Look Ma! 2/7-comma meantone with no monochord!

Hey all,

I've been busy investigating various ways to quickly approximate 2/7-comma
meantone temperament by ear without any monochord, audio aids, and beat rate
charts, but purely with proportional beating guides.

I found a remarkably accurate relationship: temporarily tune a 1/3-comma
meantone fifth C-G, and then tune C-E flat from pure so that it beats 1 for
every 2 E-G's. This 'C-E' will be a remarkably close 2/7-comma meantone third
C-E: now distribute this 'C-E' into the 4 usual meantone fifths, C-G-D-A-E;
the size of the fifth C-G will agree to 5 places after the decimal
(~1.49468733 vs. the true value of ~1.49468550)! In fact, I think this
technique might be *more* accurate than a monochord, because you are not
dealing with the error of the fret placement--it's purely an acoustic
relationship between beats...

To get 1/7-comma, take the resulting C-G, put a pure fifth G-D above it, and
then readjust C-G so that G-D beats 1.5 times faster than C-G.
This results in a 1/7-comma fifth.

Mathematically, you've of course taken the square root of C-D' (a major ninth)
to get C-G...

Two other tricks:
1/6-comma fifth= sqrt(1.5*(1/3 comma fifth))
1/8-comma fifth = sqrt(1.5*(1/4 comma fifth))

Not satisfied with that, I found a way to get a very accurate 1/5-comma
meantone fifth by ear...tune C-B as a pure 15/8. Temporarily tune C-G as a
1/4-comma meantone fifth. Then make E sharp slightly so that E-B beats at
exactly the same rate as C-G. This C-E is now a very accurate 1/5-comma
meantone third. The 'C-E' will then be divided into four fifths as any
meantone, C-G-D-A-E, as above. The new C-G after this division will be an
accurate 1/5-comma meantone fifth, again to five places after the decimal
point (~1.49627528 vs ~1.49627787)!

Cheers,
Aaron

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> Hey all,
>
> I've been busy investigating various ways to quickly approximate
2/7-comma
> meantone temperament by ear without any monochord, audio aids, and
beat rate
> charts, but purely with proportional beating guides.
>
> I found a remarkably accurate relationship...

You can get very close to 2/7 comma by setting the beat ratio to be
exactly -3/2. See

http://66.98.148.43/~xenharmo/brat.html

for discussion of beat ratios ("brats".)

Jorgensen probably has some method along these lines also.

On Thursday 25 August 2005 1:30 am, Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> > Hey all,
> >
> > I've been busy investigating various ways to quickly approximate
>
> 2/7-comma
>
> > meantone temperament by ear without any monochord, audio aids, and
>
> beat rate
>
> > charts, but purely with proportional beating guides.
> >
> > I found a remarkably accurate relationship...
>
> You can get very close to 2/7 comma by setting the beat ratio to be
> exactly -3/2. See
>
> http://66.98.148.43/~xenharmo/brat.html
>
> for discussion of beat ratios ("brats".)
>
> Jorgensen probably has some method along these lines also.

Hmmmm...I guess you didn't bother to read the rest of my post, where I go into
precisely that.

-Aaron.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> Hmmmm...I guess you didn't bother to read the rest of my post, where
I go into
> precisely that.

I looked again and still can't find it.

On Thursday 25 August 2005 5:59 pm, Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> > Hmmmm...I guess you didn't bother to read the rest of my post, where
>
> I go into
>
> > precisely that.
>
> I looked again and still can't find it.

Ok, I shouldn't have said "precisely", relating to the -3/2 ratio, but my post
(which you cut in your response, thus giving third parties the idea that it
was content-less dribble in your eyes) was full of information about beat
ratios.

You ignored the basic thrust of my post: I discovered a technique to tune
2/7-comma and 1/5 comma meantone very accurately by ear; the general theory
of 'brats' was not my concern. I found it quite condescending that you cut
out my discovery of these approximations in your response, and then proceed
to instruct me about the existence of beat ratios ('brats' as you like to
say) as if I didn't know they existed, (and I do, since obviously, I make use
of them in my kick-ass approximations)

This would have been the more fitting response:

"Wow! You found some really great approximations...thanks for sharing
that....that's a great piece of minutia for the tuning encyclopedia."

