undefined

this is probably an all-too-FAQ, but

What are the most common representations of the "neutral third" in JI?

11/9 ?
39/32 ?

CB

--- In tuning@y..., Christopher Bailey <cb202@c...> wrote:
>
> this is probably an all-too-FAQ, but
>
> What are the most common representations of the "neutral third" in
JI?
>
>
> 11/9 ?
> 39/32 ?
>
>
> CB

11/9 and 27/22 are probably the two most common in JI. They "add" up
to 3/2, so in a sense "you can't have one without the other". Another
such pair are 16/13 and 39/32, the latter which is the most common
neutral third above the root in _harmonic series tuning_, important
to a few musicians.

In 7-limit JI, the neutral third 49/40 comes up a lot, and
its "partner" 60/49. Also you will find 128/105, and its partner
315/256 (which also comes up in harmonic series tuning).

The simplest ratio-approximation to 350 cents that is more accurate
than 11/9 is 71/58, followed by 224/183, 743/607, . . .

Paul,

<<The simplest ratio-approximation to 350 cents that is more accurate
than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>

Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424, ...

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
>
> <<The simplest ratio-approximation to 350 cents that is more
accurate
> than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>
>
>
> Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424, ...
>
> --Dan Stearns

Oops! The algorithm I used allows negative terms in the continued-
fraction expansion, so skips right over some of these. Perhaps Gene
can comment over on tuning-math.

Paul,

I have a sieve that does this nicely, but I have to pick them out by
hand (this lead to the missing 38/31 a minute ago).

Robert Walker actually made an applet based on this idea sometime last
year or earlier this year I think--Robert?

--Dan Stearns

----- Original Message -----
From: "Paul Erlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, November 05, 2001 5:36 PM
Subject: [tuning] Re: Reply to Christopher Bailey

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> >
> > <<The simplest ratio-approximation to 350 cents that is more
> accurate
> > than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>
> >
> >
> > Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424,
...
> >
> > --Dan Stearns
>
> Oops! The algorithm I used allows negative terms in the continued-
> fraction expansion, so skips right over some of these. Perhaps Gene
> can comment over on tuning-math.
>
>
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> I have a sieve that does this nicely, but I have to pick them out by
> hand (this lead to the missing 38/31 a minute ago).

You missed 27/22 too.

>
> Robert Walker actually made an applet based on this idea sometime
last
> year or earlier this year I think--Robert?
>
> --Dan Stearns

One can do the same analysis by going to
http://www.uq.net.au/~zzdkeena/Music/index.htm and clicking on
_An Excel spreadsheet for calculating the size of MOS scales_.
Of course, we're not going to be calculating the size of any MOS
scale, but the mathematics is the same. Enter 1.22405354330466 (or =2^
(350/1200)) as "generator" and 1 as "period". You'll immediately see
that the solution is

11/9, 16/13, 27/22, 38/31, 49/40, 60/49,
71/58, 82/67, 153/125,
224/183, 295/241,
519/424,
734/607, 1262/1031 . . .

The leftmost column is the set of _convergents_, meaning they do
better than any simpler ratio, "better" meaning error times
complexity is lower. My algorithm looked only at the convergents, but
it skipped some because I was allowing negative terms in the CF
expansion.

The algorithm in the Excel spreadsheet is very simple -- you should
be able to see it and code it up in your favorite computer language.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > I have a sieve that does this nicely, but I have to pick them out
by
> > hand (this lead to the missing 38/31 a minute ago).
>
> You missed 27/22 too.

Oops -- I now see that some of the semiconvergents (according to
Dave's spreadsheet) are not closer approximations that their
predecessors at all! Sorry!

Gene, if there's a simple algorithm that gets us only these
semiconvergents and not the others (simpler, that is, than
calculating all the semiconvergents and then throwing some out), let
us know on tuning-math.

Hi Paul,

This is incorrect. 27/22 is not a better approximation than 11/9. The
semi-convergents on the Keenan spreadsheet are wrong as well--if what
your really looking for are successive closer approximations expressed
in the smallest RI terms (the sieve sequence I posted is correct
however).

