Yahoo Tuning Groups Ultimate Backup tuning rational convergents to ET-degrees

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> monz,
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > i've made tables of rational convergents to each degree
> > of 12edo and 19edo
> > 31edo is on the way ...
>
> What about putting all the dictionary work aside for
> the moment and focussing all your attention on your
> *product*? We can't make music with your dictionary,
> but we might be able to with Musica!
>
> Cheers,
> Jon

the Encyclopaedia will ultimately be integrated with
the software. it's all one big product.

please believe me when i tell you that we are getting
the first release of the software out as soon as we can.
we still plan to have it out this year.

we had to put our final website online sooner than
planned, and it's still quite messy so i'm busy cleaning
up the Encyclopaedia pages. as i come to a page that
always should have had more, i add the new stuff in.

in the meantime, try making use of those latest tables:

have some fun composing a new "JI" piece,
approximating these EDOs with the ratios.

-monz

🔗Jon Szanto <JSZANTO@ADNC.COM>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> the Encyclopaedia will ultimately be integrated with
> the software. it's all one big product.

I was unaware of that. I assume you will have ways for people to
upgrade the dictionary, as it goes through more revision than any
typical software I've ever seen.

> in the meantime, try making use of those latest tables:
>
> have some fun composing a new "JI" piece,
> approximating these EDOs with the ratios.

I might, if more than one or two small pieces made any sense to me.

Cheers,
Jon

🔗monz <monz@attglobal.net>

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > the Encyclopaedia will ultimately be integrated with
> > the software. it's all one big product.
>
> I was unaware of that. I assume you will have ways for people to
> upgrade the dictionary, as it goes through more revision than any
> typical software I've ever seen.

we're not exactly sure how we're going to do it yet,
but even if it just stays on the website, the graphics
(which are now all static) will eventually be virtual-3D
rotatable lattices, which will be viewable with a free
plug-in Lattice Viewer.

> > in the meantime, try making use of those latest tables:
> >
> > have some fun composing a new "JI" piece,
> > approximating these EDOs with the ratios.
>
> I might, if more than one or two small pieces made
> any sense to me.

Jon, all i did here was to find the ratios which are
closest to a particular EDO degree.

the first ratio listed (if there are more than one)
is the roughest approximation but the lowest complexity
(i.e., smallest numbers in the ratio, such as 6:5, 8:7,
etc.).

the next ratio is the next one in order of number-size
which gives a closer approximation, and so on, until
i find the best approximation within my arbitrary
denominator limit of 302.

so in other words, i start with 1:1, then try 2:1,
then 3:2, then 4:3, then 5:3, then 5:4, then 6:5, etc.,
until i reach x:302.

whichever ratios come closest to that EDO degree,
in the order of my search, go into the list.

so, as far as your potential new piece, if you want
to imitate, say, 19edo with ratios,

http://tonalsoft.com/enc/19edo.htm

and you use the typical ones like 26:25, 13:12, 9:8,
7:6, etc., then you are using only the roughest
approximations to 19edo, and your piece will not
sound much like 19edo, because it will *be*
essentially regular JI.

but if you use some of the larger ratios, then you'll
get better approximations to 19edo.

knowing somewhat where your aesthetics lie, i'm really
only jokingly suggesting this to you. you personally
would never want to use ratios to imitate an ET.

but if you do decide to give it a go, you might have
some fun, and possibly get a little more insight into
how ETs and ratios intersect.

it's kind of like the old Hammond Organ thread.
Hammond uses a rational scale to imitate 12edo.
in fact, if someone would go thru the trouble of
digging up one of those old posts, i'd bet that some
of the Hammond ratios are in my 12edo list.

http://tonalsoft.com/enc/12edo.htm

-monz

🔗Jon Szanto <JSZANTO@ADNC.COM>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> Jon, all i did here was to find the ratios which are
> closest to a particular EDO degree.

I should have gone directly to the bottom of the page. I see that
table now, and the many screenfuls of calculations above it simply
floored me.

