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

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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).

Where the two types of whole tone need be disambiguated, they can respectively be called the greater and lesser whole tones (after their size) or the Bayati and Slendric seconds (after the structures they generate).

Where the two types of whole tone need be disambiguated, they can respectively be called the greater and lesser whole tones (after their size) or the Bayati and Slendric seconds (after the structures they generate). Alternatively, their sense as whole tones may be abandoned entirely, with the lesser tone being described as a [[15edo#Porcupine Notation .28Octatonic.29|Quill]] and the greater described as a [[Interseptimal interval|Semifourth]].

=== 15edo and Carlos Alpha ===

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.

The [[Carlos Alpha|Alpha scale]] created by [[Wendy Carlos]] is an equal temperament system that contains two octave-like intervals approximately equidistant from a justly-tuned 2/1. 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.

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 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. A better approximation of Carlos Alpha in an octave-equivalent setting would likely be [[Valentine]] temperament MOS scales in [[31edo]] or [[46edo]].

=== 15edo and Mode 11 ===

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=== 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.

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 harmonically simpler due to being a rooted overtone.

=== Dual tritones ===

15edo has two different [[tritone]] intervals, each about a quartertone away from the classic [[2edo|semioctave]] tritone. These tritones may actually be considered consonances in the context of 15edo harmony, as they approximate the 11th harmonic with only approximately 10% relative error. They are quite useful as fully diminished and half diminished fifths respectively, in chords such as the [[Ptolemismic triad|Ptolemismic Triad]]. Chords containing these tritones are often useful as dominant chords for voice leading and functional harmony (see below)

15edo has two different [[tritone]] intervals, each about a quartertone away from the classic [[2edo|semioctave]] tritone. These tritones may actually be considered consonances in the context of 15edo harmony, as they approximate the 11th harmonic with only approximately 10% relative error. They are quite useful as fully diminished and half diminished fifths respectively, in chords such as the [[Ptolemismic triad|Ptolemismic Triad]]. Chords containing these tritones are often useful as dominant chords for voice leading and functional harmony (see below).

== Notation ==

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=== 3L 2M 2s===

The [[Nicetone|3L 2M 2s]] scale is often used as an analog to Diatonic in 15edo, as its step pattern resembles that of the Zarlino scale ~~that~~ was historically used as a ternary version of Diatonic ~~that~~ was considered to have more consonant thirds. Whereas the true Zarlino scale was made by alternating 5/4 and 6/5 as generators, 15edo's 3L 2M 2s scale can be made by alternating 5\15 and 4\15 generators. Rather than tempering out the [[81/80|syntonic comma]] (the difference between the two types of whole tone) as in common-practice Western music, 15edo tempers the scale such that the syntonic comma is equal to the semitone.

The [[Nicetone|3L 2M 2s]] scale is often used as an analog to Diatonic in 15edo, as its step pattern resembles that of the Zarlino scale, which was historically used as a ternary version of Diatonic and was considered to have more consonant thirds than [[3-limit|Pythagorean]] intonation, and more consonant fifths than [[Meantone]]. Whereas the true Zarlino scale was made by alternating 5/4 and 6/5 as generators, 15edo's 3L 2M 2s scale can be made by alternating 5\15 and 4\15 generators. Rather than tempering out the [[81/80|syntonic comma]] (the difference between the two types of whole tone) as in common-practice Western music, 15edo tempers the scale such that the syntonic comma is equal to the semitone.

[[File:RH Nice Ionian.mp3|thumb|<nowiki>The 4|2 mode (Ionian) of right-hand 3L2M2s</nowiki>]]

There are two versions of the 3L 2M 2s scale; the left-hand version results when one begins the sequence on a minor third, and the right-hand version results when one begins the sequence on a major third. Each of these versions has seven unique modes.

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 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).
-Where the two types of whole tone need be disambiguated, they can respectively be called the greater and lesser whole tones (after their size) or the Bayati and Slendric seconds (after the structures they generate).
+Where the two types of whole tone need be disambiguated, they can respectively be called the greater and lesser whole tones (after their size) or the Bayati and Slendric seconds (after the structures they generate).  Alternatively, their sense as whole tones may be abandoned entirely, with the lesser tone being described as a [[15edo#Porcupine Notation .28Octatonic.29|Quill]] and the greater described as a [[Interseptimal interval|Semifourth]].
 === 15edo and Carlos Alpha ===
-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.
+The [[Carlos Alpha|Alpha scale]] created by [[Wendy Carlos]] is an equal temperament system that contains two octave-like intervals approximately equidistant from a justly-tuned 2/1.  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.
-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 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.  A better approximation of Carlos Alpha in an octave-equivalent setting would likely be [[Valentine]] temperament MOS scales in [[31edo]] or [[46edo]].
 === 15edo and Mode 11 ===
@@ Line 122: / Line 122: @@
 === 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.
+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 harmonically simpler due to being a rooted overtone.
 === Dual tritones ===
-edo has two different [[tritone]] intervals, each about a quartertone away from the classic [[2edo|semioctave]] tritone.  These tritones may actually be considered consonances in the context of 15edo harmony, as they approximate the 11th harmonic with only approximately 10% relative error.  They are quite useful as fully diminished and half diminished fifths respectively, in chords such as the [[Ptolemismic triad|Ptolemismic Triad]].  Chords containing these tritones are often useful as dominant chords for voice leading and functional harmony (see below)
+edo has two different [[tritone]] intervals, each about a quartertone away from the classic [[2edo|semioctave]] tritone.  These tritones may actually be considered consonances in the context of 15edo harmony, as they approximate the 11th harmonic with only approximately 10% relative error.  They are quite useful as fully diminished and half diminished fifths respectively, in chords such as the [[Ptolemismic triad|Ptolemismic Triad]].  Chords containing these tritones are often useful as dominant chords for voice leading and functional harmony (see below).
 == Notation ==
@@ Line 485: / Line 485: @@
 === 3L 2M 2s===
-The [[Nicetone|3L 2M 2s]] scale is often used as an analog to Diatonic in 15edo, as its step pattern resembles that of the Zarlino scale that was historically used as a ternary version of Diatonic that was considered to have more consonant thirds.  Whereas the true Zarlino scale was made by alternating 5/4 and 6/5 as generators, 15edo's 3L 2M 2s scale can be made by alternating 5\15 and 4\15 generators.  Rather than tempering out the [[81/80|syntonic comma]] (the difference between the two types of whole tone) as in common-practice Western music, 15edo tempers the scale such that the syntonic comma is equal to the semitone.
+The [[Nicetone|3L 2M 2s]] scale is often used as an analog to Diatonic in 15edo, as its step pattern resembles that of the Zarlino scale, which was historically used as a ternary version of Diatonic and was considered to have more consonant thirds than [[3-limit|Pythagorean]] intonation, and more consonant fifths than [[Meantone]].  Whereas the true Zarlino scale was made by alternating 5/4 and 6/5 as generators, 15edo's 3L 2M 2s scale can be made by alternating 5\15 and 4\15 generators.  Rather than tempering out the [[81/80|syntonic comma]] (the difference between the two types of whole tone) as in common-practice Western music, 15edo tempers the scale such that the syntonic comma is equal to the semitone.
 [[File:RH Nice Ionian.mp3|thumb|<nowiki>The 4|2 mode (Ionian) of right-hand 3L2M2s</nowiki>]]
 There are two versions of the 3L 2M 2s scale; the left-hand version results when one begins the sequence on a minor third, and the right-hand version results when one begins the sequence on a major third.  Each of these versions has seven unique modes.