A Scale of two notes

Let us look at the simplest possible scale

Ls

Let us label the two primary pitches as 0 and 1

0------1---(0...

Transposing the scale to begin on 1, we arrive at

0-----1---0------1
1------0#--1

Transposing the scale downward so that the upper note of the L scale
interval is 0:
0------1---0
1b-----0---1b

This gives

0------1---0
with 0# and 1b between 0 and 1

No we ask the question : is the interval L-s larger or smaller than s?

If L-s is larger than s, then we arrive at

0--1b--0#--1---0

Otherwise

0--0#--1b--1---0

If L-s is bigger than s, then 1# and 0b are placed here in the scale:

0-1#-1b-0b-0#-1---0

Though you colud argue their exact positions dependent up on the
ratio of L-s to s.

For L-s is less than s:

0--0#--1b--1-0b-1#-0, and potentially 0b=1#

Finally, for the L-s is equal to s case

0-0#=1b-1=0b-1#=0.

For our minimal scale, we need at least three distinct pitches. For L-
s is less than s, we can have five pitches. For L-s > s, we have four
as a minimum. The most basic forms for L-s < s, we have 5 (L=3,s=2)
and 7 (L=4,s=3). For L-s > s we have 4 (L=3,s=1) and 7 (L=5,s=2). The
representation of this scale in 12EDO is L=7,s=5.

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> Let us look at the simplest possible scale
>
> Ls
>
> Let us label the two primary pitches as 0 and 1
>
> 0------1---(0...
>
> Transposing the scale to begin on 1, we arrive at
>
> 0-----1---0------1
> 1------0#--1

I must say, you lost me with the # notation, and much of the
subsequent discussion. Is there another way you could explain
this? (And what's the idea behind it?)

-Carl

THe # notation is a sharp and b is a flat sign

so you get

0 - the first pitch
1 a pitch somewhere between 0 and its octave higher note, such that
the interval from 0 up to 1 is larger than from 1 up to 0 (an octave
higher)

so we can write 0 L 1 s 0
No let's begin the scale on 1:

1 L ? s 1

The note ? is 0 raised by L-s, which traditionally is considered to
be the chromatic interval, raising by a sharp. Let's call this note 0#

so we now get

1 L 0# s 1

No let's transpose the scale so that 0 is the upper note of the L
interval:

? L 0 s ?

? is 1 lowered by the interval L-s, so this is again the chromatic
interval, so let's denote this note by 1b (1-Flat)

1b L 0 s 1b

We now have:

0 L 1 s 0, and 0# and 1b lying between 0 and 1.

Of course we can now begin the scale on 0# or set the upper note of
the L interval to be 1b. This process leads to a chain of
transpositions of the Ls scale:

0b L 1b s 0b
1b L 0 s 1b
0 L 1 s 0
1 L 0# s 1
0# L 1# s 0#

The next steps in either direction would require double sharps or
flats.

There are three interpretations of the L and s intervals

1. s is larger than L-s
2. s is equal to L-s
3. s is smaller than L-s

Interpretation 1 is analogous to the meantone interpretation of the
Diatonic scale, EDOs being 19 or 31 etc

Interpretation 2 is analogous to the twelve-tone interpretation of
the diatonic scale.

Interpretation 3 is analogous to the pythagorean interpretation of
the diatonic scale, one EDO being 17.

Of course, L and s may be interpreted such that the transpositions of
the scale do not converge on an EDO scheme, i.e. L/s or s/L can't be
written as a ratio of two integers.

The various settings of L and s I gave in my original post were to
show how this scale could be interpreted as the ancient Greek 'scale'
used for poetic recitation, consisting of just two notes, placed a
fifth apart. But, importantly, with increasing numbers of notes
(settings of L and s), other scales that approximate these relations
and are used ethnically, such as 5EDO and 7EDO (or 5 and 7 ADO -
approximate divisions of the Octave) embed this most basic scale.

This was supposed to show that this most basic scale is embedded in
all scales. (But you knew that anyway)

This was as a consequence of reading an article on factual and
conuterfactual recomposition, which explores different
interpretations of basic scale theory.

