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

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=== 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).

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).

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=== Dual tritones ===

15edo has two different [[tritone]] intervals, each about a ~~quarter-tone~~ 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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|B# = C

|}

Throughout the rest of this page, wherever notation is used, I will directly specify which ~~notation~~.

The choice of which notation system to use depends heavily on what types of structures are being emphasized. Throughout the rest of this page, wherever notation is used, I will directly specify which type.

==Chords==

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===Chords of Porcupine===

In the Porcupine scales, chords are made by stacking intervals of 4, 5, and 6 steps; this provides a 3x3 contrast for chord types, comparable to the three way distinction between Major, Minor, and Suspended in common-practice Western music.

In the Porcupine scales, chords are made by stacking intervals of 4, 5, and 6 steps; this provides a 3x3 contrast for chord types, comparable to the three-way distinction between Major, Minor, and Suspended in common-practice Western music.

{| class="wikitable"

|+Chords of Porcupine

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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 ~~3L2M2s~~ 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 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.

[[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 the number of minor third generators outnumber the major third generators, and the right hand version results when the opposite is true. Each of these versions has seven unique modes.

There are two versions of the 3L 2M 2s scale; the left-hand version results when the number of minor third generators outnumber the major third generators, and the right-hand version results when the opposite is true. Each of these versions has seven unique modes.

{| class="wikitable"

|+Modes of Right-hand 3L 2M 2s

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A recognizable Major (Pat 3) Triads occurs on the fifth degree of the scale, providing a familiar circle-of-fifths resolution as well as a leading tone from the B of the V chord into the C of the tonic chord. The subminor (Pat 1) triad on the third degree provides an interesting voice leading into the V chord if voiced correctly (with the notes E, G, and B respectively leading to D, G, and B). Finally, the major (Pat 3) triad on the fourth degree provides a leading tone from F to E and from C to B.

Ultimately, our four-chord progression looks like C - F - e - G, or I - IV - iii - V. This progression prioritizes voice leading to create a coherent and flowing sound, and provides a great framework for melodies to be written over top.

Ultimately, our four-chord progression looks like C - F - em - G, or I - IV - iii - V. This progression prioritizes voice leading to create a coherent and flowing sound, and provides a great framework for melodies to be written over top.

===Example: Chord Progression in C Starfish===

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Finally, we can select a nondominant function that emerges from the tonic at the beginning of the progression. I will use the minor chord on A, because it sounds unresolved without sounding too tense.

[[File:I - vii - VIII - iv°.mp3|thumb|C - a - B - hb° progression]]

Ultimately, our four-chord progression is C - a - B - hb°, or I - vii - VIII - vi°. This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions.

Ultimately, our four-chord progression is C - am - B - hb°, or I - vii - VIII - vi°. This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions.

== Superstructures and Modulation ==

Due to its plethora of useful structures with so few notes per octave, 15edo compositions can make great use of modulation from one structure to another; if used well, this modulation may be less comparable to Western key changes, and more so to Jins changes in [[Arabic, Turkish, Persian music|Maqam traditions]].

=== What are Superstructures? ===

I will here be using the term "superstructure" to describe any singular overarching structure that contains multiple constituent structures within it. For instance, a scale that contains a mode of 7L 1s over a given tonic, plus a second copy of that mode with its tonic a Perfect Fifth above the first, would provide a useful superstructure that allows for modulation between the two keys.

Superstructures may contain multiple copies of the same structure, multiple entirely different structures, or some combination of both. 15edo itself may additionally be taken as a single superstructure that contains all possible constituent structures over all possible roots.

=== 5L 5s as a Superstructure ===

The 5L 5s scale in 15edo contains 3L 2M 2s as a constituent structure; each note of 5L 5s is the root of several 3L 2M 2s modes, which means that the 5L 5s scale can be used as a means by which to modulate from one key of 3L 2M 2s to another.

In the bright mode of 5L 5s (C Db D Fb F Gb G A A# Cb C in Nicetone notation), we can see that the Ionian mode of 3L 2M 2s exists over the first degree (C) as well as the eighth degree (A#). Because the C Ionian scale does not contain the note A#, we would normally not be able to modulate directly from one to the other without passing through at least one other scale; however, since 5L 5s is acting as a superstructure, we can easily use the superstructure to move smoothly from the key of C to the key of A# without needing to introduce other structures in passing.

An extension of the chord progression from before (C - F - fbm - G) may be expanded to move through the 5L 5s structure and resolve to A rather than C; for instance, we might use the major chord on Fb as a transitional chord that leads into A, since it promotes circle of fifths movement.