Anyway....Gene, you wrote:

>> "You can get very close to 2/7 comma by setting the beat ratio to be
>> exactly -3/2"

I don't understand why you think that using a pure fifth would help at all:
wherever you put the third, this synchrony remains constant when your fifth
is pure, so it's completely useless for finding an accurate approximation for
a 2/7-comma meantone third by ear. Even if, as you know, you deviate the
fifth from pure, there is no way to do this systematically enough to get the
third and fifth just right except by trial and error, which is inelegant,
inefficient, and not what I was looking for, even though it's theoretically
possible.

I *was* looking for, and succesfully found, a quick and dirty way to set the
temperament without external aids which would be both easy
and accurate....2/7-comma has been throughout history considered a notorously
hard tuning to set by ear; in light of this, my discovery is no small one.

For those insterested in this discovery, my original post was here:

/tuning/topicId_59873.html#59873

Margo Schulter, (who I discussed this with on the phone), and I were both
excited by the possibility of an easy way to tune some of these important
historical meantones for the first time in an accurate way by ear. We also
had a rich discussion about the beauties of 1/7-comma meantone, which is
easily set once we have 2/7-comma meantone.

-Aaron.

Hi Aaron (and Gene),

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:

> On Thursday 25 August 2005 5:59 pm, Gene Ward Smith wrote:
>
> > --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
> > >
> > > Hmmmm...I guess you didn't bother to read the rest
> > > of my post, where I go into precisely that.
> >
> > I looked again and still can't find it.
>
> Ok, I shouldn't have said "precisely", relating to the
> -3/2 ratio, but my post (which you cut in your response,
> thus giving third parties the idea that it was content-less
> dribble in your eyes) was full of information about beat
> ratios.

Firstly, you mean "drivel" and not "dribble".

Secondly ... take it easy on Gene -- someone who contributes
to this list as much as he does has been chastised enough
by other regular posters in the past that he knows better
than to include overly-long quotes in his responses. The
archive is here, and all a reader needs to do is click on
the "Up Thread" link to find the previous post in that thread.
I'm glad that he trims his quotes.

(In case you haven't noticed, i'm so fanatical about this
that i ordinarily rearrange the text in my quotes so that
it appears neatly in my response, after Yahoo adds those
extra "greater than" signs to indicate levels of quotation,
which in turn add new line-breaks which totally disrupt
the look of the original quote.)

Anyway ... with all due respect to the rest of your post
(which, yes, i've snipped here), i have to say that,
once again on the tuning list, i'm sorry to see
personality conflicts intruding into what otherwise
is a very informative give-and-take. Just my 2 cents.

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

On Friday 26 August 2005 1:06 am, monz wrote:
> Hi Aaron (and Gene),
>
>
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
>
> wrote:
> > On Thursday 25 August 2005 5:59 pm, Gene Ward Smith wrote:
> > > --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
>
> wrote:
> > > > Hmmmm...I guess you didn't bother to read the rest
> > > > of my post, where I go into precisely that.
> > >
> > > I looked again and still can't find it.
> >
> > Ok, I shouldn't have said "precisely", relating to the
> > -3/2 ratio, but my post (which you cut in your response,
> > thus giving third parties the idea that it was content-less
> > dribble in your eyes) was full of information about beat
> > ratios.
>
> Firstly, you mean "drivel" and not "dribble".

Oy!

> Secondly ... take it easy on Gene -- someone who contributes
> to this list as much as he does has been chastised enough
> by other regular posters in the past that he knows better
> than to include overly-long quotes in his responses. The
> archive is here, and all a reader needs to do is click on
> the "Up Thread" link to find the previous post in that thread.
> I'm glad that he trims his quotes.

