Yahoo Tuning Groups Ultimate Backup tuning converting the Scale Tree to ET scale degrees

Hello Dan!
Page 8 of the scale tree generates the same scales but with different generators.
i believe this is the first area Erv examined, being tipped off by Yasser and Kornerup.
I believe the early Xenharmonikon do also.

"D.Stearns" wrote:

> Awhile back I came up with an EDO to RI model that, roughly speaking,
> bent undertone series into overtone series (and vice versa), and as
> one bent towards the other they gradually passed through what I called
> equaltone series--so this model roughly attempted to equate EDOs with
> U to O, or O to U, medians.
>
> A particularly interesting aspect of this is that it converts the
> Stern-Brocot tree--if it were seeded with 1/1 and 2/1--into EDO scale
> degrees. One example of this taken from the U to O morph would involve
> seeding the tree with 0/5 and 7/7:
>
> 0/5 7/7
> 7/12
> 7/17 14/19
> 7/22 14/29 21/31 21/26
> 7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33
>
> (etc.)
>
> This is the simplest--i.e., lowest numbered ratios as least distorted
> EDO scale degrees--way to model this that I know of. Has anyone
> noticed this before?
>
> take care,
>
> --Dan Stearns
>
>
> 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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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Kraig,

Rather than seeing these as generators, I was looking at these as
scale degrees exactly analogous to these ratios:

1/1 2/1
3/2
4/3 5/3
5/4 7/5 8/5 7/4
6/5 9/7 11/8 10/7 11/7 13/8 12/7 9/5

(etc.)

So the same algorithm would convert 5 equal into:

1/1 38/33 41/31 44/29 47/27 2/1

7 equal into:

1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1

And 12 equal into:

1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1

Roughly speaking, this is what happens when an undertone series is
bent into an overtone series. These equal series occur as one is bent
towards the other.

take care,

--Dan Stearns

----- Original Message -----
From: "Kraig Grady" <kraiggrady@anaphoria.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, June 23, 2002 2:08 AM
Subject: Re: [tuning] converting the Scale Tree to ET scale degrees

> Hello Dan!
> Page 8 of the scale tree generates the same scales but with
different generators.
> i believe this is the first area Erv examined, being tipped off by
Yasser and Kornerup.
> I believe the early Xenharmonikon do also.
>
>
> "D.Stearns" wrote:
>
> > Awhile back I came up with an EDO to RI model that, roughly
speaking,
> > bent undertone series into overtone series (and vice versa), and
as
> > one bent towards the other they gradually passed through what I
called
> > equaltone series--so this model roughly attempted to equate EDOs
with
> > U to O, or O to U, medians.
> >
> > A particularly interesting aspect of this is that it converts the
> > Stern-Brocot tree--if it were seeded with 1/1 and 2/1--into EDO
scale
> > degrees. One example of this taken from the U to O morph would
involve
> > seeding the tree with 0/5 and 7/7:
> >
> > 0/5 7/7
> > 7/12
> > 7/17 14/19
> > 7/22 14/29 21/31 21/26
> > 7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33
> >
> > (etc.)
> >
> > This is the simplest--i.e., lowest numbered ratios as least
distorted
> > EDO scale degrees--way to model this that I know of. Has anyone
> > noticed this before?
> >
> > take care,
> >
> > --Dan Stearns
> >
> >
> > You do not need web access to participate. You may subscribe
through
> > email. Send an empty email to one of these addresses:
> > tuning-subscribe@yahoogroups.com - join the tuning group.
> > tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning
group.
> > tuning-nomail@yahoogroups.com - put your email message delivery
on hold for the tuning group.
> > tuning-digest@yahoogroups.com - change your subscription to
daily digest mode.
> > tuning-normal@yahoogroups.com - change your subscription to
individual emails.
> > tuning-help@yahoogroups.com - receive general help information.
> >
> >
> > Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
>
> -- Kraig Grady
> North American Embassy of Anaphoria island
> http://www.anaphoria.com
>
> The Wandering Medicine Show
> Wed. 8-9 KXLU 88.9 fm
>
>
>
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
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> Risk Free!
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> --------------------------------------------------------------------
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>
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>
>

