cents to ratios calculator

I've had Kraig Grady post a simple Excel spreadsheet that approximates intervals in cents by ratios by means of continued fractions at http://anaphoria.com/journal.html. It is based on a decimal to ratio converter written by Kardi Teknomo. Use Sheet 2 for your computations.

In additions to continued fraction convergents, the spreadsheet also computes string lengths and frequencies using several common pitch bases such as 440, 261.63, 256, and 264 hz.

I apologize in advance for cross-posting, but I think this program will be of interest to members of all three lists and will help limit future conflicts over representing scales in decimals, cents or ratios.

--John

thanks guys, that was fun.i think a nice added feature would be one--that by some best-reasoned set criteria--defines a "best choice" for a given cents value as regards its relation to those fractions that come before it and those fractions that come after it. Seems simple enough, but it would need to be both flexible and sensible enough to be generally regarded as universally applicable.
daniel

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> I've had Kraig Grady post a simple Excel spreadsheet that approximates
> intervals in cents by ratios by means of continued fractions at
> http://anaphoria.com/journal.html. It is based on a decimal to ratio
> converter written by Kardi Teknomo. Use Sheet 2 for your computations.
>
> In additions to continued fraction convergents, the spreadsheet also
> computes string lengths and frequencies using several common pitch bases
> such as 440, 261.63, 256, and 264 hz.
>
> I apologize in advance for cross-posting, but I think this program will
> be of interest to members of all three lists and will help limit future
> conflicts over representing scales in decimals, cents or ratios.
>
> --John
>

Dear John and Kraig,

Thank you for sharing this! Could you perhaps also briefly report the formula used (say, a function taking a cent value and returning a ratio)? That would be far more easy for me to read and use then the Excel file :)

Thanks a lot!

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Mar 17, 2009, at 6:46 PM, John H. Chalmers wrote:

> I've had Kraig Grady post a simple Excel spreadsheet that approximates
> intervals in cents by ratios by means of continued fractions at
> http://anaphoria.com/journal.html. It is based on a decimal to ratio
> converter written by Kardi Teknomo. Use Sheet 2 for your computations.
>
> In additions to continued fraction convergents, the spreadsheet also
> computes string lengths and frequencies using several common pitch > bases
> such as 440, 261.63, 256, and 264 hz.
>
> I apologize in advance for cross-posting, but I think this program > will
> be of interest to members of all three lists and will help limit > future
> conflicts over representing scales in decimals, cents or ratios.
>
> --John
>
>

There is a quick and dirty, easy to use Javascript which I wrote many years ago to do this conversion in each direction on this page (about half way down the page)

http://www.lucytune.com/new_to_lt/pitch_01.html

On 17 Mar 2009, at 20:20, Torsten Anders wrote:

> Dear John and Kraig,
>
> Thank you for sharing this! Could you perhaps also briefly report the
> formula used (say, a function taking a cent value and returning a
> ratio)? That would be far more easy for me to read and use then the
> Excel file :)
>
> Thanks a lot!
>
> Best
> Torsten
>
> --
> Torsten Anders
> Interdisciplinary Centre for Computer Music Research
> University of Plymouth
> Office: +44-1752-586219
> Private: +44-1752-558917
> http://strasheela.sourceforge.net
> http://www.torsten-anders.de
>
> On Mar 17, 2009, at 6:46 PM, John H. Chalmers wrote:
>
> > I've had Kraig Grady post a simple Excel spreadsheet that > approximates
> > intervals in cents by ratios by means of continued fractions at
> > http://anaphoria.com/journal.html. It is based on a decimal to ratio
> > converter written by Kardi Teknomo. Use Sheet 2 for your > computations.
> >
> > In additions to continued fraction convergents, the spreadsheet also
> > computes string lengths and frequencies using several common pitch
> > bases
> > such as 440, 261.63, 256, and 264 hz.
> >
> > I apologize in advance for cross-posting, but I think this program
> > will
> > be of interest to members of all three lists and will help limit
> > future
> > conflicts over representing scales in decimals, cents or ratios.
> >
> > --John
> >
> >
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Charles Lucy wrote:

"There is a quick and dirty, easy to use Javascript which I wrote many years
ago to do this conversion in each direction on this page (about half way
down the page)

http://www.lucytune.com/new_to_lt/pitch_01.html"

Thanks Charles for sharing your writings with us !