Now imagine *you* posted something informative, then someone comes along and
implies that nothing you are saying is new, posts a link to a related but
off-topic subject, ignores the information in your post, fails to quote your
original post, and gives the impression that you are a sort of newbie who is
new to the concepts espoused, even when your post indicates a familiarity
with those very concepts. Doesn't it seem a bit condescending? Sorry, it sure
did to me.

Gene's link seemed a bit more like an advertisement for his 'brats' webpage
than an engagement with the content of my post. Here we are having a
meta-conversation about my post, and no-one has yet responded to its actual
content (the approximations).

You'd think that in a group about the science of tuning, the announcement of a
practical and accurate technique of tuning an important historical
temperament by ear would be given some acknowledgement or notice.

Uh, nope.

In a message dated 8/26/2005 11:39:39 AM Eastern Standard Time,
aaron@akjmusic.com writes:
You'd think that in a group about the science of tuning, the announcement of
a
practical and accurate technique of tuning an important historical
temperament by ear would be given some acknowledgement or notice.

Uh, nope.
Has anyone heard the Early CD on PITCH (P-200202)? anyone....?
any one ?

I think there is a fair amount of disconnect on this list to go around.

best, Johnny

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> You'd think that in a group about the science of tuning, the
announcement of a
> practical and accurate technique of tuning an important historical
> temperament by ear would be given some acknowledgement or notice.
>
> Uh, nope.

Well, Aaron, to be honest *I* thought it was pretty cool, but since I:

a. don't have any way to try it out
b. don't consider myself knowledgeable in the subtleties of meantone

... I didn't have a good way to respond. Maybe I'll just come back to
Chi-town and watch you tune an instrument!

Keep on, don't let the yapping get you down...

Cheers,
Jon

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

> Keep on, don't let the yapping get you down...

I don't think my response qualified as "yapping". I pointed out that a
related but not identical method would be to tune, if you could, so
that the major beat ratio was exactly -3/2. The reference to my web
page was absolutely crucial, as was shown by Aaron's misinterpretation
of a -3/2 brat as a brat of +3/2. These are not the same, and while
+3/2 does indeed give the Pythagorean beat ratio, -3/2 gives the
positive real root of 6f^4 - 10f - 15, which is very close to a 2/7
comma fifth. It is *not* the same as Aaron's approximation, but it
seems to me to be relevant to the question of tuning 2/7 comma
meantone. I have no idea how easy it would be to tune, but Jorgensen
is always doing stuff along these lines. Other algebraically defined
fifths might also be tried.

If courtesy is too much to ask for, common sense would be a good
start. Giving precise definitions is neither "advirtising" nor
"yapping". Trimming is not an evil plot. And posting responses which
involve more than saying "Jeez, cool", but follow up with relevant
related matter, is how things normally work and how things are
supposed to work.

Hi Aaron,

I really do hope that i'm not making things worse
with my comments ... i generally try as hard as i can
to be "fair and balanced" when i post here, especially
when i'm writing something negative.

anyway ...

> Now imagine *you* posted something informative, then
> someone comes along and implies that nothing you are
> saying is new, posts a link to a related but off-topic
> subject, ignores the information in your post, fails to
> quote your original post, and gives the impression that
> you are a sort of newbie who is new to the concepts espoused,
> even when your post indicates a familiarity with those very
> concepts. Doesn't it seem a bit condescending? Sorry, it
> sure did to me.

"Imagine"?!!! Don't you think this has ever happened to me?
Consider the amount of work i've posted to this list.

I've posted lots of things that got even worse than a
condescending response -- that is, no response at all.
Remember, "even bad publicity is good publicity".

> Gene's link seemed a bit more like an advertisement
> for his 'brats' webpage than an engagement with the
> content of my post. Here we are having a meta-conversation
> about my post, and no-one has yet responded to its actual
> content (the approximations).
>
> You'd think that in a group about the science of tuning,
> the announcement of a practical and accurate technique of
> tuning an important historical temperament by ear would be
> given some acknowledgement or notice.
>
> Uh, nope.

