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

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As of recent, [[15edo]] has been the subject of great debate in the xenharmonic community. Not only are many musicians skeptical of its harmonic content, but even advocates of the system disagree on how to interpret it and use it. On this page, I will present my personal experience with 15edo, and provide a potential framework that others may use to begin their own journeys through this strange and wonderful musical system.

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|1\15

|80

|[[22~~/21~~]], [[21/20]]

|[[23/22]], [[21/20]]

|[[Valentine]]

|Melodic semitone

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|3\15

|240

|[[8/7]]

|[[8/7]], [[38/33]]

|[[5edo]], [[Slendric]]

|One possible choice of whole tone (see below)

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|6\15

|480

|[[33/25]], [[4/3]]

|[[29/22]], [[4/3]]

|5edo; [[blacksmith]] period

|Highly contentious interpretation; see below

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|8\15

|640

|[[16/11~~]], [[22/15~~]]

|[[16/11]]

|Thuja

|

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|10\15

|800

|[[8/5]], [[11/7]]

|[[8/5]], [[35/22]]

|3edo; triforce period

|Same mapping as 12edo

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|12\15

|960

|7/4

|[[7/4]], [[19/11]]

|5edo, Slendric

|

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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).

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 ===

[[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.

[[11afdo|Mode 11]] of the Harmonic Series, alongside its supersets, 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 [[22afdo|mode 22]] providing 23/22 as an interpretation of 1\15, 29/22 as 6\15, and 35/22 as 10\15; additionally, [[33afdo|mode 33]] provides 38/33 as 3\15, 50/33 as 9\15, and 55/33 as 11\15.

This interpretation may also be criticized once again 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 and Mode 31 ===

[[31afdo|Mode 31]] of the Harmonic Series provides yet another lens through which to interpret 15edo. The intervals [0 2 5 6 7 8 9 10 12 15] can be interpreted as an approximation of the chord 31:34:39:41:43:45:47:49:54:62. Additionally, 11\15 can be interpreted somewhat inaccurately as 52/31; in spite of the error, this interval naturally lends itself to the structure of 15edo. Similarly, 13\15 and 14\15 can be interpreted respectively as 57/31 and 59/31. Finally, 1\15 can be said to represent 32/31 or 33/31, both with very high error but lending well to the harmonic structure of 15edo.

This yields the full chord 31:33:34:39:41:43:45:47:49:52:54:57:59:62.

=== 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 ~~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, which can be compared to Major, Minor, and Suspended chords of common-practice Western music.

{| class="wikitable"

|+Chords of Porcupine

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|LLLLLLLs

| C - D - E - F - G - H - A# - B# - C

|~~Octupus~~

|Octopus

|c°, C4

|F4/C, A#4/C

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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, 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 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 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.

{| class="wikitable"

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

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==Functional Harmony==

~~{{Todo| expand |inline=1}}~~Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize. Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 15edo, with concepts that can be extended to apply to any scale.

Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize. Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 15edo, with concepts that can be extended to apply to any scale.

Note that I will be constructing these chord progressions from back to front; this means that we will start with the resolution, then find the dominant chord, and then find a subdominant to precede it.

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Because the step size in 15edo is significantly smaller than the typical semitone, leading tones are tenser than in common-practice music. The Bayati second can be used as a useful element of voice leading, though it is not nearly as tense as the semitone. I consider voice leading to be the single most important element of 15edo harmony, because it provides a consistent sense of direction throughout melodies.

Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals. Specifically, the Major Tritone at has a tendency to resolve inward and become a Perfect Fourth, and the Minor Tritone has a tendency to resolve outwards and become a Perfect Fifth.

Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals. Specifically, the Major Tritone has a tendency to resolve inward and become a Perfect Fourth, and the Minor Tritone has a tendency to resolve outwards and become a Perfect Fifth.

=== Example: Chord Progression in C Ionian ===

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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 ==

{{Todo| expand |inline=1}}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 from G into A, since it promotes circle of fifths movement and has clear, smooth voice leading.