I have a few questions that I am sure will help to understand your writings
better:

A. You are writing about the basics of musical scales and temperaments. The
matter has been researched ever since the Middle Ages and conspicuously in
modern times, with important publications. It is agreed that a writing like
yours should quote the main sources (ancient and modern) upon which your
statements are based, yet I cannot find any list of References/Literature in
the webpage you mention in your email. Sorry cannot find it in other we
bgpages of yours, though my search was not exhaustive: could you tell us
where they are? Thanks!

B. You write in that webpage:

"this octave was subdivided in three basic ways;

1) By a geometric progression, with any number of equal intervals.Eg. 12 (as
on conventional guitars) [100 cents per semitone or interval]; 31 as
advocated by Huyghens (1629-1695) [38.71 cents per interval], or 53 by
Mercator and Bosanquet (1876 Treatise) [22.64 cents];

2) By <outbind://11/tuning/just_intonation.html> low whole number ratios.
Eg. (Just Intonation) 3:2 for the Vth; 5:4 for the major third etc.

3) By cumulative fifths. Eg. <outbind://11/tuning/pythagorean.html>
Pythagorean Tuning 3:2 for the Vth; but 81:64 for the major third.

There are also hybrids of the other three, Eg.
<outbind://11/tuning/mean_tone.html> Meantone Temperaments."

I have to say that your are the first author known to me that finds that ALL
the possible musical subdivisions of the octave (or "temperaments") are
hybrids of your basic three groups.

I fail to see how that could be the case of some specific systems belonging
to the well-known group of Circular temperaments, e.g. those using a few
sharp fifths, such as d'Alembert-Rousseau ordinaire and quite a few
Well/Good temperaments by Werckmeister and Neidhardt. Perhaps you could
explain us how they can be hybrids of your 3 basic types?

C. You then write "If the only point of agreement is that the octave ratio
should be exactly two; how do we explain the phenomenon of stretched
octaves, used by some piano tuners?"

Since you do not comment further on the matter, the non-specialist may
believe that the answer is unknown or in doubt, but the matter has been the
object of well-known and extensive research and is clearly understood. Is
there any reason why you wrote that question and left it unanswered in your
webpage?

D: Later you explain the Pi scale, originating in Harrison's writings. You
do not say which success it enjoyed in its time (surely if he had his choir
singing to it only proves his leadership or power, not the goodness of his
scale). You do not state either why should that proposal stand out among the
thousands of useless proposals written down in antiquity and in modern
times. Guess the use of the Pi ratio has some useful acoustical properties,
but your page does not show which or why. Perhaps you could give a brief
explanation about the benefits of the PI scale compared with well-known
scales such as your "basic 3"?

E. You write

"Harrison must have imagined this moment when he wrote his book for some
future generation to decipher."

Did Harrison write that he had the above intentions?

If he did, please quote the sentence.

If he did not, you are surely aware that quite a few modern writers have
assumed that the ancient - typically Bach - have left tuning clues written
as "hieroglyphics to be deciphered": Kelletat, Kellner, Lehman and many more
. Scholars have put forward strong arguments against all those writings.
What makes you believe that yours is different?

F. You write

"As hundreds of people are now using these discoveries,"

I guess I am myself , like many readers, unaware of Pi tuning and related
matters, because I fail to find which significant discoveries are you
referring to. Whether they are by yourself or by Harrison, it would be
useful to know which are they, which issues they resolve that were not
resolved by already-known scales, and if so, how they do it.

G. I also gave a look at the following webpage of yours, where you describe
your "Lucy Tuning":

http://www.lucytune.com/new_to_lt/pitch_02.html

You describe there a tuning based on the well-known Medieval-Renaissance
principle of the "Spiral of Fifths".

The peculiarity of Lucy Tuning is using fifths sized 695.49 Cents each.

Isn't this identical, to all practical purposes, to the spiral of Fifths of
the well-known 2/7 Syntonic comma tuning described by Zarlino in the 16th
century?

(its Fifths are 695.81 Cents each: it is well known that in meantone spirals
a difference of less than 1/2 Cent is not significant).

Which is the advantage of Lucy tuning over Zarlino's 2/7 S.c. temperament?

I am sure your explanations will help us to understand your writings better.

Thank you.

Kind regards,

Claudio

Hi Claudio;

I shall attempt to answer your queries.

Yet remember that the page you are referring to was written more than
20 years ago, and since then I and many others have learned much more
about tuning.