OK, then i can say at least one small thing: while i
do understand the importance of equal-beating schemes
for the ability to realize a tuning solely by ear, i
personally do not like the sound of them, particularly
for sustained chords.

IMO, when a chord is played in which two or more of
the intervals beat at the same rate, it sets up an
unwanted pulse which will probably interfere with the
rhythmic structure of the piece.

On the other hand, equal-beating tunings can be useful
for composing music in which the beats play an integral
role in the rhythmic structure ... i think of Kraig Grady's
music as a possible example.

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

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> I don't think my response qualified as "yapping".

Hate to disappoint you, but I wasn't referring to your post.

Best,
Jon

On Friday 26 August 2005 3:14 pm, monz wrote:

> OK, then i can say at least one small thing: while i
> do understand the importance of equal-beating schemes
> for the ability to realize a tuning solely by ear, i
> personally do not like the sound of them, particularly
> for sustained chords.

Monz,

My discovery is not an equal-beating scheme. I simply use, in both the
2/7-comma and the 1/5-comma techniques, an equal beating triad to get an
*inital* note (the 'E' in both cases, assuming a 'C' based tuning); from
there I expect the tuner to know what they are doing, and do it the
mathematically correct way. Read my post again carefully, and if you have any
questions, let me know.

I hope this clarifies.

-Aaron.

On Friday 26 August 2005 2:07 pm, Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
> > Keep on, don't let the yapping get you down...
>
> I don't think my response qualified as "yapping". I pointed out that a
> related but not identical method would be to tune, if you could, so
> that the major beat ratio was exactly -3/2. The reference to my web
> page was absolutely crucial, as was shown by Aaron's misinterpretation
> of a -3/2 brat as a brat of +3/2. These are not the same, and while
> +3/2 does indeed give the Pythagorean beat ratio, -3/2 gives the
> positive real root of 6f^4 - 10f - 15, which is very close to a 2/7
> comma fifth. It is *not* the same as Aaron's approximation, but it
> seems to me to be relevant to the question of tuning 2/7 comma
> meantone. I have no idea how easy it would be to tune, but Jorgensen
> is always doing stuff along these lines. Other algebraically defined
> fifths might also be tried.

Ok, I did misunderstand you with the fifth. Obviously, we both agree that
trying to find a third with a pure outer fifth would be futile, since the
beat rate between C-E and E-G would be constant wherever E was...I think a
flattened fifth fares no better, because you'd have a messy iterative
'shot-in-the-dark' situation. My technique allows high accuracy for the first
time, using a well known reference point. (Actually, I don't know if anyone
else has discovered my trick--I assume not)

> If courtesy is too much to ask for, common sense would be a good
> start. Giving precise definitions is neither "advirtising" nor
> "yapping". Trimming is not an evil plot. And posting responses which
> involve more than saying "Jeez, cool", but follow up with relevant
> related matter, is how things normally work and how things are
> supposed to work.

No hard feelings, Gene, I was being a bit sensitive to the fact that you
posted follow without the 'Jeez, cool' part....

-Aaron.

On Friday 26 August 2005 2:40 pm, wallyesterpaulrus wrote:
> Aaron,
>
> What you may not know is that exactly these beat properties of 2/7-
> comma meantone were extensively discussed on both this list and the
> tuning-math list in the not very distant past. Yes, you deserve
> congratulations for your independent discovery of this fact.

Boy, do you mis-read my post (did you read it at all?). First, I was part of
those discussions (even writing about it after the fact, restarting an old
thread in response to archival research--if you look Google in the archives,
you'll see my name pop up in those discussions). Secondly, I am well aware of
the synchronous beating properties of 2/7-comma. Thirdly, what I discovered
was not those synchronous properties (duh, I knew about them already), but an
accurate trick to find a 2/7-comma fifth without a monochord, so you could
tune the puppy up without other audio aids. (Ditto my 1/5-comma trick, and
yes, I know that 1/5-comma has great beat synchrony too, as does 1/7-comma,
etc.)

Do you understand what I'm saying here?

