Golomb Rulers

If one is looking for a scale that is maximally uneven in terms of
interval size, then I think Optimal Golomb Rulers are good candidates.
These give you (in scale terms) the highest number of different
intervals with a large variance in size, with the least number of notes.
From the site
http://members.aol.com/golomb20/intro.htm :

"A Golomb Ruler has a certain number of marks (e.g., 4). The marks are
placed at integer multiples of some fixed spacing. (Thus mark
positions are usually referred to by integer values with 0 being the
leftmost.) The objective is to achieve as many distinct measurement
distances between marks as is possible with the given number of marks.
Thus these marks must be located very efficiently, so that we avoid
redundant distances between marks - ie., the distance from any mark to
any other mark must be a unique distance for the ruler. Sample: marks
at 0-1-2-3 would not be very efficient, since three pairs are
separated by the distance 1. You could only measure the distances of
1,2, and 3 with this ruler. This is not a Golomb Ruler.
A sample Golomb Ruler with 4 marks is 0-1-3-7. It can measure six
different lengths: 1,2,3,4,6, and 7. See how no distance is measured
twice? Now, what we really want are Optimal Golomb Rulers (OGR's).
These are the shortest golomb rulers for a given number of marks. For
instance, for 4 marks, the OGR is 0-1-4-6, which measures six
distances (1,2,3,4,5,6) but is as short as possible (6 "units" in
length)."

Here are some OGR's based on the octave (or choose any other interval
or mode):
5:
70.5882 282.3529 705.8823 847.0588 2/1
6:
48.0000 192.0000 480.0000 864.0000 1104.0000 2/1
7:
35.2941 141.1765 317.6471 529.4118 776.4706 1129.4118 2/1
8:
27.2727 136.3636 327.2727 681.8182 736.3636 954.5455 1118.1818 2/1
9:
21.818 130.901 218.182 501.818 567.273 741.818 894.545 1156.364 2/1
10:
16.667 66.667 216.667 466.667 550.0 783.333 900.0 1066.667 1166.667 2/1

Manuel Op de Coul coul@ezh.nl

HI Jon,

> Just be sure to stick a "my preciousssss" in there somewhere. :)

:)

Here are a few more sound clips from recent debugging btw,
all in harmonic timbres made using various functions,
and using the Karplus Strong decay:

http://www.robertinventor.com/brassy_5o4_11o8_7o4_2o1.mp3

http://www.robertinventor.com/harmonics_to_16_xu11_plucked.mp3

(xu11 there means x^11 because that is what I used as
half wave shape to make a rounded saw tooth).

http://www.robertinventor.com/harmonics_to_16_xu11.mp3

Same notes, but each partial swells and fades and they all
melt into each other - at the mid point it consists of superimposed
rounded saw tooth curves for all the 1st 16 harmonics simultaneously.

Robert

>Here are a few more sound clips from recent debugging btw,
>all in harmonic timbres made using various functions,
>and using the Karplus Strong decay:
>
>http://www.robertinventor.com/brassy_5o4_11o8_7o4_2o1.mp3
>
>http://www.robertinventor.com/harmonics_to_16_xu11_plucked.mp3
>
>(xu11 there means x^11 because that is what I used as
>half wave shape to make a rounded saw tooth).
>
>http://www.robertinventor.com/harmonics_to_16_xu11.mp3
>
>Same notes, but each partial swells and fades and they all
>melt into each other - at the mid point it consists of superimposed
>rounded saw tooth curves for all the 1st 16 harmonics simultaneously.

Hey, these sound really cool!

-Carl