D -- 1.49323976284
2E/A -- 1.49492585512*
B/E -- 1.5
2F#/B -- 1.5050917638*
2C#/F# -- 1.5
Ab/C# -- 1.5
2Eb /Ab -- 1.49323976284
Bb/Eb -- 1.5
2F/Bb -- 1.5
2C/F -- 1.5

*Cancelling figures

Pending: Major and Minor thirds.

Hope you reach to a positive conclusion.

Thanks

Mario Pizarro

Lima, March 05, 2009

piagui@ec-red. com

🔗Mario Pizarro <piagui@...>

To Mark Rankin,

-- You can see that I didn´t consider my failed JI as a discovery:
in my message, I´didn´t mention the word "discovery"

Since I know that Claudio Di Veroli and Andreas Sparschuh are
busy men, the main address was the tuning list. The scale,
according to learned people, is having unfortunate features.

Now I will discuss your points:

---Instead of informing us, your readers, that:....
A) You are "copying this email to Claudio di Veroli who might evaluate the JI .....

I addressed the e-mail to the tuning list and copied to Claudio
Di Veroly and Andreas Sparschuh. This way both would get
direct information on the scale; it was up to them to comment
it should they wanted to do it.

What was the wrong step; I informed to the readers as well as
to the mentioned members about the basic information on this
matter, except the converted decimal numbers.

B) "The remaining tone frequencies are given in decimal numbers which wil
---Correct, instead of giving provisional decimal numbers, I´d
---have better get first www.superspace.epfl.ch.approximator
---to give common fractions rather than long decimal numbers.
---However, I couldn´t wait days for the mentioned web and
---sent the scale like it was. Finally, I got the needed tool
---(the web) to be used on similar cases, .

---Why not 1) Send a "personal" email to Claudio di Veroli asking him if he would.....
---The fact that Claudio and Andreas might be busy and
---couldn´t study the scale was a reason to also address it to
---the list where somebody might take the case.

---Right now I feel like as beeing a primary student
---giving explanations to the teacher who is having a
---hidden whip on his back!!!!!.

2) Hold off mentioning to your public that "The remaining tone frequencies are
given in decimal numbers which will be transformed to common fractions in a
matter of days"?

Hold off?. Sir, remember that you are not in your headquarters.
Stop using inapropriate terms!!!!!!

This way, X) You can avoid the possibility that, for whatever reason,
Mister Di Veroli might not be able to evaluate the JI scale you just
worked out, or that he might not be able to evaluate it any time soon,
and Y) You can avoid the possibility that the tone frequencies might
not be able to be transformed to common fractions in "a matter of days".

---- That is why I wrote: "might".

This way if, by chance, neither the evaluation of the new JI scale,
nor the transforming of the decimal numbers to fractions take place
when you thought they would, you will not have "wasted your time
and the time of your readers".

I am glad to terminate this.

Mario Pizarro

Lima, March 09

----- Original Message -----
From: Mark Rankin
To: tuning@yahoogroups.com
Sent: Sunday, March 08, 2009 1:08 AM
Subject: Re: [tuning] I just built my first JI scale

Mario,

I can see that you like to get your discoveries published on the internet tuning list, that you take pleasure in getting this information "out there" to the public, to any and all readers, as soon as possible, but I have a suggestion for you:

Instead of informing us, your readers, that:

A) You are "copying this email to Claudio di Veroli who might evaluate the JI scale that [you] just worked out", and that:

B) "The remaining tone frequencies are given in decimal numbers which will be transformed to commom fractions in a matter of days", why not approach things in the following manner:

Why not 1) Send a "personal" email to Claudio di Veroli asking him if he would evaluate the JI scale that you just worked out, and, why not:

2) Hold off mentioning to your public that "The remaining tone frequencies are given in decimal numbers which will be transformed to common fractions in a matter of days"?

This way, X) You can avoid the possibility that, for whatever reason, Mister Di Veroli might not be able to evaluate the JI scale you just worked out, or that he might not be able to evaluate it any time soon, and Y) You can avoid the possibility that the tone frequencies might not be able to be transformed to common fractions in "a matter of days".

This way if, by chance, neither the evaluation of the new JI scale, nor the transforming of the decimal numbers to fractions take place when you thought they would, you will not have "wasted your time and the time of your readers".

In other words Mario, why not hold off on publishing these things until they are complete and ready to be published on the internet?

