Where did the concept of having 8 notes in a scale originate from? Why do notes exactly an octave apart have a common tone?
I teach beginner piano lessons, and kids are extremely inquisitive. I can tell them about music theory, but eventually their questions stretch beyond the scope of my knowlege (why are there 88 keys on the piano? 
Why do intervals like 7ths sound so odd yet octaves sound ‘matched’?)
As per usual, Wikipedia is your friend.
Well…octaves sound matched because their frequencies relate to each other in a particular way. Specifically, if you have two notes one octave apart, the higher note will have twice the frequency of the lower note.
The major scale is one of the diatonic scales, which were apparently developed by a monk named Guido of Arezzo. At least, that’s what Wikipedia says.
Notes that are an octave apart are exactly double in frequency.
With 3rds and 5ths you get a similar close relationship in frequency. A 5th is the original frequency multiplied by 3 (technically it’s an octave plus a 5th), and a 3rd is the original frequency multiplied by 5. A 7th isn’t such a low even multiple of a frequency which is why it sounds a bit odd.
Dividing up the rest can be a bit arbitrary. Indian music (as in from India, not American indians) has twice as many divisions per octave as western music, so they have notes that are haflway between notes on the scales that you are used to. Western music, at leas the core of it, evolved out of italy, which is why most of the musical terms you are familiar with are italian (piano is actually short for piano-forte, or quite-loud, since a piano, unlike a harpsichord, can be played both quiet and loud). Over the years many other cultures have influenced our music, but we still use the same notes and structure that date back to Italy.
Pianos weren’t always made with 88 keys. Some had more, some had less. 88 just evolved to be the most popular, as it covers a great range of notes.
The current pitch we use (A = 440 Hz) also evolved over time. Having a standard pitch for all instruments is a 20th century invention.
Er, that should say “quiet-loud”, not “quite loud” (and someone stole the T off of the word least). I think I need to refill my caffeine before I type any more today.
Recent thread which deals with some of this. It’s not quite as simple as the divisions of frequencies described by engineer_comp_geek - the development from basic resonances to the fully-fledged chromatic scale is a long one, with origins are in religious chant.
Never trust short Wikipedia articles 
Guidonian developments were much more to do with notation, and a way of codifying a preexisting system, than about inventing new scales.
Nitpick powers, activate! 
Technically, multplying the frequency by 3 would get you what musicians would call a “perfect 12th”, or an octave plus a fifth. For example, the note with a frequency three times that of middle C would be the G above the treble clef staff. Multiplying the frequency by 5 would get you… carry the one… a “major 17th”, or two octaves plus a major third.
I always think of the “nice” frequency ratios in terms of ratios of small numbers: The frequencies of the two notes of an octave are a factor of two apart; a perfect fifth, a factor of 3/2; a perfect fourth, a factor of 4/3; a major third, a factor of 5/4; and a minor third, a factor of 6/5. Of course, under various systems of temperament, these ratios are tweaked somewhat.
Oh boy. The answer to this question involves several thousand years of history, mathematics, philosophy, Church scandal, murder, warfare and utter chaos.
The short answer is that the modern eight-tone scales are artifacts of older tuning systems that did not give us the option of having twelve equally-spaced tones per octave.
Our tuning system goes back to Pythagoras, who was one of the earliest people in the Western world to write about harmony. Pythagoras noticed that strings whose lengths were whole-number ratios sounded pleasing together. In particular, he noticed the ratios 2:1 (what we now call an octave), 3:2 (a perfect fifth), 4:3 (a perfect fourth), and others. Pythagoras also realized that there was a big problem if you tried to construct complex harmonies across several octaves. Let’s say you tune your first note to 100Hz. An octave above that has a 2:1 ratio and is therefore 200Hz. The fifth has a 3:2 ratio, so it gets tuned to 150Hz. We really like fifths, so let’s tune a bunch of fifths above that. The next one up would be
(150 * 3/2) = 225Hz, then
(225 * 3/2) = 337.5, then
506.25,
759.375,
1139.0625,
1708.59375,
2562.890625,
3844.3359375,
5766.50390625
8649.755859375
12974.633789063
Now let’s go back and tune some octaves. We have 100, 200, 400, 800, 1600, 3200, 6400, and 12800. Notice how that last one is awfully close to our last fifth? 12800 is right next to 12974. If you’ve played piano for a while you probably know that 12 fifths should be exactly equivalent to seven octaves. But there is a slight difference. The two frequencies have a ratio of about 1.0136:1. A very ugly ratio, indeed, and extremely dissonant. To give you an example, a minor-second in the pythagorean system has a ratio of 16/15, or about 1.0666:1. But the difference we notice here is much smaller than even a minor second. This difference is called the Pythagorean comma.
The comma gives us all kinds of problems. We can’t play a bunch of octaves at the same time as we’re playing a bunch of fifths. Musical instruments, and by extension entire compositions, are limited in what they can do. The western major and minor scales consisting of eight tones between an octave are developed during the middle ages, but there is no standard frequency for the tonic. The ratios behave themselves as long as your composition does not get too adventurous.
All sorts of alternate tuning systems and instruments are invented during the middle ages and the Rennaissance. The Church regards the perfect whole number ratios as divine and any attempt to use anything else is obviously heresy. This annoyed composers who wanted to make more complex music.
