The following pages contain small online executable programs for studying phenomena of relations between numbers, forms and tones. Loading these applets can take some time depending on actual internet connection (max. 1 minute). Users have to turn on Java functions in their browser preferences. Additionally the JSyn PlugIn is required. It can easily be installed under Windows, MacOS and Linux for the most common browsers (Internet Explorer, Netscape, iCab ...) from Download JSyn PlugIn. - Not least for aesthetic reasons the applets are laid out for full window display. Therefore printing out these descriptions in advance is recommended.

Klangwolken (Clouds of sound)

Every time a sound field is entered by the mouse frequency, volume and panorama values of 24 sinewave generators are defined anew by random selection. Clicking on a field with the mouse makes the sound permanent, clicking again turns the sound off.

Schwingungsbilder (Forms of vibrations)

Modifying the well known Lissajous-figures, "Schwingungsbilder" result from an attempt to visualize the complex geometric patterns, which appear in the overlapping of different movements or vibrations. The single graphics have been drawn automatically: a point moving along a circular line is an analogy to a simple sinewave (thin line). If this point itself becomes the moving center of a second circle with the same radius, on whose new circular line a second point moves along, the new point can represent as its outcome (bold line) a simple overlapping: different simple ratios of velocities of the two circular movements are analogies to different simple intervals. In this case both circular movements can go into the same direction ("gleichphasig") or not ("gegenphasig"). Combinations of three or more circular movements in a similar way show the great and beautiful variety of vibration-patterns of triads (major and minor), their inversions and other harmonic structures. Moving the mouse on single titles will load the matching graphics, the internal connection of the single variants can then be studied by toggling (graphics change by mouse moving) between the different shapes. Since relations between movements or vibrations (proportions) are presented, but not absolute frequencies, harmony can be experienced here in a sense that goes far beyond the human auditory range (20 Hz - 20 KHz).

Kreisbilder (Forms of circle movements)

Applet for drawing forms of vibrations as described above: altogether four interdependent cycloid "trajectories" are available. The single parameters can be adjusted by scrolling the number fields:

  • radius of single "trajectories" (r1 - r4, values: 1 - 300)
  • thickness of single lines (s1 - s4, values: 0 - 50; 0 = unvisible, 50 = broad line)
  • velocity of single "trajectories" relative to the others (v1 - v4; π-factor; values: +/-250)
  • direction of rotation (v1 - v4; click on "=>" changes direction (negative value))
  • density of lines (d, values 1 - 1000; 1 = lines only marked by a few points, 1000 = more complete lines - depending on complexity of relations)
Units in pixel. A common radius of 70 is default, thickness of lines: 1, equal velocity, equal direction.

To try out turn velocities to 5 : 6 : 7 : 8 (would correspond with a dominant-seventh-chord, first inversion), then 5 : -6 : 7 : -8, then 5 : 6 : -7 : -8 etc.

Klangkreise (Sounds of circle movements)

Similar to "Kreisbilder", but with animated movements of points - leaving traces, when "Bahn zeichnen" is pressed. In addition, every "trajectory" produces a sound corresponding to its velocity that grows louder when going down and dies away when going up. The respective "trajectory" sound derives from three sinewave generators: beginning in unison they get more and more detuned with the sound growing louder.

Different parameters as above can also be adjusted by scrolling the number fields:

  • basic velocity v and basic frequency (values: 1 - 1000; 1 = 1 animating impulse in a millisecond, 1000 = 1 animating impulse in a second; v = 25 is default and arbitrarily assigned to the frequency of 160 Hz)
  • velocities of single "trajectories" (v1 - v4; values: 0 - 1000; 0 = sound and movement off, 1000 = fastest movement; default is 50)
  • size of points in pixel or thickness of "trajectory" respective (s1 - s4; values: 0 - 50; 0 = unvisible, 50 = broad line; default is 1)
  • factor of detuning (vd1 - vd4; values: 1 - 1000; 1 = vibrato width one octave, 1000 = an almost vibratoless tone; numbers correlate to overtone-position; default is 32)
  • color of single "trajectories" (f1 - f4; red-, green-, blue-values: 0 - 255; 0 = color-value off, 255 = highest intensity of color)
  • change direction by clicking on "=>"
Default is equal radius and equal direction in all "trajectories". Clicking the mouse stops and starts the movement.

