User:TromboneBoi9/Approaches to weird EDOs - Revision history

TromboneBoi9: added "Relative to 24edo" and "On Blackwood's approach" sections under 23edo

2025-11-21T02:31:44Z

added "Relative to 24edo" and "On Blackwood's approach" sections under 23edo

TromboneBoi9: added "Listen for yourself" section in 23edo

2025-11-19T21:37:47Z

added "Listen for yourself" section in 23edo

@@ Line 304: / Line 304: @@
 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===
@@ Line 337: / Line 355: @@
 |}
-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.
+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.
 <!-- Next write a section about all the split primes WITH AUDIO DEMONSTRATIONS -->

TromboneBoi9: added "Blackwood's subminor" section on 13edo and started 23edo section

2025-11-19T10:11:02Z

added "Blackwood's subminor" section on 13edo and started 23edo section

Show changes

TromboneBoi9: added small "a note on the 7/4" section

2025-11-04T06:45:06Z

added small "a note on the 7/4" section

@@ Line 180: / Line 180: @@
 |*
 |}
 ==8edo==

TromboneBoi9: added small 13b edo section

2025-06-18T23:34:30Z

added small 13b edo section

← Older revision		Revision as of 23:34, 18 June 2025
Line 112:		Line 112:

	Movement down by the 7\13 minor fifth in a similar fashion is possible, but can only consonantly be done by starting on a 5\13 minor fourth, which is a well-approximated [[21/16]].		Movement down by the 7\13 minor fifth in a similar fashion is possible, but can only consonantly be done by starting on a 5\13 minor fourth, which is a well-approximated [[21/16]].

			===13''b'' edo===

			If you ''really'' want to, you can do as I once did years back when I wanted to notate 13edo: use an [[antidiatonic]] (2L5s) notation generated by the 7\13 minor fifth (13edo's ''second''-best 3/2 approximation), rather than archaeotonic or oneirotonic.

			This creates a ''very'' hard antidiatonic system, with "minor seconds" four times the size of "major seconds."

			{\| class="wikitable"
			!
			!0
			!1
			!2
			!3
			!4
			!5
			!6
			!7
			!8
			!9
			!10
			!11
			!12
			\|-
			![[6L 1s\|6L1s]] notation
			\|C
			\|*
			\|D
			\|*
			\|E
			\|*
			\|F
			\|*
			\|G
			\|*
			\|A
			\|*
			\|B
			\|-
			![[5L 3s\|5L3s]] notation
			\|C
			\|*
			\|D
			\|*
			\|E
			\|F
			\|*
			\|G
			\|*
			\|H
			\|A
			\|*
			\|B
			\|-
			![[2L 5s\|2L5s]] notation
			\|C
			\|D
			\|E
			\|*
			\|*
			\|*
			\|F
			\|G
			\|A
			\|B
			\|*
			\|*
			\|*
			\|}

	==8edo==		==8edo==

TromboneBoi9: added example to 8edo section

2025-06-15T04:55:20Z

added example to 8edo section

← Older revision		Revision as of 04:55, 15 June 2025
Line 31:		Line 31:
	!Note name		!Note name
	![[26edo]] name		![[26edo]] name
	!Pseudo-diatonic name		!Pseudo-diatonic interval name
	\|-		\|-
	\|0\13		\|0\13
Line 117:		Line 117:
	[[8edo]] is best taken free and atonally, and is far and above best notated as a [[24edo]] subset.		[[8edo]] is best taken free and atonally, and is far and above best notated as a [[24edo]] subset.

	The usage of 8edo—or rather, the three-quartertone scale—requires the acceptance of ~~dissonant~~ altered intervals, since the only pure prime it approximates with any decency is [[19/16\|19]] (in the form of 2\8, which is the same as 3\12). Even so, 8edo is so small that it is best treated as a scale within a whole ''24edo'' framework.		The usage of 8edo—or rather, the three-quartertone scale—requires the acceptance of strangely altered intervals, since the only pure prime it approximates with any decency is [[19/16\|19]] (in the form of 2\8, which is the same as 3\12). Even so, 8edo is so small that it is best treated as a scale within a whole ''24edo'' framework. [[MOS scales]] are certainly possible—albeit sparse since 8 is a small composite number—but it's already a reasonably-sized scale at its own size.

