User:BudjarnLambeth/Over the Hedge
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| ← 3746578afdo | 3746579afdo | 3746580afdo → |
Mode 3,746,579 of the harmonic series, also known as 3,746,579 afdo or the Over the Hedge scale [idiosyncratic term ], is a 3,746,579-tone octave-repeating subset of the harmonic series.
It is so named because if "hedge" is read not as a word, but as a base 26 number (where Z=0, A=1, B=2...), then converted into base 10, it would become 3,746,579.
Thus, since all intervals of 3,746,579 afdo are something-over-3,746,579, they are all "over the hedge".
The prime factorisation of 3,746,579 is actually 17 x 73 x 3019. This makes the Over the Hedge scale surprisingly interesting from a primodality perspective, because it contains a very unique set of intervals: millions of available intervals but with no Over-2, -3, -5, -7, -11 or even -13 intervals to be found anywhere.
Of course, the sheer number of notes does make the Over the Hedge scale impractical to explore in practice, at least without some way to eliminate most of the intervals and focus on a chosen few.
323afdo would make much more sense for practical use, being 17 x 19. Or if you want to keep the 73, then 1241afdo, which is 17 x 73.
Still, if you're daring and crazy enough to venture over the hedge, well, no one can stop you. 3 million intervals are waiting for you.
Approaches
Any approach to the Over the Hedge scale has to start with drastically cutting down the number of intervals in play.
One way to do that is to make a neji scale using a large edo of interest.
Another way is to try a polymicrotonal approach. The Over the Hedge scale contains AFDOs 17, 73, 1241, 3019, 51323, and 220387 above its root, without rotating to any other mode. So, you could use 2 or 3 of those AFDOs at the same time, and you would technically be playing within the Over the Hedge scale.
You could even combine this with the neji approach: make nejis from each of those chosen AFDOs, and then have music in each of those nejis playing at the same time.
| ← 3018afdo | 3019afdo | 3020afdo → |
| ← 1240afdo | 1241afdo | 1242afdo → |
1241afdo and 3019afdo are probably the best combination to use for this polymicrotonal approach, because they have a large but still sane number of notes, they are within the same order of magnitude as each other, and their lowest common multiple is 3,746,579, so they uniquely identify the Over the Hedge scale.
Algorithmic music is also one possible approach to large scales like Over the Hedge. You could have an algorithm randomly explore the pitch space of the Over the Hedge scale.
You could even use sensors to measure the electrical activity of a plant's leaves, and use that to control a modular synthesizer tuned to the Over the Hedge scale: you could have the Over the Hedge scale be played by an actual hedge.
Scales
The following scales are designed for use in the Over the Hedge scale.
They were generated using the same base-26 letters process, but putting most letters after the decimal point to keep the numbers small.
Each of these scales should be detempered to 1241afdo on half the instruments/tracks, and 3019afdo on the other half, to create a slight rub between them and imply the larger Over the Hedge tuning
- H.ammy = 8.058486660131 cET (e.g. 8.06c, 16.12c, 24.18c, …)
- H.eather = 8.19494288307244 cET
- O.zzie = 15.0005230034 cET
- R.J = 18.3846154 cET
- S.tella = 19.7773363118885 cET
- T.iger = 20.356832743952 cET
- V.erne = 22.219742393474 cET
Note that the large number of decimal places were included because without them, the last couple letters of the name will be slightly wrong.
Comparison
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.71 | -0.15 | +1.43 | +1.92 | +0.56 | -0.38 | +2.14 | -0.30 | +2.64 | -1.20 | +1.28 | -0.30 | +0.34 | +1.77 | +2.86 |
| Relative (%) | +8.9 | -1.9 | +17.7 | +23.9 | +7.0 | -4.7 | +26.6 | -3.8 | +32.7 | -14.9 | +15.8 | -3.7 | +4.2 | +22.0 | +35.5 | |
| Steps | 149 | 236 | 298 | 346 | 385 | 418 | 447 | 472 | 495 | 515 | 534 | 551 | 567 | 582 | 596 | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.54 | -0.73 | +1.12 | -0.03 | +3.93 | -0.70 | -2.42 | -1.46 | -3.57 | +3.52 | +0.39 | +1.13 | +3.95 | -0.76 | +2.24 |
| Relative (%) | -43.2 | -8.9 | +13.6 | -0.4 | +47.9 | -8.6 | -29.5 | -17.8 | -43.6 | +42.9 | +4.8 | +13.8 | +48.2 | -9.3 | +27.3 | |
| Steps | 146 | 232 | 293 | 340 | 379 | 411 | 439 | 464 | 486 | 507 | 525 | 542 | 558 | 572 | 586 | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.00 | -8.34 | +8.38 | +8.15 | +5.05 | -4.44 | +3.38 | +1.71 | +3.15 | +3.61 | +0.05 | +8.55 | +8.94 | -0.19 | -1.62 |
| Relative (%) | -27.2 | -45.4 | +45.6 | +44.3 | +27.4 | -24.2 | +18.4 | +9.3 | +17.1 | +19.6 | +0.2 | +46.5 | +48.6 | -1.0 | -8.8 | |
| Steps | 65 | 103 | 131 | 152 | 169 | 183 | 196 | 207 | 217 | 226 | 234 | 242 | 249 | 255 | 261 | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.42 | -3.33 | -6.94 | +2.29 | +3.09 | -6.68 | -0.52 | -6.66 | +8.71 | +1.92 | +9.50 | +9.37 | -0.26 | -1.04 | +5.89 |
| Relative (%) | +32.4 | -16.8 | -35.1 | +11.6 | +15.6 | -33.8 | -2.7 | -33.7 | +44.0 | +9.7 | +48.1 | +47.4 | -1.3 | -5.3 | +29.8 | |
| Steps | 61 | 96 | 121 | 141 | 157 | 170 | 182 | 192 | 202 | 210 | 218 | 225 | 231 | 237 | 243 | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.05 | -8.77 | +2.11 | +2.57 | -7.72 | -9.95 | +3.16 | +2.82 | +3.63 | +1.48 | -6.66 | -2.74 | -8.90 | -6.20 | +4.21 |
| Relative (%) | +5.2 | -43.1 | +10.3 | +12.6 | -37.9 | -48.9 | +15.5 | +13.8 | +17.8 | +7.3 | -32.7 | -13.5 | -43.7 | -30.4 | +20.7 | |
| Steps | 59 | 93 | 118 | 137 | 152 | 165 | 177 | 187 | 196 | 204 | 211 | 218 | 224 | 230 | 236 | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.13 | +8.94 | -0.27 | -8.85 | +8.81 | +8.57 | -0.40 | -4.33 | -8.98 | +3.77 | +8.68 | +3.42 | +8.44 | +0.10 | -0.54 |
| Relative (%) | -0.6 | +40.2 | -1.2 | -39.8 | +39.6 | +38.6 | -1.8 | -19.5 | -40.4 | +17.0 | +39.0 | +15.4 | +38.0 | +0.4 | -2.4 | |
| Steps | 54 | 86 | 108 | 125 | 140 | 152 | 162 | 171 | 179 | 187 | 194 | 200 | 206 | 211 | 216 | |