I leave as an exercise for the reader to perform the same experiment
on other Ls patterns.

I hope this helps.

Mark

Has anyone else heard of semiconvergents and continued fractions?
Without indulging in too much mathematics it is of the form

y = log2(ratio) = series of continued fractions

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
> > Let us look at the simplest possible scale
> >
> > Ls
> >
> > Let us label the two primary pitches as 0 and 1
> >
> > 0------1---(0...
> >
> > Transposing the scale to begin on 1, we arrive at
> >
> > 0-----1---0------1
> > 1------0#--1
>
> I must say, you lost me with the # notation, and much of the
> subsequent discussion. Is there another way you could explain
> this? (And what's the idea behind it?)
>
> -Carl

microtonalist wrote:

> Has anyone else heard of semiconvergents and continued fractions?
> Without indulging in too much mathematics it is of the form

Try

tuning continued fractions site:groups.yahoo.com

on Google.

Graham

this seem exceeding complex in description.
perhaps if you graph it out.
most of the time one will find that a L will be subdivided into a L S. while the s on the upper level become a L

>Message: 2 > Date: Fri, 23 Sep 2005 08:15:00 -0000
> From: "microtonalist" <mark@equiton.waitrose.com>
>Subject: Re: A Scale of two notes
>
>THe # notation is a sharp and b is a flat sign
>
>so you get
>
>0 - the first pitch
>1 a pitch somewhere between 0 and its octave higher note, such that >the interval from 0 up to 1 is larger than from 1 up to 0 (an octave >higher)
>
>so we can write 0 L 1 s 0
>No let's begin the scale on 1:
>
>1 L ? s 1
>
>The note ? is 0 raised by L-s, which traditionally is considered to >be the chromatic interval, raising by a sharp. Let's call this note 0#
>
>so we now get >
>1 L 0# s 1
>
>No let's transpose the scale so that 0 is the upper note of the L >interval:
>
>? L 0 s ?
>
>? is 1 lowered by the interval L-s, so this is again the chromatic >interval, so let's denote this note by 1b (1-Flat)
>
>1b L 0 s 1b
>
>We now have:
>
>0 L 1 s 0, and 0# and 1b lying between 0 and 1.
>
>Of course we can now begin the scale on 0# or set the upper note of >the L interval to be 1b. This process leads to a chain of >transpositions of the Ls scale:
>
>0b L 1b s 0b
> 1b L 0 s 1b
> 0 L 1 s 0
> 1 L 0# s 1
> 0# L 1# s 0#
>
>The next steps in either direction would require double sharps or >flats.
>
>There are three interpretations of the L and s intervals
>
>1. s is larger than L-s
>2. s is equal to L-s
>3. s is smaller than L-s
>
>Interpretation 1 is analogous to the meantone interpretation of the >Diatonic scale, EDOs being 19 or 31 etc
>
>Interpretation 2 is analogous to the twelve-tone interpretation of >the diatonic scale.
>
>Interpretation 3 is analogous to the pythagorean interpretation of >the diatonic scale, one EDO being 17.
>
>Of course, L and s may be interpreted such that the transpositions of >the scale do not converge on an EDO scheme, i.e. L/s or s/L can't be >written as a ratio of two integers.
>
>The various settings of L and s I gave in my original post were to >show how this scale could be interpreted as the ancient Greek 'scale' >used for poetic recitation, consisting of just two notes, placed a >fifth apart. But, importantly, with increasing numbers of notes >(settings of L and s), other scales that approximate these relations >and are used ethnically, such as 5EDO and 7EDO (or 5 and 7 ADO - >approximate divisions of the Octave) embed this most basic scale.
>
>This was supposed to show that this most basic scale is embedded in >all scales. (But you knew that anyway) >
>This was as a consequence of reading an article on factual and >conuterfactual recomposition, which explores different >interpretations of basic scale theory. >
>I leave as an exercise for the reader to perform the same experiment >on other Ls patterns.
>
>I hope this helps.
>
>Mark
>
>Has anyone else heard of semiconvergents and continued fractions?
>Without indulging in too much mathematics it is of the form
>
>y = log2(ratio) = series of continued fractions
>
>
>
>--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> >
>>--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>> >>
>>>Let us look at the simplest possible scale
>>>
>>>Ls
>>>
>>>Let us label the two primary pitches as 0 and 1
>>>
>>>0------1---(0...
>>>
>>>Transposing the scale to begin on 1, we arrive at
>>>
>>>0-----1---0------1
>>> 1------0#--1
>>> >>>
>>I must say, you lost me with the # notation, and much of the
>>subsequent discussion. Is there another way you could explain
>>this? (And what's the idea behind it?)
>>
>>-Carl
>> >>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>
>You can configure your subscription by sending an empty email to one
>of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>------------------------------------------------------------------------
>Yahoo! Groups Links
>
>
>
> >------------------------------------------------------------------------
>
>
>
>
>
> >

