16ed5/2
16ED5/2 is the equal division of the 5/2 interval into 16 parts of 99.1446 cents each. This is the scale which occurs as the dominant reformed Mixolydian mode tuned as an equal division of a just interval.
Intervals
| Degrees | Enneatonic | ED38\29 | Golden | ED5/2 | ED(7φ+6)\5(φ+1) | ED4\3=r¢ | ||
|---|---|---|---|---|---|---|---|---|
| 1 | 1#/2b | F#/Gb | 98.276 | 98.3795 | 99.145 | 99.2705 | 100 | |
| 2 | 2 | G | 196.552 | 196.759 | 198.289 | 198.541 | 200 | |
| 3 | 2#/3b | G#/Jb | G#/Ab | 294.828 | 295.138 | 297.433 | 297.8115 | 300 |
| 4 | 3 | J | A | 393.103 | 393.518 | 396.578 | 397.082 | 400 |
| 5 | 3#/4b | J#/Ab | A#/Bb | 491.379 | 491.897 | 495.723 | 496.3525 | 500 |
| 6 | 4 | A | B | 589.655 | 590.277 | 594.868 | 595.623 | 600 |
| 7 | 5 | B | H | 687.931 | 688.656 | 694.012 | 694.894 | 700 |
| 8 | 5#/6b | B#/Hb | H#/Cb | 786.207 | 787.036 | 793.157 | 794.164 | 800 |
| 9 | 6 | H | C | 884.483 | 885.415 | 892.3015 | 893.435 | 900 |
| 10 | 6#/7b | H#/Cb | C#/Db | 982.759 | 983.795 | 991.446 | 992.705 | 1000 |
| 11 | 7 | C | D | 1081.0345 | 1082.174 | 1090.591 | 1091.976 | 1100 |
| 12 | 7#/8b | C#/Db | D#/Sb | 1179.31 | 1180.554 | 1189.735 | 1191.246 | 1200 |
| 13 | 8 | D | S | 1277.586 | 1278.933 | 1288.88 | 1290.517 | 1300 |
| 14 | 8#/9b | D#/Eb | S#/Eb | 1375.862 | 1377.313 | 1388.0245 | 1389.787 | 1400 |
| 15 | 9 | E | 1474.138 | 1475.692 | 1487.169 | 1489.058 | 1500 | |
| 16 | 1 | F | 1572.414 | 1574.0715 | 1586.314 | 1588.328 | 1600 | |
Coincidentally, 133 steps of the pyrite EDX of this size exceed 11 octaves by only 2.978¢.
16ed5/2 as a generator
16ED5/2 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 256/255, 361/360, and 4624/4617, which is a cluster temperament with 12 clusters of notes in an octave (quintaleap temperament). This temperament is supported by Template:Val list, and 157 EDOs.
Tempering out 400/399 (equating 20/19 and 21/20) leads quintupole (12&121), and tempering out 476/475 (equating 19/17 with 28/25) leads quinticosiennic (12&145).
Another temperament related to 16ED5/2 is quintapole (12&85). It is practically identical to the Galilei tuning, which is generated by the ratios 2/1 and 18/17.
- Quintaleap (12&121)
5-limit
Comma: [37 -16 -5⟩ = 137438953472/134521003125
Mapping: [⟨1 2 1], ⟨0 -5 16]]
POTE generator: ~135/128 = 99.267
Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc
Badness: 0.444506
2.3.5.17.19 subgroup
Comma list: 256/255, 361/360, 4624/4617
Gencom: [2 18/17; 256/255 361/360 4624/4617]
Gencom mapping: [⟨1 2 1 5 4], ⟨0 -5 16 -11 3]]
POTE generator: ~18/17 = 99.276
Vals: 12, 109, 121, 133
RMS error: 0.3427 cents
- Quintupole (12&121)
7-limit
Comma list: 4000/3969, 458752/455625
Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]
POTE generator: ~135/128 = 99.175
Vals: 12, 97, 109, 121
Badness: 0.111620
11-limit
Comma list: 896/891, 1375/1372, 4375/4356
Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]
POTE generator: ~35/33 = 99.156
Vals: 12, 109, 121, 351bde, 472bdee
Badness: 0.056501
13-limit
Comma list: 352/351, 364/363, 625/624, 2704/2695
Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]
POTE generator: ~35/33 = 99.165
Vals: 12f, 109, 121
Badness: 0.038431
17-limit
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]
POTE generator: ~18/17 = 99.172
Vals: 12f, 109, 121
Badness: 0.028721
19-limit
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]
POTE generator: ~18/17 = 99.164
Vals: 12f, 109, 121
Badness: 0.023818
- Quinticosiennic (12&145)
7-limit
Comma list: 5120/5103, 395136/390625
Mapping: [⟨1 2 1 -1], ⟨0 -5 16 46]]
POTE generator: ~135/128 = 99.345
Vals: 12, 133, 145, 157, 302c, 459bcc
Badness: 0.158041
11-limit
Comma list: 441/440, 896/891, 78408/78125
Mapping: [⟨1 2 1 -1 -2], ⟨0 -5 16 46 66]]
POTE generator: ~35/33 = 99.318
Vals: 12, 133, 145
Badness: 0.080674
13-limit
Comma list: 196/195, 352/351, 364/363, 78408/78125
Mapping: [⟨1 2 1 -1 -2 -3], ⟨0 -5 16 46 66 81]]
POTE generator: ~35/33 = 99.307
Vals: 12f, 133, 145
Badness: 0.052464
17-limit
Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750
Mapping: [⟨1 2 1 -1 -2 -3 5], ⟨0 -5 16 46 66 81 -11]]
POTE generator: ~18/17 = 99.308
Vals: 12f, 133, 145
Badness: 0.037108
19-limit
Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475
Mapping: [⟨1 2 1 -1 -2 -3 5 4], ⟨0 -5 16 46 66 81 -11 3]]
POTE generator: ~18/17 = 99.303
Vals: 12f, 133, 145
Badness: 0.028440
- Quintapole (12&85)
7-limit
Comma list: 225/224, 7812500/7411887
Mapping: [⟨1 2 1 1], ⟨0 -5 16 22]]
POTE generator: ~21/20 = 98.994
Vals: 12, 73c, 85, 97d
Badness: 0.192498
11-limit
Comma list: 100/99, 225/224, 85184/84035
Mapping: [⟨1 2 1 1 0], ⟨0 -5 16 22 42]]
POTE generator: ~21/20 = 98.954
Vals: 12, 73ce, 85, 97d
Badness: 0.104353