30edo: Difference between revisions
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| Line 13: | Line 13: | ||
! | '''Step''' | ! | '''Step''' | ||
! | '''Cents''' | ! | '''Cents''' | ||
!7mus | !7mus | ||
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]][[Ups_and_Downs_Notation| notation]] | ! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]][[Ups_and_Downs_Notation| notation]] | ||
| Line 19: | Line 18: | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| style="text-align:center;" | 0¢ | | style="text-align:center;" | 0¢ | ||
|0 | |0 | ||
| style="text-align:center;" | P1 | | style="text-align:center;" | P1 | ||
| Line 27: | Line 25: | ||
| style="text-align:center;" | 1 | | style="text-align:center;" | 1 | ||
| style="text-align:center;" | 40 | | style="text-align:center;" | 40 | ||
|51.2 (33.3<sub>16</sub>) | |51.2 (33.3<sub>16</sub>) | ||
| style="text-align:center;" | ^1, ^m2 | | style="text-align:center;" | ^1, ^m2 | ||
| Line 35: | Line 32: | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| style="text-align:center;" |80 | | style="text-align:center;" |80 | ||
|102.4 (66.6<sub>16</sub>) | |102.4 (66.6<sub>16</sub>) | ||
| style="text-align:center;" | ^^1, v~2 | | style="text-align:center;" | ^^1, v~2 | ||
| Line 43: | Line 39: | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| style="text-align:center;" | 120 | | style="text-align:center;" | 120 | ||
|153.6 (99.A<sub>16</sub>) | |153.6 (99.A<sub>16</sub>) | ||
| style="text-align:center;" | ~2 | | style="text-align:center;" | ~2 | ||
| Line 51: | Line 46: | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| style="text-align:center;" | 160 | | style="text-align:center;" | 160 | ||
|204.8 (CC.D<sub>16</sub>) | |204.8 (CC.D<sub>16</sub>) | ||
| style="text-align:center;" | ^~2 | | style="text-align:center;" | ^~2 | ||
| Line 59: | Line 53: | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| style="text-align:center;" | 200 | | style="text-align:center;" | 200 | ||
|256 (100<sub>16</sub>) | |256 (100<sub>16</sub>) | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| Line 67: | Line 60: | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| style="text-align:center;" | 240 | | style="text-align:center;" | 240 | ||
|307.2 (133.3<sub>16</sub>) | |307.2 (133.3<sub>16</sub>) | ||
| style="text-align:center;" | M2, m3 | | style="text-align:center;" | M2, m3 | ||
| Line 75: | Line 67: | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
| style="text-align:center;" | 280 | | style="text-align:center;" | 280 | ||
|358.4 (166.6<sub>16</sub>) | |358.4 (166.6<sub>16</sub>) | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| Line 83: | Line 74: | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| style="text-align:center;" | 320 | | style="text-align:center;" | 320 | ||
|409.6 (199.A<sub>16</sub>) | |409.6 (199.A<sub>16</sub>) | ||
| style="text-align:center;" | v~3 | | style="text-align:center;" | v~3 | ||
| Line 91: | Line 81: | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| style="text-align:center;" | 360 | | style="text-align:center;" | 360 | ||
|460.8 (1CC.D<sub>16</sub>) | |460.8 (1CC.D<sub>16</sub>) | ||
| style="text-align:center;" | ~3 | | style="text-align:center;" | ~3 | ||
| Line 99: | Line 88: | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| style="text-align:center;" | 400 | | style="text-align:center;" | 400 | ||
|512 (200<sub>16</sub>) | |512 (200<sub>16</sub>) | ||
| style="text-align:center;" | ^~3 | | style="text-align:center;" | ^~3 | ||
