Does this mean anything?

I've just been playing with matlab.

diatonic scale measured as number of fifths from tonic

0 2 4 -1 1 3 5 positive=up negative=down

multiply by the fifth in cents, modulo 1200:

0 200 400 500 700 900 1100

take standard deviation:

386.68 ????

I've tried this for various sizes of fifth, and got

699.69356 as a reasonable guess.

I wonder if this has any meaning or impact on the choice of fifth or
scale?
and is there any other type of 'deviation' that could be of use.

apologies for the math

Mark

--- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
>
> I've just been playing with matlab.
>
> diatonic scale measured as number of fifths from tonic
>
> 0 2 4 -1 1 3 5 positive=up negative=down
>
> multiply by the fifth in cents, modulo 1200:
>
> 0 200 400 500 700 900 1100
>
> take standard deviation:
>
> 386.68 ????

The standard deviation is the RMS dispersion about the mean. The mean
is 542.8571 cents. Unless you believe this mean is meaningful, you
can't very well claim the standard deviation is meaningful, can you?
This is just a funny coincidence.

> I've tried this for various sizes of fifth, and got
>
> 699.69356 as a reasonable guess.
>
> I wonder if this has any meaning or impact on the choice of fifth
or
> scale?

I think not.

> and is there any other type of 'deviation' that could be of use.

Yes, one could use the deviation of each consonant interval from its
simple-ratio just value, and optimize the tuning to minimize the RMS
of these deviations. This has been done countless times; something
similar from 1835 you should read:

http://sonic-arts.org/monzo/woolhouse/essay.htm

Here, an optimal fifth of 696.164846 cents is derived. This is a
great tuning for the diatonic scale, and I recommend it highly for
players of Celtic diatonic harps and the like . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
> >
> > I've just been playing with matlab.
[snip]

Thanks Paul, it did seem a bit of a strange coincidence.

As for the Brun work, I can't help noticing that for two intervals it
looks like performing a kind of reverse fibbonacci series, i.e. take
two intervals that are considered adjacent members of a fibonacci
series, then divide them up, going in the direction of smaller numbers

2/1 3/2 4/3 9/8 32/27 256/243 ...

with three numbers, you choose the two larger intervals:

2/1 3/2 5/4 initial set

(apologies for the abuse of the minus sign)

2/1 - 3/2 = 4/3
3/2 - 4/3 = 9/8
now 5/4 is bigger than 9/8, so
4/3 - 5/4 = 16/15
now 9/8 is bigger than 16/15
5/4 - 9/8 = 10/9
9/8 - 10/9 = 81/80
now 16/15 is bigger than 81/80
10/9 - 16/15 = 25/24
and so on ad nauseum. The erv wilson diagram in viggo3.pdf listed in
a previous mail has a lovely diagram for this, and is also an
interesting way of generating scales ( you could choose 14/11 instead
of 5/4, for example, to see what happens)

Do these 'reverse' fibonacci series have an independent name?

Mark

--- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
> > >
> > > I've just been playing with matlab.
> [snip]
>
> Thanks Paul, it did seem a bit of a strange coincidence.
>
> As for the Brun work, I can't help noticing that for two intervals
it
> looks like performing a kind of reverse fibbonacci series, i.e.
take
> two intervals that are considered adjacent members of a fibonacci
> series, then divide them up, going in the direction of smaller
numbers
>
> 2/1 3/2 4/3 9/8 32/27 256/243 ...
>
> with three numbers, you choose the two larger intervals:
>
> 2/1 3/2 5/4 initial set
>
> (apologies for the abuse of the minus sign)
>
> 2/1 - 3/2 = 4/3
> 3/2 - 4/3 = 9/8
> now 5/4 is bigger than 9/8, so
> 4/3 - 5/4 = 16/15
> now 9/8 is bigger than 16/15
> 5/4 - 9/8 = 10/9
> 9/8 - 10/9 = 81/80
> now 16/15 is bigger than 81/80
> 10/9 - 16/15 = 25/24
> and so on ad nauseum. The erv wilson diagram in viggo3.pdf listed
in
> a previous mail has a lovely diagram for this, and is also an
> interesting way of generating scales ( you could choose 14/11
instead
> of 5/4, for example, to see what happens)
>
> Do these 'reverse' fibonacci series have an independent name?
>
> Mark

It might help if you studied up on the Euclidean algorithm first --
very straighforward, ancient mathematics. It can be "coded" as
follows:

function gcd(a, b)
while a ‚ b
if a > b
a := a - b
else
b := b - a
return a

See http://mathworld.wolfram.com/EuclideanAlgorithm.html (is this
made unnecessarily complicated here?)