🔗graham@microtonal.co.uk

🔗D.Stearns <STEARNS@CAPECOD.NET>

Graham,

The way it works you have two variables:

first, and this would follow the octave as a non distinct interval
convention, you have how many distinct tones there are in a given
series--let that be "n"

then you have the parameter that controls the degree of bend--let that
be "x"

To convert a given n into a series, 1/1 = n(x)/n(x). In the U-->O
morph, a given n undertone series is where everything starts, and in
terms of the Scale Tree to ET scale degrees, this would be:

0/1 1/1
1/2
1/3 2/3
1/4 2/5 3/5 3/4

(etc.)

So if you want to convert n into an overtone series x = 2. Now let's
say n = 5, then the series begins 10/10 and the numerator is to add
(x-2), and the denominator rule is subtract 1. So if x = 2, then x-2 =
0, and this would give us:

10/10 10/9 10/8 10/7 10/6 10/5

or:

1/1 10/9 5/4 10/7 5/3 2/1

Now what you want is an RI where x best approximates the sqrt(2)+2,
and when talking rational intonations, best approximation usually
means simplest ratios... but you also want the ET scale degrees to be
close enough as well.

Two things that satisfy the first criterion, but still fail the second
are x = 3 and x = 4. Here are the Scale Tree to ET scale degrees
(where the tree is seeded 1/1 2/1):

0/2 3/3
3/5
3/7 6/8
3/9 6/12 9/13 9/11

(etc.)

and:

0/3 4/4
4/7
4/10 8/11
4/13 8/17 12/18 12/15

(etc.)

Now Let's say n = 5 and x = 3, then you'd have:

1/1 8/7 17/13 3/2 19/11 2/1

and if n = 5 and x = 4, then you'd have:

1/1 22/19 4/3 26/17 7/4 2/1

These series are really quite wonderful and full of surprises.
However, they don't quite satisfy the search for an optimum equaltone
series.

The series that would seem to satisfy both criteria would be the
simplest rounding of sqrt(2)+2, x = 3.4. Here are the Scale Tree to ET
scale degrees:

0/12 17/17
17/29
17/41 34/46
17/53 34/70 51/75 51/63

(etc.)

Here are the n = 5, 7, and 12 series where x = 3.4:

1/1 23/20 33/25 53/35 113/65 2/1

1/1 21/19 133/109 35/26 49/33 77/47 161/89 2/1

1/1 211/199 109/97 25/21 29/23 239/179 41/29 253/169 65/41 89/53
137/77 281/149 2/1

To my mind these series satisfy the second criteria too much to the
detriment of the first. So as I said in my first post, it's my opinion
that the simplest--i.e., lowest numbered ratios as least distorted EDO
scale degrees--way to model this is where x = 3.5.

Here's the Stern-Brocot tree as if it were seeded with 1/1 and 2/1:

0/5 7/7
7/12
7/17 14/19
7/22 14/29 21/31 21/26
7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33

(etc.)

and here's the 5,6,7,8,9,10,11, and 12 series:

1/1 38/33 41/31 44/29 47/27 2/1

1/1 9/8 24/19 17/12 27/17 57/32 2/1

1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1

1/1 59/54 31/26 13/10 17/12 71/46 37/22 11/6 2/1

1/1 66/61 69/59 24/19 15/11 78/53 27/17 12/7 87/47 2/1

1/1 73/68 38/33 79/64 41/31 17/12 44/29 13/8 47/27 97/52 2/1

1/1 16/15 83/73 86/71 89/69 92/67 19/13 14/9 101/61 104/59 107/57 2/1

1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1

Increasing the value of x marches the series on a never-ending but
ever closer approximation of a given n as an overtone series, hence
the U-->O morph.

take care,

--Dan Stearns

----- Original Message -----
From: <graham@microtonal.co.uk>
To: <tuning@yahoogroups.com>
Sent: Sunday, June 23, 2002 5:06 AM
Subject: [tuning] Re: converting the Scale Tree to ET scale degrees

