15edo: Difference between revisions
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15edo can be thought of as three sets of [[5edo]] which do not connect by 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 which is radically different than a meantone system. Additionally, 15 being equal to 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 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 which is radically different than a meantone system. Additionally, 15 being equal to 3 × 5 also implies that 15edo contains five sets of [[3edo]]. | ||
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; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to approximate JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 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 [[ | 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; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to approximate JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 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, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, 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. | In the 15edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, 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. | ||
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<nowiki>*</nowiki> based on treating 15edo as an 11-limit temperament; other approaches are possible | <nowiki>*</nowiki> based on treating 15edo as an 11-limit temperament; other approaches are possible | ||
In [[ | In [[Ups and Downs Notation #Summary of EDO notation-"Pentatonic" EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the [[Ups_and_Downs_Notation #Natural Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd. For 15edo, this is also known as porcupine notation. The 15edo porcupine genchain in both absolute and relative notation: | ||
…Fx - Gx - A# - B# - C# - D# - E# - F# - G# --- A --- B --- C -- D --- E --- F --- G -- Ab -- Bb - Cb - Db - Eb - Fb - Gb - Abb - Bbb… | …Fx - Gx - A# - B# - C# - D# - E# - F# - G# --- A --- B --- C -- D --- E --- F --- G -- Ab -- Bb - Cb - Db - Eb - Fb - Gb - Abb - Bbb… | ||
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For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | ||
== | == JI approximation == | ||
15edo offers some minor improvements over | 15edo offers some minor improvements over 12et in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a [[5L 5s]] [[MOS scale]] wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as the [[blackwood]] temperament, named after [[Easley Blackwood, Jr.]], who is the first to document its existence. It has also been written on extensively by [[Igliashon Jones]] in the paper [http://www.cityoftheasleep.com/etc/5nEDOs.pdf ''Five is Not an Odd Number'']. For an in-depth treatment of harmony in 15edo based on this temperament (and its 7- and 11-limit extensions), see [[Blacksmith temperament modal harmony (in 15edo)]]. | ||
=== Selected just intervals === | === Selected just intervals === | ||
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A regular keyboard can be designed using this system placing 7 black keys as porcupine[7] and 8 whites as porcupine[8], and in fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard. | A regular keyboard can be designed using this system placing 7 black keys as porcupine[7] and 8 whites as porcupine[8], and in fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard. | ||
== Rank | == Regular temperament properties == | ||
=== Rank-2 temperaments === | |||
* [[List of 15et rank two temperaments by badness]] | * [[List of 15et rank two temperaments by badness]] | ||
* [[List of edo-distinct 15et rank two temperaments]] | * [[List of edo-distinct 15et rank two temperaments]] | ||
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| | [[5L 5s]] | | | [[5L 5s]] | ||
|} | |} | ||
== Commas == | === Commas === | ||
15edo [[tempers out]] the following [[comma]]s using the [[patent val]] {{val| 15 24 35 42 52 56 }}. | |||
{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
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|} | |} | ||
<references/> | <references/> | ||
== Scales == | |||
[[File:Screen Shot 2020-04-24 at 12.04.03 AM.png|none|thumb|986x986px|2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s]] | |||
[[File:Screen Shot 2020-04-24 at 12.04.45 AM.png|none|thumb|735x735px|4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s]] | |||
[[File:Screen Shot 2020-04-24 at 12.05.29 AM.png|none|thumb|927x927px|7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s ]] | |||
== Books == | == Books == | ||
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=== Guitar === | === Guitar === | ||
[http://www.swordguitars.com Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.] - A large repository of all known scales and temperament families in the 15edo system. 300+ examples /w chord-scale progressions. | [http://www.swordguitars.com Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.] - A large repository of all known scales and temperament families in the 15edo system. 300+ examples /w chord-scale progressions. | ||
== Diagrams == | |||
[[File:15edo_wheel.png|alt=15edo wheel.png|225x225px|15edo wheel.png]] [[File:15edo_wheel_02.png|alt=15edo wheel 02.png|250x250px|15edo wheel 02.png]] [[File:15edo_wheel_03.png|alt=15edo wheel 03.png|220x220px|15edo wheel 03.png]] | |||
=== Keyboard === | === Keyboard === | ||
[[File:Porcupine keyboard major triad shapes.png|none|thumb|500x500px|Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.]] | [[File:Porcupine keyboard major triad shapes.png|none|thumb|500x500px|Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.]] | ||
== Music == | == Music == | ||