User:TromboneBoi9/Approaches to weird EDOs: Difference between revisions

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Outside of free just intonation, most of my xenharmonic work is exclusively in [[EDO|EDOs]], generally various EDOs smaller than [[36edo]] (although I have used larger ones in the past).

I work almost exclusively in notation software, so it's important that EDOs I'm working with can be worked into traditional [[5L 2s|diatonic]] notation in some form~~. However~~, ~~as one would know, this is only~~ the ~~case for an EDO if it's approximation~~ of ~~[[3/2]] is between 686¢ (5\7)~~ and ~~720¢ (3\5), which is not always the case~~.

I work almost exclusively in notation software (although that is changing), so it's important that EDOs I'm working with can be worked into traditional [[5L 2s|diatonic]] notation in some form; from there, I can usually figure out the quirks of a tuning and its rational approximations from there.

Here are some of my own theoretical and notational approaches to various EDOs that break ~~this mold~~.

However, as one would know, this is only the case for an EDO if its approximation of [[3/2]] is between 686.71¢ (4\7) and 720.00¢ (3\5), which is not always the case. There are some methods by which they can still be worked in harmonically through various alternative rational approximations, although this ignores the notation debacle; even then, ''still'' some others would appear completely impossible with such an approach.

Here are some of my own theoretical and notational approaches to various EDOs that break these molds.

==13edo==

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Scales generated by the 2\13 will also feature 4\13—a flat but recognizable [[5/4]]—and 6\13—a dead-on [[11/8]]. In this way, a basic [[6L 1s|6L1s]] "archaeotonic" scale is produced.

Any mode can be used, but the symmetrical 3|3 "~~Holthathian~~" mode captures 13edo's best intervals:

Any mode can be used, but the symmetrical 3|3 "Horthathian" mode captures 13edo's best intervals:

{| class="wikitable"

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!Ratios

!Note name

![[26edo]] name

![[26edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name

!Pseudo-diatonic interval name

|-

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|[[11/8]]

|F

|F&sharp;

|F&sharp;, ^F

|Major fourth

|-

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|[[16/11]]

|G&flat;

|G&flat;, vG

|Minor fifth

|-

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|[[13/8]], [[8/5]]

|A&flat;

|A&flat;, vA

|Minor sixth

|-

Line 79:

Line 81:

|[[16/9]], [[9/5]]

|B&flat;

|B&flat;, vB

|Minor seventh

|-

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===A note on fifths===

13edo, of course, has notoriously bad fifths—to be specific, ''two'' bad fifths: the 7\13 minor fifth of 646¢ and the 8\13 major fifth of 738¢. While these fifths may be useless harmonically, cases can be made for their use ''melodically'', specifically for the major fifth.

13edo, of course, has notoriously bad fifths—to be specific, ''two'' bad fifths: the 7\13 minor fifth of 646¢ and the 8\13 major fifth of 738¢ (making a [[dual-fifth]] system). While these fifths may be useless harmonically, cases can be made for their use ''melodically'', specifically for the major fifth.

Consider the use of the harmonic minor scale in traditional 12edo theory. The replacement of the minor seventh by the major seventh exists in order to make the chord on the fifth degree of the minor scale a major chord rather than a minor chord. In a typical V - i cadential progression, this replacement adds tension since the third of the V chord is only a semitone below the tonic, and wants to resolve upward to complete the progression.

Line 180:

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|*

|}

===Blackwood's "subminor"===

[[Easley Blackwood]]'s short blurb on 13edo as part of the booklet packaged with the CD copy of his ''Twelve Microtonal Etudes'' states:

<blockquote>Yet even this tuning contains a strange mode best described

as “sub-minor”.<ref>https://www.cedillerecords.org/wp-content/uploads/2020/01/018-blackwood-microtonal-booklet.pdf</ref></blockquote>

He does not elaborate on the construction of this mode, but a quick analysis of his 13edo etude reveals it to be the 5|2 "Celephaïsian" mode of [[5L 3s|5L3s]] "oneirotonic". Although not in the modern sense, Blackwood aptly calls this mode "sub-minor" because its construction is ''LsLLsLLs'', which is identical to the familiar [[5L 2s|5L2s]] diatonic Aeolian (minor) mode ''LsLLsLL'' but with an extra ''s'' appended to the end, compressing all of the intervals slightly.