> Gene
> should have said something about how the link he was pointing you to
> contained a more general statement of both your (re-) discovery and
> many other related facts.

Like I said, it's not a re-discovery, but a genuine discovery of a tuning
bearing-plan trick. A 'bearing-plan', in case you didn't know, is a set of
instructions for setting a temperament (yes, I'm being facetious)

> Gene has been discourteous to you but you're also not understanding
> him. Pure fifth? You seem to be reading in haste, as someone
> (rightly) upset might. Let's love and respect one another a bit more,
> can we? I know I love you both.

(smoochie-smoochie to you to, bro)

-Aaron.

Metameantone which is a sequence that starts with whole numbers produces triads with equal beating of various sizes that converge on one that in implied by the rest.
this is best illustrated on page 4 of http://anaphoria.com/meantone-mavila.PDF
this was done back in 92-93. in fact every single diagonal and recurrent sequence of Mt Meru produces an equal beating triads

> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Hi Aaron,

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:

> On Friday 26 August 2005 3:14 pm, monz wrote:
>
> > OK, then i can say at least one small thing: while i
> > do understand the importance of equal-beating schemes
> > for the ability to realize a tuning solely by ear, i
> > personally do not like the sound of them, particularly
> > for sustained chords.
>
> Monz,
>
> My discovery is not an equal-beating scheme. I simply use,
> in both the 2/7-comma and the 1/5-comma techniques, an
> equal beating triad to get an *inital* note (the 'E' in
> both cases, assuming a 'C' based tuning); from there I
> expect the tuner to know what they are doing, and do it
> the mathematically correct way. Read my post again carefully,
> and if you have any questions, let me know.

Right ... of course i understood that, but thanks for
pointing it out anyway, because my criticism thus only
applies to that B-minor chord. Beat rates in other
chords will not be equal and thus won't create that
pulse which i find annoying, and i hadn't addressed that.

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

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> No hard feelings, Gene, I was being a bit sensitive to the fact that
you
> posted follow without the 'Jeez, cool' part....

The trouble with that is that I don't actually know what would be
practical and what wouldn't be as a tuning method, so it's all numbers
to me.

On Wed, 24 Aug 2005, Aaron Krister Johnson wrote:
> Hey all,
>
> I've been busy investigating various ways to quickly approximate 2/7-comma
> meantone temperament by ear without any monochord, audio aids, and beat
rate
> charts, but purely with proportional beating guides.
>
> I found a remarkably accurate relationship: temporarily tune a 1/3-comma
> meantone fifth C-G, and then tune C-E flat from pure so that it beats 1
for
> every 2 E-G's. This 'C-E' will be a remarkably close 2/7-comma meantone
third
> C-E: now distribute this 'C-E' into the 4 usual meantone fifths,
C-G-D-A-E;
> the size of the fifth C-G will agree to 5 places after the decimal
> (~1.49468733 vs. the true value of ~1.49468550)! In fact, I think this
> technique might be *more* accurate than a monochord, because you are not
> dealing with the error of the fret placement--it's purely an acoustic
> relationship between beats...
>
> To get 1/7-comma, take the resulting C-G, put a pure fifth G-D above it,
and
> then readjust C-G so that G-D beats 1.5 times faster than C-G.
> This results in a 1/7-comma fifth.
>
> Mathematically, you've of course taken the square root of C-D' (a major
ninth)
> to get C-G...
>
> Two other tricks:
> 1/6-comma fifth= sqrt(1.5*(1/3 comma fifth))
> 1/8-comma fifth = sqrt(1.5*(1/4 comma fifth))
>
> Not satisfied with that, I found a way to get a very accurate 1/5-comma
> meantone fifth by ear...tune C-B as a pure 15/8. Temporarily tune C-G as a
> 1/4-comma meantone fifth. Then make E sharp slightly so that E-B beats at
> exactly the same rate as C-G. This C-E is now a very accurate 1/5-comma
> meantone third. The 'C-E' will then be divided into four fifths as any
> meantone, C-G-D-A-E, as above. The new C-G after this division will be an
> accurate 1/5-comma meantone fifth, again to five places after the decimal
> point (~1.49627528 vs ~1.49627787)!
>
> Cheers,

Aaron,

Very clever! However did you happen on these "tricks"? I mean, what
kind of thinking goes on to achieve these results ... do you perhaps work
backwards from the goal?

