Logharmonic series: Difference between revisions

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If a natural number is chosen as <span><math>b</math></span>, 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 <span><math>2^1 = 2</math></span> 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 <span><math>2^2 = 4</math></span> steps instead of one. Then 7 extra steps before the 4th harmonic so it takes <span><math>2^3 = 8</math></span> steps instead of 1.
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 ==