{|class="wikitable"

! Interval !! Cents !! Note name (in 6L1s) !! Note name (in 5L3s)

! [[26edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] Names !! Pseudo-diatonic interval name

|-

| 0\13 || 0.00 || C || C || C || Perfect unison

|-

| 2\13 || 184.62 || D || D || D || Major second

|-

| 3\13 || 276.93 || E&flat; || E&flat; || E&flat;&flat;, vE&flat; || Minor third

|-

| 5\13 || 461.54 || F&flat; || F || F&flat;, vF || Major fourth

|-

| 7\13 || 646.15 || G&flat; || G || G&flat;, vG || Minor fifth

|-

| 8\13 || 738.46 || G || H&flat; || G&sharp;, ^G || Major fifth

|-

| 10\13 || 923.08 || A || A || A&sharp;, ^A || Major sixth

|-

| 12\13 || 1107.69 || B || B || B&sharp;, ^B || Major seventh

|-

| 13\13 || 1200.00 || C || C || C || Perfect octave

|}

Blackwood continues briefly on his use of the mode:

<blockquote>The first four bars of the Etude are an arrangement of this mode into consecutive thirds — a motif that recurs later in two transposed variations. The rest of the piece is comprised of chromatic resolutions of complex altered chords.</blockquote>

Indeed, the thirds of this scale as well as the aforementioned [[6L 1s|6L1s]] remain useful, since the thirds (two-mosstep intervals) in both scales are always either 3\13 minor thirds or 4\13 major thirds, which aren't all that inaccurate from JI.

===A note on the 7/4===

My favorite kinds of xenharmonic intervals are those that involve the factor of seven somehow, usually approximations of [[7/4]], [[7/6]], or [[8/7]].

13edo's approximation of the harmonic seventh 7/4 is 10\13 or 923¢, which, at a whopping '''45 cents flat''', is even more inaccurate than its approximation of [[3/2]]. Despite this, in practical use, 10\13 (to my ears, at least) appears to be a ''usable'' 7/4, at least when used sparingly among other passages that do 13edo "correctly".

My hypotheses as to why:

* I personally have more experience with [[24edo]] which has a pretty flat 7/4.

* My ears might be confusing it for [[12/7]]: 10\13 is much closer to 12/7, which is roughly in the same quartertonal interval region (supermajor sixth/subminor seventh) and is also harmonically related to 7/4 (its [[3/1|tritave]] inversion).

* My ears might be confusing it for [[55/32]]: The context in which I've used 10\13 the most often is in 0,4,6,10\13 or its subsets. In that chord, there are two 4\13 major thirds (approximating [[5/4]] well) separated by a 6\13 (approximating [[11/8]] well), so it could be said that 10\13 is acting as a 55/32 here. A just intonation chord 1/1, 5/4, 11/8 can reasonably be topped with either 7/4 ''or'' 55/32, and the two are only a [[56/55]] apart.

==8edo==

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[[File:8edo-dem1.mp3]]

Like the 13edo progression, ~~this progression largely receives its cohesion~~ melodically. This progression moves the bass by 5\8, the 750¢ superfifth, still recognizable in this instance as a fifth despite its intrusion into the minor sixth interval space. Steps are larger, so the progression must start on the 2\8 minor third rather than any kind of major third.

Like the 13edo progression, the fifth is incorporated melodically. This progression moves the bass by 5\8, the 750¢ superfifth, still recognizable in this instance as a fifth despite its intrusion into the minor sixth interval space. Steps are larger, so the progression must start on the 2\8 minor third rather than any kind of major third.

==23edo==

To me, [[23edo]] is proof that "terrible" rational approximations might not be as terrible as the theorist's approximation-centered mind might believe. Its approximations of primes 3, 5, 7, and 11 are all at least 20 cents off, and yet, even with some beating, there is a clearly-recognizable major chord 0,7,13\23 and arguably even a 4:5:6:7:9:11 in the form of 0,7,13,18,27,33\23 (although your mileage may vary with the 33\23). This along with its quartertonal step size makes it a particularly rewarding system to conquer.

23edo is technically [[dual-fifth]] like [[13edo]], the fifths being 13\23 and 14\23. [[2L 5s|Antidiatonic]] seems more natural here, especially since the minor fifth is recognizable as a perfect fifth (the major fifth is far too sharp), but this means that the 3\23 neutral seconds get the role of major seconds rather than the very good [[9/8]] approximation, 4\23. Atonally, [[46edo]] subset notation will work best.

===Listen for yourself===

That being said, here are some audio demonstrations of the various split prime approximations so you can make a decision yourself as to which version is better.