--Dan Stearns

----- Original Message -----
From: "Paul Erlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, November 05, 2001 6:12 PM
Subject: [tuning] Re: Reply to Christopher Bailey

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > I have a sieve that does this nicely, but I have to pick them out
by
> > hand (this lead to the missing 38/31 a minute ago).
>
> You missed 27/22 too.
>
> >
> > Robert Walker actually made an applet based on this idea sometime
> last
> > year or earlier this year I think--Robert?
> >
> > --Dan Stearns
>
> One can do the same analysis by going to
> http://www.uq.net.au/~zzdkeena/Music/index.htm and clicking on
> _An Excel spreadsheet for calculating the size of MOS scales_.
> Of course, we're not going to be calculating the size of any MOS
> scale, but the mathematics is the same. Enter 1.22405354330466 (or
=2^
> (350/1200)) as "generator" and 1 as "period". You'll immediately see
> that the solution is
>
> 11/9, 16/13, 27/22, 38/31, 49/40, 60/49,
> 71/58, 82/67, 153/125,
> 224/183, 295/241,
> 519/424,
> 734/607, 1262/1031 . . .
>
> The leftmost column is the set of _convergents_, meaning they do
> better than any simpler ratio, "better" meaning error times
> complexity is lower. My algorithm looked only at the convergents,
but
> it skipped some because I was allowing negative terms in the CF
> expansion.
>
> The algorithm in the Excel spreadsheet is very simple -- you should
> be able to see it and code it up in your favorite computer language.
>
>
> You do not need web access to participate. You may subscribe
through
> email. Send an empty email to one of these addresses:
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>
>

On Tue, 6 Nov 2001 08:30:06 -0800, "D.Stearns" <STEARNS@CAPECOD.NET> wrote:

>Paul,
>
>
><<The simplest ratio-approximation to 350 cents that is more accurate
>than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>
>
>
>Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424, ...
^
38/31

>
> Paul,
>
>
> <<The simplest ratio-approximation to 350 cents that is more accurate
> than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>
>
>
> Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424, ...
>
> --Dan Stearns
>

Hmmm, I get [11/9] [38/31] [49/40] [60/49] [71/58] [153/125] [224/183]

Bob Valentine

> From: Robert C Valentine <BVAL@IIL.INTEL.COM>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, November 06, 2001 3:22 AM
> Subject: [tuning] Re: Reply to Christopher Bailey
>
>
> >
> > Paul,
> >
> >
> > <<The simplest ratio-approximation to 350 cents that is more accurate
> > than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>
> >
> >
> > Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424, ...
> >
> > --Dan Stearns
> >
>
> Hmmm, I get [11/9] [38/31] [49/40] [60/49] [71/58] [153/125] [224/183]
>
> Bob Valentine

Bob is right.

The series continues with 519/424 and 743/607, and that's it for
ratios with terms under 1000.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

Hey, not that any of it really matters, but I did correct this a
second after I posted it!

----- Original Message -----
From: "monz" <joemonz@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Tuesday, November 06, 2001 8:03 AM
Subject: Re: [tuning] Re: Reply to Christopher Bailey

>
> > From: Robert C Valentine <BVAL@IIL.INTEL.COM>
> > To: <tuning@yahoogroups.com>
> > Sent: Tuesday, November 06, 2001 3:22 AM
> > Subject: [tuning] Re: Reply to Christopher Bailey
> >
> >
> > >
> > > Paul,
> > >
> > >
> > > <<The simplest ratio-approximation to 350 cents that is more
accurate
> > > than 11/9 is 71/58, followed by 224/183, 743/607, . . .>>
> > >
> > >
> > > Hmm, I get 11/9, 49/40, 60/49, 71/58, 153/125, 224/183, 519/424,
...
> > >
> > > --Dan Stearns
> > >
> >
> > Hmmm, I get [11/9] [38/31] [49/40] [60/49] [71/58] [153/125]
[224/183]
> >
> > Bob Valentine
>
>
>
> Bob is right.
>
> The series continues with 519/424 and 743/607, and that's it for
> ratios with terms under 1000.
>
>
>
> love / peace / harmony ...
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com
>
>
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