15edo/Unque's compositional approach: Difference between revisions

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 |[[22/21]], [[21/20]]
 |[[Valentine]]
-|May act as a stretched version of Carlos Alpha (see below)
+|Melodic semitone
 |-
 |2\15
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 |[[5/3]]
 |Kleismic
-|
+|Relatively accurate, though noticeably flatter than optimal kleismic
 |-
 |12\15
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 |Tuned justly
 |}
+=== 15edo as a dual-9 system ===
+The intervals 2\15 and 3\15 are both quite distant from a justly-tuned 9/8 interval; as such, some have proposed 15edo as being a "dual nines" system, in which these two intervals are both interpreted as flavors of the whole tone.  This interpretation allows for a near-1:1 correspondence between the Left and Right hand versions of Nicetone (see below).
 === 15edo and Carlos Alpha ===
-The [[Carlos Alpha|Alpha scale]] created by [[Wendy Carlos]] is a dual-octaves equal temperament system.  Because the flat octave is reached at fifteen steps, many people have offered that 15edo could be treated as a tuning of the Alpha scale that is stretched such that the octave is tuned justly.
+The [[Carlos Alpha|Alpha scale]] created by [[Wendy Carlos]] is a dual-octaves equal temperament system.  Because the flatter of the two octaves is reached at fifteen steps, many people have offered that 15edo could be treated as a tuning of the Alpha scale that is stretched such that the flat octave is tuned justly.  This interpretation provides an explanation for certain peculiarities that composers tend to converge on, such as the usage of [0 5 9 12 15] as an approximation of [[4afdo|mode 4]] of the Harmonic Series in spite of its high error.
-Treating 15edo as a stretched version of Carlos Alpha provides an interesting lens with which to view its intervals.  While some object to this view on the grounds of accuracy, it is undeniable that this interpretation helps explain many peculiarities about 15edo composition that composers tend to converge on, such as the usage of [0 5 9 12 15] as an approximation of the 4::8 segment.
-=== 15edo as a dual-9 system ===
+The connection to the Carlos Alpha scale has notably been criticized due to its poor accuracy, and the lack of clear compositional equivalence between the two, especially beyond the first octave.  Carlos Alpha in practice emphasizes 9/4 and 18/7 as fundamental consonances, whereas 15edo does not even represent either of these intervals accurately, let alone treat their approximations as fundamental.  Additionally, the characteristic [[quark]] interval provided by octave-equivalent [[Gamelismic clan|Gamelismic]] tunings (those that temper out [[1029/1024]], as Carlos Alpha does) has been tempered out in 15edo, which leads to extremely heavy error.
-The intervals 2\15 and 3\15 are both quite distant from a justly-tuned 9/8 interval; as such, some have proposed 15edo as being a "dual nines" system, in which these two intervals are both interpreted as flavors of the whole tone.  This interpretation allows for a near-1:1 correspondence between the Left and Right hand versions of Nicetone (see below).
 === 15edo and Mode 11 ===
-[[11afdo|Mode 11]] of the Harmonic Series provides another interesting way to interpret intervals of 15edo.  Notably, the intervals [0 2 5 7 8 12 13 14 15] can be interpreted as an approximation of the chord 11:12:14:15:16:19:20:21:22.
+[[11afdo|Mode 11]] of the Harmonic Series provides another interesting way to interpret intervals of 15edo.  Notably, the intervals [0 2 5 7 8 12 13 14 15] can be interpreted as an approximation of the chord 11:12:14:15:16:19:20:21:22.  The /11 logic can be extended to supersets of mode 11 to provide interpretations of other intervals, such as [[33afdo|mode 33]] providing 50/33 as an extremely accurate interpretation of 9\15, and 55/33 as an interpretation of 11\15.
-The /11 logic can be extended to supersets of mode 11 to provide interpretations of other intervals, such as [[33afdo|mode 33]] providing 50/33 as an extremely accurate interpretation of 9\15, and 55/33 as an interpretation of 11\15.
+This interpretation may also be criticized due to a lack of accuracy, but it is notably more consistent than the Carlos Alpha interpretations as the difference between the tunings does not accrue per step.
 === 15edo's fifth ===
 The interval at 9\15 is possibly the most contentious interval in the entire xenharmonic community.  Some have proposed that is represents 3/2 due to its clear function as a concordant fifth; others argue that 50/33 is more accurate and functions better alongside the other /11 intervals; still others have posited that [[97/64]] is even more accurate and simpler due to being a rooted overtone.
-Where do I stand in this debate?  I think all of these interpretations are useful in their own right, and succeed in explaining various properties of the interval - ultimately, it is all of them and none of them.
 == Notation ==