Triads, Harmonic Polyrhythms and 3D Lissajous Patterns
The original Lissajous patterns were based on diads - two note chords. But musicians often play triads (three note chords) or tetrads (four notes) or more.
Here is the 2D Lissajous pattern of the pure ratio major chord 1/1 5/4 3/2, from Tune Smithy:
Or as drawing "in progress":
To show a triad as a 2D Lissajous pattern like this - you have to have two or more frequencies in the same direction. I've done it with the 5/4 one way and the 1/1 and 3/2 the other way.
The notation here uses ratios of frequencies to describe the triad - as is normal in this context. For instance, if the 1/1 is 256 Hz then 5/4 is 5 * 256/4 i.e. 320 Hz, and 3/2 similarly is 3 * 256 / 2 = 382 Hz.
As an aternative, to bring out the different frequencies more in the pattern - you can try making it so one of the frequencies, say the 1/1, goes diagonally rather than horizontally or vertically. If you follow that up further you find what you have is a 2D drawing of a 3D pattern.
One way or another you have three axes (for the three frequencies) which you need to put into two dimensions to make your shape.
To see the Lissajous pattern of a three note chord in its fullness, really you need to show it in 3D. Here is exactly the same pattern, this time in 3D with perspective, lighting, and a tube instead of a line so you can see it's 3D shape clearly.
That's why I originally developed Lissajous 3D - though later on found it was a great tool for creating all sorts of interesting 3D patterns. Something about the Lissajous 3D pattern gives them an organic feel to them, I think that's why they work so well.
So anyway here are some more 3D patterns along the same lines. With each one I've added a musical chord with the pitches in the correct relationship for the pattern - and the notes are played using polyrhythms with the rhythms all in the same relationships too - using beats per minute as a "fequency".
For a bit of variety I've also done them with gradually changing tempo and a "buzz roll" type effect for the volumes (though steady beat for the timing). Apologies for the sound quality - hasn't turned out as well as I hoped - enough to give the idea but I will do new versions of these at some point.
7/6 1/1 3/2 - a "septimal minor" or "subminor" triad
1 2 3 - the first three harmonics
2 3 5 - second, third and fifth harmonics - if reduced to the octave, these are the pitches of a just intonation major triad
See the top of this page for 1/1 5/4 3/2 - the pitches of a just intonation major triad
Here they all are one after another as a playlist:
To find this feature in Lissajous 3D, If you start up Lissajous 3D using one of the Lissajous 3D icons in Tune Smithy or in the harmonic polyrhythms view of Bounce Metronome, then it will be automatically set up to respond to to any changes you make in the chords.
To configure in detail in Lissajous 3D, go to.
To read more about Lissajous 3D, go on to: The Screen Saver
To get the program and the screen saver, download and install Lissajous 3D.
The program comes with a Free Test drive with all the features completely unlocked (start the test drive at any time):
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© Robert Walker 2008
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