Why "Jump the gun" and give us a partially finished story one day, and then an almost finished story the next day, and perhaps, if we're lucky, a finished story on the third day?

Why not "Get all your ducks in a row" first, and then present them to us as a
"fait accompli" on the third day? Don't you see, it makes for better reading that way.

Finally, Mario, I would like to tell you, and the tuning list, that once before, many months ago, I addressed an email to both you and the tuning list. I don't remember the details, but I do remember that I tried to write it in Spanish because I had spent nine months in Peru and know some Spanish, but you got very angry and exploded.

I want you and everyone tuning in now to know that I meant nothing small minded or angry toward you then, and I mean nothing small minded or angry toward you now.

-- Mark Rankin

--- On Thu, 3/5/09, Mario Pizarro <piagui@...> wrote:

From: Mario Pizarro <piagui@...>
Subject: [tuning] I just built my first JI scale
To: "tuning yahoogroups" <tuning@yahoogroups.com>
Cc: "Mike Battaglia" <battaglia01@...>, "Claudio Di Veroli" <dvc@...>, "Norma Arias" <Norma.Arias@bmwna.com>
Date: Thursday, March 5, 2009, 7:27 PM

I am copying this e-mail to Claudio Di Veroli who
might evaluate the JI scale that I just worked out.

Three narrow fifths with equal values of
1.49323976284 let (1.5/1.49323976284) =1.0045272282
which is one third of the Pythagorean comma:
(1.0045272282) ^3 = 1.01364326477.
Additionally, the self cancelling pair are
1.5050913678 and 1.49492585512.

The remaining tone frequencies are given
in decimal numbers which will be transformed
to common fractions in a matter of days.

Tone frequencies of the JI octave follows:

C = 1
C# = 1.05826736797
D = 1.11992982213
Eb = 1.185185185= (32/27),
E = 1.25 = (5/4)
F = 1.33333..... .= (4/3)
G = 1.5 = (3/2)
Ab = 1.58740105196
A = 1.67232374199
Bb = 1.7777777... ..=(16/9)
B = 1.875 = (15/8)
2C = 2

Below, I detail the twelve perfect
and imperfect fifths. Seven pure fifths:

*Cancelling figures

Pending: Major and Minor thirds.

Hope you reach to a positive conclusion.

Thanks

Mario Pizarro

Lima, March 05, 2009

piagui@ec-red. com

;

🔗Andreas Sparschuh <a_sparschuh@...>

> > --- On Fri, 3/6/09, Claudio Di Veroli <dvc@...> wrote:
> > this temperament can be most simply described by
> > the Circle of Fifths in terms of fractions of the Pythagorean
> > Comma, thus:
>
> > 0 0
> > 0 0 -1/3 -1/3 -1/4 0
> > +1/4 0 0 -1/3
>
> Eb---Bb---F- ---C----G- ---D----A- ---E----B- ---F#---C# ---G#---D#
>
I tryed to explain Mario's scale as deviation vom JI-intervals by my

>> Rational approximation of
>> Mario Pizarro's almost ~JI, compiled by A.Sparschuh
>> 12
>> !
>> 18/17
>> 28/25 = (10/9)(126/125 {~+13.8Cents}) = (9/8)(224/225 {~7.7Cents})
>> 9/8
>> 32/27
>> 5/4
>> 4/3
>> 45/32
>> 3/2
>> 27/17 = (8/5)(135/136 {~-12.8Cents})
>> 97/58 = (5/3)(291/290 {~+5.96Cents})
>> 16/9
>> 15/8
>> 2/1

As Claudio mentioned such 1/3 PC variants have a long tradition
among classical well-temperaments:
Here an similar early 19th-century example without any wide 5th:
http://groenewald-berlin.de/tabellen/TAB-041.html
Stanhope 1801:

Ab Eb Bb F C G pc^(-1/3) D pc^(-1/3) A pc^(-1/3) E B F# C# G#

How about to combine that with
Claudio's similar recommendation
of his "French" interpretation:
http://groenewald-berlin.de/ttg/TTG_T078.html
http://groenewald-berlin.de/tabellen/TAB-078.html

" Das tempérament ordinaire nach Veroli"

http://groenewald-berlin.de/graphik-tabelle/GRA-078.html
(Sorry, unfortunately that specification there is
only available in German)

Deviations of Claudio's 5ths from pure 3/2 over the circle in Cents:

Ab +3.425((sc^+1/4)/schisma) Eb +3.910(pc^+1/6) F +1.955(pc^(+1/12) C
C - G - D - A - E - B with each lowered by -5.377 Cents := sc^(1/4)
B -3.910(pc^1/6) F# 1.955(pc^1/12) C# G#

Are that data correct so?

