TUNING SYSTEMS
As a mathematician
and Baroque music aficionado, I spent years sitting in concerts wondering
what "well-tempered" meant. Imagine my delight at discovering
that the frequencies of notes for a well-tempered tuning were spaced
apart by the 12th root of 2. That is, you can find the frequency of
any note by the formula
Frequecy=fundamental
x 2^n/12
where2^n/12 is
the 2 raised to the power of n divided by 12 and n is the number of
black and white keys above or below the fundamental. As I explored
other tuning systems, I found that many simply multiply the fundamental
by some number m (such as 3/2 or the 4th root of 5) to get the fifth
above it. The fifth then replaces the fundmental and can be multiplied
to get the fifth above the new fundamental. This recursive procedure
can be continued until some octave above the original fundamental
is obtained. By this time all notes have been tuned. Counting both
the black and white keys, a scale is a cyclic group of order 12 that
can be generated by the fifth (7 notes above the fundamental). Now
octaves can be obtained by doubling the frequency (powers of 2). The
problem is that the recursive procedure yields a note eight octaves
above the fundamental. This implies that
2^8=m^12
(2 to the 8th
power = m to the 12th power), which is impossible. For this reason,
some tuning systems sound better than well-tempering in one key but
worse for other keys. A fun exercise is use a software package such
as Maple or Mathematica to look at pictures of the way that sound
waves add for various combinations of notes in different tuning systems.
TUNING LINKS
Physics of
Music http://www.phy.mtu.edu/~suits/Physicsofmusic.html
Rachel Hall's Mathematics
of Music Page http://www.sju.edu/~rhall/newton/
Tuning, Timbre,
Spectrum, Scale http://eceserv0.ece.wisc.edu/~sethares/ttss.html
from
which you can link to Relating Tuning and Timbre http://eceserv0.ece.wisc.edu/~sethares/consemi.html
which explains mathematically why some notes sound good together
and some sound bad.
The Just Intonation
Network http://www.dnai.com/~jinetwk/
Pythagorean
Tuning http://www.aboutscotland.com/harmony/prop.html
OTHER MATHEMATICAL ASPECTS
OF MUSIC
Most people think that
mathematics is the study of number, but it is much more than that.
To me, it is the study of patterns. Certainly, there are mathematical
constructs in the tuning of instruments, in acoustics, and the development
of synthesizers, amplifiers, and other sound equipment. However, there
are patterns in the construction of music itself. I first got interested
in this when I read the book Godel,
Escher, Bach by Douglas R. Hofstadter.
LINKS
Understanding
Mathematics: Gödel, Escher, Bach (this page has lots of cool
links.)
http://www-personal.umich.edu/~jlawler/geb.html
Listen to the works of
Bach in Midi-files. http://www.bachcentral.com/midiindex.html
Well Tempered
Fractal (Using mathematics to assist in writing music.) http://www.hitsquad.com/smm/programs/Well_Tempered_Fractal/
Return
to Leslie Gardner's Research Page