To try out turn velocities to 50 : 60 : 70 : 80 (dominant-seventh-chord, first inversion) and press "Bahnen zeichnen". Then change directions 2 and 4, press "Bahnen löschen" and anew "Bahnen zeichnen". The same with direction 3 and 4 versus direction 1 and 2. Then change the basic velocity.

Patterns of circle movements

Two points moving on two concentric circles can be tuned concerning velocity and sounding interval. The connecting line of the points can be visualized, the whole pattern can be drawn, as well as the nearest and farest distance (conjunction and opposition). Stop and start again with a mouse click.

Klangstaub (Dust of sound)

The actual mouse position defines centers of sounding dust particles, envelope shapes of the individual sounds change accordingly. Stop and start again with a mouse click.


An attempt to realize the Pythagorean table, described by Albert von Thimus in the 1870's, as an online playable instrument. Especially Hans Kayser ("Akróasis", 1946, etc.) used and explained it extensive.

The 64 colored fields represent freely programmable parts of the infinite net of frequencies deriving from a given basic note as overtones and "undertones" (the base frequency is default represented in the upper left field - the displayed numbers a/b in the fields show the harmonic coordinates; a = number of overtone; b = number of "undertone").

Moving the mouse over a colored field changes its color and creates a sinewave signal according to the displayed frequency-ratio, a corresponding envelope shapes the volume. Clicking on a colored field (color changes) makes the sound permanent until clicking again (color changes back) in order to finish the sound. So you can combine all kinds of harmonic chords you like.

By typing in the base frequency (default is 220 Hz) in the parameter field "Grundfrequenz in Hz" and by pressing "enter" you can change the tuning of the whole table.

By typing in a number (1 - 999; default is 1) in the parameter fields "Oberton-Position" (overtone position) and "Unterton-Position" (undertone position) and by pressing "enter" you can "move" the whole table to any part of the lambdoma structure you want. The displayed frequency ratios will change immediately as well as the tuning of the corresponding sinewave generators. So, the higher you go in the numbers determining the table position, the more various vibrating micro intervals resulting from frequencies in low distances should be heard.

(In order to hear non-interrupted pure sounds a CPU with a minimum of 500 MHz is required)

7-note scale

Tune the single scale functions by scrolling the respective number fields. Scale functions can freely be defined as fractions in relation to the basic frequency: numerator for overtone, denominator for undertone position. Try some presets:

  • major scale (Pythagorean tuning)   1: 1/1  -  2: 9/8  -  3: 81/64  -  4: 4/3  -  5: 3/2  -  6: 27/16  -  7: 243/128  -  8: 2/1
  • minor scale (Pythagorean tuning)   1: 1/1  -  2: 9/8  -  3: 32/27  -  4: 4/3  -  5: 3/2  -  6: 128/81  -  7: 16/9  -  8: 2/1
  • major scale (just intonation - Zarlino scale) 1: 1/1  -  2: 9/8  -  3: 5/4  -  4: 4/3  -  5: 3/2  -  6: 5/3  -  7: 15/8  -  8: 2/1

12-note scale

Same principle as 7-note scale above - some more tone fields for tuning chromatic scales as well as alternative diatonic scale functions.

Equal-tempered scale

12 chromatic scale functions - fixed on semitones at 100 cent.

22-note scale

Same principle as 7-note scale above - Try two presets (according to Robert Ross):

  • extended 22-note Zarlino scale
  • 22-note Hindu scale

Discoveries in Csound

Findings on the way to real Sounddesign in mp3-format.


Combinations of different monochord-melodies sorted by scales (mp3-format).

Studies in geometry of curves

Randomlines, randomarcs, Peano-curve in 3 versions, ellipse, polynomes, cellular automatons.

last updated: 07.02.2005