User:Lhearne/Extra-Diatonic Intervals: Difference between revisions
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Mostly added sections on Miracle, 11edo and 21edo; and 6edo |
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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, Diminished, Negri and | 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, Tetracot and Slendric. | ||
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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Negri[10] 4|4: P1 Sm2 2-3 sM3 P4 A4 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] | Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8 | ||
Tetracot[7] 4|2: P1 sM2 N3 S4 P5 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 | Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 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 | |||
=== Further application in edos === | === Further application in edos === | ||
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which we have seen before as Tetracot[13] 6|6. | which we have seen before as Tetracot[13] 6|6. | ||
=== Miracle, 11edo and 21edo === | |||
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS: | |||
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 | |||
We first see S1 and s8 in the 21-note MOS: | |||
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: | |||
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. | |||
Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here. | |||
The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens: | |||
P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8. | |||
Adding neutrals gives us our primary interval names for 11edo: | |||
P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8. | |||
This is clearly not Mavila, so we don't know what's tempered out, such that we might add our alterations to arrive at a well-ordered interval name set. Let's review the 11-note scales we have encountered above: | |||
Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8 | |||
Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8 | |||
From these union of these scales we can see from P4=Sd5 that 135/128, the Mavila comma is tempered out. We apply our Mavila re-spellings to arrive at: | |||
P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set. | |||
21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations: | |||
P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8. | |||
Using the available conversion sm = Sm (and therefor SM = sM), we can confirm that 21edo supports Miracle and Whitewood temperaments. This is left as an exercise for the the inspired reader. | |||
=== 6edo === | |||
Though 6edo is normally only ever considered as a subset of 12edo, given that we have encountered 6-note MOS I'll give it a red hot go. | |||
Our P5 and P4 in 6edo is our half-octave, 4-5, so 9/8 is tempered out and our chromatic scale only covers 2edo: P1/M2/M3/A4 m2/m3/P4/4-5/P5/M6/M7 m6/m7/P8. If we want to write 6edo is a well-ordered way, we might choose: | |||
P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8. | |||
Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2. | |||
What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8? | |||
This tells us that in 6edo 81/80 is mapped to -2 steps of 6edo. This is not a problem, as we can use alterations of 64/63, mapped to 1 step, though I don't see why anyone would want to think of 6edo in this way. | |||
The primary interval names for the remaining trivial edos are trivially derived and are given along with all those described so far in the section below. | |||
== Lists of described edos and MOS scales == | == Lists of described edos and MOS scales == | ||
=== Primary interval names for edos === | === Primary interval names for edos === | ||
2edo: P1 P4/4-5/P5 P8 | |||
3edo: P1 3-4/P4 P5/5-6 P8 | |||
4edo: P1 2-3 P4/4-5/P5 6-7 P8 | |||
5edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8 | 5edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8 | ||
6edo: P1 | 6edo: P1 SM2 sm3/s4 P4/4-5/P5 S5/sM6 sm7 P8 | ||
7edo: P1 N2 N3 P4 P5 N6 N7 P8 | 7edo: P1 N2 N3 P4 P5 N6 N7 P8 | ||
8edo: P1 m3 M2 P4 | 8edo: P1 m3 M2 P4 4-5 P5 m7 M6 P8 | ||
9edo: P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 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 | 10edo: P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8 | ||
11edo: P1 | 11edo: P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8 | ||
12edo: P1 m2 M2 m3 | 12edo: P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8 | ||
13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 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 | 14edo: P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8 | ||
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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 | ||
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 4-5 s5 P5 m6 Sm6 sM6 M6 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 | ||
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Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 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[9] 4|4: P1 Sm2 2-3 sM3 P4 P5 Sm6 6-7 sM7 P8 | ||
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Semaphore[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8 | Semaphore[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 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[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8 | ||
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Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 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] | Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8 | ||
Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8 | Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8 | ||