16ed5/2: Difference between revisions
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
!Degrees | ! Degrees | ||
! colspan="3" |Enneatonic | ! colspan="3" | Enneatonic | ||
!ED38\29 | ! ED38\29 | ||
!Golden | ! Golden | ||
!ED5/2 | ! ED5/2 | ||
!ED(7φ+6)\5(φ+1) | ! ED(7φ+6)\5(φ+1) | ||
!ED4\3=''r¢'' | ! ED4\3=''r¢'' | ||
|- | |- | ||
|1 | | 1 | ||
|1#/2b | | 1#/2b | ||
| colspan="2" |F#/Gb | | colspan="2" | F#/Gb | ||
|98.276 | | 98.276 | ||
|98.3795 | | 98.3795 | ||
|99.145 | | 99.145 | ||
|99.2705 | | 99.2705 | ||
|''100'' | | ''100'' | ||
|- | |- | ||
|2 | | 2 | ||
|2 | | 2 | ||
| colspan="2" |G | | colspan="2" | G | ||
|196.552 | | 196.552 | ||
|196.759 | | 196.759 | ||
|198.289 | | 198.289 | ||
|198.541 | | 198.541 | ||
|''200'' | | ''200'' | ||
|- | |- | ||
|3 | | 3 | ||
|2#/3b | | 2#/3b | ||
|G#/Jb | | G#/Jb | ||
|''G#/Ab'' | | ''G#/Ab'' | ||
|294.828 | | 294.828 | ||
|295.138 | | 295.138 | ||
|297.433 | | 297.433 | ||
|297.8115 | | 297.8115 | ||
|''300'' | | ''300'' | ||
|- | |- | ||
|4 | | 4 | ||
|3 | | 3 | ||
|J | | J | ||
|''A'' | | ''A'' | ||
|393.103 | | 393.103 | ||
|393.518 | | 393.518 | ||
|396.578 | | 396.578 | ||
|397.082 | | 397.082 | ||
|''400'' | | ''400'' | ||
|- | |- | ||
|5 | | 5 | ||
|3#/4b | | 3#/4b | ||
|J#/Ab | | J#/Ab | ||
|''A#/Bb'' | | ''A#/Bb'' | ||
|491.379 | | 491.379 | ||
|491.897 | | 491.897 | ||
|495.723 | | 495.723 | ||
|496.3525 | | 496.3525 | ||
|''500'' | | ''500'' | ||
|- | |- | ||
|6 | | 6 | ||
|4 | | 4 | ||
|A | | A | ||
|''B'' | | ''B'' | ||
|589.655 | | 589.655 | ||
|590.277 | | 590.277 | ||
|594.868 | | 594.868 | ||
|595.623 | | 595.623 | ||
|''600'' | | ''600'' | ||
|- | |- | ||
|7 | | 7 | ||
|5 | | 5 | ||
|B | | B | ||
|''H'' | | ''H'' | ||
|687.931 | | 687.931 | ||
|688.656 | | 688.656 | ||
|694.012 | | 694.012 | ||
|694.894 | | 694.894 | ||
|''700'' | | ''700'' | ||
|- | |- | ||
|8 | | 8 | ||
|5#/6b | | 5#/6b | ||
|B#/Hb | | B#/Hb | ||
|''H#/Cb'' | | ''H#/Cb'' | ||
|786.207 | | 786.207 | ||
|787.036 | | 787.036 | ||
|793.157 | | 793.157 | ||
|794.164 | | 794.164 | ||
|''800'' | | ''800'' | ||
|- | |- | ||
|9 | | 9 | ||
|6 | | 6 | ||
|H | | H | ||
|''C'' | | ''C'' | ||
|884.483 | | 884.483 | ||
|885.415 | | 885.415 | ||
|892.3015 | | 892.3015 | ||
|893.435 | | 893.435 | ||
|''900'' | | ''900'' | ||
|- | |- | ||
|10 | | 10 | ||
|6#/7b | | 6#/7b | ||
|H#/Cb | | H#/Cb | ||
|''C#/Db'' | | ''C#/Db'' | ||
|982.759 | | 982.759 | ||
|983.795 | | 983.795 | ||
|991.446 | | 991.446 | ||
|992.705 | | 992.705 | ||
|''1000'' | | ''1000'' | ||
|- | |- | ||
|11 | | 11 | ||
|7 | | 7 | ||
|C | | C | ||
|''D'' | | ''D'' | ||
|1081.0345 | | 1081.0345 | ||
|1082.174 | | 1082.174 | ||
|1090.591 | | 1090.591 | ||
|1091.976 | | 1091.976 | ||
|''1100'' | | ''1100'' | ||
|- | |- | ||
|12 | | 12 | ||
|7#/8b | | 7#/8b | ||
|C#/Db | | C#/Db | ||
|''D#/Sb'' | | ''D#/Sb'' | ||
|1179.31 | | 1179.31 | ||
|1180.554 | | 1180.554 | ||
|1189.735 | | 1189.735 | ||
|1191.246 | | 1191.246 | ||
|''1200'' | | ''1200'' | ||
|- | |- | ||
|13 | | 13 | ||
|8 | | 8 | ||
|D | | D | ||
|''S'' | | ''S'' | ||
|1277.586 | | 1277.586 | ||
|1278.933 | | 1278.933 | ||
|1288.88 | | 1288.88 | ||
|1290.517 | | 1290.517 | ||
|''1300'' | | ''1300'' | ||
|- | |- | ||
|14 | | 14 | ||
|8#/9b | | 8#/9b | ||
|D#/Eb | | D#/Eb | ||
|''S#/Eb'' | | ''S#/Eb'' | ||
|1375.862 | | 1375.862 | ||
|1377.313 | | 1377.313 | ||
|1388.0245 | | 1388.0245 | ||
|1389.787 | | 1389.787 | ||
|''1400'' | | ''1400'' | ||
|- | |- | ||
|15 | | 15 | ||
|9 | | 9 | ||
| colspan="2" |E | | colspan="2" | E | ||
|1474.138 | | 1474.138 | ||
|1475.692 | | 1475.692 | ||
|1487.169 | | 1487.169 | ||
|1489.058 | | 1489.058 | ||
|''1500'' | | ''1500'' | ||
|- | |- | ||
|16 | | 16 | ||
|1 | | 1 | ||
| colspan="2" |F | | colspan="2" | F | ||
|1572.414 | | 1572.414 | ||
|1574.0715 | | 1574.0715 | ||
|1586.314 | | 1586.314 | ||
|1588.328 | | 1588.328 | ||
|''1600'' | | ''1600'' | ||
|} | |} | ||
Coincidentally, 133 steps of the pyrite EDX of this size exceed 11 octaves by only 2.978¢. | Coincidentally, 133 steps of the pyrite EDX of this size exceed 11 octaves by only 2.978¢. | ||
Revision as of 12:38, 13 September 2021
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: ~132/125 = 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: ~55/52 = 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