Or use even my similar draft-proposal in absolute-pitch:

5ths
-6: Ab 13 26 52 102 208 416Hz, one Hz above modern neoBaroque a'=415Hz
-5: Eb 39 = 13*3
-4: Bb (39*3=117 234 >) 234.5 (> 235)
-3: F (11 22 44 88 176 352 704 <) 705 = 235*3
-2: C 33
-1: G 99
00: D 74 148 296 (< 297 = 99*3)
+1: A (55 110 220 <) 221 (< 222 = 74*3)
+2: E 165
+3: B (247 494<) 495 = 165*3
+4: F# (185 370 740 <) 741 = 247*3
+5: C# 555 = 185*3
+6: G# 416Hz 832 1664 (< 1665 = 555*3)

concise subdistribution of the PC over the 5ths:

-6: Ab Eb 469/468 Bb 470/468 F 704/705 C G 296/297 D ...
... D 221/222 A 220/221 E B 494/495 F# 740/741 C# 1664/1665 G# :+6

or that deviations from pure 3/2 in Cents from -6:...to...+6:

-6: Ab Eb ~+3.70 Bb ~+3.69 F ~-2.46 C G ~-5.84 D ...
... D ~-7.82 A ~-7.85 E B ~-3.50 F# ~-2.34 C# ~-1.04 G# :+6

or lined up in ascending order

442 a' 120-MetronomeBeats/min above the standard tuning-fork 440Hz
469 bb'
528 c" tenor_C5
555 c#"
592 d"
624 eb"
660 e"
705 f"
741 f#"
792 g"
884 a"

or relative to 1/1

!Sparschuh442French.scl
Sparschuh's neo-Baroque French @ a=442Hz & C:E:G = 4:5:6
12
!
185/176
37/33 ! (9/8)(296/297)
39/33
5/4
235/176 ! (4/3)(705/704)
247/176
3/2
52/33
37/22 ! (5/3)(221/220) due to Scheibler's a'=440Hz choice
469/264 ! bb'/c' (16/9)(234/235)
15/8
2/1
!
!

Works nice squiggle-less for Bach.

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...> wrote/remark:
> It also seems to me that different types of 5-limit JI have the same > "make something sweeter, another thing becomes more sour" trade-off > problem...

alike in
/tuning/topicId_81835.html#81835
and a little bit less in the above case.

> ... although I admit 7-limit JI could open some doors to
> everyday musicians far as offering more expressive possibilities
> with relatively little "dissonance-penalty" vs. 12TET.
>
Agreed,
that there should appear some 7-limit intervals among 5&3-limit ones,
especially the essential "blue-notes",
containing the 7th partial, out of the overtone-series spectre:

1. blue dim.-2nd: 21/20 or 'blue-semitone'
2. blue 3rd: 7/6
3. blue tritone: 7/5 as introduced by Charlie Parker, the "Bird"
4. blue 7th: 7/4

Hence:
How about the following alternative 7-limit ~JI cycle of 5ths?

F: 11 22 44 88 := 440Hz/5 an JI-3rd below standard Normal-Pitch
C: 33
G: 99
D: (111/3 = 37 74 148 296 <) 297
A: 55 110 (< 111 = 37*3) 11*5
E: 165
B: (7.7 15.4 ... 246.4 492.8 <)495
F# 23.1
C# (1.1 2.2 ... 35.2 70.4 >) 69.3
G# 3.3
Eb (9.9 19.8 39.6 79.2 >) 77 = 11*7
Bb (87/3 = 29 58 116 232 <) 231 = 77*3
F: 11 22 44 88 (> 87 = 29*3)

or lined up in ascending absolute-pitches,
yielding blues-chromatic on the keys of the middle-octave:

c' 264 middle_C4
c# 277.2
d' 297
eb 308
e' 330
f' 352
f# 369.6
g' 396
g# 422.4
a' 440 Hz
bb 462
b' 495
c" 524 tenor_C5

or in relative ratios versus 1/1 on electric MIDI-instruments

!Spa_s_s_7_lim.scl
Sparschuh's 7-limit ~JI cycle of a dozen tempered 5ths
12
!
21/20!C# ~ 084cents
9/8 ! D. ~ 204
7/6 ! Eb ~ 267
5/4 ! E. ~ 386
4/3 ! F. ~ 498
7/5 ! F# ~ 583
3/2 ! G. ~ 702
8/5 ! G# ~ 814
5/3 ! A. ~ 884
7/4 ! Bb ~ 969
15/8! B. ~1088
2/1 ! C'
!
!
Sounds terrific on my good-old Wilhelminian-style piano.
Quest:
Who in that group here can reccomend me even more simple
7-lim. ratios for the strings in my antique instrument,
than the actual applied ones?

