Logharmonic series: Difference between revisions
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</math> | </math> | ||
If a natural number is chosen as | If a natural number is chosen as , 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 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 steps instead of one. Then 7 extra steps before the 4th harmonic so it takes steps instead of 1.<span><math>b</math></span><span><math>2^1 = 2</math></span><span><math>2^2 = 4</math></span><span><math>2^3 = 8</math></span> | ||
{| class="wikitable" | |||
|+ | |||
| colspan="6" |2-logharmonic series | |||
| colspan="6" |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 | |||
|log<sub>2</sub>2 | |||
|1.00000000 | |||
|0.00 | |||
|0.00 | |||
| - | |||
|1 | |||
|1 | |||
|1.000000 | |||
|0.00 | |||
|0.00 | |||
|1200.00 | |||
|- | |||
|3 | |||
|log<sub>2</sub>3 | |||
|1.584962501 | |||
|797.34 | |||
|797.34 | |||
|797.34 | |||
| colspan="6" | | |||
|- | |||
|4 | |||
|log<sub>2</sub>4 | |||
|2.00000000 | |||
|1200.00 | |||
|0.00 | |||
|402.66 | |||
|2 | |||
|2 | |||
|2.000000 | |||
|1200.00 | |||
|0.00 | |||
|701.96 | |||
|- | |||
|5 | |||
|log<sub>2</sub>5 | |||
|2.321928095 | |||
|1458.39 | |||
|258.39 | |||
|258.39 | |||
| colspan="6" rowspan="3" | | |||
|- | |||
|6 | |||
|log<sub>2</sub>6 | |||
|2.584962501 | |||
|1644.17 | |||
|444.17 | |||
|185.78 | |||
|- | |||
|7 | |||
|log<sub>2</sub>7 | |||
|2.807354922 | |||
|1787.05 | |||
|587.05 | |||
|142.88 | |||
|- | |||
|8 | |||
|log<sub>2</sub>8 | |||
|3.00000000 | |||
|1901.96 | |||
|701.96 | |||
|114.90 | |||
|3 | |||
|3 | |||
|3.000000 | |||
|1901.96 | |||
|701.96 | |||
|498.04 | |||
|- | |||
|9 | |||
|log<sub>2</sub>9 | |||
|3.169925001 | |||
|1997.34 | |||
|797.34 | |||
|95.38 | |||
| colspan="6" rowspan="7" | | |||
|- | |||
|1 | |||
|log<sub>2</sub>10 | |||
|3.321928095 | |||
|2078.43 | |||
|878.43 | |||
|81.09 | |||
|- | |||
|11 | |||
|log<sub>2</sub>11 | |||
|3.459431619 | |||
|2148.64 | |||
|948.64 | |||
|70.22 | |||
|- | |||
|12 | |||
|log<sub>2</sub>12 | |||
|3.584962501 | |||
|2210.35 | |||
|1010.35 | |||
|61.71 | |||
|- | |||
|13 | |||
|log<sub>2</sub>13 | |||
|3.700439718 | |||
|2265.24 | |||
|1065.24 | |||
|54.89 | |||
|- | |||
|14 | |||
|log<sub>2</sub>14 | |||
|3.807354922 | |||
|2314.55 | |||
|1114.55 | |||
|49.31 | |||
|- | |||
|15 | |||
|log<sub>2</sub>15 | |||
|3.906890596 | |||
|2359.23 | |||
|1159.23 | |||
|44.68 | |||
|- | |||
|16 | |||
|log<sub>2</sub>16 | |||
|4.00000000 | |||
|2400.00 | |||
|0.00 | |||
|40.77 | |||
|4 | |||
|4 | |||
|4.000000 | |||
|2400.00 | |||
|0.00 | |||
|386.31 | |||
|- | |||
|17 | |||
|log<sub>2</sub>17 | |||
|4.087462841 | |||
|2437.45 | |||
|37.45 | |||
|37.45 | |||
| colspan="6" rowspan="15" | | |||
|- | |||
|18 | |||
|log<sub>2</sub>18 | |||
|4.169925001 | |||
|2472.03 | |||
|72.03 | |||
|34.58 | |||
|- | |||
|19 | |||
|log<sub>2</sub>19 | |||
|4.247927513 | |||
|2504.11 | |||
|104.11 | |||
|32.09 | |||
|- | |||
|20 | |||
|log<sub>2</sub>20 | |||
|4.321928095 | |||
|2534.01 | |||
|134.01 | |||
|29.90 | |||
|- | |||
|21 | |||
|log<sub>2</sub>21 | |||
|4.392317423 | |||
|2561.98 | |||
|161.98 | |||
|27.97 | |||
|- | |||
|22 | |||
|log<sub>2</sub>22 | |||
|4.459431619 | |||
|2588.23 | |||
|188.23 | |||
|26.25 | |||
|- | |||
|23 | |||
|log<sub>2</sub>23 | |||
|4.523561956 | |||
|2612.95 | |||
|212.95 | |||
|24.72 | |||
|- | |||
|24 | |||
|log<sub>2</sub>24 | |||
|4.584962501 | |||
|2636.29 | |||
|236.29 | |||
|23.34 | |||
|- | |||
|25 | |||
|log<sub>2</sub>25 | |||
|4.64385619 | |||
|2658.39 | |||
|258.39 | |||
|22.10 | |||
|- | |||
|26 | |||
|log<sub>2</sub>26 | |||
|4.700439718 | |||
|2679.35 | |||
|279.35 | |||
|20.97 | |||
|- | |||
|27 | |||
|log<sub>2</sub>27 | |||
|4.754887502 | |||
|2699.29 | |||
|299.29 | |||
|19.94 | |||
|- | |||
|28 | |||
|log<sub>2</sub>28 | |||
|4.807354922 | |||
|2718.29 | |||
|318.29 | |||
|19.00 | |||
|- | |||
|29 | |||
|log<sub>2</sub>29 | |||
|4.857980995 | |||
|2736.43 | |||
|336.43 | |||
|18.14 | |||
|- | |||
|30 | |||
|log<sub>2</sub>30 | |||
|4.906890596 | |||
|2753.77 | |||
|353.77 | |||
|17.34 | |||
|- | |||
|31 | |||
|log<sub>2</sub>31 | |||
|4.95419631 | |||
|2770.38 | |||
|370.38 | |||
|16.61 | |||
|- | |||
|32 | |||
|log<sub>2</sub>32 | |||
|5.00000000 | |||
|2786.31 | |||
|386.31 | |||
|15.93 | |||
|5 | |||
|5 | |||
|5.000000 | |||
|2786.31 | |||
|386.31 | |||
|315.64 | |||
|} | |||
== matharmonic series == | == matharmonic series == | ||