'''Make sure your volume is set to about 50%, otherwise these may sound quite loud.''' You can alternatively listen to these comparisons in series on [https://luphoria.com/xenpaper/#%7B23edo%7D(osc%3Atriangle)(bpm%3A50)%0A%7Br220Hz%7D%0A%5B0_13%5D--._%5B0_14%5D--._%23_3%2F2_perfect_fifths%0A..%0A%5B0_7%5D--._%5B0_8%5D--._%23_5%2F4_major_thirds%0A..%0A%5B0_18%5D--._%5B0_19%5D--._%23_7%2F4_harmonic_sevenths%0A..%0A%5B0_'10%5D--._%5B0_'11%5D--._%23_11%2F4_harmonic_elevenths Xenpaper].

{|

| 5/4 || major third || 7\23 || A-vvC&sharp; || 8\23 || A-C&sharp; || [[File:Comparison-23edo-thirds.mp3]]

|-

| 3/2 || perfect fifth || 13\23 || A-vE || 14\23 || A-^E || [[File:Comparison-23edo-fifths.mp3]]

|-

| 7/4 || harmonic seventh || 18\23 || A-vvG || 19\23 || A-G || [[File:Comparison-23edo-sevenths.mp3]]

|-

| 11/4 || harmonic eleventh || 33\23 || A-^D || 34\23 || A-E&flat; || [[File:Comparison-23edo-elevenths.mp3]]

|}

My preferences lie in 0,7,13,18\23 for a 4:5:6:7, and is think either 33\23 or 34\23 could work for an 11/4. Both sound to me equally consonant, but the higher one has less beating when octave reduced (to 11\23). I personally prefer 33\23 (10\23) since I feel 34\23 (11\23) is too close to the tritone interval region; if there's any reason 11\23 has less beating, it's because it's actually a [[7/5]].

===Seconds===

Even though 23edo has a [[9/8]] only five cents off, this does not help us in devising a heptatonic notation system like it did with 13edo.

I once idiotically presumed that [[6L 1s|6L1s]] might work well here since the 9/8 is solid and that system works on major seconds, but at about 208¢, the major seconds are actually bigger than 1\6, which means they're too large. Even if such a system could exist, it wouldn't be anywhere near as useful in 23edo as it was in [[13edo]]. Three of the much flatter, 2\13 major seconds form a great [[11/8]] (6\13) whereas three of the 4\23 seconds stack to a 12\23, which is far too high to be an 11/8.

The 3\23 ''neutral'' seconds are arguably superior for a second-generated system. These are too small to generate 6L1s, however; instead, they generate [[1L 6s|1L6s]] "onyx" or "antiarchaeotonic", or [[7L 1s|7L1s]] "pine" if we add another generator. Both of these give access to the 6\23 minor third and its inverse the 17\23 major sixth, very good approximations of [[6/5]] and [[5/3]] respectively.

Using the 1L6s scale for a heptatonic system of nominals gives us a notation not unlike antidiatonic due to its emphasis of the neutral second. If we use 0|6 for the nominals (the "antiarchaeotonic" equivalent of 13edo's 6L1s notation), the symmetrical 3|3 mode will look like this:

{|class="wikitable"

! Interval !! Cents !! Ratio(s) !! 1L6s name (0|6)

! [[2L 5s|2L5s]] name<ref>I would usually notate antidiatonic with harmonic notation (where flats sharpen and sharps flatten), but I'm doing it the other way here to demonstrate the similarities between 2L5s and 1L6s.</ref>

! [[46edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name !! Pseudo-diatonic interval name

|-

| 0\23 || 0.00 || 1/1 || C || C || C || Perfect Unison

|-

| 3\23 || 156.52 || [[11/10]], [[12/11]] || D || D || vvD || Neutral second

|-

| 6\23 || 313.04 || [[6/5]] || E || E || ^E&flat; || Minor third

|-

| 9\23 || 469.57 || [[21/16]] || F || F&flat; || vF || Minor fourth

|-

| 14\23 || 730.43 || [[32/21]] || G&sharp; || G&sharp; || ^G || Major fifth

|-

| 17\23 || 886.96 || [[5/3]] || A&sharp; || A&sharp; || vA || Major sixth

|-

| 20\23 || 1043.48 || [[11/6]] || B&sharp; || B&sharp; || ^^B&flat; || Neutral seventh

|-

| 23\23 || 1200.00 || [[2/1]] || C || C || C || Perfect octave

|}

This system is similar to antidiatonic in the same way that 6L1s from earlier is similar to diatonic: B-C is the only ''large'' step here rather than it being the only small step. This 1L6s system might work well if you find yourself preferring the 14\23 major fifth rather than the 13\23 minor fifth, since it features as the fifth in this scale.

