User:TromboneBoi9/Approaches to weird EDOs

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 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 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 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 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.

The major second

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 6L1s "archaeotonic" scale is produced.

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

Interval	Cents	Ratios	Note name	26edo name	Pseudo-diatonic interval name
0\13	0.00	1/1	C	C	Perfect unison
2\13	184.62	10/9, 9/8	D	D	Major second
4\13	369.23	5/4, 16/13	E	E	Major third
6\13	553.85	11/8	F	F♯	Major fourth
7\13	646.15	16/11	G♭	G♭	Minor fifth
9\13	830.77	13/8, 8/5	A♭	A♭	Minor sixth
11\13	1015.38	16/9, 9/5	B♭	B♭	Minor seventh
13\13	1200.00	2/1	C	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♯ and F♭ 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, which will make it sound too much like an equalized whole tone scale melodically if the small step is not somehow emphasized. This is opposed to the much more popular 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.

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.

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):

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 a well-approximated 21/16.

13b 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."

	0	1	2	3	4	5	6	7	8	9	10	11	12
6L1s notation	C	*	D	*	E	*	F	*	G	*	A	*	B
5L3s notation	C	*	D	*	E	F	*	G	*	H	A	*	B
2L5s notation	C	D	E	*	*	*	F	G	A	B	*	*	*

8edo

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 strangely altered intervals, since the only pure prime it approximates with any decency is 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.

Interval	Cents	24edo name	Interval name
0\8	0	C	Perfect unison
1\8	150	vD	Neutral second
2\8	300	E♭	Minor third
3\8	450	^E, vF	Supermajor third, Subfourth
4\8	600	F♯, G♭	Tritone
5\8	750	^G, vA♭	Superfifth, Subminor sixth
6\8	900	A	Major sixth
7\8	1050	vB	Neutral seventh
8\8	1200	C	Perfect octave

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):

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.