There's an area called diatonic set theory that deals with the question you're sort of getting at -- why are fifths so important?
The most important reasons are, more or less: --A perfect fifth is seven semitones, and seven is coprime with twelve (that is, they don't share any factors). As a result, we can't divide an octave equally into fifths (unless we exhaust the whole circle), the way we can divide it into four diminished thirds. This lets us generate reasonably-sized scales based on a single interval; we can create a seven-pitch diatonic based just on the perfect fifth, but if we wanted to make a scale out of repeating augmented thirds we wouldn't get too far. --The perfect fifth builds a diatonic scale within the chromatic scale such that, if you keep following the diatonic scale, eventually you get to a fifth that's too small -- in C major, the diminished fifth between B and F. There's only one of these, though -- and because there's only one of them, we can use it to figure out which key we're in aurally. When we hear that interval, we automatically know it's between scale degree 4 and scale degree 7 (whereas with, say, a major third, there are tons of possibilities) -- and therefore we automatically know where 1 is. That's why a dominant seventh points to its key so strongly; it has both of the pitches from that key's unique interval. Supposedly, IV and V sound less stable than I for this same reason -- because IV has the sd 4 from the tritone, and V has the sd 7.
(There're other criteria dealing with the way the diatonic pitches are spaced among the chromatic pitches, with how we spell intervals and what sizes are allowed, etc, etc, but these are the big two for this conversation.)
As a result -- or so we theorize -- it's easy for us to organize information in a seven-pitch diatonic within a twelve-pitch chromatic, and to build that diatonic, we move by fifths. However, there are other sizes of diatonic and chromatic scales that work, too. More to the point, caring about the location of tonic is itself a culturally-constructed value. These mathematical benefits don't speak to the 7d/12c being intrinsically superior or more natural; it just means that system contains certain features we're culturally-disposed to exploit.
If we accept that certain intervals are more pleasing in the absence of cultural context, we're necessarily making a value judgment about music without actually dealing with the music itself. When you say that certain mathematical ratios are more "pleasing" than others, you're writing off Debussy and Bartok as less "pleasing" in their raw materials than Natalie Imbruglia. That's what Schenker was doing; his writings are meant to further the perception of common-practice tonality is intrinsically superior to everything else. Unless you're prepared to write off everything else, though, it's dangerous ground.
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timeasmymeasure
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Date: 2010-10-19 01:09 am (UTC)There's an area called diatonic set theory that deals with the question you're sort of getting at -- why are fifths so important?
The most important reasons are, more or less:
--A perfect fifth is seven semitones, and seven is coprime with twelve (that is, they don't share any factors). As a result, we can't divide an octave equally into fifths (unless we exhaust the whole circle), the way we can divide it into four diminished thirds. This lets us generate reasonably-sized scales based on a single interval; we can create a seven-pitch diatonic based just on the perfect fifth, but if we wanted to make a scale out of repeating augmented thirds we wouldn't get too far.
--The perfect fifth builds a diatonic scale within the chromatic scale such that, if you keep following the diatonic scale, eventually you get to a fifth that's too small -- in C major, the diminished fifth between B and F. There's only one of these, though -- and because there's only one of them, we can use it to figure out which key we're in aurally. When we hear that interval, we automatically know it's between scale degree 4 and scale degree 7 (whereas with, say, a major third, there are tons of possibilities) -- and therefore we automatically know where 1 is. That's why a dominant seventh points to its key so strongly; it has both of the pitches from that key's unique interval. Supposedly, IV and V sound less stable than I for this same reason -- because IV has the sd 4 from the tritone, and V has the sd 7.
(There're other criteria dealing with the way the diatonic pitches are spaced among the chromatic pitches, with how we spell intervals and what sizes are allowed, etc, etc, but these are the big two for this conversation.)
As a result -- or so we theorize -- it's easy for us to organize information in a seven-pitch diatonic within a twelve-pitch chromatic, and to build that diatonic, we move by fifths. However, there are other sizes of diatonic and chromatic scales that work, too. More to the point, caring about the location of tonic is itself a culturally-constructed value. These mathematical benefits don't speak to the 7d/12c being intrinsically superior or more natural; it just means that system contains certain features we're culturally-disposed to exploit.
If we accept that certain intervals are more pleasing in the absence of cultural context, we're necessarily making a value judgment about music without actually dealing with the music itself. When you say that certain mathematical ratios are more "pleasing" than others, you're writing off Debussy and Bartok as less "pleasing" in their raw materials than Natalie Imbruglia. That's what Schenker was doing; his writings are meant to further the perception of common-practice tonality is intrinsically superior to everything else. Unless you're prepared to write off everything else, though, it's dangerous ground.