salvatone - Posted - 01/30/2010:  05:31:28

All this Circle of fifths talk got me thinking about other interval circles. Only four of them include all twelve notes.

1. Circle of flatted seconds: Hits all twelve notes in order.
2. Circle of ninths or seconds: Whole tone scale seventh degree is an octave.
3. Circle of minor thirds: Diminished scale fifth degree is an octave.
4. Circle of thirds: Forth degree is an octave.
5. Circle of fourths: The circle of fifths backwards. All twelve tones.
6. Circle of flated fifths: The magic tritone, third degree is the octave.
7. Circle of fifths: All twelve tones.
8. Circle of augmented fifths: Fourth degree puts you two octaves up.
9. Circle of sixths: Five degrees is up three octaves. The diminished scale backwards.
10. Circle of flatted sevens: A whole tone scale backwards. 7 degrees puts you 5 octaves up.
11. Circle of sevenths: All twelve notes backwards, but up almost an octave each time.
12, Circle of octaves: I was over so quickly….

Does this have any use?

Edited by - salvatone on 01/30/2010 05:33:12

spinley - Posted - 01/30/2010:  07:02:21

It should have a lot of use but for me playing 35 years and not having a background for the technical side of music it does not help even though I am finding out that I do some of these naturally but just can not relate it technically. Wish I could. Hope that made sense?
Stuart

minstrelmike - Posted - 01/30/2010:  10:03:13

The circle of flatted seconds _is_ the chromatic scale. Thinking of it as a circle while knowing the physics of the octave and fifths demonstrates the difference between equal-tempered (making a perfect circle) and well-tempered (starting at a specific point and making the 5ths sound really good) scales.

The circle of thirds has its own wikipedia page.

salvatone - Posted - 01/31/2010:  04:43:56

Minstrelmike-Thanks. I had a tough time describing all of these without creating a long post. I really like the symmetry of this math, how the "circle of flatted 2nds" is similar to the "circle of 7ths;" and the circle of seconds is similar to the circle of flatted 7ths.

minstrelmike - Posted - 01/31/2010:  08:44:11

As far as being useful, it's hard to say. Kind of like the Group Theory aspect of mathematics which seems completely esoteric when studying yet applies to all sorts of natural phenomena.

I think your list shows the magic of tritones. Just as the circle of fourths is the inverse of the circle of fifths, all of the other circles have a separate inverse, circle of flatted 2nds is the inverse of circle of major sevenths. You could make a much shorter list and show that clockwise is one type and counterclockwise is the inverse type.

The tritone is its own inverse. I think that's an interesting chunk of music theory.

Another odd bit to study is the difference between those circles of intervals that completely map the 12 notes and those that don't. There is probably something musically interesting there, too.

salvatone - Posted - 01/31/2010:  09:01:07

The symmetry is truly lovely. I wonder if there is a way to diagram this "cycle of cycles."

The math is that any 1/2 step interval as a fraction of 12 that cannot be reduced yields a complete cycle: 1/12 5/12 7/12 11/12.

I think I should have numbered them differently in the initial list.

Banjocoltrane - Posted - 02/03/2010:  23:34:59

Take a Look at this...all you could ever want to know about cycles

http://danadler.com/misc/Cycles.pdf

Mirek Patek - Posted - 02/04/2010:  03:39:01

When I memorised the minor and major thirds from each note, I organised the 12 chromatic notes not into one or two circles, but into the "duoprism":
http://en.wikipedia.org/wiki/Duoprism

In the "Geometry of 4-dimensional duoprisms" part of this wikipedia page, there is written that "A 4-dimensional (...) duoprism is created by the product of a regular n-sided polygon and a regular m-sided polygon (...)."

Well, then imagine that one polygon is 3-sided (triangle) with the edge length of 4 because it divides the octave into three major thirds. The other polygon is 4-sided (square) with the edge length of 3 because it divides the octave into four minor thirds.

Triangular-square duoprism:
(taken from http://en.wikipedia.org/wiki/File:3-4_duoprism.png )
Note that all horizontal edges have length 3, all vertical and slant edges have length 4.


If we switch the polygons, the result is Square-triangular duoprism:
(taken from http://en.wikipedia.org/wiki/File:4-3_duoprism.png )
Note that all horizontal edges have length 4, all vertical and slant edges have length 3.


These duoprisms contain all 12 chromatic notes (as the 12 vertices) and those ones which are major or minor third apart are connected by the edge (line). Rectangle faces (3x4) contain the notes of major and minor triad from the same root, e.g. C-Eb-G-E. The square faces (3x3) contain the notes of diminished seventh chord and the triangle faces contain the notes of augmented triad.

Note that these two duoprisms are transformable to each other through the fourth dimension. You can see it also (on the torus) here:
(taken from http://en.wikipedia.org/wiki/File:I...small%29.gif )


So that is my way of taking the music into the fourth dimension - apart from fingerpicking on the four string banjo, see the links below.

Mirek
http://www.banjosessions.com/dec09/patek.html
http://www.banjosessions.com/feb10/patek.html

Edited by - Mirek Patek on 02/04/2010 05:50:43