Clearing up Confusion about Proportions

I know that some people may be having problems getting their heads around all this proportion talk and so I hope this may clear it up somewhat.
In this talk about proportions we are really talking about three similar ideas - as they say, "same, same, but different."

On the first level when we speak of these ratios we are simply referring to length of strings and the pitch that those strings would sound if they were plucked.
So if I have a string that is 6 inches long and I pluck it, it will sound a specific pitch. If I have a string 3 inches long and pluck it, it will sound the same pitch as the other string, but an octave higher - (the difference between mens and women's voices generally). Thus we could say that the ratio of the two strings - their string lengths was 2:1 and that same ratio (2:1) could represent the sound of the two strings plucked simultaneously - the interval of an octave. If one wanted they could do an experiment with rubber bands or strings. Since for the medieval musician, music was thought of as the art of measurement it would be logical for them to conceive of pitch relationships like this. When we discussed the acceptable simultaneities of pitches, we were defining those simultaneities - octaves, fifths and fourths in terms of their ratios and suggesting that these simultaneities were deemed acceptable because they could be expressed in simple superparticular - (x+1:x) - ratios.

This is one sense of the ratios.

Another sense is the physical sense - if I pluck that same 6 inch string the whole thing will vibrate. However, at the same time that whole string will vibrate each half of it vibrates, each third of it vibrates, each fourth of it vibrates and so on ad infinitum. Thus, we could concieve of the relationships of the parts of the string vibrating in ratios. Each half to the whole (1:2); each third to the whole (1:3) etc.

Finally, in the sense of the isorhythmic motet we have proportions of lengths of musical sections - self-contained, internally coherent hunks of music. Here we're talking about durations: we could be referring to beats, seconds or whatever. So the first section of music might be 60 seconds long and the next 30. Thus we could say that the sections of music were in a proportion of 2:1. There is a rhythmic component of this, known as mensuration, that complicates the matter.

You're welcome to post follow-up questions to this thread, and we can discuss it more in class on Monday.