New (old) Work - 'Self-Similar Variations'

Hi,

I've put up a midi realization of a work I wrote in July 2003 called
'Self-Similar Variations'.

http://www.akjmusic.com/audio/ssv.mp3

It is so-called because the melodic material is taken from an algorithm I
developed after being inspired by a Martin Gardner article on 'pink noise
melody'. The idea is to toss N dice, and sum them on each 'round', but only
roll the Xth die every X^2 round (where the dice are X from 0-N).

I took the idea further by making an algorithm that would create a melody that
would be the 'sum of itself' on different time scales, analogous to the way
that pink noise could be emulated by summing die rolls that 'moved' on
different time scales, giving the resulting melody(s) a 'self-similar' or
fractal feel.

The xentonality for this piece is mild; here I use the 'Equable Diatonic'
scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is so-called for
its most even superparticular 3-part division of the tetrachord...

Enjoy!

--
Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

BTW, I welcome all of your comments..... ;)

On Sunday 09 May 2004 10:47 pm, Aaron K. Johnson wrote:
> Hi,
>
> I've put up a midi realization of a work I wrote in July 2003 called
> 'Self-Similar Variations'.
>
> http://www.akjmusic.com/audio/ssv.mp3
>
> It is so-called because the melodic material is taken from an algorithm I
> developed after being inspired by a Martin Gardner article on 'pink noise
> melody'. The idea is to toss N dice, and sum them on each 'round', but only
> roll the Xth die every X^2 round (where the dice are X from 0-N).
>
> I took the idea further by making an algorithm that would create a melody
> that would be the 'sum of itself' on different time scales, analogous to
> the way that pink noise could be emulated by summing die rolls that 'moved'
> on different time scales, giving the resulting melody(s) a 'self-similar'
> or fractal feel.
>
> The xentonality for this piece is mild; here I use the 'Equable Diatonic'
> scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is so-called
> for its most even superparticular 3-part division of the tetrachord...
>
> Enjoy!

--
Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

Once again, Aaron, I'm impressed. Your algorithms are again producing
very organic results. I'd love to discuss this algorithm further --
could you give some more specific details? Anyway, this was a nice
arabic/indian/electronica piece, the alternating timbres were a nice
egyptian touch too. Tasty.

--- In MakeMicroMusic@yahoogroups.com, "Aaron K. Johnson"
<akjmicro@c...> wrote:
> BTW, I welcome all of your comments..... ;)
>
>
> On Sunday 09 May 2004 10:47 pm, Aaron K. Johnson wrote:
> > Hi,
> >
> > I've put up a midi realization of a work I wrote in July 2003
called
> > 'Self-Similar Variations'.
> >
> > http://www.akjmusic.com/audio/ssv.mp3
> >
> > It is so-called because the melodic material is taken from an
algorithm I
> > developed after being inspired by a Martin Gardner article
on 'pink noise
> > melody'. The idea is to toss N dice, and sum them on
each 'round', but only
> > roll the Xth die every X^2 round (where the dice are X from 0-N).
> >
> > I took the idea further by making an algorithm that would create
a melody
> > that would be the 'sum of itself' on different time scales,
analogous to
> > the way that pink noise could be emulated by summing die rolls
that 'moved'
> > on different time scales, giving the resulting melody(s) a 'self-
similar'
> > or fractal feel.
> >
> > The xentonality for this piece is mild; here I use the 'Equable
Diatonic'
> > scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is so-
called
> > for its most even superparticular 3-part division of the
tetrachord...
> >
> > Enjoy!
>
> --
> Aaron Krister Johnson
> http://www.dividebypi.com
> http://www.akjmusic.com

On Monday 10 May 2004 12:44 pm, Paul Erlich wrote:
> Once again, Aaron, I'm impressed. Your algorithms are again producing
> very organic results. I'd love to discuss this algorithm further --
> could you give some more specific details? Anyway, this was a nice
> arabic/indian/electronica piece, the alternating timbres were a nice
> egyptian touch too. Tasty.

Thanks, Paul !

The algorithm in this case is what I call a 'self-similar melody' algorithm
that I developed myself.

