15edo: Difference between revisions

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== Theory ==

15edo can be thought of as three sets of [[5edo]] which do not connect by [[3/2|fifths]]. The fifth at 720 cents is quite wide yet still usable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. ~~The perfect fifth~~ of ~~15edo returns to the octave if stacked~~ five ~~times. In regular temperament terms~~, ~~this means~~ the [[Pythagorean limma]] ~~is tempered out~~, ~~which~~ is radically different than a meantone system~~. This~~ has a variety of ramifications for chord progressions based on {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions~~. Additionally, 15 being equal to {{nowrap|3 × 5}} also implies that 15edo contains five sets of [[3edo]]~~.

15edo can be thought of as three sets of [[5edo]] which do not connect by [[3/2|fifths]]. The fifth at 720 cents is quite wide yet still usable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. 15edo contains 3 circles of five [[3/2]]<nowiki/>s (supporting [[blackwood]], which tempers out the [[Pythagorean limma]]), and 5 circles of three [[5/4]]<nowiki/>s (supporting [[augmented temperament]]). This is radically different than a meantone system, and has a variety of ramifications for chord progressions based on diatonic {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions.

15edo ~~can be seen as~~ a [[~~7-limit~~]] ~~temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to~~ [[11~~-limit~~]] ~~intervals~~, so it can reasonably be described as an 11-limit temperament, and is generally considered to be the first edo to work as an 11-limit system; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to represent JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity, for instance, only using 9/8 when it is being made up of two 3/2s to make its identity clear). 15edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[syntonic comma]].

A useful way to visualize the pitches and intervals of 15edo is to arrange the notes in a grid, with 3/2s on one axis and 5/4s on the other, to create a 3x5 rectangle of notes which tiles the plane.

15edo shares 5edo's 2.3.7. However, by splitting each 5edo step into three parts, reasonable approximations to [[5/4]] and [[11/8]] are obtained (as per [[valentine]] temperament), so 15edo can reasonably be described as an 11-limit temperament, and is generally considered to be the first edo to work as an 11-limit system; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to represent JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity, for instance, only using 9/8 when it is being made up of two 3/2s to make its identity clear). 15edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[syntonic comma]].

In the 15edo system, major thirds cannot be divided perfectly into two, while minor thirds, fourths, wide tritones, subminor sevenths, and supermajor sevenths can. Similarly, supermajor seconds, fourths, fifths, and subminor sevenths can all be divided into 3 equal parts, while minor thirds, tritones, and major sixths cannot.

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 == Theory ==
-edo can be thought of as three sets of [[5edo]] which do not connect by [[3/2|fifths]]. The fifth at 720 cents is quite wide yet still usable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15edo returns to the octave if stacked five times. In regular temperament terms, this  means the [[Pythagorean limma]] is tempered out, which is radically different than a meantone system. This has a variety of ramifications for chord progressions based on {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions. Additionally, 15 being equal to {{nowrap|3 × 5}} also implies that 15edo contains five sets of [[3edo]].
+edo can be thought of as three sets of [[5edo]] which do not connect by [[3/2|fifths]]. The fifth at 720 cents is quite wide yet still usable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. 15edo contains 3 circles of five [[3/2]]<nowiki/>s (supporting [[blackwood]], which tempers out the [[Pythagorean limma]]), and 5 circles of three [[5/4]]<nowiki/>s (supporting [[augmented temperament]]). This is radically different than a meantone system, and has a variety of ramifications for chord progressions based on diatonic {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions.
-edo can be seen as a [[7-limit]] temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to [[11-limit]] intervals, so it can reasonably be described as an 11-limit temperament, and is generally considered to be the first edo to work as an 11-limit system; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to represent JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity, for instance, only using 9/8 when it is being made up of two 3/2s to make its identity clear). 15edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[syntonic comma]].
+A useful way to visualize the pitches and intervals of 15edo is to arrange the notes in a grid, with 3/2s on one axis and 5/4s on the other, to create a 3x5 rectangle of notes which tiles the plane.
+edo shares 5edo's 2.3.7. However, by splitting each 5edo step into three parts, reasonable approximations to [[5/4]] and [[11/8]] are obtained (as per [[valentine]] temperament), so 15edo can reasonably be described as an 11-limit temperament, and is generally considered to be the first edo to work as an 11-limit system; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to represent JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity, for instance, only using 9/8 when it is being made up of two 3/2s to make its identity clear). 15edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[syntonic comma]].
 In the 15edo system, major thirds cannot be divided perfectly into two, while minor thirds, fourths, wide tritones, subminor sevenths, and supermajor sevenths can. Similarly, supermajor seconds, fourths, fifths, and subminor sevenths can all be divided into 3 equal parts, while minor thirds, tritones, and major sixths cannot.