===Relative to 24edo===

It might help to perceive 23edo as a system that "phases out" from [[24edo]] in the same way that [[11edo]] "phases out" from [[12edo]].

The 23edo's "quarter tone" is barely more than two cents larger than 24edo's, which means that:

* "Semitones" (2\23) are four cents larger.

* Whole tones (4\23) are eight cents larger, making them about as good of a [[9/8]] as [[12edo|12]] or [[24edo]]'s but sharp instead of flat.

* Minor thirds (6\23) are thirteen cents larger, making them very good [[6/5]]'s.

* "Neutral thirds" (7\23) are fifteen cents larger, pushing them closer to [[5/4]]'s.

* "Major thirds" (8\23) are now seventeen cents larger, making them distinctively Pythagorean.

* "Perfect fourths" (10\23) are 21 cents larger, pushing the boundary between ''perfect fourth'' and ''[[superfourth]]''.

Although by no means consistent across keys, this also introduces the idea of a notation system like [[24edo]]'s but with a missing step between F and G where the 1\2 tritone would be, which introduces the enharmonic equivalences:

* '''F''' = '''G{{sesquiflat2}}'''

* '''F{{demisharp2}}''' = '''G&flat;'''

* '''F&sharp;''' = '''G{{demiflat2}}'''

* '''F{{sesquisharp2}}''' = '''G'''

This system naturally favors the major fourth, minor fifth, and the ''proper'' major second, although this only applies for a couple of keys: The G↔D fifth, for instance, is major, and the F↔G second is obviously neutral.

===On Blackwood's approach===

The 23edo etude was the only etude of [[Easley Blackwood|Blackwood]]'s twelve etudes that uses entirely non-Western scales, specifically because, as he states:

<blockquote>…23-note tuning contains no diatonic configurations and no chromatic structures in common with any of the other tunings explored in this study. However, it does contain an intriguing arrangement of the two distinct pentatonic modes of Java and Bali, known as ''pelog'' and ''slendro'' — modes that cannot be realistically approximated in 12-note tuning.</blockquote>

With some (too) quick analysis, we see that the modes he used are as follows:

* Pelog: '''7 3 3 7 3'''

* Slendro: '''5 4 5 4 5'''

{|class="wikitable"

|+ Blackwood's ''pelog''

! Interval !! Cents

! Antidiatonic name<ref>When I use the term "antidiatonic" rather than "2L5s", I'm using '''harmonic''' notation, where flats sharpen and sharps flatten.</ref>

! [[46edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name(s)

! Pseudo-diatonic interval name

|-

| 0\23 || 0.00 || C || C || Perfect unison

|-

| 7\23 || 365.22 || E&flat; || ^D&sharp;, vvE || Neutral third

|-

| 10\23 || 521.74 || F || ^F || Major fourth

|-

| 13\23 || 678.26 || G || vG || Minor fifth

|-

| 20\23 || 1043.48 || B&flat; || ^^B&flat; || Neutral seventh

|-

| 23\23 || 1200.00 || C || C || Perfect octave

|}

{|class="wikitable"

|+ Blackwood's ''slendro''

! Interval !! Cents

! Antidiatonic name(s) !! 46edo/GDF name(s)

! Pseudo-diatonic interval name

|-

| 0\23 || 0.00 || C || C || Perfect unison

|-

| 5\23 || 260.87 || D&flat;&flat;, E&sharp; || ^^D, vE&flat; || Supermajor second, subminor third

|-

| 9\23 || 469.57 || F&sharp; || vF || Minor fourth

|-

| 14\23 || 730.43 || G&flat; || ^G || Major fifth

|-

| 18\23 || 939.13 || A&flat;&flat;, B&sharp; || ^A, vvB&flat; || Supermajor sixth, subminor seventh

|-

| 23\23 || 1200.00 || C || C || Perfect octave

|}

What's interesting about these two modes is that both kinds of fifth (major and minor) are featured in some form in each scale, and by extension, both types of fourths as well, which ostensibly demonstrates that ''both'' kinds of fourth/fifth can be used tastefully, even if in an "exotic" context.