bye
A.S.

🔗djtrancendance@...

🔗Mario Pizarro <piagui@...>

Hi Andreas,

How are you my friend.

I won´t be ashamed when I ask you a very elemental question.

¿What on earth mean 5 and 7 limits in a JI scale?

-- Five or seven tones per octave? I don´t think so.
-- Limits of the number of tones that make a chord?
-- You know that two days ago I sent to the list a JI scale data (JI ?) and created some noise too.

My kingdom for two data: ¿What on earth mean those limits and what does a JI (Just Intonation) mean or differs from any other scale?.

Once I get your information, finally, I will sleep.

Andreas: Please don´t forget to your best friend and let him sleep.

Gracias

Mario Pizarro

Lima, March 09

----- Original Message -----
From: Andreas Sparschuh
To: tuning@yahoogroups.com
Sent: Monday, March 09, 2009 3:48 PM
Subject: [tuning] Cyclic 5- & 7-limit scales, that are almost ~JI, wasRe: I just built my first JI

Ab Eb Bb F C G pc^(-1/3) D pc^(-1/3) A pc^(-1/3) E B F# C# G#

How about to combine that with
Claudio's similar recommendation
of his "French" interpretation:
http://groenewald-berlin.de/ttg/TTG_T078.html
http://groenewald-berlin.de/tabellen/TAB-078.html

" Das tempérament ordinaire nach Veroli"

http://groenewald-berlin.de/graphik-tabelle/GRA-078.html
(Sorry, unfortunately that specification there is
only available in German)

Deviations of Claudio's 5ths from pure 3/2 over the circle in Cents:

Ab +3.425((sc^+1/4)/schisma) Eb +3.910(pc^+1/6) F +1.955(pc^(+1/12) C
C - G - D - A - E - B with each lowered by -5.377 Cents := sc^(1/4)
B -3.910(pc^1/6) F# 1.955(pc^1/12) C# G#

Are that data correct so?

Or use even my similar draft-proposal in absolute-pitch:

concise subdistribution of the PC over the 5ths:

-6: Ab Eb 469/468 Bb 470/468 F 704/705 C G 296/297 D ...
... D 221/222 A 220/221 E B 494/495 F# 740/741 C# 1664/1665 G# :+6

or that deviations from pure 3/2 in Cents from -6:...to...+6:

-6: Ab Eb ~+3.70 Bb ~+3.69 F ~-2.46 C G ~-5.84 D ...
... D ~-7.82 A ~-7.85 E B ~-3.50 F# ~-2.34 C# ~-1.04 G# :+6

or lined up in ascending order

442 a' 120-MetronomeBeats/min above the standard tuning-fork 440Hz
469 bb'
528 c" tenor_C5
555 c#"
592 d"
624 eb"
660 e"
705 f"
741 f#"
792 g"
884 a"

or relative to 1/1

Works nice squiggle-less for Bach.

alike in
/tuning/topicId_81835.html#81835
and a little bit less in the above case.

1. blue dim.-2nd: 21/20 or 'blue-semitone'
2. blue 3rd: 7/6
3. blue tritone: 7/5 as introduced by Charlie Parker, the "Bird"
4. blue 7th: 7/4

Hence:
How about the following alternative 7-limit ~JI cycle of 5ths?

or lined up in ascending absolute-pitches,
yielding blues-chromatic on the keys of the middle-octave:

c' 264 middle_C4
c# 277.2
d' 297
eb 308
e' 330
f' 352
f# 369.6
g' 396
g# 422.4
a' 440 Hz
bb 462
b' 495
c" 524 tenor_C5

or in relative ratios versus 1/1 on electric MIDI-instruments

bye
A.S.

