15edo: Difference between revisions
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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 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. | ||
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. | A useful way to visualize the pitches and intervals of 15edo is to arrange the notes in a grid, with 3/2s or 7/4s 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 subgroup tuning (and thus supports [[superpyth]], [[slendric]], and [[semaphore]], like 5edo). 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]]. | 15edo shares 5edo's 2.3.7 subgroup tuning (and thus supports [[superpyth]], [[slendric]], and [[semaphore]], like 5edo). 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]]. | ||
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* [[User:Unque/15edo Composition Theory|Unque's approach]] - covers scales, chords, intervals, and functional harmony. | * [[User:Unque/15edo Composition Theory|Unque's approach]] - covers scales, chords, intervals, and functional harmony. | ||
* [[15edo/Vector's compositional guides|Vector's guides]] - covers the construction of scales, the kinds of chords found in 15edo, and a possible notation system. | * [[15edo/Vector's compositional guides|Vector's guides]] - covers the construction of scales, the kinds of chords found in 15edo, and a possible notation system. | ||
* [[User:Astaryuu/15edo Notes|Astaryuu's notes]] - covers notation, scales, modes, intervals, and chords so far. | |||
* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for. | * [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for. | ||
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===15-odd-limit interval mappings=== | ===15-odd-limit interval mappings=== | ||
{{Q-odd-limit intervals}} | {{Q-odd-limit intervals}} | ||
==Regular temperament properties== | ==Regular temperament properties== | ||