You have been reading a bunch of webpages containing some chapters
from a book. Not an academic thesis.

I did originally write a bibliography for Pitch, Pi, more than twenty
years ago.

and since the references were generally well-known writings on
microtuning etc. and things change fast, the only references that made
it onto the website were John Harrison writings.
My initial three categories seemed to me to be a reasonable way to
think about microtuning.

Maybe using the words " Mixture, combination or extension" would have
been more appropriate to hybrid in this context, yet hybrid does serve
its purpose in this context?

I did not find written (by him) firm evidence Harrison was leaving it
for future generations to find, yet it seems to me to be a reasonable
supposition.

(must have imagined) Why else would Harrison have worried to write/
publish all these ideas on tuning at the end of his life? - poetic
license?

I have a number of theories on why some people have stretched octaves,
but they don't seem to be particularly relevant to the subject of John
Harrison and tuning.

The pi tuning is yet another "hybrid" as it exhibits some
characteristics of more than one of the categories. e.g. close to 88edo.

> (its Fifths are 695.81 Cents each: it is well known that in meantone
> spirals a difference of less than 1/2 Cent is not significant).
>

WELL KNOWN by whom???

I totally disagree with this "well known" --- (fact?)
Think about what you are claiming. You get vast differences in cents
between intervals of different cent values after only a few steps of
fourths and fifths, and you are totally ignoring the beating
implications.

Yes, LucyTuning exhibits the patterns and characteristics that one
would expect from a negative meantone tuning, yet before you dismiss
it as "nothing special" explore a little further, and LISTEN to the
harmonies that it produces, try it out on some of you own
compositions, and then comment from your own experience and
experimentations.

thanks for your thoughts

On 17 Mar 2009, at 23:23, Claudio Di Veroli wrote:

>
> Charles Lucy wrote:
>
> "There is a quick and dirty, easy to use Javascript which I wrote
> many years ago to do this conversion in each direction on this page
> (about half way down the page)
>
> http://www.lucytune.com/new_to_lt/pitch_01.html"
>
>
>
> Thanks Charles for sharing your writings with us !
>
> I have a few questions that I am sure will help to understand your
> writings better:
>
>
>
> A. You are writing about the basics of musical scales and
> temperaments. The matter has been researched ever since the Middle
> Ages and conspicuously in modern times, with important publications.
> It is agreed that a writing like yours should quote the main
> sources (ancient and modern) upon which your statements are based,
> yet I cannot find any list of References/Literature in the webpage
> you mention in your email. Sorry cannot find it in other we bgpages
> of yours, though my search was not exhaustive: could you tell us
> where they are? Thanks!
>
>
>
> B. You write in that webpage:
>
> “this octave was subdivided in three basic ways;
>
> 1) By a geometric progression, with any number of equal
> intervals.Eg. 12 (as on conventional guitars) [100 cents per
> semitone or interval]; 31 as advocated by Huyghens (1629-1695)
> [38.71 cents per interval], or 53 by Mercator and Bosanquet (1876
> Treatise) [22.64 cents];
>
> 2) By low whole number ratios. Eg. (Just Intonation) 3:2 for the
> Vth; 5:4 for the major third etc.
>
> 3) By cumulative fifths. Eg. Pythagorean Tuning 3:2 for the Vth; but
> 81:64 for the major third.
>
> There are also hybrids of the other three, Eg. Meantone Temperaments."
>
> I have to say that your are the first author known to me that finds
> that ALL the possible musical subdivisions of the octave (or
> "temperaments") are hybrids of your basic three groups.
>
> I fail to see how that could be the case of some specific systems
> belonging to the well-known group of Circular temperaments, e.g. > those using a few sharp fifths, such as d'Alembert-Rousseau
> ordinaire and quite a few Well/Good temperaments by Werckmeister and
> Neidhardt. Perhaps you could explain us how they can be hybrids of
> your 3 basic types?
>
>
>
> C. You then write “If the only point of agreement is that the octave
> ratio should be exactly two; how do we explain the phenomenon of
> stretched octaves, used by some piano tuners?”
>
> Since you do not comment further on the matter, the non-specialist
> may believe that the answer is unknown or in doubt, but the matter
> has been the object of well-known and extensive research and is
> clearly understood. Is there any reason why you wrote that question
> and left it unanswered in your webpage?
>
>
>
> D: Later you explain the Pi scale, originating in Harrison’s
> writings. You do not say which success it enjoyed in its time
> (surely if he had his choir singing to it only proves his leadership
> or power, not the goodness of his scale). You do not state either
> why should that proposal stand out among the thousands of useless
> proposals written down in antiquity and in modern times. Guess the
> use of the Pi ratio has some useful acoustical properties, but your
> page does not show which or why. Perhaps you could give a brief
> explanation about the benefits of the PI scale compared with well-
> known scales such as your “basic 3”?
>
>
>
> E. You write
>
> “Harrison must have imagined this moment when he wrote his book for
> some future generation to decipher.”
>