The idea is to take a motif, represented as a set of integers. For example
[2,4,3,1], which is a four note 'motif' . Actually, this 'motif' is randomly
determined by my algorithm as the role of a six sided die, done p times,
where p is the seed phrase length.

Then, there is a depth parameter to the algorithm, so I can decide to have X
melodic strands added together. Let's make it 3 for example. Melodic Strand X
will advance every 2^X steps, and each melodic strand will be summed at every
step to give an 'output melody':

2+2+2 = 6
4+2+2 = 8
3+4+2 = 9
1+4+2 = 7
2+3+4 = 9
4+3+4 = 11
3+1+4 = 8
1+1+4 = 6

As you can see, at the slowest level of melody strand 3, ...(yes, I did this
reverse of traditional positional notation--the least significant, fastest
moving strand is represented here as the leftmost, not rightmost, column)
...on the rightmost column above, it hasn't used all of the seed phrase array
[2,4,3,1], only going as far as [2,4] because the depth parameter was 3, and
the melody length will always be 2^d, where d is the depth parameter.

The above 'melody' [6,8,9,7,9,11,8,6] is then scaled from zero to a
'scale_max' parameter, which is determined from a mapping to a set of notes.
For instance if the scale were 6 notes large [0-5], the above melody would
become [0,2,3,1,3,5,2,0]. If the range were larger or smaller, a
multiplication would take place for every element in the array as well.

I find the results of such an algorithm fairly musical, but sometimes in need
of a human touch.

I hope that explains everything...

-Aaron.
--
Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

> --- In MakeMicroMusic@yahoogroups.com, "Aaron K. Johnson"
>
> <akjmicro@c...> wrote:
> > BTW, I welcome all of your comments..... ;)
> >
> > On Sunday 09 May 2004 10:47 pm, Aaron K. Johnson wrote:
> > > Hi,
> > >
> > > I've put up a midi realization of a work I wrote in July 2003
>
> called
>
> > > 'Self-Similar Variations'.
> > >
> > > http://www.akjmusic.com/audio/ssv.mp3
> > >
> > > It is so-called because the melodic material is taken from an
>
> algorithm I
>
> > > developed after being inspired by a Martin Gardner article
>
> on 'pink noise
>
> > > melody'. The idea is to toss N dice, and sum them on
>
> each 'round', but only
>
> > > roll the Xth die every X^2 round (where the dice are X from 0-N).
> > >
> > > I took the idea further by making an algorithm that would create
>
> a melody
>
> > > that would be the 'sum of itself' on different time scales,
>
> analogous to
>
> > > the way that pink noise could be emulated by summing die rolls
>
> that 'moved'
>
> > > on different time scales, giving the resulting melody(s) a 'self-
>
> similar'
>
> > > or fractal feel.
> > >
> > > The xentonality for this piece is mild; here I use the 'Equable
>
> Diatonic'
>
> > > scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is so-
>
> called
>
> > > for its most even superparticular 3-part division of the
>
> tetrachord...
>
> > > Enjoy!
> >
> > --
> > Aaron Krister Johnson
> > http://www.dividebypi.com
> > http://www.akjmusic.com
>
>
> [MMM info]------------------------------------------------------
> More MMM music files are at http://www.microtonal.org/music.html
> ------------------------------------------------------[MMM info]
> Yahoo! Groups Links
>
>
>

--- In MakeMicroMusic@yahoogroups.com, "Aaron K. Johnson"
<akjmicro@c...> wrote:
> On Monday 10 May 2004 12:44 pm, Paul Erlich wrote:
> > Once again, Aaron, I'm impressed. Your algorithms are again
producing
> > very organic results. I'd love to discuss this algorithm further -
-
> > could you give some more specific details? Anyway, this was a nice
> > arabic/indian/electronica piece, the alternating timbres were a
nice
> > egyptian touch too. Tasty.
>
> Thanks, Paul !
>
> The algorithm in this case is what I call a 'self-similar melody'
algorithm
> that I developed myself.
>
> The idea is to take a motif, represented as a set of integers. For
example
> [2,4,3,1], which is a four note 'motif' . Actually, this 'motif' is
randomly
> determined by my algorithm as the role of a six sided die, done p
times,
> where p is the seed phrase length.
>
> Then, there is a depth parameter to the algorithm, so I can decide
to have X
> melodic strands added together. Let's make it 3 for example.
Melodic Strand X
> will advance every 2^X steps,

Is the number 2 here related to the number p mentioned above? Or is
it an additional variable? Or do you always use 2?

