This might be a silly question, but why can't we spell a diminished 3rd or an augmented 5th using only the notes in a major scale?
I think what I'm really confused is that, for example, A to F has 8 half steps, and an augmented 5th is required to have 8 half steps, so doesn't that qualify A to F as an augmented 5th?
I think what I'm really confused is that, for example, A to F has 8 half steps, an augmented 5th is required to have 8 half steps, doesn't that qualify A to F as an augmented 5th?
Well, obviously not; otherwise you would not be here asking this question. Why not? Because there are two systems for describing the size of an interval, and (in the twelve-tone system) they don't have a one-to-one correspondence. That is, the augmented fifth is not the only interval that comprises eight half steps, and as it happens A to F is a different eight-half-step interval.
One of the systems is simply to count the number of half steps. The other system is to name the interval with a quality and an ordinal number, where the quality is major, minor, perfect, augmented, or diminished, and so on, and the ordinal number is, well, an ordinal number (second, third, fourth, etcetera) except that we use "unison" instead of "first" and "octave" instead of "eighth" -- and "octave" even comes from the Latin word meaning "eighth."
We can see that these systems don't have a unique mapping by considering some examples: a major third has four half steps, but so does a diminished fourth. A minor seventh has ten half steps, but so does an augmented sixth.
You might think of half-step counting as measuring the size of an interval and the other system as providing names for the intervals. If you do, you will probably get confused because there are some names that are actually half-step-counting designations in disguise. These names do not comprise a quality and an ordinal.
The first of these, of course, is "half step" itself. Another is its sibling "whole step." In much of the English-speaking world, the preference is for the equivalent "semitone" and "tone," and when you consider these, you might recognize that "tritone" also comes from this system, because it means "three tones," or six semitones. There is also "ditone," which was used in medieval times to designate the interval between (for example) C and E because the quality-ordinal system had not yet come into being. This system counts the number of whole steps rather than half steps, but it's still a measure of the interval's size in absolute terms.
The quality-ordinal system, on the other hand, is more arbitrary than absolute. Even if you consider only the third, fourth, fifth, and sixth and ignore qualities beyond augmented and diminished (that is, doubly augmented, doubly diminished, etc.) you have fourteen intervals:
Yet the smallest of these is two half steps and the largest is ten, so there are only nine unique interval sizes among these fourteen intervals. There must therefore be some overlap; some of these quality-ordinal names must have the same size as others. We can see this by adding the sizes to the above list:
A to F is not an augmented fifth, but A to E♯ is, and so is B♭♭ to F. That's because of the heretofore unmentioned rule for determining an interval's ordinal: count the letters only, inclusively, ignoring the accidentals. So A, B. C, D, E, F: six letters, it must be some kind of sixth. What kind of sixth is it? Count the half steps to find out.
Conversely, it's impossible to know the quality-ordinal designation of an interval when all you know about the interval is its size in half steps. You need to know both its size and the letters, or, equivalently, you need to know how it is "spelled."
So why do we have this confusing state of affairs? The quality-ordinal system arose from the diatonic scale. As described in Aaron's answer, this gives us all the perfect, major, and minor intervals but only one augmented interval, the augmented fourth, and one diminished interval, the diminished fifth.
The other augmented and diminished intervals arose with chromatic alteration. In the eleventh century, there were only eight pitch classes, A, B♭, B, C, D, E, F, and G. With these pitches you can form a C major scale or an F major scale, so you can get the same set of major, minor, and perfect intervals plus the two tritones.
In fact, the quality designations largely arose from this (see my answer to the question Confusion about major and minor second intervals). For example, E to G is a third, and F to A is also a third, but E to G is smaller, and F to A is bigger. The Latin words for "smaller" and "bigger" are "minor" and "major."
This system of eight pitch classes was extended with chromatic alteration for several reasons, including for example the use of C♯ to provide a leading tone when harmonizing the E of a descending D minor scale. That C♯ gives you a diminished fourth with F. Similarly, if you have one part descending diatonically to E, and against the F you have an ascending part with its D chromatically raised to D♯, you have a diminished third or augmented sixth. This leads us to the answer to your question:
Why can't we spell a diminished 3rd or an augmented 5th using only the notes in a major scale?
Because diminished thirds and augmented fifths (and all diminished and augmented intervals apart from the diminished fifth and augmented fourth) arise only as the result of chromatic alteration to the major scale.
In the end, that's simply a "just so" answer, so it's not particularly satisfying, but I hope the answer to your follow-up question is more helpful.
For simplicity, consider C major: C D E F G A B (C)
Now let's look at all of the possible ascending intervals made using notes within the scale, extended through two octaves.
The intervals to the right of the P1 diagonal are the intervals formed by ascending from the row header pitch to the column header pitch within the first octave of the scale.