__________ Información de NOD32, revisión 3911 (20090305) __________

Este mensaje ha sido analizado con NOD32 antivirus system
http://www.nod32.com

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
Hola Mario,
>
> Hi Andreas,
>
> How are you my friend.
Well fine.
>
>....elemental question.
> ï¿½What on earth mean 5 and 7 limits in a JI scale?

http://en.wikipedia.org/wiki/Limit_(music)
http://tonalsoft.com/enc/l/limit.aspx
>
Not to be confused with:
> -- Five....
http://en.wikipedia.org/wiki/Pentatonic_scale
http://es.wikipedia.org/wiki/Escala_pentat%C3%B3nica
> .... or seven tones per octave?
http://en.wikipedia.org/wiki/Heptatonic_scale

> -- Limits of the number of tones that make a chord?
No,
a 'limit' is determinated by the highest used

http://en.wikipedia.org/wiki/Prime_number
http://es.wikipedia.org/wiki/N%C3%BAmero_primo

in both numerator versus denominator
of an musical interval-ratio:
http://en.wikipedia.org/wiki/Denominator
http://es.wikipedia.org/wiki/Fracci%C3%B3n
http://es.wikipedia.org/wiki/N%C3%BAmero_racional

> -- You know that two days ago I sent to the list
> a JI scale data (JI ?) and created some noise too.
In deed,
there had been some ado about yours ~JI? proposal,
but meanwhile the situation got rectified peacefully.

> My kingdom for two data:
>ï¿½What on earth mean those limits
see above (lat. vide supra)

> and what does a JI (Just >Intonation)
> mean....
http://en.wikipedia.org/wiki/List_of_intervals_in_5-limit_just_intonation
http://organicdesign.org/peterson/tuning/ji.html
http://www.kylegann.com/tuning.html

> or differs from any other scale?.
Example:
For assessing 5-limit Simply check, if all fractions
numerators and denominators belong to the:
http://en.wikipedia.org/wiki/Regular_number
s.
Analogous,
for 7-limit check whether they belong to
http://www.research.att.com/~njas/sequences/A002473
or even not:
Generally study the concept of
http://en.wikipedia.org/wiki/Smooth_number
s
>
> Once I get your information, finally, I will sleep.
I hope so too, that now you can go to the land of nod.
>
> Andreas:
> Please donï¿½t forget to your best friend and let him sleep.
I wish yours dreams come true.
Que te vaya muy bien.
>
> Gracias
Â¡de nada!
Â¡chao!
A.S.

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:

> 21/20!C# ~ 084cents
> 9/8 ! D. ~ 204
> 7/6 ! Eb ~ 267
> 5/4 ! E. ~ 386
> 4/3 ! F. ~ 498
> 7/5 ! F# ~ 583
> 3/2 ! G. ~ 702
> 8/5 ! G# ~ 814
> 5/3 ! A. ~ 884
> 7/4 ! Bb ~ 969
> 15/8! B. ~1088
> 2/1 ! C'
>

Hi Michael,

> ï¿½ï¿½ï¿½ Ah, a new tuning challenge (cool)! :-)
try to play on a piano in that one
the 2 extended major-7-chords:

1.Subdominant:
F:A:C:Eb:G == (F=11)*(4:5:6:7:9) == (4/3)*(4:5:6:7:9)

2.Tonic:
C:E:G:Bb:D == (C=33)*4:5:6:7:9

and finally
3. the only partial available Submediant tri-chord:
G#:C:F# == (G#=3.3)*(4:5:7) == (8/5)*(4:5:7)

>
> I came up with the following
>(which is VERY close to your suggestion,
> only I took out the 21/20 and added an 11/6)
>
> ï¿½ 1
> ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½
> (note 17/16 could fit well here if you wanted a 13-note tuning)

Sorry, negative indication:
but my old piano allows only a dozen pitches per octave.

