User:Grady/Harmonic similarity: Difference between revisions

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=== 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.

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 of the pitch classes comprising the interval class would be fixed in place, while the other 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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 === 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.
+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 of the pitch classes comprising the interval class would be fixed in place, while the other 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.