> D.Stearns wrote:
>
> > So the same algorithm would convert 5 equal into:
> >
> > 1/1 38/33 41/31 44/29 47/27 2/1
> >
> > 7 equal into:
> >
> > 1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1
> >
> > And 12 equal into:
> >
> > 1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62
2/1
> >
> > Roughly speaking, this is what happens when an undertone series is
> > bent into an overtone series. These equal series occur as one is
bent
> > towards the other.
>
> How? Where do these numbers come from?
>
>
> Graham
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
> Free $5 Love Reading
> Risk Free!
> http://us.click.yahoo.com/3PCXaC/PfREAA/Ey.GAA/RrLolB/TM
> --------------------------------------------------------------------
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>
> 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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> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
>
>

🔗Kraig Grady <kraiggrady@anaphoria.com>

Hello Dan!
I am still having some trouble getting this. I do notice that the number increase from
different direction but not quite understanding how you ( or why) seed it.

"D.Stearns" wrote:

> Graham,
>
> The way it works you have two variables:
>
> first, and this would follow the octave as a non distinct interval
> convention, you have how many distinct tones there are in a given
> series--let that be "n"
>
> then you have the parameter that controls the degree of bend--let that
> be "x"
>
> To convert a given n into a series, 1/1 = n(x)/n(x). In the U-->O
> morph, a given n undertone series is where everything starts, and in
> terms of the Scale Tree to ET scale degrees, this would be:
>
> 0/1 1/1
> 1/2
> 1/3 2/3
> 1/4 2/5 3/5 3/4
>
> (etc.)
>
> So if you want to convert n into an overtone series x = 2. Now let's
> say n = 5, then the series begins 10/10 and the numerator is to add
> (x-2), and the denominator rule is subtract 1. So if x = 2, then x-2 =
> 0, and this would give us:
>
> 10/10 10/9 10/8 10/7 10/6 10/5
>
> or:
>
> 1/1 10/9 5/4 10/7 5/3 2/1
>
> Now what you want is an RI where x best approximates the sqrt(2)+2,
> and when talking rational intonations, best approximation usually
> means simplest ratios... but you also want the ET scale degrees to be
> close enough as well.
>
> Two things that satisfy the first criterion, but still fail the second
> are x = 3 and x = 4. Here are the Scale Tree to ET scale degrees
> (where the tree is seeded 1/1 2/1):
>
> 0/2 3/3
> 3/5
> 3/7 6/8
> 3/9 6/12 9/13 9/11
>
> (etc.)
>
> and:
>
> 0/3 4/4
> 4/7
> 4/10 8/11
> 4/13 8/17 12/18 12/15
>
> (etc.)
>
> Now Let's say n = 5 and x = 3, then you'd have:
>
> 1/1 8/7 17/13 3/2 19/11 2/1
>
> and if n = 5 and x = 4, then you'd have:
>
> 1/1 22/19 4/3 26/17 7/4 2/1
>
> These series are really quite wonderful and full of surprises.
> However, they don't quite satisfy the search for an optimum equaltone
> series.
>
> The series that would seem to satisfy both criteria would be the
> simplest rounding of sqrt(2)+2, x = 3.4. Here are the Scale Tree to ET
> scale degrees:
>
> 0/12 17/17
> 17/29
> 17/41 34/46
> 17/53 34/70 51/75 51/63
>
> (etc.)
>
> Here are the n = 5, 7, and 12 series where x = 3.4:
>
> 1/1 23/20 33/25 53/35 113/65 2/1
>
> 1/1 21/19 133/109 35/26 49/33 77/47 161/89 2/1
>
> 1/1 211/199 109/97 25/21 29/23 239/179 41/29 253/169 65/41 89/53
> 137/77 281/149 2/1
>
> To my mind these series satisfy the second criteria too much to the
> detriment of the first. So as I said in my first post, it's my opinion
> that the simplest--i.e., lowest numbered ratios as least distorted EDO
> scale degrees--way to model this is where x = 3.5.
>
> Here's the Stern-Brocot tree as if it were seeded with 1/1 and 2/1:
>
> 0/5 7/7
> 7/12
> 7/17 14/19
> 7/22 14/29 21/31 21/26
> 7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33
>
> (etc.)
>
> and here's the 5,6,7,8,9,10,11, and 12 series:
>
> 1/1 38/33 41/31 44/29 47/27 2/1
>
> 1/1 9/8 24/19 17/12 27/17 57/32 2/1
>
> 1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1
>
> 1/1 59/54 31/26 13/10 17/12 71/46 37/22 11/6 2/1
>
> 1/1 66/61 69/59 24/19 15/11 78/53 27/17 12/7 87/47 2/1
>
> 1/1 73/68 38/33 79/64 41/31 17/12 44/29 13/8 47/27 97/52 2/1
>
> 1/1 16/15 83/73 86/71 89/69 92/67 19/13 14/9 101/61 104/59 107/57 2/1
>
> 1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1
>
> Increasing the value of x marches the series on a never-ending but
> ever closer approximation of a given n as an overtone series, hence
> the U-->O morph.
>
> take care,
>
> --Dan Stearns
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗graham@microtonal.co.uk

