Early attempts to understand the mathematical nature of the wave were partial and incomplete. In 1974 or earlier Royce Kelley and Leon Taylor were recruited by Terence McKenna and calculated the wave as 64 tables of 384 terms each. This work was done at the University of California at Berkeley using a FORTRAN program running on a CDC 6400 computer.
In the Acknowledgements in the 1975 The Invisible Landscape we read:
... we wish to especially thank Leon Taylor and Royce Kelley, both of whom were tireless in their dedication to the computer programming involved in the extrapolation and displaying of the I Ching graph. The material in the appendices is their work, as are illustrations 17-27.
One of the appendices contains a table of 384 lines, later known as the "Table of Intermediate and Final Values" (this table appears in the printed manuals for the MS-DOS versions published after 1993). This provides a set of 384 numbers in the range 0 - 79 (which used to be known in the jargon of Timewave Zero as the "data points") which were taken by later researchers as primitive values from which the fractal timewave was generated. The quotation above from the 1975 edition of The Invisible Landscape states that this table was made originally either by Royce Kelley or by Leon Taylor (or both). The 1975 edition of The Invisible Landscape contains some FORTRAN source code which is, unfortunately, mostly illegible, and presumably one or both of Kelley and Taylor wrote this code (no other copy of which is known to survive).
Terence McKenna has said (on the Art Bell show of 1997-05-22): "I've written computer programs which produce what I call novelty waves". In fact he has never written a computer program, and certainly not the timewave software.
In 1978 or 1979 Peter Broadwell (who had been working as a programmer for Ralph Abraham) was recruited by Terence McKenna and developed the first Timewave software to run on a microcomputer, the Apple II+. The program was written in Applesoft BASIC and proceeded from some earlier, preliminary and incomplete, code by an unknown programmer. Peter Broadwell's version was the first actually to display the wave graphically. At this point, however, little manipulation of the wave display was possible.
During the late 1970s knowledge of the Timewave began to cross the Atlantic. The first European known to have studied The Invisible Landscape was Klaus Scharff, a teacher of mathematics and physics, whose main theoretical interest was, and remains, the nature of time (see his Chronolytisches Studio). Around 1981 he read about the theory in Robert Anton Wilson's Cosmic Trigger then purchased a copy of The Invisible Landscape. He thought that the derivation of the 384 values from the King Wen Sequence, as described therein, looked rather obscure.
Soon after, basing his work on the information in Appendix II of the book, and taking the 384 values as primitive, he wrote a program (using the Pascal language) which successfully displayed the timewave in a graphical form. Klaus Scharff also found numerous interesting correspondences between the ups and downs of the timewave and historical events.
I was first exposed to the theory of the Timewave when in 1984 a girlfriend of mine gave me a copy of the original edition of The Invisible Landscape, bought from Will Noffke at Shared Visions in Berkeley.
I first met Terence McKenna in California in the summer of 1985 while visiting Occidental (where he lived at the time, on Bohemian Highway). At this time he was selling advice to people based upon the I Ching in combination with the theory of the timewave. He is not a computer programmer and when he learnt that I was he asked me to begin work on a new version of the Timewave software for the Apple //e.
Initially the the Timewave theory made little sense to me, and I found Terence's explanations less than clear. Eventually I was able to formulate a new mathematical description of the timewave (different from that presented in the appendices to the 1975 version of The Invisible Landscape) in a precise and rigorous manner. (The archaeoastronomer Simon Cassidy contributed briefly, but significantly, to this effort.) This mathematical formulation, and some timewave theorems, was published as an appendix to the 1994 edition of The Invisible Landscape, and it can be read on this website.
It was only at this stage, in the mid-1980s (some years after Mandelbrot's pioneering work on fractals) that the timewave assumed an explicitly fractal nature. The fractal nature of the timewave results from a bidirectional construction, where each point on the wave is calculated as the sum of a doubly infinite series (see The Mathematical Definition of the Timewave). The mathematical formulation given in the appendices to the 1975 edition of The Invisible Landscape captures only one direction of this construction. A timewave calculated along these lines would not have infinite complexity, though the complexity could be increased by beginning the construction at smaller basic time intervals (i.e. 1/64n of a day, for increasing n).
The programming language again was Applesoft BASIC, but this time augmented by 6502 assembly language routines. However, it was still rather slow due to the large amount of calculation involved to determine the value of each point of the wave. The Apple //e version was finally completed in February 1987. It was the first the Timewave software to allow automatic calculation of major and minor resonances.
In January 1989 I began work on an MS-DOS version in Munich, Germany. This version was written using the C programming language for speed and with eventual porting to the Macintosh in mind. The program was entirely rewritten. This was the first the Timewave software to permit multiple screens with different segments of the wave.
I completed the first MS-DOS version in August 1989 and delivered it to Lux Natura, the company formed by Terence McKenna and his (then) wife Kat Harrison mostly for the dissemination of audio tapes of McKenna's talks, which then made it available to the public.
Owing to the limitations of the Apple //e's representation of real numbers, the Apple //e version could not graph the wave back beyond about 100,000 years. The C version extended the calculation of the wave back to beyond the time of the formation of the Earth, 4,500,000,000 years ago. This at last permitted examination of the wave at such interesting times as 65,000,000 years ago, the time of the extinction of the dinosaurs.
In 1990 I founded Dolphin Software, which in 1991 took over the publication of the the Timewave software from Lux Natura, and 2012-12-21 became the default zero date used by the software (previously Terence had favored 2012-12-22). A version was released which could use a math coprocessor chip, resulting in a dramatic improvement in the speed of calculation and display of the graph.
In April 1991 a German version of the MS-DOS software was commissioned by Gaia Media in Switzerland. I converted the software to German, assisted in the translation of the text by Klaus Scharff. During the course of this work I discovered that there are parts of the wave that have the same shape but are not major resonances. It turned out that they are trigrammatic resonances, which led me to add to the software the ability to graph these resonances automatically. During 1991 - 1993 I added numerous enhancements, including the ability to calculate sets of successive trigrammatic resonances.
During 1993 - 1994 I developed the program WEN_SRCH for searching the hexagram sequence space and the program WEN_GRPH for converting hexagram sequences to the data points usable in the generation of the timewave (or an alternative timewave, if different data points — the 384 numbers — are used). I also added to the main program the ability to graph the wave in a logarithmic transformation. I released this Advanced Version early in 1994.
In January of 1996 Matthew Watkins (founder of the The RetroPsychoKinesis Project) initiated contact with Terence McKenna to discuss possible collaboration and a refining of the the Timewave theory. Watkins, having recently completed his doctoral studies in mathematics, wished to capture the theory in a formula. After studying the C code in my WEN_GRPH program he succeeded in finding a formula which maps the numbers 0, 1, ..., 383 into the 384 values - a remarkable achievement (though it should be remembered that this formula captures only the first part of the derivation of the timewave, the second part being where the actual fractal wave is generated). I later converted Matthew's original MAPLE expression into C code (see The Generation of the Data Points).
During the second half of 1997 and in early 1998 I produced new DOS Timewave software designed to be used with Windows (Fractal Time), introducing a new way to print the graph, eliminating the need for any external files, implementing a toggle between American and European date formats, facilitating the capturing of graphs in transparent GIFs, adding a twelfth screen and the ability to find "true" trigrammatic resonances. I also added the ability to switch between timewaves generated by multiple sets of 384 numbers, and included three new number sets: the Watkins, Sheliak and Huang Ti number sets.
The final version of the software is Fractal Time, Version 6.72. It can be downloaded from this web site.