If you were teaching a student Etude #22 in Vic Firth's, The
Solo Snare Drummer, Advanced Etudes and Duets, measure
#31 where you have a ratio of 7:3 in 2/4 time, what would
be your approach to teaching that?  Your explanation?

I've included the measure rendered with Sibelius,
although in the original, the ratio is 7:3, 7 1/16th notes
in the time it takes to play 3 8th notes. 

I've always been of the mindset that you have to just feel stuff like that, using checkpoints to identify whether or not you're hitting those marks. In this case, I'd say to make sure that the player has a clear understanding of 1 and the upbeat of 2 and aiming for that upbeat when metering out the 7:6.

We do it all the time with things like sixteenth note ninelets in place of eight that span two 1/4 notes or perhaps 8th note quintuplets that also span two counts. We encounter triplets much earlier in our development as percussionists, starting to learn how to subdivide and evenly space an odd grouping of notes. I've attached another good example of this in practice.

In this is example, I would have the player focus on making sure that the 1  2 & (3) & 4 was in time, while also focusing on evenly spacing out the quintuple rhythms. You're not able to always feel every down beat, as the second quintuple grouping straddles the down beat count 3, but if the accented pattern is in time, it shouldn't matter. With yours, I think beat 2 become less important as long as they have a strong sense of 1  (2) &.

Just feel it :)

Leave it to Bryan to dissect this topic in a comprehensive way!

My suggestion: I would have a student play 6 16th notes in the time that it takes to play the Septuplet. Starting on the RH, it of course ends on the RH on that 'upbeat of two' 8th note. From there, have the student think about trying to play one more note by thinking about landing on the LH.

Create a loop exercise out of it and with your hand claps or stick clicks, catch count one and the upbeat of 2. Stir in generous amounts of metronome subdividing the 8th note.

Let us know how it goes!

Something along these lines, perhaps?

To add my 2 cents, I personally like keeping tuplet ratios within the same note value  (7 1/16ths to 6 1/16ths) rather than going from one value to another (7 1/16ths to 3 1/8ths).

As for exercises, you may also try having the student play a 16th note double-paradiddle off the right, which would, of course, end on the left.  Play that a couple a times.  This will give the student the feel of a group of 6 16th notes that start on a downbeat on one hand and ends on the upbeat on the opposite hand, providing a sense of regular time within the two landmarks: 1 and the & of 2.

Then, the next pattern would be the hand-to-hand septuplet within the same space, also starting on the right hand and, of course, ending on the left.  The sense of timing gained from the first pattern, as well as the starting and stopping on opposite hands, should help facilitate the transition from a group of 6 to a group of 7.

I hope you find that helpful, as well.

EDIT: Yes, I just realized (because I didn't look at it before) this is similar to Bryan's exercise.  My version, however, stays within 2/4, which is the same time signature as the piece you're teaching.  Both versions should accomplish the goal in two very slightly different, yet similar ways.  I say, combine the two!  I think staying within 2/4 and playing one pattern immediately after the other might help retain the timing and feel since there is a lack of rests.  Then, playing Bryan's version with the rest on 4 means the student will have to play the septuplet independently, without a crutch.

No age is above a helpful word with the right amount of syllables. If you tell a kid to fit ";opportunity"; in one beat, they'll have 5lets down in no time!

In this case, try giving them a seven syllable word (";na-no-tech-no-log-i-cal"; ) or phrase (";watch out for that butt-er-fly";) with you giving heavy beats on the [1] and the [&]. This works really well, at least with a slower tempo. Maybe introduce it like this at half tempo and climb your way up?

I guess I just think on a much simpler wavelength, hope it helps :)

[quote author=Bryan Harmsen link=topic=4318.msg22554#msg22554 date=1334463352]
Something along these lines, perhaps?
[/quote]

That would be it! Thanks for ";visualizing"; (?) it!

[quote author=R Hicks link=topic=4318.msg22578#msg22578 date=1334609548]
In this case, try giving them a seven syllable word (";na-no-tech-no-log-i-cal"; ) or phrase (";watch out for that butt-er-fly";)
[/quote]

I'm with Ralph, simply due to the ease of introduction to younger students but after the student has mastered the phonics, I would introduce the explanation of the ratios. 

My colleagues and I do that same when we're first introducing odd tuplets.  We've taken to using ";hippopotamus"; for quintuplets and ";Alexander Hamilton"; for septuplets.  It definitely makes the lesson more fun and accessible.