Post by David DaltonCould you recommend good book(s) on the mathematics of music?
Such could be a book on music theory with a lot of mathematics.
I am very advanced mathematically and know a good bit about
acoustical physics and related mathematics but not much yet
about music theory including the frequencies of various notes,
the mathematical definition of musical keys and of harmonization
and more. But again I am mathematically advanced and
also have a good ear I think now, and may try to learn an
instrument soon.
It's not exactly advanced mathematics, but there are some
very nice and practical applications of group theory to
music. I recently posted some info to some other groups,
which I've combined below. The Balzano paper mentioned
below is quite good as far as pointing to the relationship
between group-theoretical structures and the physics of
waves (vibrating strings, etc.) in equal-tempered scales.
---------------------------------
I recently kludged together a bash script to print out chord and
scale diagrams. I've found these diagrams to be a useful way to
think about music theory, and helpful in composition and in analyzing
chord progressions. They are a very concise way to graphically
illustrate a lot of music-theoretical relationships. The script
can be found at
http://www.datafilter.com/software/generateChordAndScaleDiagrams
The construction is based on group theory, but you don't need to
know any group theory to use the charts. You basically just need
to be comfortable with the idea of numbering the notes in base 12,
for example as
0 1 2 3 4 5 6 7 8 9 a b
B C C# D D# E F F# G G# A A#
I got the basic idea from the paper:
Gerald J. Balzano, "The Group-theoretic Description of 12-Fold
and Microtonal Pitch Systems," _Computer Music Journal_, 4/4,
1980, p.66-84
In that paper, group theory is used to analyze chords and scales
with the idea of generalizing equal-tempered scales beyond 12-tone
scales (19-tone scales have some nice properties, for example).
I might get into that some day, but for now I just like to use
the diagrams for 12-tone music theory.
Again, using the diagrams does not require any knowledge of group
theory -- although this sort of application would be a great
introduction to group theory. (You can find a couple of such papers
on the web, and the Balzano paper above is very good.) Basically,
the 12 notes fit on the surface of a torus. In its simplest form
it looks like this:
0 4 8
9 1 5
6 a 2
3 7 b
Now mentally connect the left and right edges together and connect
the top and bottom edges together (like the screen in a game of Pong,
where the ball can go through the walls). Topologically this forms
a donut or bagel shape called a torus. The charts are just
repetitions of this basic pattern over and over, to avoid the mental
convolutions involved in mentally wrapping the numbers onto a toroidal
shape.
Notice that from left to right the sequences increase by 4 (mod 12)
and from bottom to top they increase by 3 (mod 12). [The rows and
columns are subgroup cosets of the full 12-note group, just FYI.]
Notice also that the top left to bottom right diagonals increase
by 1 (mod 12), just like counting or like the strings on fretted
instruments. The diagonals from the bottom left to the top right
increase by 7 (mod 12) and form the cycle of fifths. These are just
some of the patterns to notice, and give you an idea of what makes
these diagrams "work." I'm not going to go into the theory any more
than that; such diagrams efficiently display many important interval
relationships within (and between) chords and scales.
The basic chords and scales fit nicely onto such diagrams. Chord
progressions move along the diagrams (i.e., the surface of the
torus) in interesting ways. I think the diagram for the full
chromatic scale is useful in general, but generating diagrams for
particular scales and printing them out so you can mark them up,
etc., can be especially useful.
The bash script can optionally include a chord-pattern chart at
the bottom of of the diagrams. The basic chord patterns obviously
apply when translated anywhere on the chart. They work for minor
chords too, with the "flip over" of the chord's third. This should
be clear after studying the charts a little. Notice that the
program can also print the diagrams using the standard, lettered
C-major-scale note names.
--------------------------------------------------
Torus is the mathematical name for a donut or bagel shape.
The notes all fit neatly onto the surface of a torus. Major
and minor chords are triangles on the surface. The cycle of
fifths loops around the surface like candy-cane stripes.
You don't really need to know that to use the diagrams. The
diagrams are flattened out into 2-D so that they are easier
to comprehend.
Consider the simple example below, for the C-major scale. Note
the pattern it forms. Every triangle is a major or minor chord.
*--* * G
| / /| /|
|/ / | for example / |
x x--* C--E
x-minor x-major C-major
Knowing this, you can see all the major and minor chords that
are in the C-major scale and how they relate to each other.
For example, you can see how the C-F-G progression moves along
the diagram by locating each of those chords on it.