> knowing somewhat where your aesthetics lie, i'm really
> only jokingly suggesting this to you. you personally
> would never want to use ratios to imitate an ET.

I knew you weren't completely serious, and indeed I would use 19 if I
wanted to write in 19.

Cheers,
Jon

🔗Kurt Bigler <kkb@breathsense.com>

on 7/21/04 10:30 AM, monz <monz@attglobal.net> wrote:

> --- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
>> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>>> the Encyclopaedia will ultimately be integrated with
>>> the software. it's all one big product.
>>
>> I was unaware of that. I assume you will have ways for people to
>> upgrade the dictionary, as it goes through more revision than any
>> typical software I've ever seen.
>
>
> we're not exactly sure how we're going to do it yet,
> but even if it just stays on the website, the graphics
> (which are now all static) will eventually be virtual-3D
> rotatable lattices, which will be viewable with a free
> plug-in Lattice Viewer.

That sounds fantastic!

However... you *will* provide a viewer plug-in for the Mac also, even if
your product is just for Windows, right?

-Kurt

🔗monz <monz@attglobal.net>

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 7/21/04 10:30 AM, monz <monz@a...> wrote:
>
> > --- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> >> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >>>
> >>> the Encyclopaedia will ultimately be integrated with
> >>> the software. it's all one big product.
> >>
> >> I was unaware of that. I assume you will have ways for
> >> people to upgrade the dictionary, as it goes through
> >> more revision than any typical software I've ever seen.
> >
> >
> > we're not exactly sure how we're going to do it yet,
> > but even if it just stays on the website, the graphics
> > (which are now all static) will eventually be virtual-3D
> > rotatable lattices, which will be viewable with a free
> > plug-in Lattice Viewer.
>
> That sounds fantastic!
>
> However... you *will* provide a viewer plug-in for the
> Mac also, even if your product is just for Windows, right?
>
> -Kurt

yes, we plan to port both the full Musica program and
freebie Lattice Viewer and Piece Player to Mac at some
point in the (hopefully near) future.

one problem is that no-one at Tonalsoft uses Mac.
most likely, we'll have to enlist some outside help.

our big efforts right now are two: to get the beta version
of Musica out within the next couple of months, and to clean
up the website and get everything there working properly
and looking uniform.

everything else comes after those ...

-monz

🔗Petr Parízek <p.parizek@worldonline.cz>

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗Carl Lumma <ekin@lumma.org>

>> it's kind of like the old Hammond Organ thread.
>> Hammond uses a rational scale to imitate 12edo.
>> in fact, if someone would go thru the trouble of
>> digging up one of those old posts, i'd bet that some
>> of the Hammond ratios are in my 12edo list.
>>
>> http://tonalsoft.com/enc/12edo.htm
>
>How do you know this? Do you have some papers about Hammond
>organs? Does someone specify which ratios were used for these
>approximations? Moreover, why the hell should a Hammond organ
>be unable to play exact 12-equal?

Hammond organs use tonewheels to generate sound. They are
basically gears whose teeth change the flow of electricity
in a circuit as they pass by a certain point. Since any
pair of circular gears must have a number of teeth expressible
as a rational number, no tonewheel organ can produce truly
irrational intervals.

I don't know the ratios Hammond used. Anyone care to dig this
up? Wait, why bother, it'll just get lost again in the morass
that it Yahoo. :)

-Carl

🔗monz <monz@attglobal.net>

hi Petr,

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> Petr,
>
> --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@w...> wrote:
> > How do you know this? Do you have some papers about
> > Hammond organs?
>
> Just so that you know, about a year or so ago there was
> a long discussion, among a number of people, on this very
> list about this very subject. I didn't happen to take part,
> but it was quite in-depth. If someone could come up with
> when the thread started you could go back that far and
> read through the posts.
>
> Cheers,
> Jon