--
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

--- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> Mark
>
> Has anyone else heard of semiconvergents and continued fractions?

Yes, they've come up hundreds of times here and on the tuning-math
list. A very common and important tool around here. I think you'd find
a lot of common ground with those posting on these topics.

> Without indulging in too much mathematics it is of the form
>
> y = log2(ratio) = series of continued fractions

What's "it"? And how can the log of a single ratio be equal to a
*series* of continued fractions? You can explain on tuning-math if
you'd prefer.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:

> > y = log2(ratio) = series of continued fractions
>
> What's "it"? And how can the log of a single ratio be equal to a
> *series* of continued fractions? You can explain on tuning-math if
> you'd prefer.

I think the meaning is pretty clear; he means the sequence of
convergents, or possibly of semiconvergents, to the presumably
irrational number log2(ratio). Unless "ratio" is a power of two, and
assuming it is rational, it is irrational (in fact, transcendental.)
Hence the continued fraction is unique and infinite, and the
sequence of convergents is also unique. The convergents lead to a
corresponding infinite series, with numerators of 1, by subtracting
successive terms, but I don't know of a musical use for the series
in a strict sense since music is characteristically multiplicative
and not additive.

> The # notation is a sharp and b is a flat sign
>
> so you get
>
> 0 - the first pitch
> 1 a pitch somewhere between 0 and its octave higher note, such
> that the interval from 0 up to 1 is larger than from 1 up to
> 0 (an octave higher)
>
> so we can write 0 L 1 s 0
> No let's begin the scale on 1:
>
> 1 L ? s 1
>
> The note ? is 0 raised by L-s, which traditionally is considered
> to be the chromatic interval, raising by a sharp. Let's call this
> note 0#
>
> so we now get
>
> 1 L 0# s 1
>
> No let's transpose the scale so that 0 is the upper note of the L
> interval:
>
> ? L 0 s ?
>
> ? is 1 lowered by the interval L-s, so this is again the chromatic
> interval, so let's denote this note by 1b (1-Flat)
>
> 1b L 0 s 1b
>
> We now have:
>
> 0 L 1 s 0, and 0# and 1b lying between 0 and 1.
>
> Of course we can now begin the scale on 0# or set the upper note of
> the L interval to be 1b. This process leads to a chain of
> transpositions of the Ls scale:
>
> 0b L 1b s 0b
> 1b L 0 s 1b
> 0 L 1 s 0
> 1 L 0# s 1
> 0# L 1# s 0#
>
> The next steps in either direction would require double sharps or
> flats.
>
> There are three interpretations of the L and s intervals
>
> 1. s is larger than L-s
> 2. s is equal to L-s
> 3. s is smaller than L-s
>
> Interpretation 1 is analogous to the meantone interpretation of the
> Diatonic scale, EDOs being 19 or 31 etc
>
> Interpretation 2 is analogous to the twelve-tone interpretation of
> the diatonic scale.
>
> Interpretation 3 is analogous to the pythagorean interpretation of
> the diatonic scale, one EDO being 17.
>
> Of course, L and s may be interpreted such that the transpositions
> of the scale do not converge on an EDO scheme, i.e. L/s or s/L
> can't be written as a ratio of two integers.

Thanks Mark, that's much clearer.