In-Reply-To: <003901c21bfd$606b0f20$f772d63f@stearns>
D.Stearns wrote:

> first, and this would follow the octave as a non distinct interval
> convention, you have how many distinct tones there are in a given
> series--let that be "n"

Right, n notes.

> then you have the parameter that controls the degree of bend--let that
> be "x"

This seems to be the only definition of "x" and it isn't enough.

> To convert a given n into a series, 1/1 = n(x)/n(x). In the U-->O
> morph, a given n undertone series is where everything starts, and in
> terms of the Scale Tree to ET scale degrees, this would be:

Okay, so for n notes we seed with n/n

> 0/1 1/1
> 1/2
> 1/3 2/3
> 1/4 2/5 3/5 3/4
>
> (etc.)
>
> So if you want to convert n into an overtone series x = 2. Now let's
> say n = 5, then the series begins 10/10 and the numerator is to add
> (x-2), and the denominator rule is subtract 1. So if x = 2, then x-2 =
> 0, and this would give us:
>
> 10/10 10/9 10/8 10/7 10/6 10/5

Oh, so this is an undertone series. Where do the "numerator" and
"denominator" rules come from? Why does the series begin 10/10? I
thought from above it would be 5/5.

> or:
>
> 1/1 10/9 5/4 10/7 5/3 2/1
>
> Now what you want is an RI where x best approximates the sqrt(2)+2,
> and when talking rational intonations, best approximation usually
> means simplest ratios... but you also want the ET scale degrees to be
> close enough as well.

Do we? What's sqrt(2)+2 got to do with anything?

> Two things that satisfy the first criterion, but still fail the second
> are x = 3 and x = 4. Here are the Scale Tree to ET scale degrees
> (where the tree is seeded 1/1 2/1):
>
> 0/2 3/3
> 3/5
> 3/7 6/8
> 3/9 6/12 9/13 9/11
>
> (etc.)

But that tree's seeded with 0/2 and 3/3, not 1/1 and 2/1.

> and:
>
> 0/3 4/4
> 4/7
> 4/10 8/11
> 4/13 8/17 12/18 12/15
>
> (etc.)
>
> Now Let's say n = 5 and x = 3, then you'd have:
>
> 1/1 8/7 17/13 3/2 19/11 2/1
>
> and if n = 5 and x = 4, then you'd have:
>
> 1/1 22/19 4/3 26/17 7/4 2/1

Are these numbers supposed to come from the trees above?

