Latitude: Difference between revisions

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'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.

Latitude allows the characterization of, i.e. different "flavors" of thirds, with respect to a fifth that might be tempered, as opposed to in terms of absolute interval ranges that may be misaligned with the intervals' harmonic function. For example, one might describe 3\13, 4\13, and 5\13 in 13edo based on their size as a [[minor third|subminor third]], a [[major third|submajor third]], and an ultramajor third, but that doesn't reflect their function in triadic harmony in 13edo. Using latitude, one can see that their latitudes are -22.5°, 0°, and 22.5°, and from that they may be assigned the labels subminor, [[neutral third|neutral]], and supermajor, which better reflect their role.

Latitude allows the characterization of, i.e. different "flavors" of thirds, with respect to a fifth that might be tempered, as opposed to in terms of absolute interval ranges that may be misaligned with the intervals' harmonic function. For example, one might describe 3\13, 4\13, and 5\13 in [[13edo]] based on their size as a [[minor third|subminor third]], a [[major third|submajor third]], and an ultramajor third, but that doesn't reflect their function in [[triadic harmony]] in 13edo. Using latitude, one can see that their latitudes are -22.5°, 0°, and 22.5°, and from that they may be assigned the labels subminor, [[neutral third|neutral]], and supermajor, which better reflect their role.

They additionally allow the generalization of the triadic concept to "axis" intervals other than the fifth. For example, if one were to make a harmonic system using 5/3 as the "axis" rather than 3/2, chords built within a range of 5/3 would contain 3:4:5 and 12:15:20, and the medial intervals of the major third and perfect fourth would have a latitude of ±11.4°

They additionally allow the generalization of the triadic concept to "axis" intervals other than the fifth. For example, if one were to make a harmonic system using [[5/3]] as the "axis" rather than [[3/2]], chords built within a range of 5/3 would contain 3:4:5 and 12:15:20, and the medial intervals of the major third and perfect fourth would have a latitude of ±11.4°

Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality|arto/tendo]] chords.

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 '''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.
-Latitude allows the characterization of, i.e. different "flavors" of thirds, with respect to a fifth that might be tempered, as opposed to in terms of absolute interval ranges that may be misaligned with the intervals' harmonic function. For example, one might describe 3\13, 4\13, and 5\13 in 13edo based on their size as a [[minor third|subminor third]], a [[major third|submajor third]], and an ultramajor third, but that doesn't reflect their function in triadic harmony in 13edo. Using latitude, one can see that their latitudes are -22.5°, 0°, and 22.5°, and from that they may be assigned the labels subminor, [[neutral third|neutral]], and supermajor, which better reflect their role.
+Latitude allows the characterization of, i.e. different "flavors" of thirds, with respect to a fifth that might be tempered, as opposed to in terms of absolute interval ranges that may be misaligned with the intervals' harmonic function. For example, one might describe 3\13, 4\13, and 5\13 in [[13edo]] based on their size as a [[minor third|subminor third]], a [[major third|submajor third]], and an ultramajor third, but that doesn't reflect their function in [[triadic harmony]] in 13edo. Using latitude, one can see that their latitudes are -22.5°, 0°, and 22.5°, and from that they may be assigned the labels subminor, [[neutral third|neutral]], and supermajor, which better reflect their role.
-They additionally allow the generalization of the triadic concept to "axis" intervals other than the fifth. For example, if one were to make a harmonic system using 5/3 as the "axis" rather than 3/2, chords built within a range of 5/3 would contain 3:4:5 and 12:15:20, and the medial intervals of the major third and perfect fourth would have a latitude of ±11.4°
+They additionally allow the generalization of the triadic concept to "axis" intervals other than the fifth. For example, if one were to make a harmonic system using [[5/3]] as the "axis" rather than [[3/2]], chords built within a range of 5/3 would contain 3:4:5 and 12:15:20, and the medial intervals of the major third and perfect fourth would have a latitude of ±11.4°
 Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality|arto/tendo]] chords.