> and each melodic strand will be summed at every
> step to give an 'output melody':
>
> 2+2+2 = 6
> 4+2+2 = 8
> 3+4+2 = 9
> 1+4+2 = 7
> 2+3+4 = 9
> 4+3+4 = 11
> 3+1+4 = 8
> 1+1+4 = 6
>
> As you can see, at the slowest level of melody strand 3, ...(yes, I
did this
> reverse of traditional positional notation--the least significant,
fastest
> moving strand is represented here as the leftmost, not rightmost,
column)
> ...on the rightmost column above, it hasn't used all of the seed
phrase array
> [2,4,3,1], only going as far as [2,4] because the depth parameter
was 3, and
> the melody length will always be 2^d, where d is the depth
parameter.
>
> The above 'melody' [6,8,9,7,9,11,8,6] is then scaled from zero to a
> 'scale_max' parameter, which is determined from a mapping to a set
of notes.
> For instance if the scale were 6 notes large [0-5], the above
melody would
> become [0,2,3,1,3,5,2,0]. If the range were larger or smaller, a
> multiplication would take place for every element in the array as
well.
>
> I find the results of such an algorithm fairly musical, but
sometimes in need
> of a human touch.
>
> I hope that explains everything...

So if I hear notes of longer duration in the output melody, is that
because a certain number is repeated in the defining string?

referring to http://www.akjmusic.com/audio/ssv.mp3

On Monday 10 May 2004 03:32 pm, Paul Erlich wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Aaron K. Johnson"
>
> <akjmicro@c...> wrote:
> > On Monday 10 May 2004 12:44 pm, Paul Erlich wrote:
> > > Once again, Aaron, I'm impressed. Your algorithms are again
>
> producing
>
> > > very organic results. I'd love to discuss this algorithm further -
>
> -
>
> > > could you give some more specific details? Anyway, this was a nice
> > > arabic/indian/electronica piece, the alternating timbres were a
>
> nice
>
> > > egyptian touch too. Tasty.
> >
> > Thanks, Paul !
> >
> > The algorithm in this case is what I call a 'self-similar melody'
>
> algorithm
>
> > that I developed myself.
> >
> > The idea is to take a motif, represented as a set of integers. For
>
> example
>
> > [2,4,3,1], which is a four note 'motif' . Actually, this 'motif' is
>
> randomly
>
> > determined by my algorithm as the role of a six sided die, done p
>
> times,
>
> > where p is the seed phrase length.
> >
> > Then, there is a depth parameter to the algorithm, so I can decide
>
> to have X
>
> > melodic strands added together. Let's make it 3 for example.
>
> Melodic Strand X
>
> > will advance every 2^X steps,
>
> Is the number 2 here related to the number p mentioned above? Or is
> it an additional variable? Or do you always use 2?

I always use 2....it's a 'power of two' algorithm. Of course, there's nothing
preventing it from being otherwise. In fact, it might be fun to make
variations on a given seed where this number changes, or the strands add in a
linear, instead of a power-based fashion.

> So if I hear notes of longer duration in the output melody, is that
> because a certain number is repeated in the defining string?

Yes, I believe I made them tie....

Best,
--
Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

Hello Aaron,

I enjoy your music as ever! For me this one has a sort of North African
feel to it.

> The xentonality for this piece is mild; here I use the 'Equable Diatonic'
> scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is so-called for
> its most even superparticular 3-part division of the tetrachord...

That's a very strange third. (I thought it was a typo till I tried it.)
Can you point me to more information on this scale please?