@@ Line 3: / Line 3: @@
 Outside of free just intonation, most of my xenharmonic work is exclusively in [[EDO|EDOs]], generally various EDOs smaller than [[36edo]] (although I have used larger ones in the past).
-I work almost exclusively in notation software, so it's important that EDOs I'm working with can be worked into traditional [[5L 2s|diatonic]] notation in some form. However, as one would know, this is only the case for an EDO if it's approximation of [[3/2]] is between 686&cent; (5\7) and 720&cent; (3\5), which is not always the case.
+I work almost exclusively in notation software (although that is changing), so it's important that EDOs I'm working with can be worked into traditional [[5L 2s|diatonic]] notation in some form; from there, I can usually figure out the quirks of a tuning and its rational approximations from there.
-Here are some of my own theoretical and notational approaches to various EDOs that break this mold.
+However, as one would know, this is only the case for an EDO if its approximation of [[3/2]] is between 686.71&cent; (4\7) and 720.00&cent; (3\5), which is not always the case. There are some methods by which they can still be worked in harmonically through various alternative rational approximations, although this ignores the notation debacle; even then, ''still'' some others would appear completely impossible with such an approach.
+Here are some of my own theoretical and notational approaches to various EDOs that break these molds.
 ==13edo==
@@ Line 23: / Line 25: @@
 Scales generated by the 2\13 will also feature 4\13&mdash;a flat but recognizable [[5/4]]&mdash;and 6\13&mdash;a dead-on [[11/8]]. In this way, a basic [[6L 1s|6L1s]] "archaeotonic" scale is produced.
-Any mode can be used, but the symmetrical 3|3 "Holthathian" mode captures 13edo's best intervals:
+Any mode can be used, but the symmetrical 3|3 "Horthathian" mode captures 13edo's best intervals:
 {| class="wikitable"
@@ Line 30: / Line 32: @@
 !Ratios
 !Note name
-![[26edo]] name
+![[26edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name
 !Pseudo-diatonic interval name
 |-
@@ Line 58: / Line 60: @@
 |[[11/8]]
 |F
-|F&sharp;
+|F&sharp;, ^F
 |Major fourth
 |-
@@ Line 65: / Line 67: @@
 |[[16/11]]
 |G&flat;
-|G&flat;
+|G&flat;, vG
 |Minor fifth
 |-
@@ Line 72: / Line 74: @@
 |[[13/8]], [[8/5]]
 |A&flat;
-|A&flat;
+|A&flat;, vA
 |Minor sixth
 |-
@@ Line 79: / Line 81: @@
 |[[16/9]], [[9/5]]
 |B&flat;
-|B&flat;
+|B&flat;, vB
 |Minor seventh
 |-
@@ Line 98: / Line 100: @@
 ===A note on fifths===
-edo, of course, has notoriously bad fifths&mdash;to be specific, ''two'' bad fifths: the 7\13 minor fifth of 646&cent; and the 8\13 major fifth of 738&cent;. While these fifths may be useless harmonically, cases can be made for their use ''melodically'', specifically for the major fifth.
+edo, of course, has notoriously bad fifths&mdash;to be specific, ''two'' bad fifths: the 7\13 minor fifth of 646&cent; and the 8\13 major fifth of 738&cent; (making a [[dual-fifth]] system). While these fifths may be useless harmonically, cases can be made for their use ''melodically'', specifically for the major fifth.
 Consider the use of the harmonic minor scale in traditional 12edo theory. The replacement of the minor seventh by the major seventh exists in order to make the chord on the fifth degree of the minor scale a major chord rather than a minor chord. In a typical V - i cadential progression, this replacement adds tension since the third of the V chord is only a semitone below the tonic, and wants to resolve upward to complete the progression.
@@ Line 180: / Line 182: @@
 |*
 |}
+===Blackwood's "subminor"===
+[[Easley Blackwood]]'s short blurb on 13edo as part of the booklet packaged with the CD copy of his ''Twelve Microtonal Etudes'' states:
+<blockquote>Yet even this tuning contains a strange mode best described
+as “sub-minor”.<ref>https://www.cedillerecords.org/wp-content/uploads/2020/01/018-blackwood-microtonal-booklet.pdf</ref></blockquote>