	~~That being said, if we were to whittle a simpler scale out of 8edo,~~ [[~~3L 2s\|3L2s~~]] is ~~the natural choice, generated by 5\8, the 750~~&~~cent~~; ~~"fifthoid~~."

	{\| class="wikitable"		{\| class="wikitable"
Line 125:		Line 123:
	!Cents		!Cents
	!24edo name		!24edo name
	!~~3L2s~~ name		!Interval name
	\|-		\|-
	\|0\8		\|0\8
	\|0		\|0
	\|C		\|C
	\|C		\|Perfect unison
	\|-		\|-
	\|1\8		\|1\8
	\|150		\|150
	\|vD		\|vD
	\|~~C&sharp;~~		\|Neutral second
	\|-		\|-
	\|2\8		\|2\8
	\|300		\|300
	\|E&flat;		\|E&flat;
	\|D		\|Minor third
	\|-		\|-
	\|3\8		\|3\8
	\|450		\|450
	\|vF		\|^E, vF
	\|~~D&sharp;~~		\|Supermajor third, Subfourth
	\|-		\|-
	\|4\8		\|4\8
	\|600		\|600
	\|F&sharp;		\|F&sharp;, G&flat;
	\|E		\|Tritone
	\|-		\|-
	\|5\8		\|5\8
	\|750		\|750
	\|^G		\|^G, vA&flat;
	\|G		\|Superfifth, Subminor sixth
	\|-		\|-
	\|6\8		\|6\8
	\|900		\|900
	\|A		\|A
	\|~~G&sharp;~~		\|Major sixth
	\|-		\|-
	\|7\8		\|7\8
	\|1050		\|1050
	\|vB		\|vB
	\|A		\|Neutral seventh
	\|-		\|-
	\|8\8		\|8\8
	\|1200		\|1200
	\|C		\|C
	\|C		\|Perfect octave
	\|}		\|}

	~~The 3L2s~~ notation ~~system you see here uses~~ a ~~pentatonic CDEGA nominal scheme with~~ the ~~4\|0 mode~~.		That being said, by themselves, many of the intervals aren't terrible. Connoisseurs of 24edo ([[Wyschnegradsky]], anyone?) would obviously find 8edo rather stimulating, with a perfect one-to-one mixture of traditional intervals and quartertonally-altered intervals.

			Here's a progression similar to the "V - i"-esque progression I demonstrated in 13edo but in 8edo (24edo subset notation with Stein-Zimmerman quartertone accidentals, numbers show edosteps in 8edo):

			[[File:8edo-dem1.png\|360px]]
			[[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.