Major scale in the key of C
F---A Db F---A Db F---A Db F
| / | / | / |
| / | / | / |
|/ |/ |/ |
D Gb Bb D Gb Bb D Gb Bb D
| /| /| /|
| / | / | / |
| / | / | / |
B Eb G---B Eb G---B Eb G---B
/| / /| / /| /
/ | / / | / / | /
/ |/ / |/ / |/
Ab C---E Ab C---E Ab C---E Ab
/| / /| / /| /
/ | / / | / / | /
/ |/ / |/ / |/
F---A Db F---A Db F---A Db F
| / | / | / |
| / | / | / |
|/ |/ |/ |
D Gb Bb D Gb Bb D Gb Bb D
| /| /| /|
| / | / | / |
| / | / | / |
B Eb G---B Eb G---B Eb G---B
That's a basic example that gives a general idea of how such
diagrams can be useful. It is just a tool, though. How you use
it (or not) as far as writing and thinking about music is up to
you.
------------------------------------------------
Here are a few final comments on the chord and scale diagrams,
along with a link to the bash script to generate them.
I've noticed that in some news readers (like Google) the
diagrams don't display properly. There are some added spaces
from being sent over Usenet that throw off the alignment a
little. I hope the general structure still comes through.
The diagrams display OK for me with Mozilla. You also need
to use a fixed-width font for the diagrams to be properly
aligned.
As an aside, if you're interested in the musical torus as
it relates to the study of the brain's perception of music,
check out
http://www.dartmouth.edu/~news/releases/2002/dec/121202.html
I've put the bash script for generating the diagrams on the web at
http://www.datafilter.com/software/generateChordAndScaleDiagrams
There was obviously some feature creep, but sometimes that's
the fun part (and it didn't get out of control ;-).
The script works with
GNU bash, version 2.05b.0(1)-release (i686-pc-linux-gnu)
on my Linux system. It should be reasonably portable (or at
most require a few tweaks). The only external program it
really requires is sed, which is only used to optionally
translate to the standard C-major-scale-based note names.
Enjoy! Constructive comments welcome.
-------------------------------------------------------
Harmonic minor scale in the key of 0, with chord diagrams
0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4
| / | / | / | / | / | /
| / | / | / | / | / | /
|/ |/ |/ |/ |/ |/
9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1
| | | | | |
| | | | | |
| | | | | |
6 a 2 6 a 2 6 a 2 6 a 2 6 a 2 6 a 2 6 a
/| /| /| /| /| /|
/ | / | / | / | / | / |
/ | / | / | / | / | / |
3---7---b---3---7---b---3---7---b---3---7---b---3---7---b---3---7---b---3---7
| / | /| / | /| / | /| / | /| / | /| / | /| /
| / | / | / | / | / | / | / | / | / | / | / | / | /
|/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/
0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4
| / | / | / | / | / | /
| / | / | / | / | / | /
|/ |/ |/ |/ |/ |/
9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1
| | | | | |
| | | | | |
| | | | | |
6 a 2 6 a 2 6 a 2 6 a 2 6 a 2 6 a 2 6 a
/| /| /| /| /| /|
/ | / | / | / | / | / |
/ | / | / | / | / | / |
3---7---b---3---7---b---3---7---b---3---7---b---3---7---b---3---7---b---3---7
| / | /| / | /| / | /| / | /| / | /| / | /| /
| / | / | / | / | / | / | / | / | / | / | / | / | /
|/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/
0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4
| / | / | / | / | / | /
| / | / | / | / | / | /
|/ |/ |/ |/ |/ |/
9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1 5 9 1
| | | | | |
| | | | | |
| | | | | |
6 a 2 6 a 2 6 a 2 6 a 2 6 a 2 6 a 2 6 a
/| /| /| /| /| /|
/ | / | / | / | / | / |
/ | / | / | / | / | / |
3---7---b---3---7---b---3---7---b---3---7---b---3---7---b---3---7---b---3---7
| / | /| / | /| / | /| / | /| / | /| / | /| /
| / | / | / | / | / | / | / | / | / | / | / | / | /
|/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/ |/
0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4 8---0 4
Chords with root 0 (the pattern is the same for all chords):
9th=7th+add9
3rd 3--7 5th 7 5th 7th a 2 add9 9 7th 0--4--7
| / /| | / (-3rd=sus2) |
|/ / | |/ | augmented
0 0--4 3rd 7--b maj7 6
root root /| |
/ | |
minor major 0--4 3
/| |
/ | |
sus4 5 9 0
(-3rd) 6th
diminished
Note the cycle of 5ths along the / diagonals; counting along the \ diagonals.
Note guitar/bass fretboard pattern (minus "guitar bump") along alternate \ diags.
Standard note convention: 0=B 1=C 2=C# 3=D 4=D# 5=E 6=F 7=F# 8=G 9=G# a=A b=A#
--
Mind Control: TT&P ==> http://www.datafilter.com/mc
Home page: http://www.datafilter.com/alb
Allen Barker