> The various settings of L and s I gave in my original post were
> to show how this scale could be interpreted as the ancient
> Greek 'scale' used for poetic recitation, consisting of just two
> notes, placed a fifth apart. But, importantly, with increasing
> numbers of notes (settings of L and s), other scales that
> approximate these relations and are used ethnically, such as
> 5EDO and 7EDO (or 5 and 7 ADO - approximate divisions of the
> Octave) embed this most basic scale.
>
> This was supposed to show that this most basic scale is embedded
> in all scales. (But you knew that anyway)

Yeah.

> This was as a consequence of reading an article on factual
> and conuterfactual recomposition, which explores different
> interpretations of basic scale theory.

Hmm...

> Has anyone else heard of semiconvergents and continued fractions?
> Without indulging in too much mathematics it is of the form
>
> y = log2(ratio) = series of continued fractions

I think Graham wrote back that these have been discussed on this
list. You might also try:

http://www.math.sunysb.edu/~tony/whatsnew/column/irrational-
0799/irrational3.html

http://home.att.net/~srschmitt/rationalapprox.html

-Carl

One place where it is quite easy to see Ls patterns developing into other scales is with the horograms
http://anaphoria.com/hrgm.PDF

--
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

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "microtonalist" <mark@e...> wrote:
>
> > > y = log2(ratio) = series of continued fractions
> >
> > What's "it"? And how can the log of a single ratio be equal to a
> > *series* of continued fractions? You can explain on tuning-math if
> > you'd prefer.
>
> I think the meaning is pretty clear; he means the sequence of
> convergents, or possibly of semiconvergents, to the presumably
> irrational number log2(ratio).

Of course.

> assuming it is rational, it is irrational (in fact, transcendental.)

Hmm.

> Hence the continued fraction is unique and infinite, and the
> sequence of convergents is also unique. The convergents lead to a
> corresponding infinite series, with numerators of 1, by subtracting
> successive terms, but I don't know of a musical use for the series
> in a strict sense since music is characteristically multiplicative
> and not additive.

I'm surprised to see you, of all people, write this. Isn't this series (or should I say
sequence) exactly how you find the numbers of notes per octave in the MOS scales that
the given generator produces with a period of one octave? Is that not a "musical use"?
What is?

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

> > assuming it is rational, it is irrational (in fact, transcendental.)
>
> Hmm.

I noticed that was badly stated, but only after posting it. If r is a positive
rational number which is not an integer (positive or negative) power of
two, then log2(r) is not only irrational, it is transcendental.

> > Hence the continued fraction is unique and infinite, and the
> > sequence of convergents is also unique. The convergents lead to a
> > corresponding infinite series, with numerators of 1, by subtracting
> > successive terms, but I don't know of a musical use for the series
> > in a strict sense since music is characteristically multiplicative
> > and not additive.
>
> I'm surprised to see you, of all people, write this. Isn't this series (or
should I say
> sequence) exactly how you find the numbers of notes per octave in
the MOS scales that
> the given generator produces with a period of one octave? Is that not
a "musical use"?

But you don't need the series to get the convergents, and in fact that
isn't normally how you compute them.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > > assuming it is rational, it is irrational (in fact,
transcendental.)
> >
> > Hmm.
>
> I noticed that was badly stated, but only after posting it. If r is
a positive
> rational number which is not an integer (positive or negative)
power of
> two, then log2(r) is not only irrational, it is transcendental.

Oh.

> > > Hence the continued fraction is unique and infinite, and the
> > > sequence of convergents is also unique. The convergents lead to
a
> > > corresponding infinite series, with numerators of 1, by
subtracting
> > > successive terms, but I don't know of a musical use for the
series
> > > in a strict sense since music is characteristically
multiplicative
> > > and not additive.
> >
> > I'm surprised to see you, of all people, write this. Isn't this
series (or
> should I say
> > sequence) exactly how you find the numbers of notes per octave in
> the MOS scales that
> > the given generator produces with a period of one octave? Is that
not
> a "musical use"?
>
> But you don't need the series to get the convergents, and in fact
that
> isn't normally how you compute them.

OK, you'll have to show me an example that clarifies all this -- such
as what the series you're talking about is. Maybe on the tuning-math
list?