| Line 107: | Line 95: | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| style="text-align:center;" | 440 | | style="text-align:center;" | 440 | ||
|563.2 (233.3<sub>16</sub>) | |563.2 (233.3<sub>16</sub>) | ||
| style="text-align:center;" | vM3, v4 | | style="text-align:center;" | vM3, v4 | ||
| Line 115: | Line 102: | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| style="text-align:center;" | 480 | | style="text-align:center;" | 480 | ||
|614.4 (266.6<sub>16</sub>) | |614.4 (266.6<sub>16</sub>) | ||
| style="text-align:center;" | P4 | | style="text-align:center;" | P4 | ||
| Line 123: | Line 109: | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| style="text-align:center;" | 520 | | style="text-align:center;" | 520 | ||
|665.6 (299.A<sub>16</sub>) | |665.6 (299.A<sub>16</sub>) | ||
| style="text-align:center;" | ^4, ^d5 | | style="text-align:center;" | ^4, ^d5 | ||
| Line 131: | Line 116: | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| style="text-align:center;" | 560 | | style="text-align:center;" | 560 | ||
|716.8 (2CC.D<sub>16</sub>) | |716.8 (2CC.D<sub>16</sub>) | ||
| style="text-align:center;" | ^^4, ^^d5 | | style="text-align:center;" | ^^4, ^^d5 | ||
| Line 139: | Line 123: | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
| style="text-align:center;" | 600 | | style="text-align:center;" | 600 | ||
|768 (300<sub>16</sub>) | |768 (300<sub>16</sub>) | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4, v<span style="font-size: 90%; vertical-align: super;">3</span>5 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4, v<span style="font-size: 90%; vertical-align: super;">3</span>5 | ||
| Line 147: | Line 130: | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| style="text-align:center;" | 640 | | style="text-align:center;" | 640 | ||
|819.2 (333.3<sub>16</sub>) | |819.2 (333.3<sub>16</sub>) | ||
| style="text-align:center;" | vvA4, vv5 | | style="text-align:center;" | vvA4, vv5 | ||
| Line 155: | Line 137: | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| style="text-align:center;" | 680 | | style="text-align:center;" | 680 | ||
|870.4 (366.6<sub>16</sub>) | |870.4 (366.6<sub>16</sub>) | ||
| style="text-align:center;" | vA4, v5 | | style="text-align:center;" | vA4, v5 | ||
| Line 163: | Line 144: | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| style="text-align:center;" | 720 | | style="text-align:center;" | 720 | ||
|921.6 (399.A<sub>16</sub>) | |921.6 (399.A<sub>16</sub>) | ||
| style="text-align:center;" | P5 | | style="text-align:center;" | P5 | ||
| Line 171: | Line 151: | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| style="text-align:center;" | 760 | | style="text-align:center;" | 760 | ||
|972.8 (3CC.D<sub>16</sub>) | |972.8 (3CC.D<sub>16</sub>) | ||
| style="text-align:center;" | ^5, ^m6 | | style="text-align:center;" | ^5, ^m6 | ||
| Line 179: | Line 158: | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| style="text-align:center;" | 800 | | style="text-align:center;" | 800 | ||
|1024 (400<sub>16</sub>) | |1024 (400<sub>16</sub>) | ||
| style="text-align:center;" | v~6 | | style="text-align:center;" | v~6 | ||
| Line 187: | Line 165: | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
| style="text-align:center;" | 840 | | style="text-align:center;" | 840 | ||
|1095.2 (433.3<sub>16</sub>) | |1095.2 (433.3<sub>16</sub>) | ||
| style="text-align:center;" | ~6 | | style="text-align:center;" | ~6 | ||
