MusiNum - The Music in the Numbers

by Lars Kindermann

MusiNum is a free sonification program which turns numbers into generative fractal music. Everybody can create unique royalty free music for his homepage within few minutes. Interesting for mathematicans and other people who like to play. Fractal concepts, self-similarity and a new kind of symmetry are audible now!
Theory Sounds Compose Online Download Links Contact

Music and mathematics always had a close relationship. Since Pythagoras it is known that tonal harmony is closely related to the numerical relation of the frequencies. In the last years a new field of science and mathematics boomed. Chaos, fractals and self-similarity are topics which caught public interest not at least because of the beautiful pictures which can be generated with them. Hardly anybody does not know the colorful psychedelic pictures of the Mandelbrot-set and even people never heard of complex numbers before bought mathematical books on this topics now. Experiments which tried to extend the beauty of the fractal-art-pictures to the acoustical sense sometimes gave interesting results but usually the sound is quite strange. I think this difficulty arises from the fact that chaos theory usually works with real numbers. But our traditional music is based on discrete frequencies and simple combinations of frequencies, and the mathematical discipline which is employed with the simple numbers is number-theory. Perhaps the most fundamental entities in mathematics are the natural numbers: 1,2,3,4,5... They are something universal: It is a hard thing to imagine a mind which would count in a different way. But the style we write them down can vary: The decimal system based on the digits 0-9 is by no way the only or natural method to present numbers. It has just been arbitrarily chosen some time ago in history. The simplest notation is the binary notation which only uses the digits 0 and 1. Computers always calculate in binary notation because it can be easily mapped to electrical devices: The presence of current means 1 and no current means 0.

How the Numbers can make Music

I wrote a program that counts. And because it is running on a computer it counts in binary notation of course. Then it adds up the digits in these binary numbers (i.e. counts the ones) and plays a note: c for a sum of 1, d for 2, e for 3 etc... By doing so this melody is created!
     decimal  binary  sum of
     number   number  digits    Tone    2nd 4th   
	1       1       1       c 
	2       10      1       c       c   
	3       11      2       d       .
	4       100     1       c       c   c
	5       101     2       d       .   .
	6       110     2       d       d   .
	7       111     3       e       .   . 
	8       1000    1       c       c   c 
	9       1001    2       d       .   .
	10      1010    2       d       d   .
	11      1011    3       e       .   .
	12      1100    2       d       d   d
	13      1101    3       e       .   .
	14      1110    3       e       e   .
	15      1111    4       f       .   .
	16      10000   1       c       c   c
	...     ...     ...     ...     ... ...

Did you notice that playing only every second tone results in the same melody? This is the result of the self similarity of this series of numbers. It exploits a new kind of inherent symmetry, closely related to the scaling invariance of fractals: Within the sequence you can find one or many perfect copies of itself. Weired? Imagine a lazy pianist, leaving out every second note of a sonata and nobody would realize! You can read about the mathematical background of this sequence, closely related to the so called Morse-Thue sequence and even the Mandelbrot set in the books Number Theory in Science and Communication or Fractals, Chaos, Power Laws by M.R. Schroeder from Göttingen.

There's another surprising method to construct this Dress-sequence from "nothing": Start with zero and then repeat to append what you have, increased by one:
0 1
0 1 1 2
0 1 1 2 1 2 2 3
0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4

The sequence can even be constructed completely locally: Start with zero again and then in consecuting generations every number gives birth to a slightly grown baby: n -> (n n+1)

0 1
0 1 1 2
0 1 1 2 1 2 2 3
0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4

The sequence is also a non trivial solution of the functional equation f(x)=f(2x) on the integer domain.

Can you find other ways to generate the sequence? Or do you see why the self-similarity is not so surprising in fact?

Brain scientists listen to the cells of the brain with microelectrodes. Imagine, you were listening to the electrical currents in the circuits of your microprocessor or memory chips. What would you hear? Most changes which happen there are counting operations: Think of all the loops in the programs you wrote, all the pointers increasing or decreasing mostly by 1, 2, 4, 8, 16... The electrical current flowing in a byte corresponds to the number of active bits. This means your computer humms the Music in the Numbers all the time!
Beside playing this basic sequence, there are many possibilities to experiment around. Interesting effects are generated when the computer does not count simply 1,2,3... but is calculating different 1x1's. For example 3, 6, 9, 12... or the multiples of 5. Very interesting are the multiples of powers of two minus or plus one (e.g. 31 and 33), the baroque integers! Completely different melodies are generated! To hear what happens if we leave the binary system behind and write down these numbers in ternary, hexadecimal, our well known decimal-system or do other transformations, download the windows program!