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.11.10/120 - Release Date: 5/10/05

On Friday 07 October 2005 11:09 am, Yahya Abdal-Aziz wrote:
> On Wed, 24 Aug 2005, Aaron Krister Johnson wrote:
> > Hey all,
> >
> > I've been busy investigating various ways to quickly approximate
> > 2/7-comma meantone temperament by ear without any monochord, audio aids,
> > and beat
>
> rate
>
> > charts, but purely with proportional beating guides.
> >
> > I found a remarkably accurate relationship: temporarily tune a 1/3-comma
> > meantone fifth C-G, and then tune C-E flat from pure so that it beats 1
>
> for
>
> > every 2 E-G's. This 'C-E' will be a remarkably close 2/7-comma meantone
>
> third
>
> > C-E: now distribute this 'C-E' into the 4 usual meantone fifths,
>
> C-G-D-A-E;
>
> > the size of the fifth C-G will agree to 5 places after the decimal
> > (~1.49468733 vs. the true value of ~1.49468550)! In fact, I think this
> > technique might be *more* accurate than a monochord, because you are not
> > dealing with the error of the fret placement--it's purely an acoustic
> > relationship between beats...
> >
> > To get 1/7-comma, take the resulting C-G, put a pure fifth G-D above it,
>
> and
>
> > then readjust C-G so that G-D beats 1.5 times faster than C-G.
> > This results in a 1/7-comma fifth.
> >
> > Mathematically, you've of course taken the square root of C-D' (a major
>
> ninth)
>
> > to get C-G...
> >
> > Two other tricks:
> > 1/6-comma fifth= sqrt(1.5*(1/3 comma fifth))
> > 1/8-comma fifth = sqrt(1.5*(1/4 comma fifth))
> >
> > Not satisfied with that, I found a way to get a very accurate 1/5-comma
> > meantone fifth by ear...tune C-B as a pure 15/8. Temporarily tune C-G as
> > a 1/4-comma meantone fifth. Then make E sharp slightly so that E-B beats
> > at exactly the same rate as C-G. This C-E is now a very accurate
> > 1/5-comma meantone third. The 'C-E' will then be divided into four fifths
> > as any meantone, C-G-D-A-E, as above. The new C-G after this division
> > will be an accurate 1/5-comma meantone fifth, again to five places after
> > the decimal point (~1.49627528 vs ~1.49627787)!
> >
> > Cheers,
>
> Aaron,
>
> Very clever! However did you happen on these "tricks"? I mean, what
> kind of thinking goes on to achieve these results ... do you perhaps work
> backwards from the goal?

I was fooling around with equal-beating equations in a spreadsheet, just
looking for simple integer relationships, and seeing if I could find any that
would help for an ear-based bearing plan...

-Aaron.

Hi Aaron,
On Fri, 7 Oct 2005, you wrote:
>
> On Friday 07 October 2005 11:09 am, Yahya Abdal-Aziz wrote:
> > On Wed, 24 Aug 2005, Aaron Krister Johnson wrote:
> > > Hey all,
> > >
> > > I've been busy investigating various ways to quickly approximate
> > > 2/7-comma meantone temperament by ear without any monochord, audio
aids,
> > > and beat rate charts, but purely with proportional beating guides.
... [snipt]

> > Very clever! However did you happen on these "tricks"? I mean, what
> > kind of thinking goes on to achieve these results ... do you perhaps
work
> > backwards from the goal?
>
> I was fooling around with equal-beating equations in a spreadsheet, just
> looking for simple integer relationships, and seeing if I could find any
that
> would help for an ear-based bearing plan...

Ah so! Very much of our best work is done while playing ...

Best,
Yahya

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