> > Does someone specify which ratios were used for these
> > approximations? Moreover, why the hell should a Hammond
> > organ be unable to play exact 12-equal?
> > Petr

i don't have time to try and find it in the Yahoo archives,
but here's a Wiki article that explains it:

http://www.dairiki.org/HammondWiki/WhatIsAHammondOrgan

the Hammond Organ is based on an "additive synthesis"
type of concept, so everything about its tuning is rational.

it uses higher-complexity ratios to closely approximate
12edo. these would be the rational convergents on my
webapges which are further to the right on each row.

please understand ... the rational tuning used by the
Hammond is *audibly indistinguishable* from 12edo,
so essentially it *is* 12edo ... but not mathematically
exact 12edo.

in the discussion on this list, someone did provide the
actual ratios ... i think it was Dave Keenan.

and the discussion goes back far longer than a year
... IIRC, it started around November or December 2000,
when there was a big argument over the exact meaning
of "just intonation".

we decided to create a new category called "RI"
(rational intonation) for tunings which are rational
but do not sound like JI, and the Hammond tuning is
a perfect example.

-monz

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗monz <monz@attglobal.net>

--- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> monz,
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > we decided to create a new category called "RI" (rational
intonation)
>
> Um, just so you know, the term "rational intonation" was in use back
> in the very first days of the Mills College list; you can find it in
> the archives there. I don't know if your definition is the same, but
I
> don't think you can lay claim to "creating a new category".
>
> Cheers,
> Jon

hmmm ... ok. but i'd have to read those old Mills posts
to see exactly what those using the term back then meant by it.

if it wasn't used specifically to separate rational tunings
which can by consensus be called "JI" from those which cannot,
then in 2000 we still did indeed create a new category.

thanks for the heads up ... i'll be waiting for Robert to
make the Mills archive so i can check it out, unless i can
do the search with the copies i have on my own hard drive.
(now, where the heck are they?)

-monz

🔗monz <monz@attglobal.net>

--- In tuning@yahoogroups.com, "asarkiss" <asarkiss@y...> wrote:

> --- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
> > Petr,
> >
> > --- In tuning@yahoogroups.com, Petr Parízek <p.parizek@w...>
wrote:
> > > How do you know this? Do you have some papers about
> > > Hammond organs?
> >
> > Just so that you know, about a year or so ago there was
> > a long discussion, among a number of people, on this very
> > list about this very subject. I didn't happen to take part,
> > but it was quite in-depth.
> > If someone could come up with when the thread started you
> > could go back that far and read through the posts.
> >
> > Cheers,
> > Jon
>
>
> msg. #47238
>
>
> the hammond organ uses ratios of two-digit numbers to
> get within about 1 cent of A-440 12-equal:
>
> 1/1 = 320 Hz (internal gear only; does not get heard)
>
> ratio freqen. ET freq deviation (cents)
> 71/82 277.073 277.183 -0.68
> 67/73 293.699 293.665 +0.20
> 35/36 311.111 311.127 -0.09
> 69/67 329.552 329.628 -0.40
> 12/11 349.091 349.228 -0.68
> 37/32 370.000 369.994 +0.03
> 49/40 392.000 391.995 +0.02
> 48/37 415.135 415.305 -0.71
> 11/08 440.000 440.000 0.000
> 67/46 466.087 466.164 -0.29
> 54/35 493.714 493.883 -0.59
> 85/52 523.077 523.251 -0.58

thanks for digging that out.

having all the ratios refer to an unheard 1/1 of
320 Hz doesn't make it easy to do comparisons.

i reworked that table so that it is a bit more
meaningful ...

Hammond Organ tuning
--------------------

"abs-rat" (absolute ratio) is the ratios of the notes
to the unheard 320 Hz

"rel-rat" (relative ratio) is the ratios of the top
11 notes to the one on the bottom (277.073 Hz).

