## somemath.htm

Some Just Intonation Math

# Some Just Intonation Math

### by Mark Nowitzky,

9/2/98

## 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

out why.

## 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

corresponding ratios:

Interval | Ratio | Example |
---|---|---|

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) |
---|---|

C | 4096 |

G | 6144 |

D | 4608 |

A | 6912 |

E | 5184 |

B | 7776 |

F# | 5832 |

C# | 4374 |

G# | 6561 |

D# | 4920.75 |

A# | 7381.125 |

E# | 5535.84375 |

B# | 4151.8828125 |

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!