I have seen chords used where they will show a major a or minor interval with a kind of 6/4 or 6/5 fraction after it - what sort of chord is this, and when are these chords most commonly used and why?

Many thanks

Michal
^ could you possibly mean something like Am 3/5 ?
It might be figured bass, which relates chord inversions by giving you a root note and naming the intervals from that note to the other notes in the chord. Example: a chord in second inversion will have the fifth in the bass. The other two notes, the root and third, are a fourth and a sixth above that bass. This is often used for shorthand for roman numeral chord notation. For example, I 6/4 is a tonic chord in first inversion.
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sirpsycho is right, it's most likely figured bass. The roman numeral gives you the root, and the figured bass tells you the intervals above the bass note (note that the root and the bass note are not necessarily the same).

So, _ 6/4 is a triad in second inversion, that is, the fifth of the chord is the bass note. If we count up from the fifth, we see that the root of the chord would be a fourth away, and the third would be a sixth away. That isn't to say that the voicing couldn't be changed, but the important thing is that the lowest note is the fifth.

I'll give a short list of triads and sevenths as they may be notated:

Root Position: Could be 5/3, but it is common for nothing to be written, with the 5 and 3 assumed.

First Inversion (third in the bass): 6/3, but 6 might be more common, with the 3 assumed.

Second Inversion (fifth in the bass): 6/4

Seventh Chords

Root Position: 7/5/3, but 7 is more common.

First Inversion: 6/5/3, with 6/5 being more common.

Second Inversion: 6/4/3, with 4/3 being more common.

Third Inversion (seventh in bass): 6/4/2, with 4/2 being more common.

As you can see, the numbers are often reduced to whatever is necessary to distinguish one chord from another. I may have been a bit unclear on some parts, so please ask if I messed up or something confused you.
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Psychodelia, I understand the example that you have explained, but I don't quite understand the others. I would really appreciate it if you could expand on them slightly? Also I have noticed you said 6/4 is 2nd inversion, which i understood, until I noticed Sirpsycho said 6/4 is 1st inversion - is there any good websites or books that could clear it up for me, or is there any chance of you guys expanding more?

THankyou for the information so far, I actually understand the basic concept of what it is for, just not which symbol is what inversion.

Michal
It's an (imo) really dumb way of notating inversions that I had to learn for entrance exams.
Let's look at the C major chord. If you see it simply written as a C, that's as good as it being written C 5/3 (indicating that the fifth and the third [in reverse order... see: stupid] come after the note C), coming out as C E G. The 5 and 3 are always left out and are merely implied.
The omission of the 5 and 3 are what cause things to get complicated. Once again looking at our C major chord, we now look at the second inversion - E G C. This would be notated E 6, with the 5 being replaced by a 6 and the 3 being ommited as it is implied. This is not to be confused with E6, which is an E major sixth chord.
Finally, the second inversion. The second inversion of C major, being G C E, would be written as G 6/4, indicating that the sixth and the fourth are to be sounded with the G.

The fundamental flaw in this is that it's inconsistant with pretty much every other aspect of theoretical music and is relatively hard to get used to reading. In jazz notation it's not uncommon to see things like C 9/6 and single notes followed by lines of figured bass. I hate it. Good luck~
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Well, 6/4 is second inversion.

As for the others...

Triad, root position: 5/3. Makes sense: the fifth is a fifth above the root and the 3 is a 3rd above. However, 5/3 is often not written down. So if you see a numeral with no figured bass, it's a triad in root position.

Triad, 1st inversion: 6/3. The third is in the bass, so the fifth is a third above it, and the root is a 6th above it. We usually assume the 3, so 6 is all that we see.

7th chord, root position: 7/5/3. Just the intervals between the root and the chord tones. However, all that's needed is a 7, since, as we'll see, none of the 7th chord inversions contain an interval of a 7th between the bass and any other note.

7th chord, 1st inversion: 6/5/3. The third is in the bass, so it's a third up to the fifth, a fifth up to the seventh, and a 6th up to the root. Now, we can't drop the 6 when naming it, because it'd look like a triad: 5/3. We also can't drop the fifth, because it'd look like a triad in 1st inversion: 6/3. But, we can drop the three and have a unique name, so often it is labeled 6/5.

7th chord, 2nd inversion: 6/4/3. The fifth is in the bass, so it's a third to the 7th, a 4th to the root, and a 7th to the third. We can't drop either the 4 or the 3, because the resulting notation would look like a triad. So, we drop the 6, and get 4/3.

7th chord, 3rd inversion: 6/4/2. The seventh is in the bass, so it's a second to the root, a fourth to the third, and a 6th to the fifth. We drop the 6, and so we often see this as 4/2.

Note that when I say "drop" the note is not actually removed; it's just that only the numbers that are necessary to let the player know the type of chord are written. I ii7 V4/2 I6 is a bit easier than I5/3 ii7/5/3 V6/4/2 I6/3
(Slightly outdated) Electronic and classical compositions by m'self: Check 'em out
yep 6/4 is second, error on my part.
"I see my light come shining from the west down to the east
Any day now, any day now I shall be released"

Know any good teachers in NY, especially skilled in teaching ear training? Tell me