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Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8


=== Further application in edos ===
The primary interval names are shown below for some larger edos:
19edo: P1 1-2 m2 M2 2-3 m3 M3 3-4 P4 A4 d5 P5 5-6 m6 M6 6-7 m7 M7 7-8 P8.
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 m7 Sm7 sM7 M7 P8.
24edo: P1 1-2 m2 N2 M2 2-3 m3 N3 M3 3-4 P4 N4 4-5 N5 P5 5-6 m6 N6 M6 6-7 m7 N7 M7 7-8 P8.
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8.
27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8.
29edo: P1 1-2 m2 Sm2 sM2 M2 2-3 m3 Sm3 sM3 M3 3-4 P4 S4 d5 A4 s5 P5 5-6 m6 Sm6 sM6 M6 6-7 m7 Sm7 sM7 M7 7-8 P8.
31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8.
34edo: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8.
41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8.
46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8.
Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names)
=== Formal summary ===
=== Formal summary ===
We will sum up our definitions and corollary's for the divergent system:
We will sum up our definitions and corollary's for the divergent system:
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'''Definition 1c.''' The single size of 1 and 8 is labelled P.
'''Definition 1c.''' The single size of 1 and 8 is labelled P.


'''Definition 2.''' A chroma above M or P is A, a chroma below m or P is d.
'''Definition 2.''' A chroma above M or P is A and below m or P is d.


'''Definition 3.''' Within generic interval classes, half way between M and m is N.
'''Definition 3a.''' Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.
 
'''Definition 3b'''. Within generic interval classes 1 and 4, half way between P and A is N.
 
'''Definition 3c.''' Within generic interval classes 5 and 8, half way between P and d is N.  


'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.
'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.
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'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.
'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.


'''Definition 5a.''' 'S', when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.
'''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.


'''Definition 5b.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 36/35 respectively.
'''Definition 5b.''' 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.


'''Definition 5c.''' 'S' when applied to P1 and P4 (leading to S1 and S4) and 's' when applied to P8 and P5 (leading to s8 and and s5) raise and lower by 33/32 respectively.
'''Definition 5c.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.


'''Definition 5d.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.
'''Definition 5d.''' 'S' when applied to P4 (leading to S4) and 's' when applied to P5 (leading to s5) raise and lower by 81/80 respectively.


=== Further application in edos ===
'''Definition 5e.''' 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.
The primary interval names are shown below for some larger edos:


19edo: P1 1-2 m2 M2 2-3 m3 M3 3-4 P4 A4 d5 P5 5-6 m6 M6 6-7 m7 M7 7-8 P8.
'''Definition 6a.''' When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.
 
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 m7 Sm7 sM7 M7 P8.


24edo: P1 1-2 m2 N2 M2 2-3 m3 N3 M3 3-4 P4 S4 4-5 s5 P5 5-6 m6 N6 M6 6-7 m7 N7 M7 7-8 P8.
'''Definition 6b.''' When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.


26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8.
'''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.


27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 S4 d6 A3 s5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8.
=== What about Mavila? ===
 
In Mavila
29edo: P1 1-2 m2 Sm2 sM2 M2 2-3 m3 Sm3 sM3 M3 3-4 P4 S4 d5 A4 s5 P5 5-6 m6 Sm6 sM6 M6 6-7 m7 Sm7 sM7 M7 7-8 P8.
 
31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8.
 
Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names)


== Further divergent scheme ==
== Further divergent scheme ==
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'''Corollary:''' Sm1 is 81/80 and sM1 is 64/63. sM8 is 81/40 and sm8 is 63/36.
'''Corollary:''' Sm1 is 81/80 and sM1 is 64/63. sM8 is 81/40 and sm8 is 63/36.


Using this system 34edo and 41edo can be labelled (without resorting to extended diatonic interval names):
Using this scheme 41edo is labelled:
 
34edo: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 m4 Sm4 N4 4-5 N5 sM5 M5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8.


41edo: P1 SM1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 sm4 m4 Sm4 N4 m5 M4 N5 sM5 M5 SM5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 sm8 P8.
41edo: P1 SM1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 sm4 m4 Sm4 N4 m5 M4 N5 sM5 M5 SM5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 sm8 P8.
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For both of these options N4 can be used like it is in the further divergent scheme to indicated an 11/8, 33/32 above 4/3 (and N5 to indicate 33/32 below 3/2). For both options, as in the further divergent scheme, the ability to name the interval 33/32 is lost while the ability to name the intervals 81/80 and 64/63 is gained. This is because N1, between M1 and m1, is P1 rather than 33/32 above P1. This can easily be solved. We redefine N so that it is between either Major and minor, or between perfect and augmented (for P1 and P4) or perfect and diminished (for P8 and P5).
For both of these options N4 can be used like it is in the further divergent scheme to indicated an 11/8, 33/32 above 4/3 (and N5 to indicate 33/32 below 3/2). For both options, as in the further divergent scheme, the ability to name the interval 33/32 is lost while the ability to name the intervals 81/80 and 64/63 is gained. This is because N1, between M1 and m1, is P1 rather than 33/32 above P1. This can easily be solved. We redefine N so that it is between either Major and minor, or between perfect and augmented (for P1 and P4) or perfect and diminished (for P8 and P5).
41edo using these alternative schemes:
P1 S1 sm2 m2 Lm2 N2 zM2 M2 SM2 sm3 m3 Lm3 N3 zM3 M3 SM3 s4 P4 L4 N4 d5 A4 N5 z5 P5 S5 sm6 m6 Lm6 N6 zM6 M6 SM6 sm7 m7 Lm7 N7 zM7 M7 SM7 s8 P8.
P1 S1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 s8 P8.
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