+He does not elaborate on the construction of this mode, but a quick analysis of his 13edo etude reveals it to be the 5|2 "Celephaïsian" mode of [[5L 3s|5L3s]] "oneirotonic". Although not in the modern sense, Blackwood aptly calls this mode "sub-minor" because its construction is ''LsLLsLLs'', which is identical to the familiar [[5L 2s|5L2s]] diatonic Aeolian (minor) mode ''LsLLsLL'' but with an extra ''s'' appended to the end, compressing all of the intervals slightly.
+{|class="wikitable"
+! Interval !! Cents !! Note name (in 6L1s) !! Note name (in 5L3s)
+! [[26edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] Names !! Pseudo-diatonic interval name
+|-
+| 0\13 || 0.00 || C || C || C || Perfect unison
+|-
+| 2\13 || 184.62 || D || D || D || Major second
+|-
+| 3\13 || 276.93 || E&flat; || E&flat; || E&flat;&flat;, vE&flat; || Minor third
+|-
+| 5\13 || 461.54 || F&flat; || F || F&flat;, vF || Major fourth
+|-
+| 7\13 || 646.15 || G&flat; || G || G&flat;, vG || Minor fifth
+|-
+| 8\13 || 738.46 || G || H&flat; || G&sharp;, ^G || Major fifth
+|-
+| 10\13 || 923.08 || A || A || A&sharp;, ^A || Major sixth
+|-
+| 12\13 || 1107.69 || B || B || B&sharp;, ^B || Major seventh
+|-
+| 13\13 || 1200.00 || C || C || C || Perfect octave
+|}
+Blackwood continues briefly on his use of the mode:
+<blockquote>The first four bars of the Etude are an arrangement of this mode into consecutive thirds — a motif that recurs later in two transposed variations. The rest of the piece is comprised of chromatic resolutions of complex altered chords.</blockquote>
+Indeed, the thirds of this scale as well as the aforementioned [[6L 1s|6L1s]] remain useful, since the thirds (two-mosstep intervals) in both scales are always either 3\13 minor thirds or 4\13 major thirds, which aren't all that inaccurate from JI.
+===A note on the 7/4===
+My favorite kinds of xenharmonic intervals are those that involve the factor of seven somehow, usually approximations of [[7/4]], [[7/6]], or [[8/7]].
+edo's approximation of the harmonic seventh 7/4 is 10\13 or 923&cent;, which, at a whopping '''45 cents flat''', is even more inaccurate than its approximation of [[3/2]]. Despite this, in practical use, 10\13 (to my ears, at least) appears to be a ''usable'' 7/4, at least when used sparingly among other passages that do 13edo "correctly".
+My hypotheses as to why:
+* I personally have more experience with [[24edo]] which has a pretty flat 7/4.
+* My ears might be confusing it for [[12/7]]: 10\13 is much closer to 12/7, which is roughly in the same quartertonal interval region (supermajor sixth/subminor seventh) and is also harmonically related to 7/4 (its [[3/1|tritave]] inversion).
+* My ears might be confusing it for [[55/32]]: The context in which I've used 10\13 the most often is in 0,4,6,10\13 or its subsets. In that chord, there are two 4\13 major thirds (approximating [[5/4]] well) separated by a 6\13 (approximating [[11/8]] well), so it could be said that 10\13 is acting as a 55/32 here. A just intonation chord 1/1, 5/4, 11/8 can reasonably be topped with either 7/4 ''or'' 55/32, and the two are only a [[56/55]] apart.
 ==8edo==
@@ Line 246: / Line 297: @@
 [[File:8edo-dem1.mp3]]
-Like the 13edo progression, this progression largely receives its cohesion melodically. This progression moves the bass by 5\8, the 750&cent; superfifth, still recognizable in this instance as a fifth despite its intrusion into the minor sixth interval space. Steps are larger, so the progression must start on the 2\8 minor third rather than any kind of major third.
+Like the 13edo progression, the fifth is incorporated melodically. This progression moves the bass by 5\8, the 750&cent; superfifth, still recognizable in this instance as a fifth despite its intrusion into the minor sixth interval space. Steps are larger, so the progression must start on the 2\8 minor third rather than any kind of major third.
+==23edo==
+To me, [[23edo]] is proof that "terrible" rational approximations might not be as terrible as the theorist's approximation-centered mind might believe. Its approximations of primes 3, 5, 7, and 11 are all at least 20 cents off, and yet, even with some beating, there is a clearly-recognizable major chord 0,7,13\23 and arguably even a 4:5:6:7:9:11 in the form of 0,7,13,18,27,33\23 (although your mileage may vary with the 33\23). This along with its quartertonal step size makes it a particularly rewarding system to conquer.
