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 ==

Revision as of 02:03, 3 February 2020

introduction

2-logharmonic series vs. harmonic series

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 , 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.[math]\displaystyle{ b }[/math][math]\displaystyle{ 2^1 = 2 }[/math][math]\displaystyle{ 2^2 = 4 }[/math][math]\displaystyle{ 2^3 = 8 }[/math]

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, ...

Retrieved from "https://en.xen.wiki/index.php?title=Logharmonic_series&oldid=43369"