Sound Examples

You listened to the pure sound of some numbers so far. But the cool stuff is done by the combination of several numbers mapped to different instruments. Here are some compositions created by users of MusiNum around the world:

The term baroque integers is perfectly explained by Bark1. Some "classical" scores: Slow, Komposition 14c and Moonrise. Like pretty easy listening sounds Komposition 6. Very interesting are the percussive patterns in: Burundi, Rhythm2 and Afloopende zaak. Guitareros should try Classical Plasmas. Most abundant are ambient sounds like Task, Na002 and Mello1 Slightly jazzy pieces were also generated like Just Think it Over (Side by Side) and Vor dem Hirschen. Many pieces are best described as minimal music: Minimal, Glass 1, 6 Marimbas Again or NumQr. The Techno era has arrived with Track1 and Zeittest.

These examples and many more can be found on the user page, together with the MusiNum files and infos about their composers.

Click to enlarge the screenshot of MusiNum

Download MusiNum!

It'sFreeware! The current version is 2.08 (Beta), released at Oct-19-1999. It will run on Windows 2.x, 3.x, 9x, ME, NTx, 2000, XP and Vista, but sadly no more on Windows 7 and Windows 8 because Microsoft dropped the support for old 16Bit software.

But it works perfectly under Linux with Wine and Timidity!

Click to download musinum208.exe (635.077 Bytes). To install MusiNum, just run this self-extracting archive!

If you don't trust the .exe, rename it to, extract it to a temporary directory and run setup.exe.

Sorry, the online-help of Version 2 is not yet up to date, so visit again in few days for an upgrade. And please report bugs to me!

Known Problems in Beta 2.08:

Installation does not work on Windows 7 and Windows 8 because Microsoft does not support old 16Bit software anymore.
Use Linux with Wine and Timidity!

The Program does not run on XP/Vista immediately after installation.
You have to reboot your computer after installation!

The installation hangs on "creating program manager icon". This happend on some systems with ME after installing IE 6.0.
Workaround: In fact, MusiNum installation has been completed - except creating the "Start Menu" entry. Just start MUSINUM.EXE manually from Windows Explorer, it resides in the installation folder you specified during install. You may also create a shortcut yourself. Maybe you have to reboot manually after install.

If installation fails for any other reason:
MusiNum can easily be installed manually: Rename MUSINUM208.EXE to MUSINUM208.ZIP and extract it's contents into some directory. You can just run MUSINUM.EXE from there.

If you save files with voice 1 turned off, after reloading it will be turned on again. (Occured in 2.08)
Workaround: Use voice 1 only as a startup voice, or set the volume to zero, or turn it off in script 1!

On some systems "save to midi" does not work and MusiNum seems to hang. What actually happens, is the "save as midi" dialog window hides BEHIND the main window and in not visible.
Workaround: Sometimes it helps to click the icon "Save as Midi" in the taskbar!

On some systems MusiNum crashes when opening too complex settings with an "out of memory" error.
Workaround: MusiNum uses 16 Bit memory. Look if there are uneccessary drivers in config.sys or autoexec.bat. This problem appears not to exist under Windows XP.
I need some replies to this problem as I cannot reproduce it on any of my systems...

Under XP with the freeware "PowerToys xMouse" installed, the drop-down menues work strange when using the mouse.
Workaround: Click into the box and use the keyboard arrow keys to select an entry (or disable xMouse).

Read the Release Notes for recent changes...

Some Freeware Utilities helpful with MusiNum

A Knot in the Braid of Mathemathics


Contacting the Author

Comments? Questions? Suggestions? Have a look at the guestbook or you can reach me online on my homepage or by email to

MusiNum, The Music in the Numbers ©1995-2014 Lars Kindermann. Linking to this page is definitely allowed and wellcome! The algorithm is not patented - feel free to implement it yourself! The source code will be published when I forget to be ashamed about my awful programming style... Using music from these pages (unless created by yourself) may require authorisation by the composer!