Time signatures are not fractions. Thus 34is pronounced “three four time,” not “three fourths” or “three quarters time”; similarly 68is pronounced “six eight time,” not “six eighths time.”
Time signatures answer the question: “How many of what?” Time signatures consist of two, independent numerical labels, which we will call “numerator” (the top number, which enumerates the counted objects) and “denominator” (the bottom number, which denotes which objects are being counted). Admittedly this can be confusing, since fractions also have numerators and denominators. The crucial difference between a fraction and a time signature is that whereas fractions quantify proportions, time signatures count things, namely beat units. (A fraction is unitless; but counts have units.) Consider: “3 apples” is not a fraction; it is nonsense to say “3 appleths of an orange.” It simply means “There are three objects, namely apples.” Likewise, 34 simply means “There are three beats in the bar, namely quarter-note beats.”
Time signatures are not proportions (in and of themselves), but can be proportional (one to another). The time signature 24means “There are two quarter-note beats in the bar.” Importantly, it does not intrinsically specify how large or small those quarter-note beats are, just that they are quarter-note beats and not eighth-note or half-note beats. Thus 22 may be equivalent with 24 if we arbitrarily define the half-note beat length of the former to be exactly the same as the quarter-note beat length of the latter. The proportional relationships between the different note values only hold within a given time signature. In other words, half notes in 24 have twice the duration that quarter notes do in 24, but not necessarily twice the duration of quarter notes in other time signatures. The proportions between notes need not apply across different time signatures. Thus, it is possible to have two pieces in precisely the same tempo, even if one is in 22 and the other is in 24—all we have to do is arbitrarily define the 24 quarter note as equal to the 22 half note. However, time signatures often change within a single piece; and if there is no explicit tempo indication (Allegro, Adagio, etc.) for the new time signature, there would be no way to know its tempo unless we assume by convention that some proportion between notes—for example, that half notes have twice the duration that quarter notes have—is true across the time-signature boundary. Such an assumption would imply that the 24bars are half as long as the 22bars, or, equivalently, that the tempo of 22is twice as fast as the tempo of 22. In this way we can see that although time signatures are not themselves proportions, they can be in proportion to other time signatures if and only if we assume or the score stipulates that the proportional relationships between at least some note values is the same for two given time signatures. (For certain styles and periods of Western music, the proportional relationship between time signatures is conventionalized.)
Continue to the article on Meter.