Re: [tuning] Golomb Rulers

http://home.snafu.de/djwolf/fragment.mp3

is a short (30") exercise in this five-tone scale using two timbres derived
(Sethares-style) from the tuning.

Daniel Wolf

> From: <MANUEL.OP.DE.COUL@EZH.NL>
>
> (cut)
>
> 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

Manuel!
How about the fibonacci series!

MANUEL.OP.DE.COUL@EZH.NL wrote:

> From: <MANUEL.OP.DE.COUL@EZH.NL>
>
> 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 :
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

>> From: <MANUEL.OP.DE.COUL@EZH.NL>
>>
>> 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.

For what _musical reason would one want such a thing?

John Link

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Kraig,

Yes, the Fibonacci series also gives scales with many different
intervals. In fact _all_ intervals are different, while this isn't
guaranteed in Optimal Golomb Rulers. However with the Fibonacci
series a lot of intervals can be very close together, while the
differences in OGRs are evenly spaced. Here's an example of a
Fibonacci scale, by Brian McLaren, Xenharmonikon 13, 1991.
First 9 Fibonacci terms reduced by 2/1.
17/16 5/4 21/16 89/64 3/2 13/8 55/32 2/1

John Link wrote:
> For what _musical reason would one want such a thing?

Well if there is a musical reason for wanting maximally even scales I
don't see why there couldn't be one for wanting maximally uneven
scales. But this raises a philosophical question: what is considered
a musical reason and what not? Clearly choosing a particular scale
influences the music, so there's a musical reason. I think even if a
composer can legitimise his/her choices in a musical meaningful way,
there is no need for it. History is full of composers who have put
things in their music without apparent musical effect. For example
Bach, who has put numbers like 3, 14, 41, etc. in his music but this
can only be counted and not heard.

Manuel Op de Coul coul@ezh.nl

[Manuel Op de Coul:]
>But this raises a philosophical question: what is considered a
musical reason and what not? Clearly choosing a particular scale
influences the music, so there's a musical reason. I think even if a
composer can legitimise his/her choices in a musical meaningful way,
there is no need for it. History is full of composers who have put
things in their music without apparent musical effect.

Thanks Manuel, I think these are really fine points, and refreshing
ones to see here at the TD... made my day in fact...

Dan

Manuel wrote,

>> 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.

John Link wrote,

>For what _musical reason would one want such a thing?

Well, in Rothenberg's theory these scales would be maximally efficient
(please correct my terminology, Carl). Meaning that if a piece is based on
one of these scales, with transpositions to different keys allowed, you can
tell which key you're in from hearing only two notes, since any interval
occurs only once in the scale.

Manfred Schoeder, in _Number Theory in Science and Communication_, p. 299,
mentions more acoustical applications of Golomb rulers, such as
pitch-detection and compression for speech signals, and design of
loudspeaker arrays for active sound absorption in airconditioning ducts.

I wrote,

>if a piece is based on
>one of these scales, with transpositions to different keys allowed, you can
>tell which key you're in from hearing only two notes, since any interval
>occurs only once in the scale.

I should amplify that: you can tell what key you're in from hearing _any_
two notes, and only those notes!