>
> Did Harrison write that he had the above intentions?
>
> If he did, please quote the sentence.
>
> If he did not, you are surely aware that quite a few modern writers
> have assumed that the ancient - typically Bach - have left tuning
> clues written as “hieroglyphics to be deciphered”: Kelletat,
> Kellner, Lehman and many more . Scholars have put forward strong
> arguments against all those writings. What makes you believe that
> yours is different?
>
>
>
> F. You write
>
> “As hundreds of people are now using these discoveries,”
>
> I guess I am myself , like many readers, unaware of Pi tuning and
> related matters, because I fail to find which significant
> discoveries are you referring to. Whether they are by yourself or by
> Harrison, it would be useful to know which are they, which issues
> they resolve that were not resolved by already-known scales, and if
> so, how they do it.
>
>
>
> G. I also gave a look at the following webpage of yours, where you
> describe your "Lucy Tuning":
>
> http://www.lucytune.com/new_to_lt/pitch_02.html
>
> You describe there a tuning based on the well-known Medieval-
> Renaissance principle of the "Spiral of Fifths".
>
> The peculiarity of Lucy Tuning is using fifths sized 695.49 Cents
> each.
>
> Isn't this identical, to all practical purposes, to the spiral of
> Fifths of the well-known 2/7 Syntonic comma tuning described by
> Zarlino in the 16th century?
>
> (its Fifths are 695.81 Cents each: it is well known that in meantone
> spirals a difference of less than 1/2 Cent is not significant).
>
> Which is the advantage of Lucy tuning over Zarlino's 2/7 S.c.
> temperament?
>
>
>
> I am sure your explanations will help us to understand your writings
> better.
>
> Thank you.
>
>
>
> Kind regards,
>
>
>
> Claudio
>
>
>
>
>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Hi Charles,

Thanks for your prompt reply

> I did not find written (by him) firm evidence Harrison was leaving it for
future generations to find, yet it seems to me to be a reasonable
supposition.
We differ here I'm sorry to say ...

> I have a number of theories on why some people have stretched octaves, but
they don't seem to be particularly relevant to the subject of John Harrison
and tuning.
As far as present scholars is concerned, it has to do with trying to cope
with the piano notorious inharmonicity in the lowermost and uppermost
octaves (caused by the strings having a much larger thickness to length
ratio than in the middle of the range).
But you are absolutely right, it is not relevant to the subject at all.

> I totally disagree with this "well known" --- (fact?) Think about what you
are claiming. You get vast differences in cents between intervals of
different cent values after only a few steps of fourths and fifths, and you
are totally ignoring the beating implications.
I have indeed argued the need to perform Cents calculations with precision
of 2 decimals minimum, to avoid error propagation!
HOWEVER, as for determining whether two temperaments are audibly different,
having 0.4 Cents of different in fifths means 1.6 C. of difference in major
thirds. No author known to me considers this a significant, let alone
audible, difference. [Of course there will be higher differences in absolute
pitches of accidentals, but that is irrelevant if all the musicians in an
ensemble agree to tune to the same temperament].
I understand that your opinion here differs from mine.

Thanks again for your kind explanations.

Yours,

Claudio
http://harps.braybaroque.ie <http://harps.braybaroque.ie/> /

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Charles Lucy
Sent: 18 March 2009 00:28
To: tuning@yahoogroups.com
Subject: Re: [tuning] cents to ratios calculator - quick and dirty - easy to
use FYI
Importance: High

Hi Claudio;

I shall attempt to answer your queries.

Yet remember that the page you are referring to was written more than 20
years ago, and since then I and many others have learned much more about
tuning.

You have been reading a bunch of webpages containing some chapters from a
book. Not an academic thesis.

