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
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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.)
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[ 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 } ]