Some Just Intonation Math
Some Just Intonation Math
by Mark Nowitzky,
Inspired by this question (excerpted from an email):
Date: Wed, 02 Sep 1998 14:10:36 -0500
Subject: b # c natural
…asked [whether] in the Just intonation system a B sharp is the same
as a C natural.. I say no.. but still don’t totally understand the
system… my math is not the best!! …
The short answer:
The short answer is as expected: In just intonation, B sharp
is not the same as C natural.
It’s even worse than that. Sometimes D is not the same as D. (Yes;
those are both the same letter, “D”.) But you’ll have to read on to find
Background for “the long answer”:
The math behind all this is based on a discovery, attributed to
Pythagoras, a zillion or so years ago. The discovery was that two notes,
when played at the same time, sounded pleasant if they were mathematically
related by a ratio of small integers. In Pythagoras’ case, he was dealing
with something like strings being plucked.
If you have two strings, with one string being half the length of the
other, the shorter string will sound exactly one octave higher (provided
everything else is the same, such as the thickness and tension of the
strings). So the ratio in this case is 1/2 (or 2/1, depending on how you
want to look at it). Here’s some common musical intervals, and their
|Octave (also known as a “perfect eighth”)||2/1||from C, to C an octave higher|
|Fifth (also known as a “perfect fifth”)||3/2||from C to G|
|Third (also known as a “major third”)||5/4||from C to E|
The physics of it is that as you halve the length of the string, you
double the frequency at which it vibrates (i.e., it vibrates twice as
fast). If you’re using a wind instrument instead of a string instrument,
you would halve the length of the tubing to double the frequency. So, for
example, trumpets, which play about an octave higher than trombones, are
about half the size of trombones (half the length, that is).
Frequency is usually measured in “Hertz” (abbreviated “Hz”), which is
the same thing as measuring in “cycles per second” (abbreviated “cps”).
And now, “the long answer”:
Okay, with all that background, here’s why C (natural) is not the same
as B sharp. Presume we start at a relatively high C (I’m “arbitrarily”
choosing 4096 Hz, which is a high power of 2, to keep the numbers from
getting too weird). In the following, I start at C 4096 Hz, and go thru
the “cycle of fifths”, staying within a one-octave range. From the
interval table above, to go up a fifth, the frequency is multiplied by
3/2. So, the G above C is 4096*3/2, which is 6144 Hz. (By the way, for
the notationally challenged, typical computer nerd notation is to use “*”
for times, and “/” for divide.)
To go up another fifth, multiply by 3/2 again. So D is 6144*3/2, which
is 9216 Hz. But then to stay within the same octave range (from C 4096 Hz
to C 8192 Hz), bring the D down an octave, by halving its
frequency: 9216/2 = 4608 Hz.
Anyway, here’s the whole cycle, from C (natural) to B sharp:
|Note||Frequency (in Hertz)|
Ah hah! The starting note (C) is 4096 Hz, but the ending note (B#)
is 4151.8828125. Close, but no cigar. (I suppose a Clinton joke
could be added here. Feel free to add your own.) In this case, the C is
a little flatter (add another joke) than the B#.
Okay, fine, but when is a D not the same as a D?:
This question is actually easier. In this example, we’re starting with
C 1800 Hz, and working our way to D by two different paths:
|Do this||To get this|
|Start first path:||C 1800 Hz|
|*2/3||F 1200 Hz|
|*2/3||Bb 800 Hz|
|*5/4||D 1000 Hz|
|*2/1||D 2000 Hz|
|Start second path:||C 1800 Hz|
|*3/2||G 2700 Hz|
|*3/2||D 4050 Hz|
|*1/2||D 2025 Hz|
Notice that D 2000 Hz is not equal to D 2025 Hz. The second D is
a little sharper!
But does this occur frequently in music? I would argue that it happens
all the time. An extremely common chord progression is ii, V, I.
In the key of C major, I’d tune it a little something like this:
|D minor||G Major||C Major|
|A 3000 Hz||B 3375 Hz||C 3600 Hz|
|F 2400 Hz||G 2700 Hz||G 2700 Hz|
|D 2000 Hz||D 2025 Hz||E 2250 Hz|
See how when going from the first chord (D minor) to the second chord
(the G Major), the bottom note (D 2000 Hz) has to move up a little
(to D 2025 Hz).
I’ve rambled enough:
Any questions? Thanks!