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Logharmonic series: Difference between revisions

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5,172 edits
Line 21: Line 21:
|'''frequency multiplier (decimal)'''
|'''frequency multiplier (decimal)'''
|'''pitch (¢)'''
|'''pitch (¢)'''
|'''pitch Δ (¢)'''
|'''octave reduced pitch (¢)'''
|'''octave reduced pitch (¢)'''
|'''pitch Δ (¢)'''
|'''pitch #'''
|'''pitch #'''
|'''frequency multiplier (definition)'''
|'''frequency multiplier (definition)'''
|'''frequency multiplier (decimal)'''
|'''frequency multiplier (decimal)'''
|'''pitch (¢)'''
|'''pitch (¢)'''
|'''pitch Δ (¢)'''
|'''octave reduced pitch (¢)'''
|'''octave reduced pitch (¢)'''
|'''pitch Δ (¢)'''
|-
|-
|2
|2
Line 34: Line 34:
|1.00000000
|1.00000000
|0.00
|0.00
| -
|0.00
|0.00
| -
|1
|1
|1
|1
|1.000000
|1.000000
|0.00
|0.00
| -
|0.00
|0.00
|1200.00
|-
|-
|3
|3
Line 55: Line 55:
|2.00000000
|2.00000000
|1200.00
|1200.00
|402.66
|0.00
|0.00
|402.66
|2
|2
|2
|2
|2.000000
|2.000000
|1200.00
|1200.00
|1200.00
|0.00
|0.00
|701.96
|-
|-
|5
|5
Line 76: Line 76:
|2.584962501
|2.584962501
|1644.17
|1644.17
|185.78
|444.17
|444.17
|185.78
|-
|-
|7
|7
Line 83: Line 83:
|2.807354922
|2.807354922
|1787.05
|1787.05
|142.88
|587.05
|587.05
|142.88
|-
|-
|8
|8
Line 90: Line 90:
|3.00000000
|3.00000000
|1901.96
|1901.96
|114.90
|701.96
|701.96
|114.90
|3
|3
|3
|3
Line 97: Line 97:
|1901.96
|1901.96
|701.96
|701.96
|498.04
|701.96
|-
|-
|9
|9
Line 103: Line 103:
|3.169925001
|3.169925001
|1997.34
|1997.34
|95.38
|797.34
|797.34
|95.38
| colspan="6" rowspan="7" |
| colspan="6" rowspan="7" |
|-
|-
Line 111: Line 111:
|3.321928095
|3.321928095
|2078.43
|2078.43
|81.09
|878.43
|878.43
|81.09
|-
|-
|11
|11
Line 118: Line 118:
|3.459431619
|3.459431619
|2148.64
|2148.64
|70.22
|948.64
|948.64
|70.22
|-
|-
|12
|12
Line 125: Line 125:
|3.584962501
|3.584962501
|2210.35
|2210.35
|61.71
|1010.35
|1010.35
|61.71
|-
|-
|13
|13
Line 132: Line 132:
|3.700439718
|3.700439718
|2265.24
|2265.24
|54.89
|1065.24
|1065.24
|54.89
|-
|-
|14
|14
Line 139: Line 139:
|3.807354922
|3.807354922
|2314.55
|2314.55
|49.31
|1114.55
|1114.55
|49.31
|-
|-
|15
|15
Line 146: Line 146:
|3.906890596
|3.906890596
|2359.23
|2359.23
|44.68
|1159.23
|1159.23
|44.68
|-
|-
|16
|16
Line 153: Line 153:
|4.00000000
|4.00000000
|2400.00
|2400.00
|40.77
|0.00
|0.00
|40.77
|4
|4
|4
|4
|4.000000
|4.000000
|2400.00
|2400.00
|498.04
|0.00
|0.00
|386.31
|-
|-
|17
|17
Line 174: Line 174:
|4.169925001
|4.169925001
|2472.03
|2472.03
|34.58
|72.03
|72.03
|34.58
|-
|-
|19
|19
Line 181: Line 181:
|4.247927513
|4.247927513
|2504.11
|2504.11
|32.09
|104.11
|104.11
|32.09
|-
|-
|20
|20
Line 188: Line 188:
|4.321928095
|4.321928095
|2534.01
|2534.01
|29.90
|134.01
|134.01
|29.90
|-
|-
|21
|21
Line 195: Line 195:
|4.392317423
|4.392317423
|2561.98
|2561.98
|27.97
|161.98
|161.98
|27.97
|-
|-
|22
|22
Line 202: Line 202:
|4.459431619
|4.459431619
|2588.23
|2588.23
|26.25
|188.23
|188.23
|26.25
|-
|-
|23
|23
Line 209: Line 209:
|4.523561956
|4.523561956
|2612.95
|2612.95
|24.72
|212.95
|212.95
|24.72
|-
|-
|24
|24
Line 216: Line 216:
|4.584962501
|4.584962501
|2636.29
|2636.29
|23.34
|236.29
|236.29
|23.34
|-
|-
|25
|25
Line 223: Line 223:
|4.64385619
|4.64385619
|2658.39
|2658.39
|22.10
|258.39
|258.39
|22.10
|-
|-
|26
|26
Line 230: Line 230:
|4.700439718
|4.700439718
|2679.35
|2679.35
|20.97
|279.35
|279.35
|20.97
|-
|-
|27
|27
Line 237: Line 237:
|4.754887502
|4.754887502
|2699.29
|2699.29
|19.94
|299.29
|299.29
|19.94
|-
|-
|28
|28
Line 244: Line 244:
|4.807354922
|4.807354922
|2718.29
|2718.29
|19.00
|318.29
|318.29
|19.00
|-
|-
|29
|29
Line 251: Line 251:
|4.857980995
|4.857980995
|2736.43
|2736.43
|18.14
|336.43
|336.43
|18.14
|-
|-
|30
|30
Line 258: Line 258:
|4.906890596
|4.906890596
|2753.77
|2753.77
|17.34
|353.77
|353.77
|17.34
|-
|-
|31
|31
Line 265: Line 265:
|4.95419631
|4.95419631
|2770.38
|2770.38
|16.61
|370.38
|370.38
|16.61
|-
|-
|32
|32
Line 272: Line 272:
|5.00000000
|5.00000000
|2786.31
|2786.31
|15.93
|386.31
|386.31
|15.93
|5
|5
|5
|5
Line 279: Line 279:
|2786.31
|2786.31
|386.31
|386.31
|315.64
|386.31
|}
|}


