14edo: Difference between revisions
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== Theory == | == Theory == | ||
14edo is the double of [[7edo]], and thus contains its flat 686{{C}} fifth, while adding new intervals halfway between each 7edo step. The intervals of 14edo not found in 7edo are [[Backslash notation|1\14]] = 86{{C}}, 3\14 = 257{{C}}, 5\14 = 429{{C}}, 7\14 = 600{{C}}, and their [[octave complement]]s. The 1\14 interval is a small semitone, and its inversion a major seventh, which is suitable for a {{w|leading tone}}. The 3\14 interval can be considered a small [[subminor third]] (or inframinor third), thus bringing a new, distinct flavor from the [[neutral third]] of 7edo, which is 4 steps of 14edo. The 5\14 interval is the [[fifth complement]] of 3\14, and can be considered a [[supermajor third]], so that stacking 3\14 and 5\14 gives the triad 0–3–8 steps (0–257–686{{C}}). Finally, the 7\14 interval is the familiar [[tritone]] found in [[12edo]], as well as every even-numbered [[edo]]. | |||
In terms of [[just intonation]], 14edo contains the approximation of [[3/2]] from 7edo. 14edo does not do well in the [[5-limit]], with [[5/4]] being close to halfway between its steps, so that 14edo does not approximate the [[4:5:6]] major triad or the [[10:12:15|1/(6:5:4)]] minor triad accurately. The closest approximation of [[7/4]] is very flat at 11\14 (943{{C}}), so that two of them stack to [[3/1]], meaning that [[49/48]] is [[tempering out|tempered out]], so that 14edo [[support]]s the [[semaphore]] temperament. However, since the 3rd harmonic is flat, the [[7/6]] and [[9/7]] intervals are approximated much more accurately, so that the 0–3–8 steps triad is a usable approximation of [[6:7:9]], and the 0–5–8 steps (0–429–686{{C}}) triad approximates [[14:18:21|1/(9:7:6)]]. The semaphore temperament notably generates the [[mos scale]] with pattern [[5L 4s]] (named ''semiquartal''), which contains many [[~]]6:7:9 and ~1/(9:7:6) triads. In the [[11-limit]], the [[11/8]] interval is tuned very flat and equated with [[4/3]]. However, [[11/9]] is tuned rather accurately, being represented with the 4\14 interval (343{{C}}), so that the [[Neutral (interval quality)|neutral]] triad formed by dividing the perfect fifth in two can be interpreted as a stack of two [[11/9]]'s, thus tempering out [[243/242]]. The neutral third can also be stacked on the supermajor third to get a ~[[7:9:11]] chord. | |||
14edo contains an [[omnidiatonic]] scale that can replace the standard diatonic scale, allowing for recognizable triadic harmony using the chords [[6:7:9]] and [[14:18:21]], as well as a neutral chord which can be seen as [[2:sqrt(6):3]]. | While prime [[5/1|5]] is poorly approximated, the [[7/5]] and [[11/10]] intervals are approximated fairly well. If we accept these approximations in addition to the ones described earlier, then we end up with a low-complexity, high-damage full [[11-limit]] temperament where many rather large [[comma]]s are tempered out. This leads to some of the bizarre equivalences described in the second "Approximate ratios" column in the table. This mapping uses the (barely) second-best mapping of prime 5, so it is notated with [[wart notation]] as "14c", where c is the 3rd letter of the alphabet, and 5 is the 3rd prime number. While not very accurate as a temperament, this mapping can be used to classify 11-limit intervals, which conveniently tempers out the [[square superparticular]]s of odds 5, 7, 9, and 11, and is the unique mapping to do so. | ||
14et has quite a bit of [[xenharmonic]] appeal, in a similar way to [[17edo|17et]], on account of having three types of 3rd and three types of 6th, rather than the usual two of [[12et]]. Since 14et also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. The 5L 4s mos scale is rich in triads, wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad. | |||
14edo also contains an [[omnidiatonic]] scale that can replace the standard diatonic scale, allowing for recognizable triadic harmony using the chords [[6:7:9]] and [[14:18:21]], as well as a neutral chord which can be seen as [[2:sqrt(6):3]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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This notation uses the same sagittal sequence as [[9edo#Sagittal notation|9-EDO]], is a subset of the notations for EDOs [[28edo#Sagittal notation|28]] and [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]]. | This notation uses the same sagittal sequence as [[9edo#Sagittal notation|9-EDO]], is a subset of the notations for EDOs [[28edo#Sagittal notation|28]] and [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]]. | ||
{{Sagittal chart|}} | |||
=== Ivor Darreg's notation === | === Ivor Darreg's notation === | ||
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==== Selected 13-limit intervals ==== | ==== Selected 13-limit intervals ==== | ||
[[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
=== Interval mappings === | |||
{{Q-odd-limit intervals|14}} | |||
{{Q-odd-limit intervals|13.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 14c val mapping}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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[[File:14edo_mus2.jpg|thumb]] | [[File:14edo_mus2.jpg|thumb]] | ||
== Instruments == | |||
[[Lumatone mapping for 14edo|Lumatone mappings for 14edo]] are available. | |||
== Music == | == Music == | ||
{{Main|Music | {{Main|14edo/Music}} | ||
{{Catrel|14edo tracks}} | {{Catrel|14edo tracks}} | ||
== See also == | == See also == | ||
* [[MisterShafXen’s take on 14edo harmony]] | * [[MisterShafXen’s take on 14edo harmony]] | ||