> ï¿½ 9/8ï¿½ï¿½ 1.125ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½
> ï¿½ 7/6 = 1.1666666666
> ï¿½ 5/4 = 1.25
> ï¿½ 4/3 = 1.3333
> ï¿½ 7/5 = 1.4
> ï¿½ 3/2 = 1.5
> ï¿½ 8/5 = 1.6ï¿½
> ï¿½ 5/3 = 1.66666
> ï¿½ 7/4 = 1.75

> !!!11/6 = 1.83333!!! (new note related to 5/3 which = 10/6)
That's opens the doors into 11-limit,
above the 7-limit concept of 'blue-notes' on the Swiss:
http://silvertone.princeton.edu/~ted/alphorn.html
Quote:
'Intervallic relationships'...
"Very important: the seventh partial, written Bb, middle line treble clef, is a lowered 7th -- it sounds flat [as it should]. Also, the Alphorn FA, the 11th partial, written F#, top line treble clef, is a raised 4th leading to the G [written]: in the key of F# it sounds in-between B natural and C natural; it is a very distinct sound. These notes are obviously not out of tune but part of a natural tuning which western music has trained musicians to think is out of tune! [just intonation junkies can come back now. -t.]"

http://www.people.iup.edu/rahkonen/ilwm/Switzerland.bib.htm
"The instrument restricts players to the overtone series of the 11th partial (F in a C scale) characteristically sharp, is commonly called alphorn fa."

http://en.wikipedia.org/wiki/Harmonic_series_(music)

> ï¿½ 15/8 = 1.875
>
> ï¿½ï¿½ Of course, it would be nice to have something that would fit
> nicely near the midpoint between 1 and 9/8 for my scale...

Otherwise the tension due to overstressing the strings would lead
to the risk of break.

>but that would involve making the denominator something about 14 or >larger.ï¿½
No problem!
Even
http://plato.stanford.edu/entries/archytas/#Mus
begins all his 3 genera commonly with 28/27 each.
http://www.ex-tempore.org/ARCHYTAS/ARCHYTAS.html
"
Enharmonic: 28/27 36/36 5/4
Diatonic: 28/27 8/7 9/8
Chromatic: 28/27 243/224 32/27
"
>17/16 wouldn't be a bad option in that case. :-)
Why not?
http://www.bambooweb.com/articles/s/e/Semitonium.html
http://www.xs4all.nl/~huygensf/doc/stevinsp.html
Quote:
"
...There is a continuous range of semitones with the values
14 : 15 : 16 : 17 : 18 : 19 : 20 : 21,
from major semitones 14:15 to minor semitones 20:21.
Vincenzo Galilei (Dialogo della musica antica e moderna, Firenze, 1581) disclaimed such a variety of semitones. Before Stevin, he wanted them all to be equal, and he chose the value 17:18, which happens to be midway...."

My view:
I see no problem as long as we stay persisting within
the old variety in range of semitones before Stevin & Galilei.

bye
A.S.

🔗djtrancendance@...

🔗Mark Rankin <markrankin95511@...>

Mario,

You are amazing! I have looked in my English-Espanol Dictionary to present the English and Spanish words to explain your logic. Here they are:

Terco - Stubborn
Testarudo - Obstinate
Porfiado - Intractable

I tried carefully to explain to you that I think it is better to compose an idea completely, and present a "finished product" to your readers, than it is to send your readers 1/3 of an idea one day, then another 1/3 of an idea another day and then a final 1/3 of an idea on a third day.

Why not present a nice finished product, hermano?

Su amigo,

Marco

--- On Sun, 3/8/09, Mario Pizarro <piagui@...> wrote:

From: Mario Pizarro <piagui@...>
Subject: Re: [tuning] I just built my first JI scale
To: tuning@yahoogroups.com
Cc: "Claudio Di Veroli" <dvc@...>, "Andreas Sparschuh" <a_sparschuh@...>
Date: Sunday, March 8, 2009, 2:44 PM

To Mark Rankin,

-- You can see that I didn´t consider my failed JI as a discovery:
in my message, I´didn´t mention the word "discovery"

Since I know that Claudio Di Veroli and Andreas Sparschuh are
busy men, the main address was the tuning list. The scale,
according to learned people, is having unfortunate features.

Now I will discuss your points:

---Instead of informing us, your readers, that:....
A) You are "copying this email to Claudio di Veroli who might evaluate the JI .....

I addressed the e-mail to the tuning list and copied to Claudio
Di Veroly and Andreas Sparschuh. This way both would get
direct information on the scale; it was up to them to comment
it should they wanted to do it.

What was the wrong step; I informed to the readers as well as
to the mentioned members about the basic information on this
matter, except the converted decimal numbers.

B) "The remaining tone frequencies are given in decimal numbers which wil
---Correct, instead of giving provisional decimal numbers, I´d
---have better get first www.superspace. epfl.ch.approxim ator
---to give common fractions rather than long decimal numbers.
---However, I couldn´t wait days for the mentioned web and
---sent the scale like it was. Finally, I got the needed tool
---(the web) to be used on similar cases, .

