Matrices for beginners

My first exposition of interval matrices was intended to be as
brief as possible, in the hope that someone would say "Oh yes, so
and so explained this in such and such a paper. We all know that."
It looks like this is new to everyone, though, so here's an
explanation that requires the minimum of background knowledge.

First, let's start with 7-limit JI. What the hell! A frequency
ratio can be expressed as:

f = 2^i * 3^j * 5^k * 7^l.

Where i, j, k and l are integers. A pitch difference p=log(f) can
be expressed:

p = i*log(2) + j*log(3) + k*log(5) + l*log(7).

Some people may not understand logarithms but, don't worry, you
may be able to pick up the thread later on. Sums like this can
be written using a matrix product as follows:

a*b + c*d + e*f = (a c e)(b)
(d)
(f)

The b, d and f should all be enclosed in one big bracket, but I'm
compromising with three little brackets in this medium. If b, d
and f aren't in the same column on your screen, choose a font so
that they are.

Matrices are arrays of numbers that can be added, multiplied and
subtracted. There should be books in your local library that can
tell you about matrix arithmetic. If you're at a University, you
may have access to software like Maple or Mathematica. If so,
look up matrices in the Help index.

A pitch difference can then be written in matrix form as follows:

p = (i j k l)(log(2)) = (i j k l)H
(log(3))
(log(5))
(log(7))

So H is a column matrix containing logarithms of prime numbers.
The number of rows is ... as many as you need. It could be
described as the metric of pitch space. If you set i, j, k and l
to a 4-D lattice, a pitch difference is the vector pointing from
one note to the other. People who use ratio space are now on
familiar ground. As this is a general way of writing pitch, H has
to be octave specific. H can also be approximated as:

(1200 )
H = (1901.955) cents
(2786.314)
(3363.826)


The next concept to introduce is that of the basis of a scale.
This is one of the smallest sets of intervals that define a scale.
A basis for 5-limit JI is H with 3 rows. Another is:

(t) (1 -2 1)
(s) = (4 -1 -1)H
(p) (-4 4 -1)

This is a definition of three intervals using H. The first is a
minor tone t=(1 -2 1)H which has a frequency ratio 10/9. We also
have a diatonic semitone s=(4 -1 -1)H or 16/15 and a syntonic
comma (-4 4 -1)H or 81/80.


Now for some precise definitions. A scale is defined on a basis
using a matrix containing only integers. Each note in a scale
must have a unique definition in terms of the basis. That means
that no basis interval can be defined in terms of the other ones
using rational multiples. A scale by this definition is a
discrete vector space. Look up vector spaces to see how close the
analogy is. The concept of basis of a musical scale has been
defined before, although not in terms of matrices. This is all
very similar to group theory. I have defined the basic dimension
of a scale as the number of intervals in its basis. I think this
is the same as the dimension of the vector space, but I didn't
take good enough notes in my linear algebra classes to be sure.


A C major scale (not the only one) can be defined using t, s and p
as follows:

C D E F G A B C
t+p t s t+p t p+t s

This can be written using matrices:

(C) (5 3 2)
(B) (5 3 1)
(A) (4 2 1)
(G) (3 2 1)(t)
(F) = (2 1 1)(s)
(E) (2 0 1)(p)
(D) (1 0 1)
(C) (0 0 0)



That covers JI. For temperaments, the basis is defined on H using
fractions. By "temperament" I mean a scale that approximates JI
using fewer notes. Temperaments can be treated like JI if you
use an approximate matrix H' that pretends to be H, but is slightly
different. As an example, 1/4 comma meantone can be defined as
follows:

(1 0 0 ) (1200)
H' = (1 0 0.25)H = (1897) cents
(0 0 1 ) (2786)

You can then redefine t, s and p in terms of H':

(t) (1 -2 1) (-1 0 0.5 )
(s) = (4 -1 -1)H' = ( 3 0 -1.25)H
(p) (-4 4 -1) ( 0 0 0 )

You can get the last matrix using standard matrix arithmetic.

Not surprisingly, the approximate syntonic comma comes out the
same as a unison in 1/4 comma meantone. As t, s and p constitute
a basis for JI, t and s are a basis for a negative meantone.
Hence a meantone, or linear temperament, is a 2-D scale. The
first two rows of H' also constitute a basis for a negative
meantone, so the third row can be defined in terms of these:

( 1 0)
H' = ( 0 1)H'
(-4 4)

I explained how to define this matrix back in TD1132 (the second
of the digests numbered 1132). Using it, you can approximate any
5-limit interval as a Pythagorean interval. Any negative meantone
can be defined by the basis:

(t) = (-3 2)H'
(s) ( 8 -5)



Matrices can also be used to describe scales that are not even
approximations of JI. Charles Lucy assures us that LucyTuning is
such a scale. It uses the following basis:

(L) = ( 1/(2pi)) oct
(s) (1/2 - 5/(4pi))



SMTPOriginator: tuning@eartha.mills.edu
From: adam.silverman@yale.edu
Subject: JI Reference
PostedDate: 27-08-97 17:27:45
SendTo: CN=coul1358/OU=AT/O=EZH
ReplyTo: tuning@eartha.mills.edu
$UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH
RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH
RouteTimes: 28-08-97 10:21:57-28-08-97 10:21:58,28-08-97 10:19:31-28-08-97 10:19:31
DeliveredDate: 28-08-97 10:19:31
Categories:
$Revisions:

Received: from ns.ezh.nl by notesrv2.ezh.nl (Lotus SMTP MTA v1.1 (385.6 5-6-1997)) with SMTP id
C1256501.002DD747; Thu, 28 Aug 1997 10:20:42 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02326; Wed, 27 Aug 1997 17:27:45 +0200
Date: Wed, 27 Aug 1997 17:27:45 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA02324
Received: (qmail 20531 invoked from network); 27 Aug 1997 15:27:39 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 27 Aug 1997 15:27:39 -0000
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu