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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.
===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.
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.
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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.
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, 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.
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.
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===A note on fifths===
===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.
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.
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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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.
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]].
 
==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 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.
 
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¢ "fifthoid."
 
{| class="wikitable"
!Interval
!Cents
!24edo name
!3L2s name
|-
|0\8
|0
|C
|C
|-
|1\8
|150
|vD
|C♯
|-
|2\8
|300
|E♭
|D
|-
|3\8
|450
|vF
|D♯
|-
|4\8
|600
|F♯
|E
|-
|5\8
|750
|^G
|G
|-
|6\8
|900
|A
|G♯
|-
|7\8
|1050
|vB
|A
|-
|8\8
|1200
|C
|C
|}
 
The 3L2s notation system you see here uses a pentatonic CDEGA nominal scheme with the 4|0 mode.
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