---Why not 1) Send a "personal" email to Claudio di Veroli asking him if he would.....
---The fact that Claudio and Andreas might be busy and
---couldn´t study the scale was a reason to also address it to
---the list where somebody might take the case.

---Right now I feel like as beeing a primary student
---giving explanations to the teacher who is having a
---hidden whip on his back!!!!!.

2) Hold off mentioning to your public that "The remaining tone frequencies are
given in decimal numbers which will be transformed to common fractions in a
matter of days"?

Hold off?. Sir, remember that you are not in your headquarters.
Stop using inapropriate terms!!!!!!

This way, X) You can avoid the possibility that, for whatever reason,
Mister Di Veroli might not be able to evaluate the JI scale you just
worked out, or that he might not be able to evaluate it any time soon,
and Y) You can avoid the possibility that the tone frequencies might
not be able to be transformed to common fractions in "a matter of days".

---- That is why I wrote: "might".

This way if, by chance, neither the evaluation of the new JI scale,
nor the transforming of the decimal numbers to fractions take place
when you thought they would, you will not have "wasted your time
and the time of your readers".

I am glad to terminate this.

Mario Pizarro

Lima, March 09

----- Original Message -----
From: Mark Rankin
To: tuning@yahoogroups. com
Sent: Sunday, March 08, 2009 1:08 AM
Subject: Re: [tuning] I just built my first JI scale

--- On Thu, 3/5/09, Mario Pizarro <piagui@ec-red. com> wrote:

From: Mario Pizarro <piagui@ec-red. com>
Subject: [tuning] I just built my first JI scale
To: "tuning yahoogroups" <tuning@yahoogroups. com>
Cc: "Mike Battaglia" <battaglia01@ gmail.com>, "Claudio Di Veroli" <dvc@braybaroque. ie>, "Norma Arias" <Norma.Arias@ bmwna.com>
Date: Thursday, March 5, 2009, 7:27 PM

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
> ï¿½ï¿½ï¿½ï¿½ Ok, so take your pick: 17/16 vs.ï¿½ 11/6...
> which do you think fits in your scale better vs. 21/20
> (and I'd be interested to know why you pick which of those two)?
> :-)
>
Hi Michael,
in order to have all that two 21/20 & 17/16 both available
at the same time,
consider an 53-scale that is located somewhere inbetween:

http://en.wikipedia.org/wiki/53_equal_temperament
and the approximated ratios inhttp://www.microtonal-synthesis.com/scale_53tet.html
.
The new cyle consists of
53 times 5ths taken upwards
and 31 octaves downwards, due to Mercator's-Comma

(3/2)^53 / 2^31

when applying the usual
http://commator.googlepages.com/BOSANQUET.pdf
'comma-shift' notation system for any 53-tone scales:

00 C- 1 C\\
01 G- 3 G\\
02 D- 9 D\\
03 A- 27 A\\
04 E- 81 E\\
05 B- 243 B\\
06 GB 729 = 3^6 Gb\
07 DB547 1094 2188 ( > 2187 = 3^7 ) Db\
08 AB 821 1642 ( > 1641 = DB*3 ) Ab\
09 EB 77 154 318 1232 2464 ( > 2463 = AB*3) Eb\
10 BB 221
11 F\ 693
12 C\ 65 130 260 520 1040 2080 ( > 2079 = F\*3 )
13 G\ 195
14 D\ 293 586 ( > 585 = G\*3 )
15 A\ 55 110 220 440Hz 880 ( > 879 = D\*3 )
16 E\ 165
17 B\ 495
18 Gb 1485
19 Db 2227 4454 ( < 4455 = Gb*3 )
20 Ab 835 1670
21 Eb 313 626 1252 2504 ( < 2505 = Ab*3 )
22 Bb 939
23 F. 11 22 44 88 176 352 704 1408 2816 ( < 2817 = Bb*3 )
24 C. 33
25 G. 99
26 D. 297
27 A. 445Hz 890 ( < 891 = D.*3 )
28 E. 667 1334 ( < 1335 = A.*3 )
29 B. 125 250 500 1000=1kHz 2000 ( < 2001 = E.*3 ) 5^3
30 F# 375
31 C# 281 562 1124 ( < 1125 = F#*3 )
32 G# 843
33 D# 79 158 316 632 1264 2528 ( < 2529 = D#*3 )
34 A# 237
35 F/ 365 710 ( < 711 = A#*3 )
36 C/ 133 266 532 1064 ( < 1063 = F/*3 )
37 G/ 399
38 D/ 299 598 1196 ( < 1197 = G/*3 )
39 A/ 7 14 28 56 112 224 448 896 ( < 897 = D/*3 )
40 E/ 21
41 B/ 63
42 F& 189 F#/
43 C& 71 142 284 568 ( > 567 = F&*3 ) C#/
44 G& 213 G#/
45 D& 5 10 20 40 80 160 320 640 ( > 639 = 3*G& ) D#/
46 A& 15
47 F+ 45 F// = E& = E#/
48 C+ 135 C//
49 G+ 405
50 D+ 1215
51 A+ 911 1822 3644 ( < 3545 = 3*D+ )
52 E+ 683 1366 2732 ( < 2733 = 3*A+ )
53 B+ 1 2 4... 2048 ( < 2049 = 3*E+ ) = C- = C\\

