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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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, 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, ~~and~~ 29/22 as 6\15; additionally, [[33afdo|mode 33]] provides 50/33 as 9\15, and 55/33 as 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 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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===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 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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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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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:~~Method~~]]

[[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 107: / Line 109: @@
 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, 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, and 29/22 as 6\15; additionally, [[33afdo|mode 33]] provides 50/33 as 9\15, and 55/33 as 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 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 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 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.
@@ Line 597: / Line 604: @@
 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 ===
@@ Line 634: / Line 641: @@
 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:Method]]
+[[Category:15edo]]
+[[Category:Approaches to tuning systems]]