Re: [keykit] Changing keys

[Michael Ellis]

Interesting way to look at the math.  For playing purposes,  most musicians just memorize the circle of fifths (CFBEADG) and use the fact that modulating by a fifth up (7 half steps)  or a fourth down (5 half steps) alters only one tone.  To modulate up a fifth, sharp the fourth degree of the original scale, to modulate down a fifth, flat the seventh.

So if I'm currently playing in Eb major,  then Bb major and Ab major (and their relative natural minors) have scales that differ by only one pitch.  Similarly, moving by two positions on the circle to F major or Db major alter only two pitches, and so on.

Your method is neat in that it works for any scale constructed from any larger set of pitch classes, so it would useful for exploring quarter tone scales and the like.

On Fri, May 15, 2009 at 12:26 PM, ncg <greni...@gmail.com> wrote:

Hello. I just found a nice trick and I thought it would be nice to
share it. (I'm aware someone somewhere at some time must have found
this before me!)

I heard it's interesting, when you're changing key, to only change one
note in order to preserve continuity.

I found a way to calculate the number of notes that are the same
between different keys of the same scale.

I use bit strings to represent a scale and build a symmetric matrix by
rotating the bit string left on each line.

Here's the symmetric matrix for the major scale :

1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1
0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1
1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0
0 , 1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1
1 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0
1 , 0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1
0 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 1
1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 0
0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1
1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0
0 , 1 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1
1 , 1 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 0 , 1 , 0

Then, I multiply that matrix by itself. Here's the result again for
the major scale :

7 , 2 , 5 , 4 , 3 , 6 , 2 , 6 , 3 , 4 , 5 , 2
2 , 7 , 2 , 5 , 4 , 3 , 6 , 2 , 6 , 3 , 4 , 5
5 , 2 , 7 , 2 , 5 , 4 , 3 , 6 , 2 , 6 , 3 , 4
4 , 5 , 2 , 7 , 2 , 5 , 4 , 3 , 6 , 2 , 6 , 3
3 , 4 , 5 , 2 , 7 , 2 , 5 , 4 , 3 , 6 , 2 , 6
6 , 3 , 4 , 5 , 2 , 7 , 2 , 5 , 4 , 3 , 6 , 2
2 , 6 , 3 , 4 , 5 , 2 , 7 , 2 , 5 , 4 , 3 , 6
6 , 2 , 6 , 3 , 4 , 5 , 2 , 7 , 2 , 5 , 4 , 3
3 , 6 , 2 , 6 , 3 , 4 , 5 , 2 , 7 , 2 , 5 , 4
4 , 3 , 6 , 2 , 6 , 3 , 4 , 5 , 2 , 7 , 2 , 5
5 , 4 , 3 , 6 , 2 , 6 , 3 , 4 , 5 , 2 , 7 , 2
2 , 5 , 4 , 3 , 6 , 2 , 6 , 3 , 4 , 5 , 2 , 7

The cell aij (row i, column j) is the number of notes that are the
same between the key i and j.

Notice how the diagonal is equal to the number of notes in the scale
because a particular key is the same as itself.

--
Cheers,
Mike

[Michael Ellis]

Oops! should read "to modulate down a fourth, flat the seventh."

--
Cheers,
Mike

[Tony]

Is this kind of "autocorrelation of the scale space". Here the link to
Autocorrelation in Wikipedia: http://en.wikipedia.org/wiki/Autocorrelation

Which would mean you could apply this method even to different scale spaces
just by multiplying their matrix representations.

It would be interesting to plot the resulting matrices visually using a
color gradient between min..max values and see what kind of patterns you
will get.

Kind regards,
Tony

> I use bit strings to represent a scale and build a symmetric
> matrix by rotating the bit string left on each line.
>

[Tony]

Hi Tim,

 

your Galaxy - live solo performance sounds and looks great!

http://www.youtube.com/watch?v=8-04KrT2hRM&feature=related

 

The more "psychedelity" you can build into your visuals, the better it would get imo, at least for dance floors.

 

Kind regards,

Tony

[Michael Ellis]

Very nice, indeed!

--
Cheers,
Mike