I did originally write a bibliography for Pitch, Pi, more than twenty years
ago.

and since the references were generally well-known writings on microtuning
etc. and things change fast, the only references that made it onto the
website were John Harrison writings.
My initial three categories seemed to me to be a reasonable way to think
about microtuning.

Maybe using the words " Mixture, combination or extension" would have been
more appropriate to hybrid in this context, yet hybrid does serve its
purpose in this context?

I did not find written (by him) firm evidence Harrison was leaving it for
future generations to find, yet it seems to me to be a reasonable
supposition.

(must have imagined) Why else would Harrison have worried to write/publish
all these ideas on tuning at the end of his life? - poetic license?

I have a number of theories on why some people have stretched octaves, but
they don't seem to be particularly relevant to the subject of John Harrison
and tuning.

The pi tuning is yet another "hybrid" as it exhibits some characteristics of
more than one of the categories. e.g. close to 88edo.

(its Fifths are 695.81 Cents each: it is well known that in meantone spirals
a difference of less than 1/2 Cent is not significant).

WELL KNOWN by whom???

I totally disagree with this "well known" --- (fact?)
Think about what you are claiming. You get vast differences in cents between
intervals of different cent values after only a few steps of fourths and
fifths, and you are totally ignoring the beating implications.

Yes, LucyTuning exhibits the patterns and characteristics that one would
expect from a negative meantone tuning, yet before you dismiss it as
"nothing special" explore a little further, and LISTEN to the harmonies
that it produces, try it out on some of you own compositions, and then
comment from your own experience and experimentations.

thanks for your thoughts

On 17 Mar 2009, at 23:23, Claudio Di Veroli wrote:

Charles Lucy wrote:

"There is a quick and dirty, easy to use Javascript which I wrote many years
ago to do this conversion in each direction on this page (about half way
down the page)

http://www.lucytune <http://www.lucytune.com/new_to_lt/pitch_01.html>
.com/new_to_lt/pitch_01.html"

Thanks Charles for sharing your writings with us !

I have a few questions that I am sure will help to understand your writings
better:

A. You are writing about the basics of musical scales and temperaments. The
matter has been researched ever since the Middle Ages and conspicuously in
modern times, with important publications. It is agreed that a writing like
yours should quote the main sources (ancient and modern) upon which your
statements are based, yet I cannot find any list of References/Literature in
the webpage you mention in your email. Sorry cannot find it in other we
bgpages of yours, though my search was not exhaustive: could you tell us
where they are? Thanks!

B. You write in that webpage:

"this octave was subdivided in three basic ways;

1) By a geometric progression, with any number of equal intervals.Eg. 12 (as
on conventional guitars) [100 cents per semitone or interval]; 31 as
advocated by Huyghens (1629-1695) [38.71 cents per interval], or 53 by
Mercator and Bosanquet (1876 Treatise) [22.64 cents];

2) By <outbind://11/tuning/just_intonation.html> low whole number ratios.
Eg. (Just Intonation) 3:2 for the Vth; 5:4 for the major third etc.

3) By cumulative fifths. Eg. <outbind://11/tuning/pythagorean.html>
Pythagorean Tuning 3:2 for the Vth; but 81:64 for the major third.

There are also hybrids of the other three, Eg.
<outbind://11/tuning/mean_tone.html> Meantone Temperaments."

I have to say that your are the first author known to me that finds that ALL
the possible musical subdivisions of the octave (or "temperaments") are
hybrids of your basic three groups.

I fail to see how that could be the case of some specific systems belonging
to the well-known group of Circular temperaments, e.g. those using a few
sharp fifths, such as d'Alembert-Rousseau ordinaire and quite a few
Well/Good temperaments by Werckmeister and Neidhardt. Perhaps you could
explain us how they can be hybrids of your 3 basic types?

C. You then write "If the only point of agreement is that the octave ratio
should be exactly two; how do we explain the phenomenon of stretched
octaves, used by some piano tuners?"

Since you do not comment further on the matter, the non-specialist may
believe that the answer is unknown or in doubt, but the matter has been the
object of well-known and extensive research and is clearly understood. Is
there any reason why you wrote that question and left it unanswered in your
webpage?

D: Later you explain the Pi scale, originating in Harrison's writings. You
do not say which success it enjoyed in its time (surely if he had his choir
singing to it only proves his leadership or power, not the goodness of his
scale). You do not state either why should that proposal stand out among the
thousands of useless proposals written down in antiquity and in modern
times. Guess the use of the Pi ratio has some useful acoustical properties,
but your page does not show which or why. Perhaps you could give a brief
explanation about the benefits of the PI scale compared with well-known
scales such as your "basic 3"?

