What is the meaning of a sixths, thirds, and a octave?

A 3rd is 4 semitones away from a note, or 3 notes away.
A 6th is 9 semitones away from a note, or 6 notes away.
An Octave is 12 semitones away from a note, or 8 whole notes away. An Octave up from a note, means it has doubled in frequency, and octave down means that the frequency has halved.

So, for example, on C , one third up would be E, and one third down, would be A.

C' E' C' A
One 6th up would be A, and one 6th down would be E

C' A' C' E
One Octave up would be C, and one down would be C again.

C' C'' C' C


N.B Pedants, yes, I have assumed they mean in a Major Scale, and yes, I am aware it varies according to the type of scale

Thanks to Solari (aka Lala) for producing these noticons. Click this text to find out more!

Oh thank you. Sounds simple now :]

These are intervals - i.e the distance between two notes

When you first start to learn about intervals it's probably easier to work them out using the letter names of the notes, and use counting semitones when you get to a more advanced stage of working out intervals.

so, the distance between the two notes C and E will be a third because there are three letter names - C, D, E - always remember to count the lower note of the interval and the upper one, and I always teach to count from the lower note. Between A and C is also a 3rd (A, B, C - remember you are counting from the lower note).
Between C and A will be a 6th (C, D, E, F, G, A - 6 notes), and between C and C will be an octave (C, D, E, F, G, A, B, C - 8 notes).

If all you want to know about the interval is the number part of the name, this is all you need. Thirds and sixths have four other qualifiers that you will meet with varying frequencies: major, minor, augmented and diminished. The common ones are major and minor. The seconds, thirds, sixths and sevenths above the tonic in the major scale are all major. If you raise the lower note or lower the upper note by a semitone, a major interval becomes minor; by two semitones and it becomes diminished. If you enlarge a major interval by a semitone, it becomes augmented. In the long run, this is easier than counting semitones, as the following illustrates:

C-E: major 3rd, 4 semitones;
C-Eb: minor 3rd, 3 semitones;
C#-Eb: diminished 3rd, 2 semitones;
C#-Ebb: doubly diminished 3rd, 1 semitone;
C-E#: augmented 3rd, 5 semitones;
Cb-E#: doubly augmented 3rd, 6 semitones.

You can see from the above that, in principle, taking all the notes from Cbb to Cx and from Ebb to Ex, a third can be anything from 0 to 8 semitones.

The table is slightly different when you start with a perfect interval (unison, 4th, 5th and octave etc. above the tonic in a major scale): the augmented interval is a semitone larger then the perfect one, the diminished interval a semitone smaller.

C-G: perfect 5th, 7 semitones;
C-G#: augmented 5th, 8 semitones;
C-Gb: diminished 5th, 6 semitones, etc.

The extreme intervals are rare, but I have come across a doubly augmented unison (Fb to F#) in Brahms, and a doubly diminished 4th in Elgar, both melodic.

Careful here. The interval of a third would be two 'notes' higher or lower than the original note, and a sixth, five.

Also, an octave is comprised not of eight whole tones, but six (or five tones and two semitones). This is obviously different to the number of notes in a one-octave scale.

I'm sure you know what you mean, but it is very important to be precise and accurate when trying to explain a new concept.

Exactly. This is why I said that counting semitones (if you're going to use that method) is best left til a bit later otherwise you get very confused. C (lower note) to E (upper note) is always, always a 3rd of some kind whether the two notes are sharps/flats/double flats etc. When I start teaching the major/minor/augmented/diminished intervals I use kenm's method as I think it's the best, but if you are at the stage of asking what a 3rd, 6th or octave is then I guess you're right at the very beginning of learning about intervals.

I agree. Working out intervals by counting semitones is to almost entirely remove the all-important context of key and (in my opinion) should only be used to double-check. It can also, as you say, obfuscate the process. The whole point of the terms 'diminished' and 'augmented' is, to my mind, to define intervals which deviate from how they would appear within a particular major or minor tonality (assuming the lower note of the interval to be the tonic). (Yes, of course there is the case of atonal music, but that isn't really relevant at this level.)

too - it seems crazy to me that it's so often suggested that people START by counting semitones. As well as the fact that a 3rd can be a huge variety of different semitones, it also ignores the fact that the same apparent interval can be called by a different name even though it has the same number of semitones. Perfect way to completely confuse a student, and it always infuriates me when books state this as the basic method.

Part of the problem is that we no longer make much of the two sorts of semitones: diatonic (e.g. B-C) and chromatic (e.g. B-B#). In equal temperament they are the same size, but in other tuning systems they are different. In this context you will find that two intervals with the same total number of semitones but with different names have different numbers of the two sorts, e.g.

C-E, maj. 3rd: 2 chromatic (C-C#, D-D#), 2 diatonic (C#-D, D#-E)
C-Fb, dim. 4th: 1 chromatic (C-C#), 3 diatonic (C#-D, D-Eb, Eb-Fb).

If you spell the C# as Db, the sub-totals come out the same. If you spell the Eb of the dim. 4th as D#, the last semitone (D#-Fb) is neither diatonic nor a normal chromatic but a strange compound of + 2 diatonic - 1 chromatic.