. Hz ... abs-rat . rel-rat .. cents ...... cents-error

523.077 . 85/52 .. 17/9 .. 1100.108645 ... 0.108645107 x
493.714 . 54/35 .. 98/55 . 1000.091309 ... 0.091309308 x
466.087 . 67/46 .. 37/22 .. 900.3997717 .. 0.399771695 x
440 ..... 11/8 ... 27/17 .. 800.6848713 . -0.315128717 x
415.135 . 48/37 ... 3/2 ... 699.9773275 . -0.022672524 x
392 ..... 49/40 .. 58/41 .. 600.705028 .. -0.294972009
370 ..... 37/32 ... 4/3 ... 500.7109677 . -0.289032327 x
349.091 . 12/11 .. 63/50 .. 400.0044383 .. 0.004438261 x
329.552 . 69/67 . 113/95 .. 300.287994 ... 0.287993985
311.111 . 35/36 .. 64/57 .. 200.5959292 . -0.404070777
293.699 . 67/73 .. 53/50 .. 100.886667 .. -0.113332998
277.073 . 71/82 ... 1/1 ..... 0 .......... 0

"x" marks ratios which appear in my "table of rational
convergents" to 12edo (at the bottom of the webpage):

http://tonalsoft.com/enc/12edo.htm

in fact, the maximum error from true 12edo is just over
0.4 cent.

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_54741.html#54766

> >> it's kind of like the old Hammond Organ thread.
> >> Hammond uses a rational scale to imitate 12edo.
> >> in fact, if someone would go thru the trouble of
> >> digging up one of those old posts, i'd bet that some
> >> of the Hammond ratios are in my 12edo list.
> >>
> >> http://tonalsoft.com/enc/12edo.htm
> >
> >How do you know this? Do you have some papers about Hammond
> >organs? Does someone specify which ratios were used for these
> >approximations? Moreover, why the hell should a Hammond organ
> >be unable to play exact 12-equal?
>
> Hammond organs use tonewheels to generate sound. They are
> basically gears whose teeth change the flow of electricity
> in a circuit as they pass by a certain point. Since any
> pair of circular gears must have a number of teeth expressible
> as a rational number, no tonewheel organ can produce truly
> irrational intervals.
>
> I don't know the ratios Hammond used. Anyone care to dig this
> up? Wait, why bother, it'll just get lost again in the morass
> that it Yahoo. :)
>
> -Carl

***I'm sure Paul Erlich has or can find this. It seems like he's on
vacation right now or some such...

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_54741.html#54776
>
> hmmm ... ok. but i'd have to read those old Mills posts
> to see exactly what those using the term back then meant by it.
>
> if it wasn't used specifically to separate rational tunings
> which can by consensus be called "JI" from those which cannot,
> then in 2000 we still did indeed create a new category.
>
> thanks for the heads up ... i'll be waiting for Robert to
> make the Mills archive so i can check it out, unless i can
> do the search with the copies i have on my own hard drive.
> (now, where the heck are they?)
>
>
>
> -monz

***This discussion, with the various definitions of RI and JI was one
of the most interesting ones I've seen in my four or so years on this
list...

🔗monz <monz@attglobal.net>

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_54741.html#54784

> hi Joseph,
>
>
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> > ***I'm sure Paul Erlich has or can find this. It seems like he's
on
> > vacation right now or some such...
> >
> > JP
>
>
> he's busy, finishing his paper for the next _Xenharmonikon_.
>
>
>
> -monz

***Hey Monz!

Got it. When's the next big deadline??

🔗Kurt Bigler <kkb@breathsense.com>

on 7/22/04 12:01 AM, monz <monz@attglobal.net> wrote:

