Here's the dekany tumbling in 4 dimensions (Excel chart 150k). You will need to cancel any error message regarding circular references (it needs them), enable macros, and the first time you may need to click the "MIDI out" button, choose your MIDI output device, and save the spreadsheet.
Here are the program notes for when this was shown at a concert of algorithmic compositions organised by David Cope at University of California, Santa Cruz on 25-Apr-2001. For this concert it was entitled "Polyhedron" and was used to open and close the show.
Dave Keenan is a software engineer, electronics designer, sustainable systems designer, teacher, nonviolent activist and tuning theorist, living in Brisbane Australia. See http://dkeenan.com [URL updated] for more information.
Andy Fillebrown, a recent graduate of Berklee College of Music, is a
composer/programmer living in New England. The co-composers have never met, but collaborated over the internet.
The algorithm consists in mapping the notes of an octave-specific 4,5,6,7,9-Dekany (a type of justly intoned decatonic scale due to Erv Wilson) onto the vertices of a dispentachoron (a regular 4 dimensional figure). The notes are arranged so that nearest neighbours have various consonant relationships, including the exotic ratios of 7 (subminors and supermajors). This 4D figure is then rotated in 5 different planes simultaneously, each one "geared" to the next by the golden-ratio, so that that it never exactly repeats any position, and takes as long as possible to almost repeat. A fixed point in 4D space, just outside the figure, represents the listener, and the volume of each note depends on its distance from the listener in four dimensions. As well as hearing the resulting continuously-mutating harmonic progression in stereo, the audience sees, with the aid of simple red/blue 3D glasses, a mutating 3 dimensional projection of the 4D figure as it rotates, with the sounding notes nearest to them.
Some other experimental compositions by Andy can be found at http://groups.yahoo.com/group/tuning/files/AMiltonF/
Before we released the first tumbling dekany with sound (folks had
only seen the graphics), I asked Andy if we could easily get 5 or 10
minutes of it (without graphics) into a MIDI file or MP3 (using Choir
Aahs and varying the speed and volume over the course of it) because I
wanted to try an experiment.
I wanted to know how much people's perception of the artistic merit
an algorithmic composition could be altered if they believed it to
have been created by non-mathematical means and without machine
assistance, particularly if the fictitious artist claimed to have
composed it to express emotional experiences in his or her own life.
I intended to put it up on a website that noone would associate with
me, and post the following message to the list from an unfamiliar
To whom it may concern,
It has been a year now since the death of my friend Peter Hindemith
(no relation to Paul Hindemith). I believe his music deserves to be
more widely known.
Please listen to
Two years before his death he lost both his wife and young son in a
car accident. This piece was written in those final two years in which
I was priveleged to know him. He called it "Life in 10 voices".
As Peter described it to me, it was intended to be performed by a
choir of ten "voices" where each voice only ever sings a single note.
At a pinch it could be performed by as few as six singers, since
certain notes were never required to be sung together. The singers
were free to vocalise in any way, except that the sounds must never
"make sense", i.e. not be recognisable words in the language of the
audience. His handwritten score with its strange curves, simply
indicates when and how loudly each singer should sing his or her note.
Although the singers need to be synchronised by a conductor and the
scores show bar lines for that purpose, these are entirely arbitrary
and are not intended to introduce any "beat" into the music.
Having spent his childhood in central Australia before moving to the
east coast to study both music and mathematics, I suspect there may be
some influence from aboriginal music, in particular the digeridoo with
its harmonic variations on a basic drone.
I understand that he called it "Life in 10 voices" because it was
intended to represent the 10 most significant people in his life with
their varying distance from him and their periodic blending together,
sometimes harmonious, sometimes not. But he would never say who they
were for fear of offending someone.
I know it is rather a pathetic imitation to try reproduce this in MIDI
format, and I wrestled with my conscience over this. But I don't have
access to a choir and, as I said, I think his work deserves to be
It was intended to be performed in a form of "just intonation", and
although I am myself not a mathematician (nor am I much of a
musician), with the help of a friend I believe we understood what
Peter wrote regarding how much each note departs from standard tuning
and we hope we have faithfully reproduced it using MIDI pitch bends.
At least it sounds absolutely awesome to us.
Please let me know if you think it has any merit. And please pass this
message on to anyone you think might appreciate it.
We would have waited a few weeks before posting
and then waited until someone put 2 and 2 together.
Unfortunately it was all just too difficult at the time. But I
encourage you to select the "Choir Aahs" patch and maybe bump the
Zero-Vol-Distance to 180 and close your eyes and listen, and think
about the life of Peter Hindemith. Isn't it more enjoyable that way?
Does it really matter that he never existed?
You can find discussion of the Tumbling Dekany in the archives of the
alternative tuning list.