E. You write

"Harrison must have imagined this moment when he wrote his book for some
future generation to decipher."

Did Harrison write that he had the above intentions?

If he did, please quote the sentence.

If he did not, you are surely aware that quite a few modern writers have
assumed that the ancient - typically Bach - have left tuning clues written
as "hieroglyphics to be deciphered": Kelletat, Kellner, Lehman and many more
. Scholars have put forward strong arguments against all those writings.
What makes you believe that yours is different?

F. You write

"As hundreds of people are now using these discoveries,"

I guess I am myself , like many readers, unaware of Pi tuning and related
matters, because I fail to find which significant discoveries are you
referring to. Whether they are by yourself or by Harrison, it would be
useful to know which are they, which issues they resolve that were not
resolved by already-known scales, and if so, how they do it.

G. I also gave a look at the following webpage of yours, where you describe
your "Lucy Tuning":

http://www.lucytune <http://www.lucytune.com/new_to_lt/pitch_02.html>
.com/new_to_lt/pitch_02.html

You describe there a tuning based on the well-known Medieval-Renaissance
principle of the "Spiral of Fifths".

The peculiarity of Lucy Tuning is using fifths sized 695.49 Cents each.

Isn't this identical, to all practical purposes, to the spiral of Fifths of
the well-known 2/7 Syntonic comma tuning described by Zarlino in the 16th
century?

(its Fifths are 695.81 Cents each: it is well known that in meantone spirals
a difference of less than 1/2 Cent is not significant).

Which is the advantage of Lucy tuning over Zarlino's 2/7 S.c. temperament?

I am sure your explanations will help us to understand your writings better.

Thank you.

Kind regards,

Claudio

Charles Lucy
lucy@lucytune. <mailto:lucy@...> com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune <http://www.lucytune.com> .com

For LucyTuned Lullabies go to:
http://www.lullabie <http://www.lullabies.co.uk> s.co.uk

If your interest is limited to 11-limit ratios, i have
computed the cents values for a very large subset of
the theoretically-infinite 11-limit lattice, on this webpage:

http://tonalsoft.com/enc/i/interval-list.aspx

Most of this list has a rather messy format ... i was
formatting it by hand and got as far as 150 cents, and
also included the "tina" calculation (my preferred EDO
unit of measurement) for each ratio up to that point.

I plan to eventually write a python script to finish
the formatting ... but can't say when ...

-monz

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> I've had Kraig Grady post a simple Excel spreadsheet
> that approximates intervals in cents by ratios by means
> of continued fractions at http://anaphoria.com/journal.html.
> It is based on a decimal to ratio converter written by
> Kardi Teknomo. Use Sheet 2 for your computations.
>
> In additions to continued fraction convergents, the
> spreadsheet also computes string lengths and frequencies
> using several common pitch bases such as 440, 261.63, 256,
> and 264 hz.
>
> I apologize in advance for cross-posting, but I think this
> program will be of interest to members of all three lists
> and will help limit future conflicts over representing scales
> in decimals, cents or ratios.
>
> --John
>

John,
 
When I try to download Kraig's usual anaphoria link in order to access your new Excel spreadsheet all I get is PAGE NOT FOUND.
 
-- Mark Rankin

--- On Tue, 3/17/09, John H. Chalmers <JHCHALMERS@...> wrote:

From: John H. Chalmers <JHCHALMERS@...>
Subject: [tuning] cents to ratios calculator
To: "tuning@yahoogroups.com" <tuning@yahoogroups.com>, "Tuning Math" <tuning-math@yahoogroups.com>, "MakeMicroMusic" <MakeMicroMusic@yahoogroups.com>
Date: Tuesday, March 17, 2009, 11:46 AM

I've had Kraig Grady post a simple Excel spreadsheet that approximates
intervals in cents by ratios by means of continued fractions at
http://anaphoria. com/journal. html. It is based on a decimal to ratio
converter written by Kardi Teknomo. Use Sheet 2 for your computations.

In additions to continued fraction convergents, the spreadsheet also
computes string lengths and frequencies using several common pitch bases
such as 440, 261.63, 256, and 264 hz.