+edo is technically [[dual-fifth]] like [[13edo]], the fifths being 13\23 and 14\23. [[2L 5s|Antidiatonic]] seems more natural here, especially since the minor fifth is recognizable as a perfect fifth (the major fifth is far too sharp), but this means that the 3\23 neutral seconds get the role of major seconds rather than the very good [[9/8]] approximation, 4\23. Atonally, [[46edo]] subset notation will work best.
+===Listen for yourself===
+That being said, here are some audio demonstrations of the various split prime approximations so you can make a decision yourself as to which version is better.
+'''Make sure your volume is set to about 50%, otherwise these may sound quite loud.''' You can alternatively listen to these comparisons in series on [https://luphoria.com/xenpaper/#%7B23edo%7D(osc%3Atriangle)(bpm%3A50)%0A%7Br220Hz%7D%0A%5B0_13%5D--._%5B0_14%5D--._%23_3%2F2_perfect_fifths%0A..%0A%5B0_7%5D--._%5B0_8%5D--._%23_5%2F4_major_thirds%0A..%0A%5B0_18%5D--._%5B0_19%5D--._%23_7%2F4_harmonic_sevenths%0A..%0A%5B0_'10%5D--._%5B0_'11%5D--._%23_11%2F4_harmonic_elevenths Xenpaper].
+{|
+| 5/4 || major third || 7\23 || A-vvC&sharp; || 8\23 || A-C&sharp; || [[File:Comparison-23edo-thirds.mp3]]
+|-
+| 3/2 || perfect fifth || 13\23 || A-vE || 14\23 || A-^E || [[File:Comparison-23edo-fifths.mp3]]
+|-
+| 7/4 || harmonic seventh || 18\23 || A-vvG || 19\23 || A-G || [[File:Comparison-23edo-sevenths.mp3]]
+|-
+| 11/4 || harmonic eleventh || 33\23 || A-^D || 34\23 || A-E&flat; || [[File:Comparison-23edo-elevenths.mp3]]
+|}
+My preferences lie in 0,7,13,18\23 for a 4:5:6:7, and is think either 33\23 or 34\23 could work for an 11/4. Both sound to me equally consonant, but the higher one has less beating when octave reduced (to 11\23). I personally prefer 33\23 (10\23) since I feel 34\23 (11\23) is too close to the tritone interval region; if there's any reason 11\23 has less beating, it's because it's actually a [[7/5]].
+===Seconds===
+Even though 23edo has a [[9/8]] only five cents off, this does not help us in devising a heptatonic notation system like it did with 13edo.
+I once idiotically presumed that [[6L 1s|6L1s]] might work well here since the 9/8 is solid and that system works on major seconds, but at about 208&cent;, the major seconds are actually bigger than 1\6, which means they're too large. Even if such a system could exist, it wouldn't be anywhere near as useful in 23edo as it was in [[13edo]]. Three of the much flatter, 2\13 major seconds form a great [[11/8]] (6\13) whereas three of the 4\23 seconds stack to a 12\23, which is far too high to be an 11/8.
+The 3\23 ''neutral'' seconds are arguably superior for a second-generated system. These are too small to generate 6L1s, however; instead, they generate [[1L 6s|1L6s]] "onyx" or "antiarchaeotonic", or [[7L 1s|7L1s]] "pine" if we add another generator. Both of these give access to the 6\23 minor third and its inverse the 17\23 major sixth, very good approximations of [[6/5]] and [[5/3]] respectively.
+Using the 1L6s scale for a heptatonic system of nominals gives us a notation not unlike antidiatonic due to its emphasis of the neutral second. If we use 0|6 for the nominals (the "antiarchaeotonic" equivalent of 13edo's 6L1s notation), the symmetrical 3|3 mode will look like this:
+{|class="wikitable"
+! Interval !! Cents !! Ratio(s) !! 1L6s name (0&vert;6)
+! [[2L 5s|2L5s]] name<ref>I would usually notate antidiatonic with harmonic notation (where flats sharpen and sharps flatten), but I'm doing it the other way here to demonstrate the similarities between 2L5s and 1L6s.</ref>
+! [[46edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name !! Pseudo-diatonic interval name
+|-
+| 0\23 || 0.00 || 1/1 || C || C || C || Perfect Unison
+|-
+| 3\23 || 156.52 || [[11/10]], [[12/11]] || D || D || vvD || Neutral second
+|-
+| 6\23 || 313.04 || [[6/5]] || E || E || ^E&flat; || Minor third
+|-
+| 9\23 || 469.57 || [[21/16]] || F || F&flat; || vF || Minor fourth
+|-
+| 14\23 || 730.43 || [[32/21]] || G&sharp; || G&sharp; || ^G || Major fifth
+|-
+| 17\23 || 886.96 || [[5/3]] || A&sharp; || A&sharp; || vA || Major sixth
+|-
