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

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

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 "Horthathian" mode captures 13edo's best intervals:

Interval	Cents	Ratios	Note name	26edo/GDF 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♯, ^F	Major fourth
7\13	646.15	16/11	G♭	G♭, vG	Minor fifth
9\13	830.77	13/8, 8/5	A♭	A♭, vA	Minor sixth
11\13	1015.38	16/9, 9/5	B♭	B♭, vB	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¢ (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.

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	*	*	*

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:

Yet even this tuning contains a strange mode best described as “sub-minor”.

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 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 5L2s diatonic Aeolian (minor) mode LsLLsLL but with an extra s appended to the end, compressing all of the intervals slightly.

Interval	Cents	Note name (in 6L1s)	Note name (in 5L3s)	26edo/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♭	E♭	E♭♭, vE♭	Minor third
5\13	461.54	F♭	F	F♭, vF	Major fourth
7\13	646.15	G♭	G	G♭, vG	Minor fifth
8\13	738.46	G	H♭	G♯, ^G	Major fifth
10\13	923.08	A	A	A♯, ^A	Major sixth
12\13	1107.69	B	B	B♯, ^B	Major seventh
13\13	1200.00	C	C	C	Perfect octave

Blackwood continues briefly on his use of the mode:

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.

Indeed, the thirds of this scale as well as the aforementioned 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 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

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, 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. 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 Xenpaper.

5/4	major third	7\23	A-vvC♯	8\23	A-C♯
3/2	perfect fifth	13\23	A-vE	14\23	A-^E
7/4	harmonic seventh	18\23	A-vvG	19\23	A-G
11/4	harmonic eleventh	33\23	A-^D	34\23	A-E♭

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 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 1L6s "onyx" or "antiarchaeotonic", or 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:

Interval	Cents	Ratio(s)	1L6s name (0|6)	2L5s name	46edo/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♭	Minor third
9\23	469.57	21/16	F	F♭	vF	Minor fourth
14\23	730.43	32/21	G♯	G♯	^G	Major fifth
17\23	886.96	5/3	A♯	A♯	vA	Major sixth
20\23	1043.48	11/6	B♯	B♯	^^B♭	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 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
• F⁠ ⁠ = G♭
• F♯ = G⁠ ⁠
• F⁠ ⁠ = 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 Blackwood's twelve etudes that uses entirely non-Western scales, specifically because, as he states:

…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.

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

Blackwood's pelog

Interval	Cents	Antidiatonic name	46edo/GDF name(s)	Pseudo-diatonic interval name
0\23	0.00	C	C	Perfect unison
7\23	365.22	E♭	^D♯, vvE	Neutral third
10\23	521.74	F	^F	Major fourth
13\23	678.26	G	vG	Minor fifth
20\23	1043.48	B♭	^^B♭	Neutral seventh
23\23	1200.00	C	C	Perfect octave

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♭♭, E♯	^^D, vE♭	Supermajor second, subminor third
9\23	469.57	F♯	vF	Minor fourth
14\23	730.43	G♭	^G	Major fifth
18\23	939.13	A♭♭, B♯	^A, vvB♭	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.