> These series are really quite wonderful and full of surprises.
> However, they don't quite satisfy the search for an optimum equaltone
> series.
>
> The series that would seem to satisfy both criteria would be the
> simplest rounding of sqrt(2)+2, x = 3.4. Here are the Scale Tree to ET
> scale degrees:
>
> 0/12 17/17
> 17/29
> 17/41 34/46
> 17/53 34/70 51/75 51/63
>
> (etc.)
>
> Here are the n = 5, 7, and 12 series where x = 3.4:
>
> 1/1 23/20 33/25 53/35 113/65 2/1
>
> 1/1 21/19 133/109 35/26 49/33 77/47 161/89 2/1
>
> 1/1 211/199 109/97 25/21 29/23 239/179 41/29 253/169 65/41 89/53
> 137/77 281/149 2/1

You're making the same conceptual leaps there as before. Can you fill in
the intermediate steps?

> To my mind these series satisfy the second criteria too much to the
> detriment of the first. So as I said in my first post, it's my opinion
> that the simplest--i.e., lowest numbered ratios as least distorted EDO
> scale degrees--way to model this is where x = 3.5.
>
> Here's the Stern-Brocot tree as if it were seeded with 1/1 and 2/1:
>
> 0/5 7/7
> 7/12
> 7/17 14/19
> 7/22 14/29 21/31 21/26
> 7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33
>
> (etc.)

As it includes ratios not in their simplest terms, it can't be a
Stern-Brocot tree, can it?

> and here's the 5,6,7,8,9,10,11, and 12 series:
>
> 1/1 38/33 41/31 44/29 47/27 2/1
>
> 1/1 9/8 24/19 17/12 27/17 57/32 2/1
>
> 1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1
>
> 1/1 59/54 31/26 13/10 17/12 71/46 37/22 11/6 2/1
>
> 1/1 66/61 69/59 24/19 15/11 78/53 27/17 12/7 87/47 2/1
>
> 1/1 73/68 38/33 79/64 41/31 17/12 44/29 13/8 47/27 97/52 2/1
>
> 1/1 16/15 83/73 86/71 89/69 92/67 19/13 14/9 101/61 104/59 107/57 2/1
>
> 1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1
>
> Increasing the value of x marches the series on a never-ending but
> ever closer approximation of a given n as an overtone series, hence
> the U-->O morph.

You'll have to explain what this x does.

Graham

🔗genewardsmith <genewardsmith@juno.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Kraig,

Another way to look at this aside from the U-->O morphs, is just as a
way of converting the Scale Tree to ET scale degrees that simply shows
what a given ET seeding looks like when it is grafted onto this JI
grid:

1/1 2/1
3/2
4/3 5/3
5/4 7/5 8/5 7/4

(etc.)

The closer your ET 3/2 is to just, then the closer the grid is to
just.

Here's 41 ET as an example:

0/17 24/24
24/41
24/58 48/65
24/75 48/99 72/106 72/89

(etc.)

Obviously, the simpler the ET approximating the 3/2, the simpler the
RI series approximating the ETs, and that's why 12 works so well--hope
this helps!

take care,

--Dan Stearns

----- Original Message -----
From: "Kraig Grady" <kraiggrady@anaphoria.com>
To: <tuning@yahoogroups.com>
Sent: Monday, June 24, 2002 9:36 PM
Subject: Re: [tuning] Re: converting the Scale Tree to ET scale
degrees

