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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Line 97:

Using a subset of 26edo as a notation system, as you can see above, is also an option, and works best for modal or atonal music in 13edo, since it provides a much more intuitive grasp of 13edo's intervals outside of any particular scale.

===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”.</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 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.

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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.

===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 alike to the 6L1s system shown earlier for 13edo but with the step sizes swapped: B-C is now the only ''large'' step. 1L6s 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.

@@ 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 95: / Line 97: @@
 Using a subset of 26edo as a notation system, as you can see above, is also an option, and works best for modal or atonal music in 13edo, since it provides a much more intuitive grasp of 13edo's intervals outside of any particular scale.
+===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”.</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 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 257: / 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.
+===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 alike to the 6L1s system shown earlier for 13edo but with the step sizes swapped: B-C is now the only ''large'' step. 1L6s 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.
+<!-- Next write a section about all the split primes WITH AUDIO DEMONSTRATIONS -->