> hi Kurt,
>
>
> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 7/21/04 10:30 AM, monz <monz@a...> wrote:
>>
>>> --- In tuning@yahoogroups.com, "Jon Szanto" <JSZANTO@A...> wrote:
>>>> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>>>>>
>>>>> the Encyclopaedia will ultimately be integrated with
>>>>> the software. it's all one big product.
>>>>
>>>> I was unaware of that. I assume you will have ways for
>>>> people to upgrade the dictionary, as it goes through
>>>> more revision than any typical software I've ever seen.
>>>
>>> we're not exactly sure how we're going to do it yet,
>>> but even if it just stays on the website, the graphics
>>> (which are now all static) will eventually be virtual-3D
>>> rotatable lattices, which will be viewable with a free
>>> plug-in Lattice Viewer.
>>
>> That sounds fantastic!
>>
>> However... you *will* provide a viewer plug-in for the
>> Mac also, even if your product is just for Windows, right?
>>
>> -Kurt
>
> yes, we plan to port both the full Musica program and
> freebie Lattice Viewer and Piece Player to Mac at some
> point in the (hopefully near) future.
>
> one problem is that no-one at Tonalsoft uses Mac.
> most likely, we'll have to enlist some outside help.
>
> our big efforts right now are two: to get the beta version
> of Musica out within the next couple of months, and to clean
> up the website and get everything there working properly
> and looking uniform.
>
> everything else comes after those ...

At risk of belaboring the obvious, I thought that the implications of your
original post were that you would need the plug-in to view your website, but
no problem since it is free. But problem if it is not Mac because then Mac
people can't use your website anymore. No?

Anyway I figure getting the plugin ported to Mac is a lot smaller deal than
porting Musica.

Thanks,
Kurt

> -monz

🔗monz <monz@attglobal.net>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> At risk of belaboring the obvious, I thought that
> the implications of your original post were that you
> would need the plug-in to view your website, but
> no problem since it is free. But problem if it is
> not Mac because then Mac people can't use your website
> anymore. No?

visitors will not be able to see the 3-D rotatable
graphics without the Viewer ... but the Encyclopaedia
is already huge, and always growing larger ...
transforming all of the graphics is going to take
a long time, and we'll certainly have Mac and Linux
versions of the Lattice Viewer up and running long
before everything is converted.

> Anyway I figure getting the plugin ported to Mac is a
> lot smaller deal than porting Musica.

indeed, it will be.

i expect that porting Musica to the Mac will be a
gigantic deal, because it's a big and very complex
program. but i'm sure that some Mac programmer out
there will be interested enough to want the job.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> thanks for digging that out.
>
> having all the ratios refer to an unheard 1/1 of
> 320 Hz doesn't make it easy to do comparisons.
>
> i reworked that table so that it is a bit more
> meaningful ...
>
>
> Hammond Organ tuning
> --------------------
>
> "abs-rat" (absolute ratio) is the ratios of the notes
> to the unheard 320 Hz
>
> "rel-rat" (relative ratio) is the ratios of the top
> 11 notes to the one on the bottom (277.073 Hz).
>
>
> . Hz ... abs-rat . rel-rat .. cents ...... cents-error
>
> 523.077 . 85/52 .. 17/9 .. 1100.108645 ... 0.108645107 x
> 493.714 . 54/35 .. 98/55 . 1000.091309 ... 0.091309308 x
> 466.087 . 67/46 .. 37/22 .. 900.3997717 .. 0.399771695 x
> 440 ..... 11/8 ... 27/17 .. 800.6848713 . -0.315128717 x
> 415.135 . 48/37 ... 3/2 ... 699.9773275 . -0.022672524 x
> 392 ..... 49/40 .. 58/41 .. 600.705028 .. -0.294972009
> 370 ..... 37/32 ... 4/3 ... 500.7109677 . -0.289032327 x
> 349.091 . 12/11 .. 63/50 .. 400.0044383 .. 0.004438261 x
> 329.552 . 69/67 . 113/95 .. 300.287994 ... 0.287993985
> 311.111 . 35/36 .. 64/57 .. 200.5959292 . -0.404070777
> 293.699 . 67/73 .. 53/50 .. 100.886667 .. -0.113332998
> 277.073 . 71/82 ... 1/1 ..... 0 .......... 0
>
>
> "x" marks ratios which appear in my "table of rational
> convergents" to 12edo (at the bottom of the webpage):
> http://tonalsoft.com/enc/12edo.htm

Monz,

I think perhaps your "rel-rat" column should have been called
the "smell-a-rat" column. :-)

It's a deep mystery how you managed to get those ratios. They are
all quite wrong.