+| 20\23 || 1043.48 || [[11/6]] || B&sharp; || B&sharp; || ^^B&flat; || Neutral seventh
+|-
+| 23\23 || 1200.00 || [[2/1]] || C || C || C || Perfect octave
+|}
+This system is similar to antidiatonic in the same way that 6L1s from earlier is similar to diatonic: B-C is the only ''large'' step here rather than it being the only small step. This 1L6s system might work well if you find yourself preferring the 14\23 major fifth rather than the 13\23 minor fifth, since it features as the fifth in this scale.
+===Relative to 24edo===
+It might help to perceive 23edo as a system that "phases out" from [[24edo]] in the same way that [[11edo]] "phases out" from [[12edo]].
+The 23edo's "quarter tone" is barely more than two cents larger than 24edo's, which means that:
+* "Semitones" (2\23) are four cents larger.
+* Whole tones (4\23) are eight cents larger, making them about as good of a [[9/8]] as [[12edo|12]] or [[24edo]]'s but sharp instead of flat.
+* Minor thirds (6\23) are thirteen cents larger, making them very good [[6/5]]'s.
+* "Neutral thirds" (7\23) are fifteen cents larger, pushing them closer to [[5/4]]'s.
+* "Major thirds" (8\23) are now seventeen cents larger, making them distinctively Pythagorean.
+* "Perfect fourths" (10\23) are 21 cents larger, pushing the boundary between ''perfect fourth'' and ''[[superfourth]]''.
+Although by no means consistent across keys, this also introduces the idea of a notation system like [[24edo]]'s but with a missing step between F and G where the 1\2 tritone would be, which introduces the enharmonic equivalences:
+* '''F''' = '''G{{sesquiflat2}}'''
+* '''F{{demisharp2}}''' = '''G&flat;'''
+* '''F&sharp;''' = '''G{{demiflat2}}'''
+* '''F{{sesquisharp2}}''' = '''G'''
+This system naturally favors the major fourth, minor fifth, and the ''proper'' major second, although this only applies for a couple of keys: The G&harr;D fifth, for instance, is major, and the F&harr;G second is obviously neutral.
+===On Blackwood's approach===
+The 23edo etude was the only etude of [[Easley Blackwood|Blackwood]]'s twelve etudes that uses entirely non-Western scales, specifically because, as he states:
+<blockquote>&hellip;23-note tuning contains no diatonic configurations and no chromatic structures in common with any of the other tunings explored in this study. However, it does contain an intriguing arrangement of the two distinct pentatonic modes of Java and Bali, known as ''pelog'' and ''slendro'' — modes that cannot be realistically approximated in 12-note tuning.</blockquote>
+With some (too) quick analysis, we see that the modes he used are as follows:
+* Pelog: '''7 3 3 7 3'''
+* Slendro: '''5 4 5 4 5'''
+{|class="wikitable"
+|+ Blackwood's ''pelog''
+! Interval !! Cents
+! Antidiatonic name<ref>When I use the term "antidiatonic" rather than "2L5s", I'm using '''harmonic''' notation, where flats sharpen and sharps flatten.</ref>
+! [[46edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name(s)
+! Pseudo-diatonic interval name
+|-
+| 0\23 || 0.00 || C || C || Perfect unison
+|-
+| 7\23 || 365.22 || E&flat; || ^D&sharp;, vvE || Neutral third
+|-
+| 10\23 || 521.74 || F || ^F || Major fourth
+|-
+| 13\23 || 678.26 || G || vG || Minor fifth
+|-
+| 20\23 || 1043.48 || B&flat; || ^^B&flat; || Neutral seventh
+|-
+| 23\23 || 1200.00 || C || C || Perfect octave
+|}
+{|class="wikitable"
+|+ Blackwood's ''slendro''
+! Interval !! Cents
+! Antidiatonic name(s) !! 46edo/GDF name(s)
+! Pseudo-diatonic interval name
+|-
+| 0\23 || 0.00 || C || C || Perfect unison
+|-
+| 5\23 || 260.87 || D&flat;&flat;, E&sharp; || ^^D, vE&flat; || Supermajor second, subminor third
+|-
+| 9\23 || 469.57 || F&sharp; || vF || Minor fourth
+|-
+| 14\23 || 730.43 || G&flat; || ^G || Major fifth
+|-
+| 18\23 || 939.13 || A&flat;&flat;, B&sharp; || ^A, vvB&flat; || Supermajor sixth, subminor seventh
+|-
+| 23\23 || 1200.00 || C || C || Perfect octave
+|}
+What's interesting about these two modes is that both kinds of fifth (major and minor) are featured in some form in each scale, and by extension, both types of fourths as well, which ostensibly demonstrates that ''both'' kinds of fourth/fifth can be used tastefully, even if in an "exotic" context.