Yahoo Tuning Groups Ultimate Backup tuning-math "Music and Mathematics" Schroeder & Kindermann

Here's an interesting short paper I just happened to notice

Manfred R. Schroeder:Music and Mathematics,
Nova Acta Leopoldina 92, 341 (2005) 9-15:
http://www.physik3.gwdg.de/~mrs/Vortraege/Music+Math.pdf
Its author recently died.
It includes some discussion of "quadratic residue diffusers" and
"tritave" based music scales -- he says 13 notes
per tritave is a good scale and references
H.Bohlen: 13 Tonstufen in der Duodezime, Acustica 39, pp 7686, S.Hirzel Stuttgart (1978) J.R.Pierce: The Science of Musical Sound, New York: Scientific American Library, W. H. Freeman 1983.

--------

Schroeder also pointed me to this web site:
http://www.reglos.de/musinum/
which creates "fractal music" on demand that you can listen to.
This page apparently is an online version of the following paper:
Lars Kindermann: MusiNum - The Music in the Numbers, in: E.R. Miranda, Composing with Computers. Music Technology Series, Focal Press, Oxford 2001.

For example, letting F(x) = the number of "1"s in the binary representation of the integer x,
we have F(1,2,3...)=1,1,2,1,2,2,3,1,2,2,3,2...
which is a "fractal" sequence in the sense that the even numbered term
subsequence is the sequence itself. (Also other senses too.)
That is F(x)=F(2*x) for integer x.
He lets 1=C, 2=D, 3=E and so on to get a melody:
http://www.reglos.de/musinum/demo1.mid
That didn't seem like a good melody to me, but taking every 3rd term results
in a melody which actually is not so bad:
http://www.reglos.de/musinum/3.mid
By interaction of human composers and his freeware program, quite good pieces can be
created:
http://www.reglos.de/musinum/midi/bark1.mid
http://www.reglos.de/musinum/midi/moonrise.mid
http://www.reglos.de/musinum/midi/track1.mid
http://www.reglos.de/musinum/midi/kompo14c.mid
Over 100 such pieces are available at his website, although a lot of them
do not work when I try to play them. He also has a list of other useful music-related freeware programs.

🔗WarrenS <warren.wds@gmail.com>

http://en.wikipedia.org/wiki/BohlenPierce_scale

discusses the 13 notes per tritave scale.
It also was discovered by
Kees van Prooijen in the 1970s.
(You all quite likely already know about all this... if so, sorry)

It has good approximations to
1/1, 9/7, 7/5, 5/3, 9/5, 7/3, and 3/1.
Note that all of these integers are odd.
The role of the "major triad" 4:5:6 has to be replaced by 3:5:7
and you can get a 3:5:7:9 tetrachord.

Heinz Bohlen's original paper is online here:
Bohlen, Heinz: 13 Tonstufen in der Duodezime.
Acustica 39,2 (January 1978) 76-86
S.Hirzel Verlag, Stuttgart.
http://www.huygens-fokker.org/bpsite/publication0178.html
It also is available translated into English:
Bohlen, Heinz: 13 Tone Steps in the Twelfth,
Acustica 87,5 (Sept/Oct 2001) 617-624
http://www.ingentaconnect.com/content/dav/aaua/2001/00000087/00000005/art00011

Pierce's original paper is
Mathews, Max V., L. A. Roberts and John R. Pierce: Four New Scales Based on Nonsuccessive-Integer-Ratio Chords.
J. Acoust. Soc. Amer. 75 (1984) S10(A)
and you can download PDF file for free:
http://asadl.org/jasa/resource/1/jasman/v75/iS1/pS10_s3

van Prooijen's original paper was:
Kees van Prooijen: A Theory of Equal-Tempered Scales. Interface, Vol. 7 (1978), pp. 45-56. Swets & Zeitlinger B.V. - Amsterdam.
free online:
http://www.kees.cc/tuning/interface.html

The Bohlen-Pierce scale also is discussed in
Dave Benson:Music: a Mathematical Offering,
Cambridge University Press, Nov 2006, 426 pages. ISBN: 0521853877
book available online here:
http://www.abdn.ac.uk/~mth192/html/maths-music.html