[[Category:15edo]]

[[Category:Approaches to tuning systems]]

@@ Line 1: / Line 1: @@
+{{breadcrumb|15edo}}
 As of recent, [[15edo]] has been the subject of great debate in the xenharmonic community.  Not only are many musicians skeptical of its harmonic content, but even advocates of the system disagree on how to interpret it and use it.  On this page, I will present my personal experience with 15edo, and provide a potential framework that others may use to begin their own journeys through this strange and wonderful musical system.
@@ Line 15: / Line 17: @@
 |1\15
 |80
-|[[22/21]], [[21/20]]
+|[[23/22]], [[21/20]]
 |[[Valentine]]
 |Melodic semitone
@@ Line 27: / Line 29: @@
 |3\15
 |240
-|[[8/7]]
+|[[8/7]], [[38/33]]
 |[[5edo]], [[Slendric]]
 |One possible choice of whole tone (see below)
@@ Line 45: / Line 47: @@
 |6\15
 |480
-|[[33/25]], [[4/3]]
+|[[29/22]], [[4/3]]
 |5edo; [[blacksmith]] period
 |Highly contentious interpretation; see below
@@ Line 57: / Line 59: @@
 |8\15
 |640
-|[[16/11]], [[22/15]]
+|[[16/11]]
 |Thuja
 |
@@ Line 69: / Line 71: @@
 |10\15
 |800
-|[[8/5]], [[11/7]]
+|[[8/5]], [[35/22]]
 |3edo; triforce period
 |Same mapping as 12edo
@@ Line 81: / Line 83: @@
 |12\15
 |960
-|7/4
+|[[7/4]], [[19/11]]
 |5edo, Slendric
 |
@@ Line 105: / Line 107: @@
 === 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).
+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 ===
-[[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.
+[[11afdo|Mode 11]] of the Harmonic Series, alongside its supersets, 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 [[22afdo|mode 22]] providing 23/22 as an interpretation of 1\15, 29/22 as 6\15, and 35/22 as 10\15; additionally, [[33afdo|mode 33]] provides 38/33 as 3\15, 50/33 as 9\15, and 55/33 as 11\15.
 This interpretation may also be criticized once again 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 and Mode 31 ===
+[[31afdo|Mode 31]] of the Harmonic Series provides yet another lens through which to interpret 15edo.  The intervals [0 2 5 6 7 8 9 10 12 15] can be interpreted as an approximation of the chord 31:34:39:41:43:45:47:49:54:62.  Additionally, 11\15 can be interpreted somewhat inaccurately as 52/31; in spite of the error, this interval naturally lends itself to the structure of 15edo.  Similarly, 13\15 and 14\15 can be interpreted respectively as 57/31 and 59/31.  Finally, 1\15 can be said to represent 32/31 or 33/31, both with very high error but lending well to the harmonic structure of 15edo.
+This yields the full chord 31:33:34:39:41:43:45:47:49:52:54:57:59:62.
 === 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 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 238: @@
 |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 244: @@
 ===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, which can be compared to Major, Minor, and Suspended chords of common-practice Western music.
 {| class="wikitable"
 |+Chords of Porcupine
@@ Line 340: / Line 347: @@
 |LLLLLLLs
 | C - D - E - F - G - H - A# - B# - C
-|Octupus
+|Octopus
 |c°, C<sup>4</sup>
 |F<sup>4</sup>/C, A#<sup>4</sup>/C
@@ Line 483: / Line 490: @@
 === 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, 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 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 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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 |+Modes of Right-hand 3L 2M 2s
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 ==Functional Harmony==
-{{Todo| expand |inline=1}}Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize.  Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 15edo, with concepts that can be extended to apply to any scale.
+Useful harmonic progressions may arise in a number of ways depending on the scale being used and depending on what chord the composer wishes to tonicize.  Here, I will document some examples of how functional harmonic progressions may be created in the different scales of 15edo, with concepts that can be extended to apply to any scale.
 Note that I will be constructing these chord progressions from back to front; this means that we will start with the resolution, then find the dominant chord, and then find a subdominant to precede it.
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 Because the step size in 15edo is significantly smaller than the typical semitone, leading tones are tenser than in common-practice music.  The Bayati second can be used as a useful element of voice leading, though it is not nearly as tense as the semitone.  I consider voice leading to be the single most important element of 15edo harmony, because it provides a consistent sense of direction throughout melodies.
-Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals.  Specifically, the Major Tritone at has a tendency to resolve inward and become a Perfect Fourth, and the Minor Tritone has a tendency to resolve outwards and become a Perfect Fifth.
+Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals.  Specifically, the Major Tritone has a tendency to resolve inward and become a Perfect Fourth, and the Minor Tritone has a tendency to resolve outwards and become a Perfect Fifth.
 === Example: Chord Progression in C Ionian ===
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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 ==
+{{Todo| expand |inline=1}}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 from G into A, since it promotes circle of fifths movement and has clear, smooth voice leading.
+[[Category:15edo]]
+[[Category:Approaches to tuning systems]]