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
> > > >
> > > > I've just been playing with matlab.
> > [snip]
> >
> > Thanks Paul, it did seem a bit of a strange coincidence.
> >
> > As for the Brun work, I can't help noticing that for two
intervals
> it
> > looks like performing a kind of reverse fibbonacci series, i.e.
> take
> > two intervals that are considered adjacent members of a fibonacci
> > series, then divide them up, going in the direction of smaller
> numbers
> >
> > 2/1 3/2 4/3 9/8 32/27 256/243 ...
> >
> > with three numbers, you choose the two larger intervals:
> >
> > 2/1 3/2 5/4 initial set
> >
> > (apologies for the abuse of the minus sign)
> >
> > 2/1 - 3/2 = 4/3
> > 3/2 - 4/3 = 9/8
> > now 5/4 is bigger than 9/8, so
> > 4/3 - 5/4 = 16/15
> > now 9/8 is bigger than 16/15
> > 5/4 - 9/8 = 10/9
> > 9/8 - 10/9 = 81/80
> > now 16/15 is bigger than 81/80
> > 10/9 - 16/15 = 25/24
> > and so on ad nauseum. The erv wilson diagram in viggo3.pdf listed
> in
> > a previous mail has a lovely diagram for this, and is also an
> > interesting way of generating scales ( you could choose 14/11
> instead
> > of 5/4, for example, to see what happens)
> >
> > Do these 'reverse' fibonacci series have an independent name?
> >
> > Mark
>
> It might help if you studied up on the Euclidean algorithm first --
> very straighforward, ancient mathematics. It can be "coded" as
> follows:
>
> function gcd(a, b)
> while a ‚ b

That's supposed to be "a does not equal b".

> if a > b
> a := a - b
> else
> b := b - a
> return a
>
> See http://mathworld.wolfram.com/EuclideanAlgorithm.html (is this
> made unnecessarily complicated here?)
>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > [snip]
> > See http://mathworld.wolfram.com/EuclideanAlgorithm.html (is this
> > made unnecessarily complicated here?)
> >
>
Unnecessarily complicated - it goes right over my head completely

I need an idiots guide.

Mark

--- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > > [snip]
> > > See http://mathworld.wolfram.com/EuclideanAlgorithm.html (is this
> > > made unnecessarily complicated here?)
> > >
> >
> Unnecessarily complicated - it goes right over my head completely
>
> I need an idiots guide.
>
> Mark

Does this make sense

http://www.math.umn.edu/~garrett/crypto/a01/Euclid.html

?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Mark" <mark@e...> wrote:
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > > > [snip]
> > > > See http://mathworld.wolfram.com/EuclideanAlgorithm.html (is
this
> > > > made unnecessarily complicated here?)
> > > >
> > >
> > Unnecessarily complicated - it goes right over my head completely
> >
> > I need an idiots guide.
> >
> > Mark
>
> Does this make sense
>
> http://www.math.umn.edu/~garrett/crypto/a01/Euclid.html
>
> ?

Also, read this article:

http://www.sciencenews.org/articles/20000212/mathtrek.asp

It explains the Euclidean algorithm and also bring you up-to-date on
the subject. I had no idea Ferguson was a sculptor!

I'd be interested to see Gene talk about the connection with lattice
reduction (on the tuning-math list, of course).

John Chalmers might know.

--- Mark <mark@equiton.waitrose.com> wrote:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Mark" <mark@e...>
> wrote:
> > >
> > > I've just been playing with matlab.
> [snip]
>
> Thanks Paul, it did seem a bit of a strange
> coincidence.
>
> As for the Brun work, I can't help noticing that for
> two intervals it
> looks like performing a kind of reverse fibbonacci
> series, i.e. take
> two intervals that are considered adjacent members
> of a fibonacci
> series, then divide them up, going in the direction
> of smaller numbers
>
> 2/1 3/2 4/3 9/8 32/27 256/243 ...
>
> with three numbers, you choose the two larger
> intervals:
>
> 2/1 3/2 5/4 initial set
>
> (apologies for the abuse of the minus sign)
>
> 2/1 - 3/2 = 4/3
> 3/2 - 4/3 = 9/8
> now 5/4 is bigger than 9/8, so
> 4/3 - 5/4 = 16/15
> now 9/8 is bigger than 16/15
> 5/4 - 9/8 = 10/9
> 9/8 - 10/9 = 81/80
> now 16/15 is bigger than 81/80
> 10/9 - 16/15 = 25/24
> and so on ad nauseum. The erv wilson diagram in
> viggo3.pdf listed in
> a previous mail has a lovely diagram for this, and
> is also an
> interesting way of generating scales ( you could
> choose 14/11 instead
> of 5/4, for example, to see what happens)
>
> Do these 'reverse' fibonacci series have an
> independent name?
>
> Mark
>
>
>
>
>
>
>
>

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