That's lined up modulo-octave in ascending order
as absolute pitches in 53 consecutive commata steps

@ 00 C- 512Hz sopran_c-'
A 01 C\ 520
B 02 C. 528
C 03 C/ 532
D 04 C+ 540
E 05 DB 547
F 06 Db 556.75
G 07 C# 562
H 08 C& 568
I 09 D- 576
J 10 D\ 586
K 11 D. 594
L 12 D/ 598
M 13 D+ 607.5
N 14 EB 616
O 15 Eb 626
P 16 D# 632
Q 17 D& 640
R 18 E- 648
S 19 E\ 660
T 20 E. 667
U 21 E/ 672
V 22 E+ 683 F-
W 23 F\ 693
X 24 F. 704
Y 25 F/ 710
Z 26 F+ 720
a 27 GB 729
b 28 Gb 742.5
c 29 F# 750
d 30 F& 756
e 31 G- 768
f 32 G\ 780
g 33 G. 792
h 34 G/ 798
i 35 G+ 810
j 36 AB 821
k 37 Ab 835
l 38 G# 843
m 39 G& 852
n 40 A- 864
o 41 A\ 880 = 440Hz*2
p 42 A. 890
q 43 A/ 896
r 44 A+ 911
s 45 BB 924
t 46 Bb 939
u 47 A# 948
v 48 A& 960
w 49 B- 972
x 50 B\ 990
y 51 B. 1000
z 52 B/ 1008
@'53 B+ 1024 sopran_c-"

or when expressed as
http://www.xs4all.nl/~huygensf/scala/scl_format.html
in
http://en.wikipedia.org/wiki/Dyadic_fraction
terms

!Sp_11_lim_53.scl
!
Sparschuh's 11-limit cyclic 53-tone in terms of 'Dyadic fractions'
53
! 1/1 ! @
65/64 ! A
33/32 ! B
133/128 ! C
135/128 ! D
547/512 ! E
2227/2048 ! F
281/256 ! G
71/64 ! H
9/8 ! I
293/256 ! J
297/256 ! K
299/256 ! L
1215/1024 ! M
77/64 ! N
313/256 ! O
79/64 ! P
5/4 ! Q
81/64 ! R
165/128 ! S
667/512 ! T
21/16 ! U
683/512 ! V
693/512 ! W
11/8 ! X
355/256 ! Y
45/32 ! Z
729/512 ! a
1485/1024 ! b
375/256 ! c
189/128 ! d
3/2 ! e
195/128 ! f
99/64 ! g
399/256 ! h
405/256 ! i
821/512 ! j
835/512 ! k
843/512 ! l
213/128 ! m
27/16 ! n
55/32 ! o = 440Hz/256Hz := (A\):(C-)
445/256 ! p
7/4 ! q
911/512 ! r
231/128 ! s
939/512 ! t
237/128 ! u
15/8 ! v
243/128 ! w
495/256 ! x
125/64 ! y
63/32 ! z
2/1 ! @'

Attend that:
There appear some JI overtone-series multi-chords alike:

@ : Q : e : q : I : X == 4:5:6:7:9:11 == C- : D& : G- : A/ : D- : F.

while most other chords do deviate from JI
more or less from that only by small tolerable error of departure.
That intended bias in key-characteristics variation
depends on how far one dares to modulate away from the basic root 1/1 relative in distance according to the harmonic excursion.

bye
A.S.