| Line 195: | Line 172: | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| style="text-align:center;" | 880 | | style="text-align:center;" | 880 | ||
|1126.4 (466.6<sub>16</sub>) | |1126.4 (466.6<sub>16</sub>) | ||
| style="text-align:center;" | ^~6 | | style="text-align:center;" | ^~6 | ||
| Line 203: | Line 179: | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| style="text-align:center;" | 920 | | style="text-align:center;" | 920 | ||
|1177.6 (499.A<sub>16</sub>) | |1177.6 (499.A<sub>16</sub>) | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| Line 211: | Line 186: | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| style="text-align:center;" | 960 | | style="text-align:center;" | 960 | ||
|1228.8 (4CC.D<sub>16</sub>) | |1228.8 (4CC.D<sub>16</sub>) | ||
| style="text-align:center;" | M6. m7 | | style="text-align:center;" | M6. m7 | ||
| Line 219: | Line 193: | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| style="text-align:center;" | 1000 | | style="text-align:center;" | 1000 | ||
|1280 (500<sub>16</sub>) | |1280 (500<sub>16</sub>) | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| Line 227: | Line 200: | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
| style="text-align:center;" | 1040 | | style="text-align:center;" | 1040 | ||
|1331.2 (533.3<sub>16</sub>) | |1331.2 (533.3<sub>16</sub>) | ||
| style="text-align:center;" | v~7 | | style="text-align:center;" | v~7 | ||
| Line 235: | Line 207: | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| style="text-align:center;" | 1080 | | style="text-align:center;" | 1080 | ||
|1382.4 (566.6<sub>16</sub>) | |1382.4 (566.6<sub>16</sub>) | ||
| style="text-align:center;" | ~7 | | style="text-align:center;" | ~7 | ||
| Line 243: | Line 214: | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| style="text-align:center;" | 1120 | | style="text-align:center;" | 1120 | ||
|1433.6 (599.A<sub>16</sub>) | |1433.6 (599.A<sub>16</sub>) | ||
| style="text-align:center;" | ^~7, vv8 | | style="text-align:center;" | ^~7, vv8 | ||
| Line 251: | Line 221: | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| style="text-align:center;" | 1160 | | style="text-align:center;" | 1160 | ||
|1484.8 (5CC.D<sub>16</sub>) | |1484.8 (5CC.D<sub>16</sub>) | ||
| style="text-align:center;" | vM7, v8 | | style="text-align:center;" | vM7, v8 | ||
| Line 259: | Line 228: | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| style="text-align:center;" | 1200 | | style="text-align:center;" | 1200 | ||
|1536 (600<sub>16</sub>) | |1536 (600<sub>16</sub>) | ||
| style="text-align:center;" | P8 | | style="text-align:center;" | P8 | ||
Revision as of 13:58, 12 December 2019
The 30 equal division divides the octave into 30 equal steps of precisely 40 cents each. Its patent val is a doubled version of the patent val for 15edo through the 11-limit, so 30 can be viewed as a contorted version of 15. In the 13-limit it supplies the optimal patent val for quindecic temperament.
However, 5\30 is 200 cents, which is a good (and familiar) approximation for 9/8, and hence 30 can be viewed inconsistently, as having a 9' at 95\30 as well as a 9 at 96\30. Instead of the 18\30 fifth of 720 cents, 30 also makes available a 17\30 fifth of 680 cents. This is an ideal tuning for pelogic (5-limit mavila), which tempers out 135/128. When 30 is used for pelogic, 5\30 can again be used inconsistently as a 9/8.
Below is a plot of the Z function around 30:
Intervals
| Step | Cents | 7mus | ups and downs notation | ||
|---|---|---|---|---|---|
| 0 | 0¢ | 0 | P1 | unison, minor 2nd | D, Eb |