Here's what it really looks like when you take C# as 1/1 as you have
done.

Name . Hz ... abs-rat . rel-rat

C .. 523.077 . 85/52 . 3485/1846
B .. 493.714 . 54/35 . 4428/2485
Bb . 466.087 . 67/46 . 2747/1633
A .. 440 ..... 11/8 ... 451/284
G# . 415.135 . 48/37 . 3936/2627
G .. 392 ..... 49/40 . 2009/1420
F# . 370 ..... 37/32 . 1517/1136
F .. 349.091 . 12/11 .. 984/781
E .. 329.552 . 69/67 . 5658/4757
Eb . 311.111 . 35/36 . 1435/1278
D .. 293.699 . 67/73 . 5494/5183
C# . 277.073 . 71/82 .... 1/1

A more sensible choice for 1/1 in this case is A, since it is the
only note at concert pitch, and it gives you smaller integers in the
relative ratios, as follows.

Name . Hz ... abs-rat . rel-rat

C .. 523.077 . 85/52 . 170/143
B .. 493.714 . 54/35 . 432/385
Bb . 466.087 . 67/46 . 268/253
A .. 440 ..... 11/8 .... 1/1
G# . 415.135 . 48/37 . 384/407
G .. 392 ..... 49/40 .. 49/55 .. c
F# . 370 ..... 37/32 .. 37/44 .. c
F .. 349.091 . 12/11 .. 96/121
E .. 329.552 . 69/67 . 552/737
Eb . 311.111 . 35/36 .. 70/99 .. c
D .. 293.699 . 67/73 . 536/803
C# . 277.073 . 71/82 . 284/451

Only three are convergents. None are semi-convergents.

> in fact, the maximum deviation from true 12edo is just over
> 0.4 cent.

Wrong again I'm afraid. In fact, the maximum error from 12-equal
Concert Pitch is -0.7 cents (G#), as it says on this page.
http://www.bikexprt.com/tunings/tunings2.htm.

The maximum error in the size of any interval, relative to its 12-
equal size, is 0.9 cents (between G# and D).

🔗monz <monz@attglobal.net>

hi guys,

sorry about those errors. that what happens sometimes
when i stay up all night working on webpages.

i'm not sure what happened ... i'll try it again and
see if it matches yours. actually, since this is
rational and not exact-12edo, it would be interesting
to see what the ratios are when taking each of the
12 degrees as 1/1 in turn.