The intervals to the left of the P1 diagonal are the intervals formed by ascending from the row header pitch to the column header pitch across the octave and into the scale's second octave.
| C | D | E | F | G | A | B | (C) | |
|---|---|---|---|---|---|---|---|---|
| C | P1 | M2 | M3 | P4 | P5 | M6 | M7 | P8 |
| D | m7 | P1 | M2 | m3 | P4 | P5 | M6 | m7 |
| E | m6 | m7 | P1 | m2 | m3 | P4 | P5 | m6 |
| F | P5 | M6 | M7 | P1 | M2 | M3 | A4 | P5 |
| G | P4 | P5 | M6 | m7 | P1 | M2 | M3 | P4 |
| A | m3 | P4 | P5 | m6 | m7 | P1 | M2 | m3 |
| B | m2 | m3 | P4 | d5 | m6 | m7 | P1 | m2 |
There are no d3 or A5 intervals created between any two ascending notes in the C major scale.
The chart, and the conclusions from it, would remain the same no matter which major scale was chosen.
Here is the same chart using scale degrees rather than absolute pitches.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | (8) | |
|---|---|---|---|---|---|---|---|---|
| 1 | P1 | M2 | M3 | P4 | P5 | M6 | M7 | P8 |
| 2 | m7 | P1 | M2 | m3 | P4 | P5 | M6 | m7 |
| 3 | m6 | m7 | P1 | m2 | m3 | P4 | P5 | m6 |
| 4 | P5 | M6 | M7 | P1 | M2 | M3 | A4 | P5 |
| 5 | P4 | P5 | M6 | m7 | P1 | M2 | M3 | P4 |
| 6 | m3 | P4 | P5 | m6 | m7 | P1 | M2 | m3 |
| 7 | m2 | m3 | P4 | d5 | m6 | m7 | P1 | m2 |
And here is the complete list of intervals that can be found within the major scale:
P1 m2 M2 m3 M3 P4 A4 d5 P5 m6 M6 m7 M7 P8.
Because those intervals don't occur between the notes of a major scale.
You might as well ask 'Why can't we spell 'frog' using the letters 'abcde'.'
To your follow-up question: Yes, A to F is 8 half-steps. But it is also 6 letter-names, A, B, C, D, E, F. And spelling matters when naming intervals. 6 letters, it's going to be some sort of a 6th. That's just how naming intervals works.
If I can put the cart before the horse for a bit, I think there's another way of conceptualizing this.
The Western harmonic system is, simply put, a system of stacking thirds one on top of the other. And these stacked thirds, especially as music history progressed, were typically taken from the current scale collection.
So it's interesting that our Western harmonic system only stacks major and minor thirds and never diminished or augmented thirds; see Triads with thirds that aren't major or minor? The simple reason is that those intervals didn't appear in the major scales that these composers were using.
There are some problems with this answer: it's anachronistic at points, and it simplifies the relationship between scale and chord building. But the fact is that, if the major scale had a diminished third, our theory of Western harmony would have developed very differently than it did.
I think what I'm really confused is that, for example, A to F has 8 half steps, and an augmented 5th is required to have 8 half steps, so doesn't that qualify A to F as an augmented 5th?
No, it doesn't qualify. A to F is a sixth. As spelled it's a minor sixth, but if you add sharps or flats to either pitch, you can get other qualities of sixth.
Probably you just need to read up on how musical intervals are "spelled." The Wikipedia page on intervals is a good review. In a nutshell, you first get the number part of the intervals by the spacing of the letters (ex. A to F is six steps, you count the A as 1), and then you get the quality, in this case minor, from the number of half steps.
You did count the half steps correctly: 8 half steps. A to E♯, an augmented fifth, is 8 half steps and A to F, a minor sixth, is also 8 half steps. When two different intervals share the same number of half steps you can call them enharmonic equivalents.
Choosing one or another enharmonically equivalent intervals does matter. Some reasons to choose one or another are simplicity such as using the tones of the key signature, or to visually show step-wise voice leading such as F descending to E which would be a space/line change on the staff rather than E♯ to E♮ which would remain on the same space/line. These are not a complete list of reasons. Suffice to say that with the study of harmony there are certain conventions about how to "spell" pitches for scales, chord, intervals, etc.
Why can't we spell a diminished 3rd or an augmented 5th using only the notes in a major scale?
All the thirds of the major (or diatonic) scale are either minor thirds or major thirds.
All the fifths of the major scale are either perfect fifths or diminished fifths.
One way to "prove" that is by just listing out all the scale degrees by thirds or fifths...
If you look at those two series, all scale degrees are given along with the thirds or fifths either above or below each degree, and you will notice all the thirds are minor or major and all the fifths are perfect or diminished. You can't make any other movement by third or fifth and stay within the scale.
There are two ways of specifying a musical interval:
Recall that the C-major scale, in terms of semitones above the tonic "C", consists of the notes:
If you enumerate all possible intervals between notes in the C-major scale, then you get (up to an octave):
You'd get the same results with any major scale, but I chose C-major for its simplicity in not using sharps or flats in its note names.
Point is, there are only certain intervals that you can make using the notes of the major scale, and these just don't include the 2-semitone (diminished) third or 8-semitone (augmented) fifth.
You can, however, construct such intervals using notes of the chromatic scale. For example:
Which raises the obvious question: What difference does it make whether I call a note “E” or “F♭”? Aren't they the same thing? But that's been discussed elsewhere on this site.
