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User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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=== Neutrals ===
=== Neutrals ===
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds and [[Whitewood]] temperaments, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds temperament, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.


Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.
Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.
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=== Intermediates ===
=== Intermediates ===
To provide native support for Barbados temperament, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.
To provide native support for [[The Archipelago|Barbados]] and Diminishes temperaments, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.


‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth<!-- plural?!
‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth<!-- plural?!
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P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.
P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.


The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’.
The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. It is necessary for [[Diminished]] temperament, where the half-octave cannot be represented as any alteration of A4 or d5. Diminished temperament has a period of 1/4 of an octave, an approximation of 6/5. Therefore, the difference between four 6/5's and 2, 648/625, is tempered out.


Diminished[8] 1|0 (4) can be written as P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8, and
[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8,
== [[12edo]] and a problem ==
The primary interval names for 12edo are as we are familiar:
The primary interval names for 12edo are as we are familiar:


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but the interval labeled A4/d5 has the secondary name '4-5'.
but the interval labeled A4/d5 has the secondary name '4-5'.
We know that 12edo supports Diminished temperament, but we can't see that 12edo as equivalent to Diminished[12]: If we follow the rules from our premise, the notes of Diminished[12] in 12edo would give: P1 M2 m2 M3 m3 P4 A4/d5 P5 M6 m6 M7 m7 P8, with the majors and minor flipped. To be equivalent, sM must equal M and Sm must equal m.


=== [[10edo]], Pajara and a problem ===
=== [[10edo]], Pajara and a problem ===
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Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8
[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 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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=== [[Mavila]] ===
=== [[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.  
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  
Mavila[7] 3|3 can be written  
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We say that our first Mavila[9] interval names are not ''well-ordered'', where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).  
We say that our first Mavila[9] interval names are not ''well-ordered'', where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).  
Using the fact that 64/63 is represented by a single step in 9edo, it is left as an exercise for the reader to prove that 9edo supports Negri temperament (by replacing some names with sub and super prefixed names to arrive at Negri[9] 4|4).


The primary interval names for Augmented[9] 1|1 (3) are well-ordered.  
The primary interval names for Augmented[9] 1|1 (3) are well-ordered.  
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 M2 M3 m3 P4 P5 M6 m6 m7 P8
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  
Mavila[16] 8|7 can be writtten  


P1 A2 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 d7 P8.  
P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 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.
The enharmonic equivalences for 16edo can be generated by: M=Sm=sm or m=sM=SM


Following the same path as in 9edo, the primary well-ordered interval name set for 16edo is:  
This leads us to our primary well-ordered interval name set for 16edo:  


P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8,  
P1 S1 Sm2 sM2 Sm3 sM3 s4 P4 4-5 P5 S5 Sm6 sM6 Sm7 sM7 s8 P8,  


wherein we can see that it supports Diminished temperament.  
wherein we can see that it supports Diminished temperament.  
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P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8
P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8


The primary interval names for [[23edo]] are the same but for the inclusion of intermediates:
And the primary well-ordered interval names for [[23edo]] are appear as:


P1 d1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 A8 P8
P1 S1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 d8 P8


=== [[Father]] ===
=== [[Father]] ===
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[[Blackjack|Miracle[21]]] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,  
[[Blackjack|Miracle[21]]] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,  


the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to  small/supra, we can write their short-hand instead as 'sl' for 'small' and 'SR' for supra. Miracle[21] would then be re-written:
the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to  small/supra, we can write their short-hand instead as 'sl' for 'small' and 'Sa' for supra. Miracle[21] would then be re-written:


P1 S1 SRm2 N2 SM2 sm3 N3 slM3 s4 P4 SRd5 slA4 P5 S5 SRm6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.
P1 S1 Sam2 N2 SM2 sm3 N3 slM3 s4 P4 Sad5 slA4 P5 S5 Sam6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.


Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.
Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.
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Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8  
Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 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[10] 1|0 (5): P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7/s8 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
 
Machine[5] 2|2: P1 M2 M3 m6 m7 P8
 
Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8
 
Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 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 A4 P4 A5 P5 d5 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 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8
 
Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8 
 
Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 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 sA4/Sd5 P5 Sm6 m7 sM7 P8
 
Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 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 SM2/sm3 P4 P5 SM6/sm7 P8
 
Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8
 
Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8
 
Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8
 
Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 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 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8
 
Tetracot[6] 3|2: P1 sM2 N3 S4 N6 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
 
== Conclusion ==
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.
 
== Appendix ==
 
=== Primary well-ordered interval names for edos, without intermediates ===
2edo: P1 P4/P5 P8
 
3edo: P1 P4 P5 P8
 
4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8
 
5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8
 
6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8
 
7edo: P1 N2 N3 P4 P5 N6 N7 P8
 
8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8
 
9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8
 
10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8
 
11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8
 
12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
 
13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8
 
14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8
 
15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8
 
16edo: P1 S1 Sm2 sM2 Sm3 sM3 A4 P4 S4/s5 P5 d5 Sm6 sM6 Sm7 sM7 s8 P8 (not quite well-ordered without intermediates)
 
17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
 
19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8
 
21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8
 
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 sA4/Sd5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8
 
23edo: P1 d1 A2 M2 m2 d2/A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6/A7 M7 m7 d7 A8 P8 (can't get close to well-ordered without intermediates, so not bothering)
 
24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8
 
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/dd5 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
 
28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8 (can't complete without intermediates)
 
29edo: P1 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 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 S1 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 s8 P8 (can't complete without intermediates)
 
38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 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


43edo: P1 d2 AA7 A1/sm2 m2 dd3 AA1 M2 SM2/d3 AAA1/ddd4 A2/sm3 m3 dd4 AA2 M3 SM3/d4 AAA2/ddd5 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 AAA4/ddd7 A5/sm6 m6 dd7 AA5 M6 SM6/d7 AAA5/ddd8 A6/sm7 m7 dd8 AA6 M7 SM7/d8 dd2 s8 P8
Blackwood[15] 1|1 (5): P1 S1/Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7/s8 P8
 
Without intermediates 43edo in particular is very unruly, however things began to break down in 43edo anyway, where there is no available well-ordered interval name set.
 
46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 Sd5/sA4 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8
 
(not possible to find a well-ordered interval name set)
 
=== Interval names for MOS scales, without intermediates ===
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/m2 sM2 M2/m3 sM3 M3/P4 s5 P5/m6 sM6 M6/m7 sM7 M7/P8
 
Blackwood[15] 1|1 (5): P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8


Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8
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Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8
Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8


Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 sA4/Sd5 P5 sm6 M6 m7 SM7 P8
Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8


Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 sA4/Sd5 P5 sm6 M6 SM6 m7 SM7 P8
Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8


Machine[5] 2|2: P1 M2 M3 m6 m7 P8
Machine[5] 2|2: P1 M2 M3 m6 m7 P8
Line 809: Line 645:
Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 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 A4 P4 A5 P5 d5 M6 m6 M7 m7 d7 P8  
Mavila[16] 8|7: P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8  


Meantone[5] 2|2: P1 M2 P4 P5 m7 P8  
Meantone[5] 2|2: P1 M2 P4 P5 m7 P8  
Line 864: Line 700:


Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
== Conclusion ==
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.
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