Download Lecture 14a: Additional Remarks on Tuning Systems In previous

Math in Music
Spring 2009
Lecture 14a: Additional Remarks on Tuning Systems
In previous lectures, I’ve discussed tuning systems like just intonation
and meantone tuning which were developed to solve problems with
previous systems, each of which were widely used at some point in
musical history. (In fact, just intonation and meantone temperament
are still used today to construct keyboard instruments that are
specifically dedicated to perform music from the Renaissance and
baroque periods.) Collectively, these tuning systems might be called
traditional Western systems. But the twentieth century has seen
much more exploration of alternatives to these systems, derived
either from mathematical analysis, from exploration of musical
traditions of other cultures, or from “liberating” oneself of the
assumptions behind Western tuning systems. I want to briefly
describe the results of these explorations.
1. The 19- and 53-note equal temperament scales
The 12-note equal tempered scale is based on the approximation
3/2 ≈ 2^(7/12)
or, taking base-2 logs of both sides
log_2(3/2) ≈ 7/12
How good is this approximation?
log_2(3/2) = .5849625 while 7/12 = .5833333
Now, it follows from unique factorization into primes that log_2(3/2)
is irrational; otherwise, if log_2(3/2) = p/q for integers p and q, then
3/2 = 2^(p/q) and 3^q = 2^(p+q), which is impossible unless p and q
are zero. (Thus, we will never be able to express some number of
octaves as a power of 3/2.) However, mathematicians have
systematically studied the approximation of irrational numbers by
sequences of rational numbers, and can produce such approximations
by continued fractions. (A discussion of continued fractions is beyond
the scope of this course, but you can find a good treatment in Benson
section 6.2.) As the approximation gets better, the denominators of
Math in Music
Spring 2009
these rational numbers grow, which is problematic because the
denominator (like the 12 in 7/12) is the number of notes you end up
dividing the octave into. However, later in the sequence of
approximations to log_2(3/2) there is a better approximation that has
not too large a denominator:
31/53 = .58490566..
(The approximating sequence is shown on page 224.)
This means that 53 perfect fifths is close to 31 octaves; equivalently,
if we divide the octave into 53 equal steps, then 31 of these steps is
close to a perfect fifth. How close? Well,
2^(31/53) = 1.49994,
so that the difference between this and a perfect fifth (1.5) is about
.07 cents. This approximation seems to have been known to the
ancient Greeks. However, there are drawbacks to dividing the octave
into 53 notes; Figure 6.3 shows the keyboard of a harmonium, built in
the 19th century, tuned to a 53-note scale.
Another approach is to try to construct an equal temperament system
that gives a good approximation to some other just interval. For
example, a just minor third is a ratio of 6/5=1.2, and in the 12-tone
equal temperament system this is approximated by 2^(3/12) = 1.189.
Because log_2(6/5) =.26303 while log_2(2^(1/4))=.25, these differ by
about 15.6 cents. Now, a sequence of rational numbers approximating
log_2(6/5) is given by 1/3, 1/4, 5/19, 111/422, …
So, the next number along is 5/19 = .2631. If we write
log_2(6/5) ≈ 5/19,
this is equivalent to 6/5 = 2^(5/19); in other words, 19 just minor
thirds is approximately 5 octaves. If we use this idea to divide the
octave into 19 equal steps, then 5 of these steps differs from a just
minor third by only about .15 cents. However, the approximation of
the perfect fifth is slightly worse than in 12-tone equal temperament.
The 19-tone system was advocated by the physicist Christian Huygens
in the 17th century, and is still being explored by composers today.
(See link on my web page for a sound sample.)
Math in Music
Spring 2009
Similarly, one can base a 31-note scale on approximating the
meantone fifth 5^(1/4).
2. Expanding Just Intonation Another route that leads to an increased
number of notes in the octave is to allow rational ratios involving
prime factors greater than 5. (Recall that just intonation involves
multiples only of ratios like 3/2 and 5/4.) Benson calls just intonation
systems involving ratios up to and including p (where p is a prime
number) “p-limit systems”.
For example, we could get a 7-limit system by using the interval
of 7/4, which might be called a just minor seventh, and comes from
taking the 7th harmonic transposed down two octaves. The “major
7th chord”, which uses the root, plus a major third, fifth, and minor
7th above the root, is extremely common in classical harmony.
In section 6.1 several examples of 7-limit system are given,
together with an 11-limit system (dividing the octave into 43 tones)
used by the American composer Harry Partch. There is also a 5-limit
scale from classical Indian music, called the Sruti scale, which has 22
notes per octave (see Benson p.213). For example, the first 5 notes of
this scale have ratios 1, 256:243, 16:15, 10:9, 9:8.
Noting that 256/243 = 2^3 (2/3)^5, we see that the second note is
exactly 5 perfect fifths down from first one, while the third note
16/15 is 81/80 times the second one. This means we can describe
these notes in Eitz notation as
C0, Dflat0,
Dflat+1, D-1, D0