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phew. Learnt a lot writing this edit! Mavila and Father added. edo and scales Lists added. Crap it's 4am again and I told my housemate I'd do the dishes tonight...
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Neutral[17] 8|8 may be written as
Neutral[17] 8|8 may be written as


P1 N1 N2 M2 m3 N3 P4 N4 N5 P5 m6 N6 M6 m7 N7 s8 P8,  
P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,  


which is almost equivalent to the primary interval names of 17edo,
which is almost equivalent to the primary interval names of 17edo,


P1 m2 N2 M2 m3 N3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 P8,
P1 m2 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 P8,


consisting of the neutrals, perfects, majors and minors.
consisting of the neutrals, perfects, majors and minors.
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But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to 16/11, and S4 11/8. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to 21/16, and ‘S5’ to 32/21, we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.  
But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to 16/11, and S4 11/8. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to 21/16, and ‘S5’ to 32/21, we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.  


Blacksmith[15] 1|1 (5) can be written as all the intermediates, supras minors and smalls:
Blacksmith[15] 1|1 (5) can be written as all the intermediates, supras and smalls:


P1/1-2 S1/Sm2 sM2 2-3 Sm3 sM3 3-4/P4 S4 s5 P5/5-6 Sm6 sM6 6-7 Sm7 sM7/s8 7-8/P8,
P1/1-2 S1/Sm2 sM2 2-3 Sm3 sM3 3-4/P4 S4 s5 P5/5-6 Sm6 sM6 6-7 Sm7 sM7/s8 7-8/P8,
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=== Other rank-2 temperaments' MOS scales ===
=== Other rank-2 temperaments' MOS scales ===
Other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including Augmented, Porcupine and Diminished.
Other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including Augmented, Porcupine, Diminished, Negri and Tetracot


Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
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Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8


Diminished[12] 1|1 (4): P1 Sm1 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8
Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8


Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
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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
Negri[9] 4|4: P1 Sm2 2-3 sM3 P4 P5 Sm6 6-7 sM7 P8
Negri[10] 4|4: P1 Sm2 2-3 sM3 P4 A4 P5 Sm6 6-7 sM7 P8
Tetracot[6] 4|1: P1 sM2 N3 S4 P5 Sm7 P8
Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8
Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8


=== Further application in edos ===
=== Further application in edos ===
The primary interval names are shown below for some larger 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.
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.
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 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.
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.
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.
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.
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.
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.
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.
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.
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)
Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names)
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'''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'.
'''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'.


=== What about Mavila? ===
=== Mavila and 16edo ===
In Mavila  
In Mavila, the perfect 5th is flatter than in 7edo, so major intervals are below minor and augmented below major. In Mavila the major third approximates 6/5 and the minor third 5/4, tempering out 135/128. This presents no problem to the scheme however, and the rules are applied just the same. The small major third, 81/80 below 6/5 or 81/64 comes to 32/27, the minor 3rd, and the sub minor 3rd remains 7/6.
 
Mavila[7] 3|3 can be written
 
P1 M2 m3 P4 P5 M6 m7 P8,
 
the same as the diatonic scale.
 
Mavila[9] 4|4 can be written 
 
P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,
 
If we don't have major being below minor, we can hide it with some secondary interval names:
 
P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].
 
We might think that the primary interval names of 9edo are the same as our first spelling of Mavila[9] 4|4 above, however we note that the third step is half-way between M2 and m3, and so is primarily a serd. The fourth step, half-way between M3 and P4 is a thourth. If the M3 was a serd, it is also a m2, meaning that the second step is half-way between P1 and m2, and is primarily a unicond. Our primary interval names for 9edo are as follows:
 
P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8. 
 
Rewriting M2 as Sm2 and m7 as sM7 and putting sM3 and Sm6 back gives us Negri[9]. 
 
Mavila[16] 8|7 can be writtten
 
P1 A2 M2 m2 M3 m3 d3 P4 A5 P5 A6 M6 m6 M7 m7 d7 P8.
 
The primary interval names for 16edo are the same, but for the inclusion of intermediates:
 
P1 1-2 M2 m2 M3 m3 3-4 P4 4-5 P5 5-6 M6 m6 M7 m7 7-8 P8.
 
Following the same path as in 9edo, we could also write 16edo as:
 
P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8,
 
wherein we can see that it supports Diminished temperament.
 
=== Father ===
In the other direction, the fifths of Father temperament are sharper than those of 5edo, leading to minor second going backwards. In Father temperament, 5/4 and 4/3 are tempered to a unison, along with 9/7. As 64/63 is tempered out, alterations of 64/63 act as identity alterations.The M2, larger than m3 is also a Sm3. sM2, then, returns to m3.
 
The primary interval names for Father[5] 2|2,
 
P1 M2 P4 P5 m7 P8,
 
present no problems.
 
In the primary interval names for Father[8] 4|3, however:
 
P1 m3 M2 P4 M3 P5 m7 M6 P8,
 
which are the same as the primary intervals for 8edo, but with M3 rather than 4-5, we see our diatonic interval names begin to cross over. If this is a problem, we may use some secondary interval names to address it, i.e.
 
P1 sM2 M2 P4 S4 P5 m7 Sm7 P8.
 
If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.
 
Using some secondary interval names to 'fix' the order leads us to
 
P1 sM2 M2 P4 4-5 P5 m7 M6 P8
 
The primary interval names for Father[13] 6|6:
 
P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,
 
look very unruly.
 
We will fix up the ordering again with secondary interval names:
 
P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.
 
The primary interval names for 13edo are similar:
 
P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.
 
Minimally fixing the order leads us to
 
P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,
 
which we have seen before as Tetracot[13] 6|6.
 
