Not quite. They’re evenly spaced by frequency ratio. The differences is rather profound. For instance: the difference between A below middle C to the A above is 220 hertz. But the difference between A above middle C and the A above that is 440 hertz.
But a chromatic scale has 12 steps. This only has 7. Does it matter which type of scale you use or are they are derived from the same thing?
I get this.
Greek to me. Is this because the diatonic scale uses steps and half steps?
You don’t have to go across octaves for the whole-number ratios to break down; they fall apart rather quickly: e.g. a major second is 9/8 in Pythagorean tuning, and a major third is 5/4. But two major seconds, which should add up to a major third, would give you 81/64. So (in C), either C and E or D and E must be out of tune with each other.
On an even tempered scale, the frequency ratio between adjacent half steps is the 12th root of two (1.0595)
The two is because an octave is a doubling of frequency. The 12 is because there are 12 half steps in the octave.
“Concert pitch” is defined as Middle A = 440Hz, but if the piano is not there for example, the other instruments are tuned to match the piano, rather than to a specific frequency.
Long papers have been written on that subject. Here is my cliff notes version:
A “key” is the starting and ending note of the scale. Frequently a tune will end on the named note of the key in which it is written (no hard rule there though) Major keys put the half steps between the third and fourth, and the seventh and eight notes. Various minor keys put the half steps at different locations, giving a different “feel”.
The 12 note even tempered scale is a compromise that produces third and fifth intervals that are near integer ratios for any key. Many will claim that this compromise results in music that sounds equally bad in any key.
“Natural” scales have exact integer ratios between many of the notes. Chords thus do not have dissonant beating between the various notes.
To do this on a piano, however, would require that the piano be re-tuned for each key, and it would be impossible to change keys in the middle of a performance, which is common…thus the equal temperament compromise.
While descriptions of scales themselves often ends up oriented around the physics of oscillation, the ‘rules’ which make up western harmony (the chords and so on which are being talked about) aren’t rules but a description. It’s a system which evolved over many centuries. For the first recognisable vertical harmony (i.e. music constructed out of chords, with the intention that each part moves to the next at corresponding times), look to the Notre Dame School, which is the music produced in Paris in the decades around 1200. Listen to some samples here. Over hundreds of years, the use of chords, scales, etc. all evolved. The descriptions of what was going on followed suit.
If you can visualize a piano keyboard, think of it like this: there are several octaves; each one can be broken down visually by looking at the pattern of black notes–one group of two and one group of three. Concentrate on just one octave (start with a white note immediately to the right of the group of two black notes)–there will be twelve keys: five black and seven white. The white note immediately to the left of the group of two black notes is C. The seven white notes make up a diatonic major scale (specifically, C-major). The five black notes make a pentatonic scale (F# pentatonic, although you should really start with a group of three to get F# pentatonic). All twelve of them make up the chromatic scale.
So, various scales (there are many types of scales) are created by taking the chromatic scale and subtracting notes.
To get a diatonic scale, you miss out some of the 12 chromatic pitches. A semitone is the interval from one chromatic pitch and its neighbour, a tone is a leap of two. A major scale goes tone-tone-semitone-tone-tone-tone-semitone. For the C major scale on a piano, the semitones correspond to the white notes which have no intervening black notes. For other keys, the black notes are needed to keep this sequence intact.
Basically, yes. There are all kinds of crazy scales, and composers have tried to build chords based on most of them. The diatonic scale is the basic building block of Western music, so that’s where our chord vocabulary comes from.
There are only two real patterns of symmetrical scales, and only one is build on identical intervals–the whole tone scale: CDEF#G#A#C
The other symmetrical scale is often called the diminished scale. It is made up of alternating whole and half steps: CDEbFGbAbABC
Neither of these has traditionally been used to build chords.
And GorillaMan’s right. None of this stuff has much to do with science. It has to do with what sounds pleasant to our (cultural) ears.
I’ve never heard that called a diminished scale. It’s always been octatonic to me.
Jazz musicians will call it a diminished scale (either whole-half diminished scale or half-whole diminished scale).
