15edo: Difference between revisions
→Notation: categorized notations more accurately. expanded on the disadvantages of porcupine notation. expanded on interval names in eef notation. cleaned up the writing. |
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== Theory == | == 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. 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 or 7/4s on one axis and 5/4s on the other, to create a 3x5 rectangle of notes which tiles the plane. | |||
15edo | 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]]. | ||
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. | 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. | ||
This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[5L 2s|5L 2s diatonic scale]] and of a standard tritone, this tuning can be disorienting at first. Nonetheless, 15edo is notable for being the next-smallest edo after 9edo, 12edo and 14edo that contains recognizable major and minor triads. Under a stricter definition excluding 9edo and 14edo, this is a property noted in the works of theorists like [[Ivor Darreg]] and [[Easley Blackwood]]. In addition, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo. | This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[5L 2s|5L 2s diatonic scale]] and of a standard tritone, this tuning can be disorienting at first. Nonetheless, 15edo is notable for being the next-smallest edo after 9edo, 12edo and 14edo that contains recognizable major and minor triads. Under a stricter definition excluding 9edo and 14edo, this is a property noted in the works of theorists like [[Ivor Darreg]] and [[Easley Blackwood]]. In addition, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo. | ||
A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L 6s]] (onyx) and [[5L 5s]] (blackwood) scales are also an option. | A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L 6s]] (onyx) and [[5L 5s]] (blackwood) scales are also an option. | ||
15edo is also the second-smallest edo (after [[10edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | 15edo is also the second-smallest edo (after [[10edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]]. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|15}} | {{Harmonics in equal|15}} | ||
=== Composition theory === | |||
* [[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. | |||
* [[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. | |||
==Intervals== | ==Intervals== | ||
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==Notation == | ==Notation == | ||
There are many ways to notate 15edo, and the choice of notation depends heavily on which temperament or scale one wishes to focus on. | There are many ways to notate 15edo, and the choice of notation depends heavily on which temperament or scale one wishes to focus on. | ||
=== Notations generated by the fifth === | === Notations generated by the fifth === | ||
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|J | |J | ||
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One advantage of this notation is that its notated D major scale, D E F# G A B C# D, directly corresponds to 15edo’s zarlino LH Ionian scale. However, this only holds true for the key of D. Furthermore the perfect 4th and/or 5th of most other keys | One advantage of this notation is that its notated D major scale, D E F# G A B C# D, directly corresponds to 15edo’s zarlino LH Ionian scale. However, this only holds true for the key of D. Furthermore, the perfect 4th and/or 5th of most other keys is notated the same way a diminished or augmented fourth or fifth is in standard diatonic. For example, in the key of A the perfect fifth is E#. | ||
==== Zarlino notation (heptatonic) ==== | ==== Zarlino notation (heptatonic) ==== | ||
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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== | ||
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*Blackwood[10] [[5L 5s]] (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic) | *Blackwood[10] [[5L 5s]] (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic) | ||
[[File:BlackwoodMajor 15edo.mp3]] | [[File:BlackwoodMajor 15edo.mp3]] | ||
Blackwood decatonic, major mode, in 15edo | Blackwood decatonic, major mode, in 15edo | ||