MathOfTheWesternMusicalScale

There's a decent article in issue 4 of the Napalm e-zine

on Just Intonation and the western scale in general:

http://napalm.firest0rm.org/issue4.txt

It describes what makes a chord sound in tune to our ear, the

logarithmic nature of our perception of sound, and how we

went from just intonation to the less precise compromise of the

modern scale. With ajax's permission,

I've posted the text of the entire article here.

Everything You Never Wanted To Know About Just Intonation

or

Western Scales Ate My Brain

First, some introduction: according to napalm.firest0rm.org:80, "Napalm

is an e-zine devoted to computer security, with a healthy dose of music,

news, and ethics." Well, maybe we oughtta live up to that, so here's some

music theory in your ear. (Or your eye, unless you're text-to-speeching

this.)

Generally speaking, a scale is a progression of notes that takes you from

one octave to the next. (Although it can actually be much larger than

just one octave, generally we don't do that.) Notes have pitches, which

can be measured in oscillations per second, or hertz. The human ear can

perceive from about 20Hz at the low end to around 20,000Hz at the high

end. Now, it turns out that the human ear differentiates pitch

logarithmically, and that doubling any given pitch raises it an

octave. Thus the human ear has a range of a little under ten octaves.

Now it also turns out that when the ear hears any two notes in

combination, the interval, whether major or minor, is perceived to be in

tune when the ratio between the two notes approaches a ratio of two low

integers. The degenerate case would be a 1:1 ratio, which would be two

notes in unison; a 2:1 ratio, as above, is an octave. Roughly, the

intervals for the twelve-tone scale used in western music are:

1:1 unison

21:20 half step / minor second

9:8 whole step / major second

6:5 minor third

5:4 major third

4:3 perfect fourth

7:5 tritone / augmented fourth / diminished fifth

3:2 perfect fifth

8:5 minor sixth

5:3 major sixth

9:5 dominant seventh / flat seventh

17:9 major seventh / natural seventh

2:1 octave

The note that a scale centers around is called the tonic. A scale in

which the ratio of any note to the tonic is an integer ratio is called a

scale of just intonation. These scales have a very natural-sounding

quality to them.

The problem is, they're murder to implement in any stopped or fretted

instrument. The difficulty is subtle, but it means big headaches. It's

hard to explain, though, so I'll give an example.

For purposes of tuning we need a reference pitch, something all the

instruments can agree on. Usually a 440hz sine wave is used as the

reference pitch, as an A natural. Now, according to our table above, we

can calculate the pitch of any other note by setting up a simple ratio

relationship. For example, if I wanted to calculate the pitch of a

perfect fifth from an A440, I would write:

(X / 440) = (3 / 2)

and solve for X. Simple algebra, right? In the above, X comes to 660.

Let's calculate two more:

(X / 440) = (9 / 8); major second = 495

(X / 440) = (5 / 4); major third = 550

Now, assume that what you know about scale theory is right and that the

second of a second (of a tonic) is the same note (and pitch) as the major

third of the tonic. If that were true, then in:

(X / 495) = (9 / 8)

X should equal 550. This is the subtle bit, so make sure you understand

it. The above equation takes the second of 440 (495) and finds the note a

second up from it (9 / 8). I'll let you all punch that into xcalc and see

what happens.

Didn't work, did it? You got 556.875, right?

This is the problem. Any given scale of just intonation must be tuned to

a tonic, which is fine if you only want to play in one key. However, you

have to retune the instrument every time you modulate keys. As many

classical composers (and pop ballad writers) will tell you, this has a way

of limiting your expressive power.

So what's a keyboard manufacturer to do? The answer is simple: make one

note in tune, and space all the other notes equally (logarithmically

equally, anyway). This is what happens on most fretted instruments and

keyboard instruments. Now, instead of calculating pitch with integer

ratios, we just plug an interval into the following equation:

P = 440 * 2 ^ (n / 12)

where n is the number of half steps sharp you want to go (and hey, guess

what, negative numbers work as expected; (n == -3) finds the pitch of the

major sixth below A440). This is called a scale of even (or equal)

temperament, since the distance to any other note is independent of (and

consistent across) key centers.

Now, waitasec, doesn't this put everything out of tune? Well, yes it

does, but not by very much. Observe:

Note Just Pitch E.T. Pitch (approx.) Error (%)

A 440 440 0.0

Bb 462 466.16 +0.9

B 495 493.88 -0.2

C 528 523.25 -0.8

C# 550 554.37 +0.7

D 586.6- 587.33 +0.1

D# 616 622.25 +1.0

E 660 659.26 -0.1

F 704 698.46 -0.7

F# 733.3- 739.99 +0.9

G 792 783.99 -1.0

Ab 831.1- 830.61 -0.1

As you can see, we're never more than 1% out of pitch, which most people

can't hear. However, you *are* out of tune inherently, so when your

instrument then goes further out of tune you sound *really* bad. The

advantage, though, is that you get stopped instruments, which makes

composition and playing much easier.

(Of course, you'll notice I cheated, and listed a just intonation scale

that happens to be close to the 12-step equal temperament scale. Sue

me. There are also Indian raga scales and other non-standard scales that

are "just intonation" scales, but I wanted to present something

understandable.)

---

[ one thing i neglected to mention in the original article was that many classical composers wrote compositions for just-intonated instruments (wind instruments in particular). however, since these instruments couldn't re-tune to a new tonic, modulating the key of the piece created a tension; it sounded like you were still playing in the original key and wanted to return to it. at least, that's what some people say, and they're usually quite insistent that playing the piece on a JI instrument is the only way to truly hear what the composer intended. i personally don't believe that, but judge for yourself. { ajax } ]