Our modern notation system and the idea of the chromatic scale with sharps and flats is a vestigial product of this time, and we still use it even though there’s no reason to today. You can easily see that, depending on where you begin to calculate your ratios, C-sharp and D-flat were not actually the same thing. Instead, they were two notes separated by a tiny comma, and so could never be used together in the same composition. Our current system of key signatures reflects this; you either use C-sharp or D-flat, but never both, even though now they are the same.
So, the big discovery that gave us our modern, much more convenient system, is called equal temperament. This system abandons the perfect Pythagorean ratios, (except for the octave) and thoroughly pissed off the Church. To give an approximation of all the scales that were in common use in that period, a 12-tone equal-tempered scale was set up. This consists of twelve notes, spaced equally between an octave. To calculate the next note up, you multiply the first one by some constant. Since we want twelve steps to culminate in multiplying the first step by two, the constant we use is the twelfth root of two (about 1.059463). A 12-tone equal-tempered scale starting on 100Hz looks like this:
100Hz
105.9463
112.2461848369
118.920679725857
125.992060104395
133.483925974383
141.421280664598
149.830614276757
158.739992093495
168.179148243351
178.179584935345
188.774677594356
200
Notice that our perfect fifth isn’t so perfect anymore. Instead of exactly 150Hz, we get about 149.83Hz. This difference is so slight that most people agree it’s not even noticeable most of the time. And even if it is, the advantages of equal temperament outweight the slight dissonence.
And there’s no particular reason we have to use twelves tones, either. A 24-tone equal tempered scale can be easily constructed by using the 24th root of two instead.
So, to sum up: There are eight notes in a scale because there were eight commonly used Pythagorean ratios before we had equal temperament. This is of course why we call it an octave, even though there are twelve notes instead of eight in the entire thing. We didn’t get all those extra notes until all the various keys and tuning systems were folded into this big compromise system we use now.
Music is fun.
And, indeed, such scales are consistently used by some contemporary composers.
I’d hesitate to call it a discovery - rather, a solution. The difficulty with non-equal tuning systems is not that they sound ‘wrong’ in themselves, i.e. in the key they are contructed from, but that they restrict freedom for modulation - move from one key to certain anothers mid-piece, and you can start getting some very strange sounds. Equal temperament makes every key sound ‘the same’…and this was the very reason that some musicians disliked it. Some preferred to accept the restrictions of other tuning systems, because of the added benefit of each key having unique characteristics.
Yep, that’s exactly the point I was trying to make in a lot more words. Before we figured out equal temperament we were pretty much restricted to one key and tuning system for every thing we did.
Oh! I forgot to reccomend a truly excellent little book:
Temperament: How Music Became a Battleground for the Great Minds of Western Civilization by Stuart Isacoff.
Not one key, but a restricted number, yes. And certainly not only one tuning system, but an enormous variety of them! Even Bach didn’t adhere to equal temperament, yet finds his way around quite a few keys
I meant we were restricted to one key for a given composition. Obviously we could (and still do) invent whatever system we want for the compositions we write. But the equal-tempered system allows us to jump around between keys which closely match ones that were previously incompatible. So we can write compositions in many keys, or, even more exciting play works in completely different styles with the same instruments at the same concert!
There are different notational systems.
A note with a frequency just double another note is considered the same note, an octave higher, because plucking a string tuned to one of the notes will tend to make the other string vibrate. It also sounds that way to our ear, as it has developed.
A note three times the frequency will also be a harmonic of the first, but it is not two octaves higher–it is only 1.5 octaves. One half of three times the frequency is the note an octave lower, or just a half octave up. Because (1.5)^12 is very nearly 2^(7), if you go up a half octave each time, you will end up with twelve tones and be very nearly seven octaves higher than where you started from. Those are the twelve semitones of our chromatic scale. If you started on an F, the first seven of them (FGABCDE) are our seven notes of the A major scale, the other five are the semitones that represent sharps/flats in that scale.
It doesn’t work out exactly, it sometimes gets fudged a little, but it’s close enough for rock and roll.
Is there an award for the most deceptively simple OP question?
I think this might be a good one for Cecil to put in his column.
Well, this isn’t quite true. Like GorillaMan mentioned, Bach did not use equal temperament. Neither did Mozart. Neither did Beethoven. It’s not until the late Romantics and early 20th musicians that equal temperament gains popularity. Some caddish young classical composer remarked that equal temperament is a tuning system in which all keys sounds equally bad. Mozart was said to have explicitly forbidden anyone from playing his works in equal temperament (which was around at his time), favoring unequal temperaments which bring out certain characters from keys that are missing today.
Now, it’s not like the Baroque composers were using pure major and minor scales, but they certainly were not using equal temperament.
Inadvertently, you made an excellent point. Blues (and therefore jazz and rock and roll and all that jazz) doesn’t only originate from a Western tradition. Tunings used don’t always correspond to western systems - we can say “oh, well, it’s nearly A-flat, so we’ll call it that”, but that doesn’t do it justice. If it’s not A-flat, then it’s part of a different system.
Likewise, you can hear folk music in just about any part of Europe that thoroughly fails to get anywhere near Pythagoras, yet is very close to the chromatic scale. Go figure. (Last week, I heard a piece on BBC Radio 3 - a Swedish fiddle tune - which was introduced as using different scales in each octave.)
Can these several thousand years be condensed any?
(Or at least give us a nudge to read more on the Church and warfare bit?)
Check out the book I reccomended a few posts up. It’s pretty short and touches on all these subjects, and provides references for further reading.