TromboneBoi9: Small edits in 13edo section, started 8edo section

2025-06-14T22:01:47Z

Small edits in 13edo section, started 8edo section

@@ Line 16: / Line 16: @@
 Thus, while 11edo and 13edo may be lost causes for traditional approaches, there is still potential from the JI approximation perspective.
 For instance, the 2\13 interval is about 184&cent;, which is a fine major second; it's two cents flat of [[10/9]] and can act as a [[9/8]] if need be. This makes it a reasonable "fundamental consonance," taking the place of the fifth in traditional theory.
@@ Line 90: / Line 92: @@
 You can also see demonstrated in the table above a useful notation system based on 6L1s, specifically the 6|0 "Ryonian" mode. This notation scheme is identical to the traditional 12edo notation system, except there is an extra step between E and F; E&sharp; and F&flat; become enharmonics.
-A potential downside to the compositional and notational use of 6L1s as a tonal system is that its small steps are too sparse, making it sound too much like an equalized [[6edo|whole tone scale]] melodically if the small step is not somehow emphasized. This is opposed to the much more popular [[5L 3s|5L3s]] "oneirotonic" system which, while generated on the dissonant 8\13 "major fifth," is far more melodically captivating; it also happens to support [[18edo]], another problematic EDO. 5L3s, however, is octatonic rather than heptatonic, which sacrifices clarity in staff notation ''greatly'' (since octaves will appear like ninths).
+A potential downside to the compositional and notational use of 6L1s as a tonal system is that its small steps are too sparse, which will make it sound too much like an equalized [[6edo|whole tone scale]] melodically if the small step is not somehow emphasized. This is opposed to the much more popular [[5L 3s|5L3s]] "oneirotonic" system which. While generated on the dissonant 8\13 "major fifth," it's capable of creating more diatonic-like melody. (It also happens to support [[18edo]], another problematic EDO.) 5L3s, however, is octatonic rather than heptatonic, which sacrifices clarity in staff notation ''greatly'' (since octaves will appear like ninths).
 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.
@@ Line 96: / Line 98: @@
 ===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;. (I personally perceive the minor fifth to be more consonant, but only ''very'' marginally so.) 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;. 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 109: / Line 111: @@
 This can also be done with the 6\13 major fourth instead of the 4\13 major third; both are consonances one step from 5\13.
-Movement down by the 7\13 minor fifth in a similar fashion is possible, but can only consonantly be done by starting on a 5\13 minor fourth, which is very close to [[21/16]]; even then, this is best suited as a modulation.
+Movement down by the 7\13 minor fifth in a similar fashion is possible, but can only consonantly be done by starting on a 5\13 minor fourth, which is a well-approximated [[21/16]].

TromboneBoi9: some edits, "note on fifths" for 13edo section

2025-06-14T04:24:27Z

some edits, "note on fifths" for 13edo section

← Older revision		Revision as of 04:24, 14 June 2025
Line 11:		Line 11:
	I consider [[13edo]] to be one of two EDOs which "phases out" from [[12edo]], the other being [[11edo]]. These EDOs have specific properties in relation to 12edo:		I consider [[13edo]] to be one of two EDOs which "phases out" from [[12edo]], the other being [[11edo]]. These EDOs have specific properties in relation to 12edo:

	* Intervals in the second and seventh interval regions are relatively close to their 12edo equivalents; consequently, various intervals like [[9/8]], [[10/9]], [[9/5]], [[16/9]], etc. may find good approximations.		* Intervals in the second and seventh interval regions are relatively close to their 12edo equivalents; consequently, various intervals like [[8/7]], [[9/8]], [[10/9]], [[9/5]], [[16/9]], etc. may find good approximations.
	* Intervals around the third and sixth interval regions are somewhat detuned; consequently, various intervals like [[5/4]], [[7/6]], [[13/8]], [[5/3]], etc. may be approximated depending on the EDO.		* Intervals around the third and sixth interval regions are somewhat detuned; consequently, various intervals like [[5/4]], [[7/6]], [[13/8]], [[5/3]], etc. may be approximated depending on the EDO.
	* Intervals around the fourth and fifth interval regions are nearly a quartertone off from their 12edo equivalents; consequently, [[11/8]] and [[16/11]] are approximated relatively well.		* Intervals around the fourth and fifth interval regions are nearly a quartertone off from their 12edo equivalents; consequently, [[11/8]] and [[16/11]] are approximated relatively well, and there is more or less no [[4/3]] or [[3/2]].

	Thus, while 11edo and 13edo may be lost causes for traditional approaches, there is still potential from the JI approximation perspective.		Thus, while 11edo and 13edo may be lost causes for traditional approaches, there is still potential from the JI approximation perspective.
Line 19:		Line 19:
	For instance, the 2\13 interval is about 184¢, which is a fine major second; it's two cents flat of [[10/9]] and can act as a [[9/8]] if need be. This makes it a reasonable "fundamental consonance," taking the place of the fifth in traditional theory.		For instance, the 2\13 interval is about 184¢, which is a fine major second; it's two cents flat of [[10/9]] and can act as a [[9/8]] if need be. This makes it a reasonable "fundamental consonance," taking the place of the fifth in traditional theory.