-monz

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > thanks for digging that out.
> >
> > having all the ratios refer to an unheard 1/1 of
> > 320 Hz doesn't make it easy to do comparisons.
> >
> > i reworked that table so that it is a bit more
> > meaningful ...
> >
> >
> > Hammond Organ tuning
> > --------------------
> >
> > "abs-rat" (absolute ratio) is the ratios of the notes
> > to the unheard 320 Hz
> >
> > "rel-rat" (relative ratio) is the ratios of the top
> > 11 notes to the one on the bottom (277.073 Hz).
> >
> >
> > . Hz ... abs-rat . rel-rat .. cents ...... cents-error
> >
> > 523.077 . 85/52 .. 17/9 .. 1100.108645 ... 0.108645107 x
> > 493.714 . 54/35 .. 98/55 . 1000.091309 ... 0.091309308 x
> > 466.087 . 67/46 .. 37/22 .. 900.3997717 .. 0.399771695 x
> > 440 ..... 11/8 ... 27/17 .. 800.6848713 . -0.315128717 x
> > 415.135 . 48/37 ... 3/2 ... 699.9773275 . -0.022672524 x
> > 392 ..... 49/40 .. 58/41 .. 600.705028 .. -0.294972009
> > 370 ..... 37/32 ... 4/3 ... 500.7109677 . -0.289032327 x
> > 349.091 . 12/11 .. 63/50 .. 400.0044383 .. 0.004438261 x
> > 329.552 . 69/67 . 113/95 .. 300.287994 ... 0.287993985
> > 311.111 . 35/36 .. 64/57 .. 200.5959292 . -0.404070777
> > 293.699 . 67/73 .. 53/50 .. 100.886667 .. -0.113332998
> > 277.073 . 71/82 ... 1/1 ..... 0 .......... 0
> >
> >
> > "x" marks ratios which appear in my "table of rational
> > convergents" to 12edo (at the bottom of the webpage):
> > http://tonalsoft.com/enc/12edo.htm
>
> Monz,
>
> I think perhaps your "rel-rat" column should have been called
> the "smell-a-rat" column. :-)
>
> It's a deep mystery how you managed to get those ratios. They are
> all quite wrong.
>
> Here's what it really looks like when you take C# as 1/1 as you have
> done.
>
> Name . Hz ... abs-rat . rel-rat
>
> C .. 523.077 . 85/52 . 3485/1846
> B .. 493.714 . 54/35 . 4428/2485
> Bb . 466.087 . 67/46 . 2747/1633
> A .. 440 ..... 11/8 ... 451/284
> G# . 415.135 . 48/37 . 3936/2627
> G .. 392 ..... 49/40 . 2009/1420
> F# . 370 ..... 37/32 . 1517/1136
> F .. 349.091 . 12/11 .. 984/781
> E .. 329.552 . 69/67 . 5658/4757
> Eb . 311.111 . 35/36 . 1435/1278
> D .. 293.699 . 67/73 . 5494/5183
> C# . 277.073 . 71/82 .... 1/1
>
> A more sensible choice for 1/1 in this case is A, since it is the
> only note at concert pitch, and it gives you smaller integers in the
> relative ratios, as follows.
>
> Name . Hz ... abs-rat . rel-rat
>
> C .. 523.077 . 85/52 . 170/143
> B .. 493.714 . 54/35 . 432/385
> Bb . 466.087 . 67/46 . 268/253
> A .. 440 ..... 11/8 .... 1/1
> G# . 415.135 . 48/37 . 384/407
> G .. 392 ..... 49/40 .. 49/55 .. c
> F# . 370 ..... 37/32 .. 37/44 .. c
> F .. 349.091 . 12/11 .. 96/121
> E .. 329.552 . 69/67 . 552/737
> Eb . 311.111 . 35/36 .. 70/99 .. c
> D .. 293.699 . 67/73 . 536/803
> C# . 277.073 . 71/82 . 284/451
>
> Only three are convergents. None are semi-convergents.
>
> > in fact, the maximum deviation from true 12edo is just over
> > 0.4 cent.
>
> Wrong again I'm afraid. In fact, the maximum error from 12-equal
> Concert Pitch is -0.7 cents (G#), as it says on this page.
> http://www.bikexprt.com/tunings/tunings2.htm.
>
> The maximum error in the size of any interval, relative to its 12-
> equal size, is 0.9 cents (between G# and D).

🔗monz <monz@attglobal.net>

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

i've made what i hope are correct tables of the
Hammond Organ ratios with each of the 12 notes in turn
as 1/1.

http://tonalsoft.com/enc/rational-intonation.htm

i'd appreciate someone checking my work. thanks.

-monz

> hi guys,
>
>
> sorry about those errors. that what happens sometimes
> when i stay up all night working on webpages.
>
> i'm not sure what happened ... i'll try it again and
> see if it matches yours. actually, since this is
> rational and not exact-12edo, it would be interesting
> to see what the ratios are when taking each of the
> 12 degrees as 1/1 in turn.
>
>
> -monz
>
>
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Monz,
> >
> > I think perhaps your "rel-rat" column should have been called
> > the "smell-a-rat" column. :-)
> >
> > It's a deep mystery how you managed to get those ratios. They are
> > all quite wrong.
> >
> > <snip>