| 1 | 40 | 51.2 (33.316) | ^1, ^m2 | up unison, upminor 2nd | D^, Eb^ |
| 2 | 80 | 102.4 (66.616) | ^^1, v~2 | double-up unison, downmid 2nd | D^^, Eb^^ |
| 3 | 120 | 153.6 (99.A16) | ~2 | mid 2nd | Ev3 |
| 4 | 160 | 204.8 (CC.D16) | ^~2 | upmid 2nd | Evv |
| 5 | 200 | 256 (10016) | vM2 | downmajor 2nd | Ev |
| 6 | 240 | 307.2 (133.316) | M2, m3 | major 2nd, minor 3rd | E, F |
| 7 | 280 | 358.4 (166.616) | ^m3 | upminor 3rd | F^ |
| 8 | 320 | 409.6 (199.A16) | v~3 | downmid 3rd | F^^ |
| 9 | 360 | 460.8 (1CC.D16) | ~3 | mid 3rd | F^3 |
| 10 | 400 | 512 (20016) | ^~3 | upmid 3rd | F#vv |
| 11 | 440 | 563.2 (233.316) | vM3, v4 | downmajor 3rd, down 4th | F#v, Gv |
| 12 | 480 | 614.4 (266.616) | P4 | major 3rd, perfect 4th | F#, G |
| 13 | 520 | 665.6 (299.A16) | ^4, ^d5 | up 4th, updim 5th | G^, Ab^ |
| 14 | 560 | 716.8 (2CC.D16) | ^^4, ^^d5 | double-up 4th, double-up dim 5th | G^^, Ab^^ |
| 15 | 600 | 768 (30016) | ^34, v35 | triple-up 4th, triple-down 5th | G^3, Av3 |
| 16 | 640 | 819.2 (333.316) | vvA4, vv5 | double-down aug 4th, double-down 5th | G#vv, Avv |
| 17 | 680 | 870.4 (366.616) | vA4, v5 | downaug 4th, down 5th | G#v, Av |
| 18 | 720 | 921.6 (399.A16) | P5 | perfect 5th, minor 6th | A, Bb |
| 19 | 760 | 972.8 (3CC.D16) | ^5, ^m6 | up 5th, upminor 6th | A^, Bb^ |
| 20 | 800 | 1024 (40016) | v~6 | downmid 6th | Bb^^ |
| 21 | 840 | 1095.2 (433.316) | ~6 | mid 6th | Bv3 |
| 22 | 880 | 1126.4 (466.616) | ^~6 | upmid 6th | Bvv |
| 23 | 920 | 1177.6 (499.A16) | vM6 | downmajor 6th | Bv |
| 24 | 960 | 1228.8 (4CC.D16) | M6. m7 | major 6th, minor 7th | B, C |
| 25 | 1000 | 1280 (50016) | ^m7 | upminor 7th | C^ |
| 26 | 1040 | 1331.2 (533.316) | v~7 | downmid 7th | C^^ |
| 27 | 1080 | 1382.4 (566.616) | ~7 | mid 7th | C^3 |
| 28 | 1120 | 1433.6 (599.A16) | ^~7, vv8 | upmid 7th, double-down 8ve | C#vv, Dvv |
| 29 | 1160 | 1484.8 (5CC.D16) | vM7, v8 | downmajor 7th, down 8ve | C#v, Dv |
| 30 | 1200 | 1536 (60016) | P8 | major 7th, 8ve | C#, D |
Commas
30 EDO tempers out the following commas. (Note: This assumes the val < 30 48 70 84 104 111 | .)
| Comma | Monzo | Value (Cents) | Name 1 | Name 2 | Name 3 |
|---|---|---|---|---|---|
| 256/243 | | 8 -5 > | 90.22 | Limma | Pythagorean Minor 2nd | |
| 250/243 | | 1 -5 3 > | 49.17 | Maximal Diesis | Porcupine Comma | |
| 128/125 | | 7 0 -3 > | 41.06 | Diesis | Augmented Comma | |
| 15625/15552 | | -6 -5 6 > | 8.11 | Kleisma | Semicomma Majeur | |
| 1029/1000 | | -3 1 -3 3 > | 49.49 | Keega | ||
| 49/48 | | -4 -1 0 2 > | 35.70 | Slendro Diesis | ||
| 64/63 | | 6 -2 0 -1 > | 27.26 | Septimal Comma | Archytas' Comma | Leipziger Komma |
| 64827/64000 | | -9 3 -3 4 > | 22.23 | Squalentine | ||
| 875/864 | | -5 -3 3 1 > | 21.90 | Keema | ||
| 126/125 | | 1 2 -3 1 > | 13.79 | Septimal Semicomma | Starling Comma | |
| 4000/3969 | | 5 -4 3 -2 > | 13.47 | Octagar | ||
| 1029/1024 | | -10 1 0 3 > | 8.43 | Gamelisma | ||
| 6144/6125 | | 11 1 -3 -2 > | 5.36 | Porwell | ||
| 250047/250000 | | -4 6 -6 3 > | 0.33 | Landscape Comma | ||
| 100/99 | | 2 -2 2 0 -1 > | 17.40 | Ptolemisma | ||
| 121/120 | | -3 -1 -1 0 2 > | 14.37 | Biyatisma | ||
| 176/175 | | 4 0 -2 -1 1 > | 9.86 | Valinorsma | ||
| 65536/65219 | | 16 0 0 -2 -3 > | 8.39 | Orgonisma | ||
| 385/384 | | -7 -1 1 1 1 > | 4.50 | Keenanisma | ||
| 441/440 | | -3 2 -1 2 -1 > | 3.93 | Werckisma | ||
| 4000/3993 | | 5 -1 3 0 -3 > | 3.03 | Wizardharry | ||
| 3025/3024 | | -4 -3 2 -1 2 > | 0.57 | Lehmerisma |
Music
Fifteen Short Pieces by Todd Harrop