	Scales generated by the 2\13 will also feature 4\13&~~ndash~~;a flat but recognizable [[5/4]]&~~ndash~~;and 6\13&~~ndash~~;a dead-on [[11/8]]~~–~~. In this way, a basic [[6L 1s\|6L1s]] "archaeotonic" scale is produced.		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 "Holthathian" mode captures 13edo's best intervals:
Line 28:		Line 28:
	!Ratios		!Ratios
	!Note name		!Note name
			![[26edo]] name
	!Pseudo-diatonic name		!Pseudo-diatonic name
	\|-		\|-
Line 33:		Line 34:
	\|0.00		\|0.00
	\|[[1/1]]		\|[[1/1]]
			\|C
	\|C		\|C
	\|Perfect unison		\|Perfect unison
Line 39:		Line 41:
	\|184.62		\|184.62
	\|[[10/9]], [[9/8]]		\|[[10/9]], [[9/8]]
			\|D
	\|D		\|D
	\|Major second		\|Major second
Line 45:		Line 48:
	\|369.23		\|369.23
	\|[[5/4]], [[16/13]]		\|[[5/4]], [[16/13]]
			\|E
	\|E		\|E
	\|Major third		\|Major third
Line 52:		Line 56:
	\|[[11/8]]		\|[[11/8]]
	\|F		\|F
			\|F&sharp;
	\|Major fourth		\|Major fourth
	\|-		\|-
Line 57:		Line 62:
	\|646.15		\|646.15
	\|[[16/11]]		\|[[16/11]]
			\|G&flat;
	\|G&flat;		\|G&flat;
	\|Minor fifth		\|Minor fifth
Line 63:		Line 69:
	\|830.77		\|830.77
	\|[[13/8]], [[8/5]]		\|[[13/8]], [[8/5]]
			\|A&flat;
	\|A&flat;		\|A&flat;
	\|Minor sixth		\|Minor sixth
Line 69:		Line 76:
	\|1015.38		\|1015.38
	\|[[16/9]], [[9/5]]		\|[[16/9]], [[9/5]]
			\|B&flat;
	\|B&flat;		\|B&flat;
	\|Minor seventh		\|Minor seventh
Line 75:		Line 83:
	\|1200.00		\|1200.00
	\|[[2/1]]		\|[[2/1]]
			\|C
	\|C		\|C
	\|Perfect octave		\|Perfect octave
Line 81:		Line 90:
	You can also see demonstrated in the table above a useful notation system based on 6L1s, specifically the 6\|0 "Ryonian" mode. This notation scheme is identical to the traditional 12edo notation system, except there is an extra step between E and F; E&sharp; and F&flat; become enharmonics.		You can also see demonstrated in the table above a useful notation system based on 6L1s, specifically the 6\|0 "Ryonian" mode. This notation scheme is identical to the traditional 12edo notation system, except there is an extra step between E and F; E&sharp; and F&flat; become enharmonics.

	A potential downside to the compositional and notational use of 6L1s as a tonal system is that its small steps are too sparse, making it sound too much like an equalized [[6edo\|whole tone scale]] melodically. This is opposed to the much more popular [[5L 3s\|5L3s]] "oneirotonic" system which, while generated on the dissonant 8\13 "major fifth," is far more melodically captivating; it also happens to support [[18edo]], another problematic EDO. 5L3s, however, is octatonic rather than heptatonic, which sacrifices clarity in staff notation ''greatly''.		A potential downside to the compositional and notational use of 6L1s as a tonal system is that its small steps are too sparse, making it sound too much like an equalized [[6edo\|whole tone scale]] melodically if the small step is not somehow emphasized. This is opposed to the much more popular [[5L 3s\|5L3s]] "oneirotonic" system which, while generated on the dissonant 8\13 "major fifth," is far more melodically captivating; it also happens to support [[18edo]], another problematic EDO. 5L3s, however, is octatonic rather than heptatonic, which sacrifices clarity in staff notation ''greatly'' (since octaves will appear like ninths).