> Hello Dan!
> I am still having some trouble getting this. I do notice that
the number increase from
> different direction but not quite understanding how you ( or why)
seed it.
>
> "D.Stearns" wrote:
>
> > Graham,
> >
> > The way it works you have two variables:
> >
> > first, and this would follow the octave as a non distinct interval
> > convention, you have how many distinct tones there are in a given
> > series--let that be "n"
> >
> > then you have the parameter that controls the degree of bend--let
that
> > be "x"
> >
> > To convert a given n into a series, 1/1 = n(x)/n(x). In the U-->O
> > morph, a given n undertone series is where everything starts, and
in
> > terms of the Scale Tree to ET scale degrees, this would be:
> >
> > 0/1 1/1
> > 1/2
> > 1/3 2/3
> > 1/4 2/5 3/5 3/4
> >
> > (etc.)
> >
> > So if you want to convert n into an overtone series x = 2. Now
let's
> > say n = 5, then the series begins 10/10 and the numerator is to
add
> > (x-2), and the denominator rule is subtract 1. So if x = 2, then
x-2 =
> > 0, and this would give us:
> >
> > 10/10 10/9 10/8 10/7 10/6 10/5
> >
> > or:
> >
> > 1/1 10/9 5/4 10/7 5/3 2/1
> >
> > Now what you want is an RI where x best approximates the
sqrt(2)+2,
> > and when talking rational intonations, best approximation usually
> > means simplest ratios... but you also want the ET scale degrees to
be
> > close enough as well.
> >
> > Two things that satisfy the first criterion, but still fail the
second
> > are x = 3 and x = 4. Here are the Scale Tree to ET scale degrees
> > (where the tree is seeded 1/1 2/1):
> >
> > 0/2 3/3
> > 3/5
> > 3/7 6/8
> > 3/9 6/12 9/13 9/11
> >
> > (etc.)
> >
> > and:
> >
> > 0/3 4/4
> > 4/7
> > 4/10 8/11
> > 4/13 8/17 12/18 12/15
> >
> > (etc.)
> >
> > Now Let's say n = 5 and x = 3, then you'd have:
> >
> > 1/1 8/7 17/13 3/2 19/11 2/1
> >
> > and if n = 5 and x = 4, then you'd have:
> >
> > 1/1 22/19 4/3 26/17 7/4 2/1
> >
> > These series are really quite wonderful and full of surprises.
> > However, they don't quite satisfy the search for an optimum
equaltone
> > series.
> >
> > The series that would seem to satisfy both criteria would be the
> > simplest rounding of sqrt(2)+2, x = 3.4. Here are the Scale Tree
to ET
> > scale degrees:
> >
> > 0/12 17/17
> > 17/29
> > 17/41 34/46
> > 17/53 34/70 51/75 51/63
> >
> > (etc.)
> >
> > Here are the n = 5, 7, and 12 series where x = 3.4:
> >
> > 1/1 23/20 33/25 53/35 113/65 2/1
> >
> > 1/1 21/19 133/109 35/26 49/33 77/47 161/89 2/1
> >
> > 1/1 211/199 109/97 25/21 29/23 239/179 41/29 253/169 65/41 89/53
> > 137/77 281/149 2/1
> >
> > To my mind these series satisfy the second criteria too much to
the
> > detriment of the first. So as I said in my first post, it's my
opinion
> > that the simplest--i.e., lowest numbered ratios as least distorted
EDO
> > scale degrees--way to model this is where x = 3.5.
> >
> > Here's the Stern-Brocot tree as if it were seeded with 1/1 and
2/1:
> >
> > 0/5 7/7
> > 7/12
> > 7/17 14/19
> > 7/22 14/29 21/31 21/26
> > 7/27 14/39 21/46 21/41 28/43 35/50 35/45 28/33
> >
> > (etc.)
> >
> > and here's the 5,6,7,8,9,10,11, and 12 series:
> >
> > 1/1 38/33 41/31 44/29 47/27 2/1
> >
> > 1/1 9/8 24/19 17/12 27/17 57/32 2/1
> >
> > 1/1 52/47 11/9 58/43 61/41 64/39 67/37 2/1
> >
> > 1/1 59/54 31/26 13/10 17/12 71/46 37/22 11/6 2/1
> >
> > 1/1 66/61 69/59 24/19 15/11 78/53 27/17 12/7 87/47 2/1
> >
> > 1/1 73/68 38/33 79/64 41/31 17/12 44/29 13/8 47/27 97/52 2/1
> >
> > 1/1 16/15 83/73 86/71 89/69 92/67 19/13 14/9 101/61 104/59 107/57
2/1
> >
> > 1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62
2/1
> >
> > Increasing the value of x marches the series on a never-ending but
> > ever closer approximation of a given n as an overtone series,
hence
> > the U-->O morph.
> >
> > take care,
> >
> > --Dan Stearns
> >
>
> -- Kraig Grady
> North American Embassy of Anaphoria island
> http://www.anaphoria.com
>
> The Wandering Medicine Show
> Wed. 8-9 KXLU 88.9 fm
>
>
>
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