Peter Frazer
www.midicode.com

Hey Peter and MMM friends,

I don't know, but I reckon that Kraig is correct about this type of scale
historically preceding the 5/4 third-type diatonic scale, etc....certainly we
hear a gritty, earthy 'xentonal' third in a lot of pure folk music (and I
don't mean Peter Paul and Mary)...

I know of it from literature describing it as an 'Equable Diatonic scale of
Ptolemy', it goes back to ancient Greek theory, and of the 26 3-step
divisions of a perfect fourth into superparticular ratios, this is the most
'even', hence the name 'Equable'....

And thanks for listening, and for your comments...yes the African feel is
certainly there...I always imagine a big caravan of elephants swinging their
trunks in the heat when I hear this one.

Best,
Aaron.

Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

On Sunday 16 May 2004 03:53 am, Peter Frazer wrote:
> Hello Aaron,
>
> I enjoy your music as ever! For me this one has a sort of North African
> feel to it.
>
> > The xentonality for this piece is mild; here I use the 'Equable
> > Diatonic' scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is
> > so-called
>
> for
>
> > its most even superparticular 3-part division of the tetrachord...
>
> That's a very strange third. (I thought it was a typo till I tried it.)
> Can you point me to more information on this scale please?
>
>
> Peter Frazer
> www.midicode.com
>
>
>
> [MMM info]------------------------------------------------------
> More MMM music files are at http://www.microtonal.org/music.html
> ------------------------------------------------------[MMM info]
> Yahoo! Groups Links
>
>
>

Aaron K. Johnson wrote:

>Hey Peter and MMM friends,
>
>I don't know, but I reckon that Kraig is correct about this type of scale >historically preceding the 5/4 third-type diatonic scale, etc....certainly we >hear a gritty, earthy 'xentonal' third in a lot of pure folk music (and I >don't mean Peter Paul and Mary)...
>
Yep. A neutral thinrd.

--
* David Beardsley
* microtonal guitar
* http://biink.com/db

Kraig,

Thank you for you comment. I was unaware of this 'neutral' third in historical
use and I shall research this some more as and when I find time.

Peter.

Kraig wrote

> The use of this type of diatonic is very wide spread in various folk musics all
> over. It seems the push toward major and minor is a development FROM such a
> scale.

> Peter Frazer wrote:

>> Hello Aaron,
.>
.> I enjoy your music as ever! For me this one has a sort of North African
.> feel to it.
.>
.> > The xentonality for this piece is mild; here I use the 'Equable Diatonic'
.> > scale of Ptolemy (1/1 10/9 11/9 4/3 3/2 5/3 11/6 2/1) which is so-called
.> for
.> > its most even superparticular 3-part division of the tetrachord...
.>
.> That's a very strange third. (I thought it was a typo till I tried it.)
.> Can you point me to more information on this scale please?
.>
.> Peter Frazer
.> www.midicode.com

Aaron,

Thank you for your feedback. Do you have in mind any particular
'literature' which would get me started on this? A Google search
is not finding me what I'm looking for.

I don't know about caravans of elephants, I was thinking more of
Arab bazaars.

Best,
Peter.

> Hey Peter and MMM friends,

> I don't know, but I reckon that Kraig is correct about this type of scale
> historically preceding the 5/4 third-type diatonic scale, etc....certainly we
> hear a gritty, earthy 'xentonal' third in a lot of pure folk music (and I
> don't mean Peter Paul and Mary)...

> I know of it from literature describing it as an 'Equable Diatonic scale of
> Ptolemy', it goes back to ancient Greek theory, and of the 26 3-step
> divisions of a perfect fourth into superparticular ratios, this is the most
> 'even', hence the name 'Equable'....

> And thanks for listening, and for your comments...yes the African feel is
> certainly there...I always imagine a big caravan of elephants swinging their
> trunks in the heat when I hear this one.

> Best,
> Aaron.

> Aaron Krister Johnson
> http://www.dividebypi.com
> http://www.akjmusic.com

Kraig wrote

> I believe it is a thing you are going to find in the music and not the textbooks
> as historically , the music of the people is usually ignored

Good point Kraig.