The number Phi (AKA the Golden Ratio,
1.618033988749894848204586834...) is the ratio to which the Fibonacci
series converges and is often linked to music in terms of
form, but I thought I'd use its facility to endlessly and smoothly fill
gaps*. I first applied it to just rhythm, with my Phi Delay Clap, which
is
one
clap per second going through a delay plug-in set to 1.61803398875**
seconds and with infinite repeats. You can hear the pattern recur at
13, 21,
34, 55, 89 (1:29), 144 (2:24) and 233 (3:53) seconds: Fibonacci
numbers, of course! It approaches even-ness just prior to each
recurrence. The next one is 377 seconds (6:17), but it's barely
audible, so I've saved you the agony of waiting for it.

I then tried it with a chromatic scale through the same (Phi) delay and
one note per second. It's two guitar tracks panned,
each starting half way through the other, playing up a chromatic scale
2 octaves, fading in the low octave and fading out high octave, so it
seems endless. This effect is known as a "Shepard Tone". Like the clap
version, you can hear the rhythmic pattern recur in the same manner,
but it's not as clear.

Next I applied Phi to the pitch too! Each note
rises a 1/Phi of an octave (full details below***), also as a "Shepard
tone"(see above) and through the same (Phi) delay at one note per
second. The
rhythmic recurrances are also blurry compared to the clap version, but
I've faded the track out and back in at 89, and again at 144, 233 and
377 seconds so you can hear how the smoothing progresses.

Following that, I applied the Phi delay to the harmonic series, using
sine wave blips starting at 80Hz (just because it made the maths
easy) and finish at around 19kHz (which is well up into my tinnitus
clouds). Note the cute
spectrograph visuals!

Having explore Phi, my first-cousin-once-removed, David Pfefferle,
suggested I explore some other irrational numbers, hence my Harmonic
Series Delays of Other Irrational Numbers.
My
favourite of these is the first one here, the number e (Euler number), which it
creates an 19/8 groove with many 7 groupings over the top of it.

*To find out more about Phi, watch Vi Hart's glorious video about it.
https://www.youtube.com/watch?v=lOIP_... , which is part 2 of her 3
videos on Fibonacci, etc. Watch them all!
Or for a drier version, try Wikipedia!
**Note that if I'd set the delay to 1.625 (which is 13/8) then it would
be straight 1/8 notes and after 13 beats the rhythm would land back on
the one.

***Calculating the 1/Phi of an octave pitch shifts hurt my brain a
little and the way I've done it means the range expands upward a
little. But it was the only way I could think to do it while pitch
shifting the smallest ranges as possible. Here's the maths!

C
G + 41.640786499874 (cents)
Eb -16.718427000252
Bb +24.92235949962
F# -33.436854000504
C# +08.20393249937
Ab +49.844718999244
E -8.514494500882
B +33.12629199899
G -25.232921501134
D +16.40786499874
Bb -41.951348501386
F -0.310562001512
C +41.33022449836
Ab -17.028989001766
Eb +24.611797498108
B -33.747416002018
F# +7.893370497856
C# +49.53415699773
A -8.825056502396
E +32.815729997478
C -25.543483502648
From where I start the above pattern 25.543483502648 cents down. Then
at the other C (a bit more than half way through) I restart again, but
41.33022449836 cents up. And from there, the whole 34 note pattern
rises by 15.786740995712 cents each time.