== Lists of described edos and MOS scales ==
 
=== Primary interval names for edos ===
5edo: P1/1-2 2-3 3-4/P4 P5/5-6 7-8/P8
 
6edo: P1 M2 M3 4-5 m6 m7 P8
 
7edo: P1 N2 N3 P4 P5 N6 N7 P8
 
8edo: P1 m3 M2 P4 S4 P5 m7 M6 P8
 
9edo: P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8.
 
10edo: P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8
 
12edo: P1 m2 M2 m3 M4 P4 4-5 P5 m6 M6 m7 M7 P8
 
13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8
 
14edo: P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8
 
15edo: P1/1-2 S1/Sm2 sM2 2-3 Sm3 sM3 3-4/P4 S4 s5 P5/5-6 Sm6 sM6 6-7 Sm7 sM7/s8 7-8/P8
 
16edo: P1 1-2 M2 m2 M3 m3 3-4 P4 4-5 P5 5-6 M6 m6 M7 m7 7-8 P8
 
17edo: P1 m2 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 P8
 
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 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 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
 
=== Interval names for MOS scales ===
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
 
Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8
 
Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8
 
Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
 
Blackwood[10] 1|0 (5): P1/1-2 sM2 2-3 sM3 3-4/P4 s5 P5/ 5-6 sM6 6-7 sM7 7-8/P8
 
Blackwood[15] 1|1 (5): P1/1-2 S1/Sm2 sM2 2-3 Sm3 sM3 3-4/P4 S4 s5 P5/5-6 Sm6 sM6 6-7 Sm7 sM7/s8 7-8/P8
 
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8
 
Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8
 
Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 4-5 P5 sm6 M6 m7 SM7 P8
 
Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 4-5 P5 sm6 M6 SM6 m7 SM7 P8
 
Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
 
Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8
 
Mavila[16] 8|7: P1 A2 M2 m2 M3 m3 d3 P4 A5 P5 A6 M6 m6 M7 m7 d7 P8
 
Meantone[5] 2|2: P1 M2 P4 P5 m7 P8
 
Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
 
Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
 
Meantone[19] 9|9: P1 A1/d2 m2 M2 A2/d3 m3 M3 A3/d4 P4 A4 d5 P5 A5/d6 m6 M6 A6/d7 m7 M7 A7/d8 P8
 
Negri[9] 4|4: P1 Sm2 2-3 sM3 P4 P5 Sm6 6-7 sM7 P8
 
Negri[10] 4|4: P1 Sm2 2-3 sM3 P4 A4 P5 Sm6 6-7 sM7 P8
 
Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8
 
Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8
 
Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
 
Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 4-5 P5 Sm6 m7 sM7 P8
 
Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 m7 sM7 P8
 
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
 
Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8
 
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
 
Semaphore[5] 2|2: P1 2-3 P4 P5 6-7 P8
 
Semaphore[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8
 
Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
 
Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
 
Superpyth[17] 8|8: P1 m2 A1/d3 M2 m3 A2/d4 M3 P4 A3/d5 A4/d6 P5 m6 A5/d7 M6 m7 A6/d8 M7 P8


== Further divergent scheme ==
== Further divergent scheme ==
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'''Definition 1a.''' M and m imply two sizes of the generic interval in the Pythagorean diatonic scale (M larger, m smaller),  
'''Definition 1a.''' M and m imply two sizes of the generic interval in the Pythagorean diatonic scale (M larger, m smaller),  


'''Definition 1b.''' P implies a single size. Corollary: Only 1 and 8 are P.
'''Definition 1b.''' P implies a single size.  
 
'''Corollary:''' Only 1 and 8 are P.


'''Lemma:''' P intervals are also M as well as m.
'''Lemma:''' P intervals are also M as well as m.
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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, with long-form 'super' and 'sub' respectively.


'''Definition 5b.''' '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 81/80 respectively, with long-form 'supra' and 'small' respectively.


'''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 scheme 41edo is labelled:
'''Definition 6.''' 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'.
 
Using this scheme 41edo can be written:


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.
== Summary and further options ==
So far the goals in the premise have been achieved. Looking especially at the interval names for edos presented here, we can see they are simple and align with informal interval names. One argument that might be made against this system is that (unlike the interval size indicators), 'S' and 's' mean different things in different places. This complicates interval arithmetic, where an S may not be cancelled by an s, or carry over as would be expected after addition to the diatonic intervals (P, M, m, A and d). However, this is only true for scales or edos where more than one class of 'S' and 's' indicators is present and not represented by the same interval / number of steps. This is not true for any of the above examples, which cover most common tunings. It was never necessary to provide unique labels for edos much larger than 41edo anyway, above which ups and downs / the original premise becomes inconsistent in it's mappings anyway, where, for example, in 72edo, a 9/7, a 64/63 above 81/64, is labelled 'SSM3' instead of 'SM3'. In my scheme the inconsistency manifests in that in 72edo 'S' before M raises by 2 steps, but 'S' before m raises by 1. In this way they remain comparable, but my scheme may be better applied to rank-2 temperaments.
If one desires that this sort of complication be avoided, one can substitute s for z (small) and S for L (large) when alterations of 81/80 are desired (although obviously small begins with 's', 's' is already being used as 'sub').
Alternatively as a middle ground possibility, one could keep 'S' and 's' for both 81/80 and 64/63, but where for 81/80 the long-form for 's' in this case is small, and for 'S' remains 'supra'. Then we can supposedly have it both ways, where the long-form shows us that they are different, but no short-form qualifiers are added, so these differences can be ignored in almost all scales and edos, but are there when needed. If for some reason you need an interval 64/63 above m3, or you need both a 81/80 and a 64/63 above 4/3, then for indicators 'small' and 'supra' could be written 'sl' and 'SR'.
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).
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