@@ Line 105: / Line 105: @@
 === 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).
+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).
@@ Line 123: / Line 123: @@
 === Dual tritones ===
-edo has two different [[tritone]] intervals, each about a quarter-tone 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 231: / Line 231: @@
 |B# = C
 |}
-Throughout the rest of this page, wherever notation is used, I will directly specify which notation.
+The choice of which notation system to use depends heavily on what types of structures are being emphasized.  Throughout the rest of this page, wherever notation is used, I will directly specify which type.
 ==Chords==
@@ Line 237: / Line 237: @@
 ===Chords of Porcupine===
-In the Porcupine scales, chords are made by stacking intervals of 4, 5, and 6 steps; this provides a 3x3 contrast for chord types, comparable to the three way distinction between Major, Minor, and Suspended in common-practice Western music.
+In the Porcupine scales, chords are made by stacking intervals of 4, 5, and 6 steps; this provides a 3x3 contrast for chord types, comparable to the three-way distinction between Major, Minor, and Suspended in common-practice Western music.
 {| class="wikitable"
 |+Chords of Porcupine
@@ Line 483: / Line 483: @@
 === 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 3L2M2s 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 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.
 [[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 the number of minor third generators outnumber the major third generators, and the right hand version results when the opposite is true.  Each of these versions has seven unique modes.
+There are two versions of the 3L 2M 2s scale; the left-hand version results when the number of minor third generators outnumber the major third generators, and the right-hand version results when the opposite is true.  Each of these versions has seven unique modes.
 {| class="wikitable"
 |+Modes of Right-hand 3L 2M 2s
@@ Line 606: / Line 606: @@
 A recognizable Major (Pat 3) Triads occurs on the fifth degree of the scale, providing a familiar circle-of-fifths resolution as well as a leading tone from the B of the V chord into the C of the tonic chord.  The subminor (Pat 1) triad on the third degree provides an interesting voice leading into the V chord if voiced correctly (with the notes E, G, and B respectively leading to D, G, and B).  Finally, the major (Pat 3) triad on the fourth degree provides a leading tone from F to E and from C to B.
-Ultimately, our four-chord progression looks like C - F - e - G, or I - IV - iii - V.  This progression prioritizes voice leading to create a coherent and flowing sound, and provides a great framework for melodies to be written over top.
+Ultimately, our four-chord progression looks like C - F - em - G, or I - IV - iii - V.  This progression prioritizes voice leading to create a coherent and flowing sound, and provides a great framework for melodies to be written over top.
 ===Example: Chord Progression in C Starfish===
@@ Line 617: / Line 617: @@
 Finally, we can select a nondominant function that emerges from the tonic at the beginning of the progression.  I will use the minor chord on A, because it sounds unresolved without sounding too tense.
 [[File:I - vii - VIII - iv°.mp3|thumb|C - a - B - hb° progression]]
-Ultimately, our four-chord progression is C - a - B - hb°, or I - vii - VIII - vi°.  This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions.
+Ultimately, our four-chord progression is C - am - B - hb°, or I - vii - VIII - vi°.  This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions.
+== Superstructures and Modulation ==
+Due to its plethora of useful structures with so few notes per octave, 15edo compositions can make great use of modulation from one structure to another; if used well, this modulation may be less comparable to Western key changes, and more so to Jins changes in [[Arabic, Turkish, Persian music|Maqam traditions]].
+=== What are Superstructures? ===
+I will here be using the term "superstructure" to describe any singular overarching structure that contains multiple constituent structures within it.  For instance, a scale that contains a mode of 7L 1s over a given tonic, plus a second copy of that mode with its tonic a Perfect Fifth above the first, would provide a useful superstructure that allows for modulation between the two keys.
+Superstructures may contain multiple copies of the same structure, multiple entirely different structures, or some combination of both.  15edo itself may additionally be taken as a single superstructure that contains all possible constituent structures over all possible roots.
+=== 5L 5s as a Superstructure ===
+The 5L 5s scale in 15edo contains 3L 2M 2s as a constituent structure; each note of 5L 5s is the root of several 3L 2M 2s modes, which means that the 5L 5s scale can be used as a means by which to modulate from one key of 3L 2M 2s to another.
+In the bright mode of 5L 5s (C Db D Fb F Gb G A A# Cb C in Nicetone notation), we can see that the Ionian mode of 3L 2M 2s exists over the first degree (C) as well as the eighth degree (A#).  Because the C Ionian scale does not contain the note A#, we would normally not be able to modulate directly from one to the other without passing through at least one other scale; however, since 5L 5s is acting as a superstructure, we can easily use the superstructure to move smoothly from the key of C to the key of A# without needing to introduce other structures in passing.
+An extension of the chord progression from before (C - F - fbm - G) may be expanded to move through the 5L 5s structure and resolve to A rather than C; for instance, we might use the major chord on Fb as a transitional chord that leads into A, since it promotes circle of fifths movement.