User:Grady/Harmonic similarity: Difference between revisions

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=== Usage of the circle of fifths ===

In Western music theory, the [[Chain of fifths|circle of fifths]] is often touted as a tool for determining which chords and keys are related to each other. The circle or chain of fifths is essentially an abstraction of the idea of harmonic similarity: it first assumes octave equivalence by implying that notes within the same pitch class can be treated as equivalent, and then denotes which pitch classes are harmonically related to each other. Chord progressions often involve successive chords whose root notes are adjacent on the chain of fifths; this is viewed as a smooth harmonic movement, which can be explained by the fact that the two root notes are harmonically similar to each other. The full theory of harmonic similarity correctly demonstrates that an even smoother harmonic movement can be achieved by moving from one chord to another chord whose root note is an octave away from that of the first, although this of course results in a change of character that's so minimal that Western music theory wouldn't even consider it to be a different chord at all.

=== Interval quality ===

Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one pitch class would be fixed in place, while another pitch class would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.

As an important note, the concept of [[interval class]] is usually defined to refer to undirected interval class: for example, a major third and minor sixth would be considered the same interval class due to being octave complements. However, this theory requires establishing the concept of directed interval class, in which octave complements are separate classes. In order for this to work, we need to define intervals as being directed in general; in other words, that an upward major third and a downward major third be considered two separate intervals. The downward major third falls into the same directed interval class as the upward minor sixth, but the upward major third does not. If the upward major third can be thought of as the ratio 5/4, then the downward major third can be thought of as 4/5. In the absence of a specifier for upward or downward direction, it should be assumed that an interval points in the upward direction.

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 === Usage of the circle of fifths ===
 In Western music theory, the [[Chain of fifths|circle of fifths]] is often touted as a tool for determining which chords and keys are related to each other. The circle or chain of fifths is essentially an abstraction of the idea of harmonic similarity: it first assumes octave equivalence by implying that notes within the same pitch class can be treated as equivalent, and then denotes which pitch classes are harmonically related to each other. Chord progressions often involve successive chords whose root notes are adjacent on the chain of fifths; this is viewed as a smooth harmonic movement, which can be explained by the fact that the two root notes are harmonically similar to each other. The full theory of harmonic similarity correctly demonstrates that an even smoother harmonic movement can be achieved by moving from one chord to another chord whose root note is an octave away from that of the first, although this of course results in a change of character that's so minimal that Western music theory wouldn't even consider it to be a different chord at all.
+=== Interval quality ===
+Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one pitch class would be fixed in place, while another pitch class would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.
+As an important note, the concept of [[interval class]] is usually defined to refer to undirected interval class: for example, a major third and minor sixth would be considered the same interval class due to being octave complements. However, this theory requires establishing the concept of directed interval class, in which octave complements are separate classes. In order for this to work, we need to define intervals as being directed in general; in other words, that an upward major third and a downward major third be considered two separate intervals. The downward major third falls into the same directed interval class as the upward minor sixth, but the upward major third does not. If the upward major third can be thought of as the ratio 5/4, then the downward major third can be thought of as 4/5. In the absence of a specifier for upward or downward direction, it should be assumed that an interval points in the upward direction.