Logharmonic series
introduction
A logharmonic series is a variation of the harmonic series. Like the harmonic series, it is an infinitely long series of pitches from which scales can be drawn. But where the harmonic series is a linear series of pitches, with the formula [math]\displaystyle{ f(n) = n }[/math], the formula for a b-logharmonic series is:
[math]\displaystyle{ \qquad f(n) = log_b{n} }[/math]
If a natural number is chosen as [math]\displaystyle{ b }[/math], the resulting series will be a superset of the harmonic series, inserting extra pitches. For example, the 2-logharmonic series inserts an extra step in between the fundamental and the 2nd harmonic, so that it takes [math]\displaystyle{ 2^1 = 2 }[/math] steps to reach the 2nd harmonic instead of one. Then it inserts 3 extra steps in between the 2nd harmonic and 3rd harmonic so that it takes [math]\displaystyle{ 2^2 = 4 }[/math] steps instead of one. Then 7 extra steps before the 4th harmonic so it takes [math]\displaystyle{ 2^3 = 8 }[/math] steps instead of one.
| 2-logharmonic series | harmonic series | ||||||||||
| pitch # | frequency multiplier (definition) | frequency multiplier (decimal) | pitch (¢) | octave reduced pitch (¢) | pitch Δ (¢) | pitch # | frequency multiplier (definition) | frequency multiplier (decimal) | pitch (¢) | octave reduced pitch (¢) | pitch Δ (¢) |
| 2 | log22 | 1.00000000 | 0.00 | 0.00 | - | 1 | 1 | 1.000000 | 0.00 | 0.00 | 1200.00 |
| 3 | log23 | 1.584962501 | 797.34 | 797.34 | 797.34 | ||||||
| 4 | log24 | 2.00000000 | 1200.00 | 0.00 | 402.66 | 2 | 2 | 2.000000 | 1200.00 | 0.00 | 701.96 |
| 5 | log25 | 2.321928095 | 1458.39 | 258.39 | 258.39 | ||||||
| 6 | log26 | 2.584962501 | 1644.17 | 444.17 | 185.78 | ||||||
| 7 | log27 | 2.807354922 | 1787.05 | 587.05 | 142.88 | ||||||
| 8 | log28 | 3.00000000 | 1901.96 | 701.96 | 114.90 | 3 | 3 | 3.000000 | 1901.96 | 701.96 | 498.04 |
| 9 | log29 | 3.169925001 | 1997.34 | 797.34 | 95.38 | ||||||
| 1 | log210 | 3.321928095 | 2078.43 | 878.43 | 81.09 | ||||||
| 11 | log211 | 3.459431619 | 2148.64 | 948.64 | 70.22 | ||||||
| 12 | log212 | 3.584962501 | 2210.35 | 1010.35 | 61.71 | ||||||
| 13 | log213 | 3.700439718 | 2265.24 | 1065.24 | 54.89 | ||||||
| 14 | log214 | 3.807354922 | 2314.55 | 1114.55 | 49.31 | ||||||
| 15 | log215 | 3.906890596 | 2359.23 | 1159.23 | 44.68 | ||||||
| 16 | log216 | 4.00000000 | 2400.00 | 0.00 | 40.77 | 4 | 4 | 4.000000 | 2400.00 | 0.00 | 386.31 |
| 17 | log217 | 4.087462841 | 2437.45 | 37.45 | 37.45 | ||||||
| 18 | log218 | 4.169925001 | 2472.03 | 72.03 | 34.58 | ||||||
| 19 | log219 | 4.247927513 | 2504.11 | 104.11 | 32.09 | ||||||
| 20 | log220 | 4.321928095 | 2534.01 | 134.01 | 29.90 | ||||||
| 21 | log221 | 4.392317423 | 2561.98 | 161.98 | 27.97 | ||||||
| 22 | log222 | 4.459431619 | 2588.23 | 188.23 | 26.25 | ||||||
| 23 | log223 | 4.523561956 | 2612.95 | 212.95 | 24.72 | ||||||
| 24 | log224 | 4.584962501 | 2636.29 | 236.29 | 23.34 | ||||||
| 25 | log225 | 4.64385619 | 2658.39 | 258.39 | 22.10 | ||||||
| 26 | log226 | 4.700439718 | 2679.35 | 279.35 | 20.97 | ||||||
| 27 | log227 | 4.754887502 | 2699.29 | 299.29 | 19.94 | ||||||
| 28 | log228 | 4.807354922 | 2718.29 | 318.29 | 19.00 | ||||||
| 29 | log229 | 4.857980995 | 2736.43 | 336.43 | 18.14 | ||||||
| 30 | log230 | 4.906890596 | 2753.77 | 353.77 | 17.34 | ||||||
| 31 | log231 | 4.95419631 | 2770.38 | 370.38 | 16.61 | ||||||
| 32 | log232 | 5.00000000 | 2786.31 | 386.31 | 15.93 | 5 | 5 | 5.000000 | 2786.31 | 386.31 | 315.64 |
matharmonic series
The e-logharmonic series can be approximated by pitches taken from the mathematical harmonic series:
1/1, 3/2, 11/6, 25/12, 137/60, 49/20, ...