Really? I think I probably saw it as octatonic a few times in books and the like, but certainly both usages are out there.
http://www.malletjazz.com/lessons/dim_les1.html
http://www.music.vt.edu/musicdictionary/texto/Octatonicscale.html
Just never heard it that way. It’s possible, in light of pulykamell’s post, that it is a classical vs. jazz thing. At any rate, consider ignorance fought.
I’m just amazed that I don’t recall hearing it called octatonic. I’ve seen in discussed in the context of it being an eight-note scale, but I just don’t remember seeing the word. So ignorance was fought bidirectionally. 
You guys have give me a great start with lots of links to look at. I’ve learned tons already. Once I get done reading all this stuff again, I’m sure I’ll have more questions.
Er… that should’ve been “immediately to the left”.
Just for fun, I’ve graphed out the functions for certain chords to see what their shape would be on an oscilloscope. I have a program on my computer that does it. I don’t think the typical calculator will have sufficient resolution to see the shape well. I wish I could post screenshots, but I’ll have to settle for listing some example functions for those who have their own program.
Two notes one octave apart:
y=sin(x)+1/2*sin(2x)
Major chord using perfect ratios:
y=sin(x)+4/5sin(5x/4)+2/3sin(3x/2)
Major chord using actual even-tempered scale:
g(x)=sin(x)+2^(-4/12)*sin(2^(4/12)*x)+2^(-7/12)*sin(2^(7/12)*x)
To see a minor chord, change 4/5 to 5/6, 5/4 to 6/5, 2^(-4/12) to 2^(-3/12), and 2^(4/12) to 2^(3/12).
I graphed the major chord with both perfect ratios and the even-tempered scale with different colors. With the x-axis displaying -20 to 20, they have a very nearly identical shape. By zooming out to -100 to 100 though, they start to diverge. I think I’ll play around with my tone generator to see what the audible difference is between the two.
Except that - in C major - D to E is a minor second, 10/9. (On a natural - valveless - trumpet built in C - that is, where the lowest note fits one wavelength into the pipe, and this note is C - the first D you can play is the one where you get nine wavelengths in the pipe, and the next E above that is ten wavelengths.) So that keeps C, D and E in tune with each other in C major.
Trouble arises when you want to construct a scale in D major. On a piano, you don’t get to change the tuning. Now D to E is supposed to be a 9/8 ratio, but if it was tuned correctly for C major it’s now appreciably out of tune. (On a natural trumpet in D, the first E you can play is the nine-wavelengths note. Put two skilled trumpeters next to each other with a C trumpet and a D trumpet and get them both to play D, then E, and they will not be in tune with each other.)
A major scale built on whole-number ratios goes like this:
C - 9/8 - D - 10/9 - E - 16/15 - F - 9/8 - G - 9/8 - A - 10/9 - B - 16/15 - C. That is, the frequency of the D is 9/8 of the frequency of the low C, and so on. Multiply all the ratios together and they come to exactly 2, i.e. the high C is twice the frequency of the low one.
(Aside to Diogenes - you and I both know, of course, that A to A is not a minor scale, unless you’re going downhill. 
)
First of all, 10/9 is a minor tone, not a minor second. The rest is exactly my point: the intervals within each octave are not consistent. Note, by the way, that in your example, D to F, F to A, and A to C are out of tune. You could fix some of that by making G to A a minor tone (and A to B a major tone), but then the fifth from D to A would be out of tune (D to F would still be out of tune, rendering the ii chord rather hideous).
:smack: I’ve only known that for thirty-odd years!
Yes. Which, of course, is the point of equal temperament: it’s a compromise that leaves everyone unsatisfied in that now no interval except for exact octaves will be quite in tune - major thirds aren’t 5/4, minor thirds aren’t 6/5, even perfect fifths aren’t 3/2 - but at least every interval is equally compromised, and C - E is no more (and no less) out of tune than F - A.
Thirty very, very odd years.
I’m fairly certain that we’re arguing the exact same point. I was merely pointing out that friedo’s assertion that whole number ratios break down over the course of octaves was not entirely correct. (I suspect he may have been referring to the overtone series, which does yield out of tune partials (particularly the 7th, 11th, 13th and 14th), but I won’t presume to speak for him).