And in this paper
Dave Benson:Musical scales and the Baker's Dozen,
Matilde, Nyhedsbrev for Dansk Matematisk Forening, Nr. 28, September 2006, pp.15-16
http://www.matilde.mathematics.dk/arkiv/M28/M28-DB.pdf
where he argues this scale is best suited to instruments with a lot of odd harmonics, such as clarinets.
You can buy Bohlen-Pierce clarinets for about $750:
http://www.sfoxclarinets.com/BP_sale.htm

In this psychoacoustic paper, people rate the different triads in the BP scale for lovability:
Mathews, Max V., John R. Pierce, A. Reeves and L. A. Roberts: Theoretical and Experimental Explorations of the Bohlen-Pierce Scale.
J. Acoust. Soc. Amer. 84 (1988) 1214-1222.

Here is another online book "harmony and tonality" including a
paper about the BP scale:
http://www.speech.kth.se/music/publications/kma/papers/kma54-ocr.pdf

Some musical instruments have been constructed for it, and some music
composed for it, but both are very rare.

Wikipedia has links that ultimately lead to pages where you can listen to music in BP scale such as
http://www.kees.cc/music/oddpiano/Kees%20van%20Prooijen%20-%20Odd%20Piano.mp3
http://www.ziaspace.com/elaine/BP/BPmusic/DrB/IKnowOfNoGeometry_RB.mp3
http://www.seraph.it/dep/det/Bohl-en%20Roll.mp3
http://www.transpectra.org/Wanderer_demo.mp3 (Played on BP clarinets)
http://www.transpectra.org/Calypso_demo.mp3 (BP clarinets+electronics+percussion)

BP scale has a very weird sound, to me. Don't think I like it as much as usual scale,
but makes an interesting change.

🔗Carl Lumma <carl@lumma.org>

The paper contains excerpts from his book, Fractals, Chaos and
Power Laws. MusiNum is a classic in algocomp. I still have many
MIDIs in my collection from the '90s.

-Carl

At 09:47 AM 10/24/2011, you wrote:
>Here's an interesting short paper I just happened to notice
>
>Manfred R. Schroeder:Music and Mathematics,
>Nova Acta Leopoldina 92, 341 (2005) 9-15:
> http://www.physik3.gwdg.de/~mrs/Vortraege/Music+Math.pdf
>Its author recently died.
>It includes some discussion of "quadratic residue diffusers" and
>"tritave" based music scales -- he says 13 notes
>per tritave is a good scale and references
>H.Bohlen: 13 Tonstufen in der Duodezime, Acustica 39, pp 7686,
>S.Hirzel Stuttgart (1978) J.R.Pierce: The Science of Musical Sound,
>New York: Scientific American Library, W. H. Freeman 1983.
>
>--------
>
>Schroeder also pointed me to this web site:
> http://www.reglos.de/musinum/
>which creates "fractal music" on demand that you can listen to.
>This page apparently is an online version of the following paper:
>Lars Kindermann: MusiNum - The Music in the Numbers, in: E.R. Miranda,
>Composing with Computers. Music Technology Series, Focal Press, Oxford 2001.
>
>For example, letting F(x) = the number of "1"s in the binary
>representation of the integer x,
>we have F(1,2,3...)=1,1,2,1,2,2,3,1,2,2,3,2...
>which is a "fractal" sequence in the sense that the even numbered term
>subsequence is the sequence itself. (Also other senses too.)
>That is F(x)=F(2*x) for integer x.
>He lets 1=C, 2=D, 3=E and so on to get a melody:
> http://www.reglos.de/musinum/demo1.mid
>That didn't seem like a good melody to me, but taking every 3rd term results
>in a melody which actually is not so bad:
> http://www.reglos.de/musinum/3.mid
>By interaction of human composers and his freeware program, quite good
>pieces can be
>created:
> http://www.reglos.de/musinum/midi/bark1.mid
> http://www.reglos.de/musinum/midi/moonrise.mid
> http://www.reglos.de/musinum/midi/track1.mid
> http://www.reglos.de/musinum/midi/kompo14c.mid
>Over 100 such pieces are available at his website, although a lot of them
>do not work when I try to play them. He also has a list of other
>useful music-related freeware programs.
>
>