			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.

			===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¢. (I personally perceive the minor fifth to be more consonant, but only ''very'' marginally so.) 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.

			A similar progression can be rendered in 13edo, since there is a recognizable major third: 4\13. Starting on a simple major third dyad, 0\13 and 4\13, if the major third moves up one step in the same way it would in 12edo, we land on 5\13. The bass note, then, would shoot from 0\13 to 5\13 an octave below—a distance of a 8\13 major fifth.

			Here's a demonstration in 26edo subset notation (Numbers show edosteps in 13edo):

			[[File:13edo-dem1.png\|240px]]
			[[File:13edo-dem1.mp3]]

			This can also be done with the 6\13 major fourth instead of the 4\13 major third; both are consonances one step from 5\13.

			Movement down by the 7\13 minor fifth in a similar fashion is possible, but can only consonantly be done by starting on a 5\13 minor fourth, which is very close to [[21/16]]; even then, this is best suited as a modulation.

TromboneBoi9: Created page, started 13edo section

2025-06-13T06:00:06Z

Created page, started 13edo section

New page

'''THIS PAGE IS A WORK IN PROGRESS AND WILL DEVELOP AS MY THEORIES DEVELOP'''

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.

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

==13edo==

I consider [[13edo]] to be one of two EDOs which "phases out" from [[12edo]], the other being [[11edo]]. These EDOs have specific properties in relation to 12edo:

* Intervals in the second and seventh interval regions are relatively close to their 12edo equivalents; consequently, various intervals like [[9/8]], [[10/9]], [[9/5]], [[16/9]], etc. may find good approximations.
* Intervals around the third and sixth interval regions are somewhat detuned; consequently, various intervals like [[5/4]], [[7/6]], [[13/8]], [[5/3]], etc. may be approximated depending on the EDO.
* Intervals around the fourth and fifth interval regions are nearly a quartertone off from their 12edo equivalents; consequently, [[11/8]] and [[16/11]] are approximated relatively well.

Thus, while 11edo and 13edo may be lost causes for traditional approaches, there is still potential from the JI approximation perspective.

For instance, the 2\13 interval is about 184¢, which is a fine major second; it's two cents flat of [[10/9]] and can act as a [[9/8]] if need be. This makes it a reasonable "fundamental consonance," taking the place of the fifth in traditional theory.

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:

{| class="wikitable"
!Interval
!Cents
!Ratios
!Note name
!Pseudo-diatonic name
|-
|0\13
|0.00
|[[1/1]]
|C
|Perfect unison
|-
|2\13
|184.62
|[[10/9]], [[9/8]]
|D
|Major second
|-
|4\13
|369.23
|[[5/4]], [[16/13]]
|E
|Major third
|-
|6\13
|553.85
|[[11/8]]
|F
|Major fourth
|-
|7\13
|646.15
|[[16/11]]
|G&flat;
|Minor fifth
|-
|9\13
|830.77
|[[13/8]], [[8/5]]
|A&flat;
|Minor sixth
|-
|11\13
|1015.38
|[[16/9]], [[9/5]]
|B&flat;
|Minor seventh
|-
|13\13
|1200.00
|[[2/1]]
|C
|Perfect octave
|}

You can also see demonstrated in the table above a useful notation system based on 6L1s, specifically the 6|0 "Ryonian" mode. This notation scheme is identical to the traditional 12edo notation system, except there is an extra step between E and F; E&sharp; and F&flat; become enharmonics.

A potential downside to the compositional and notational use of 6L1s as a tonal system is that its small steps are too sparse, making it sound too much like an equalized [[6edo|whole tone scale]] melodically. This is opposed to the much more popular [[5L 3s|5L3s]] "oneirotonic" system which, while generated on the dissonant 8\13 "major fifth," is far more melodically captivating; it also happens to support [[18edo]], another problematic EDO. 5L3s, however, is octatonic rather than heptatonic, which sacrifices clarity in staff notation ''greatly''.