I apologize in advance for cross-posting, but I think this program will
be of interest to members of all three lists and will help limit future
conflicts over representing scales in decimals, cents or ratios.

--John

--- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> wrote:
>
> Dear John and Kraig,
>
> Thank you for sharing this! Could you perhaps also briefly
> report the formula used (say, a function taking a cent value
> and returning a ratio)? That would be far more easy for me
> to read and use then the Excel file :)
>
> Thanks a lot!
>
> Best
> Torsten
>

I did some investigating on this a while back. Here is
the algorithm I settled on:

;; Returns the simplest rational within a given factor (range) of
;; target, where complexity is defined as numerator*denominator
;; and 1 < range < target.
;; US$100 bounty for a proof of this statement! carl@...

(define gear
(lambda (target range)
(letrec
((loop (lambda (left right target range)
(let ((mediant (middle left right)))
(if (<= (/ (max mediant target)
(min mediant target)) range)
mediant
(loop
(if (< mediant target) mediant left)
(if (> mediant target) mediant right)
target
range))))))
(if (or (<= range 1) (>= range target))
'inputerror
(loop
(inexact->exact (floor (/ target range)))
(inexact->exact (ceiling (* target range)))
target
range)))))

;; Returns the mediant of two fractions.

(define middle
(lambda (left right)
(if (not (and (exact? left) (exact? right)))
'inputerror
(/ (+ (numerator left) (numerator right))
(+ (denominator left) (denominator right))))))

-Carl

Hi Carl and everyone else in this thread,

That is Lisp, right?
Would have been a good idea for you to say so.
:-)

Calculating 2^(cents/1200) will give you a decimal
value for the ratio, then it's a simple matter to
find the fractions which best approximate that.

Of course "best" is entirely subjective, depending
on what criteria you set -- such as prime-limit etc.

BTW, Tonescape does this when you choose "notation"
from the Lattice pop-up menu and select "ratio.
Tonescape puts each note of a tuning into the Lattice
according to the generators you define which produce
the Lattice. Then selecting "ratio" notation will show
the ratios.

For example:

* If you set up 19-edo as a meantone in 3,5-space,
Tonescape will show all the typical 5-limit ratios,
and selecting "closed curved" geometry will give you
the cool twisted helical Lattice.

* If you set up 46-edo as a pseudo-JI in 3,5,7,11,13-space,
Tonescape will map the pitches of 46-edo to the nearest
13-limit ratio according to the defined maps of the
prime-factors, using Tenny-height.

No need for spreadsheets -- Tonescape does it all.
;-)

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

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@> wrote:
> >
> > Dear John and Kraig,
> >
> > Thank you for sharing this! Could you perhaps also briefly
> > report the formula used (say, a function taking a cent value
> > and returning a ratio)? That would be far more easy for me
> > to read and use then the Excel file :)
> >
> > Thanks a lot!
> >
> > Best
> > Torsten
> >
>
> I did some investigating on this a while back. Here is
> the algorithm I settled on:
>
> ;; Returns the simplest rational within a given factor (range) of
> ;; target, where complexity is defined as numerator*denominator
> ;; and 1 < range < target.
> ;; US$100 bounty for a proof of this statement! carl@...
>
> (define gear
> (lambda (target range)
> (letrec
> ((loop (lambda (left right target range)
> (let ((mediant (middle left right)))
> (if (<= (/ (max mediant target)
> (min mediant target)) range)
> mediant
> (loop
> (if (< mediant target) mediant left)
> (if (> mediant target) mediant right)
> target
> range))))))
> (if (or (<= range 1) (>= range target))
> 'inputerror
> (loop
> (inexact->exact (floor (/ target range)))
> (inexact->exact (ceiling (* target range)))
> target
> range)))))
>
> ;; Returns the mediant of two fractions.
>
> (define middle
> (lambda (left right)
> (if (not (and (exact? left) (exact? right)))
> 'inputerror
> (/ (+ (numerator left) (numerator right))
> (+ (denominator left) (denominator right))))))
>
> -Carl
>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:

> * If you set up 19-edo as a meantone in 3,5-space,
> Tonescape will show all the typical 5-limit ratios,
> and selecting "closed curved" geometry will give you
> the cool twisted helical Lattice.

Oops ... actually, 19-edo (or any edo) will give you
a toroidal Lattice, not a helix.

For a helical Lattice you need an open-ended chain,
such as any fraction-of-a-comma meantone.

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