Line 296: Line 296:
We can call this approximating series the ''matharmonic series''.
We can call this approximating series the ''matharmonic series''.


The difference between pitches of the e-logharmonic series and the matharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the mathematical harmonic series.
The difference between pitches of the e-logharmonic series and the matharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], , which represents the difference between the natural logarithm and the mathematical harmonic series.
 
{| class="wikitable"
|+
| rowspan="2" |'''pitch #'''
| colspan="5" |'''e-logharmonic series'''
| colspan="5" |'''matharmonic series'''
| rowspan="2" |difference between frequency multipliers
|-
|'''frequency multiplier (definition)'''
|'''frequency multiplier (decimal)'''
|'''pitch (¢)'''
|'''pitch Δ (¢)'''
|'''octave reduced pitch (¢)'''
|'''frequency multiplier (definition)'''
|'''frequency multiplier (decimal)'''
|'''pitch (¢)'''
|'''pitch Δ (¢)'''
|'''octave reduced pitch (¢)'''
|-
|1
|ln(1)
|0
|N/A
|N/A
|N/A
|H(1)
|1
|0.00
|701.96
|0.00
|1
|-
|2
|ln(2)
|0.6931471806
| -634.52
| -
|565.48
|H(2)
|1.5
|701.96
|347.41
|701.96
|0.8068528194
|-
|3
|ln(3)
|1.098612289
|162.82
|797.34
|162.82
|H(3)
|1.833333333
|1049.36
|221.31
|1049.36
|0.7347210447
|-
|4
|ln(4)
|1.386294361
|565.48
|402.66
|565.48
|H(4)
|2.083333333
|1270.67
|158.70
|70.67
|0.6970389722
|-
|5
|ln(5)
|1.609437912
|823.87
|258.39
|823.87
|H(5)
|2.283333333
|1429.37
|121.97
|229.37
|0.6738954209
|-
|6
|ln(6)
|1.791759469
|1009.65
|185.78
|1009.65
|H(6)
|2.45
|1551.34
|98.11
|351.34
|0.6582405308
|-
|7
|ln(7)
|1.945910149
|1152.53
|142.88
|1152.53
|H(7)
|2.592857143
|1649.45
|81.51
|449.45
|0.6469469938
|-
|8
|ln(8)
|2.079441542
|1267.44
|114.90
|67.44
|H(8)
|2.717857143
|1730.96
|69.37
|530.96
|0.6384156012
|-
|9
|ln(9)
|2.197224577
|1362.82
|95.38
|162.82
|H(9)
|2.828968254
|1800.33
|60.14
|600.33
|0.6317436766
|-
|10
|ln(10)
|2.302585093
|1443.91
|81.09
|243.91
|H(10)
|2.928968254
|1860.47
|52.92
|660.47
|0.626383161
|-
|11
|ln(11)
|2.397895273
|1514.12
|70.22
|314.12
|H(11)
|3.019877345
|1913.39
|47.13
|713.39
|0.6219820721
|-
|12
|ln(12)
|2.48490665
|1575.83
|61.71
|375.83
|H(12)
|3.103210678
|1960.51
|42.39
|760.51
|0.6183040284
|-
|13
|ln(13)
|2.564949357
|1630.72
|54.89
|430.72
|H(13)
|3.180133755
|2002.90
|38.45
|802.90
|0.6151843977
|-
|14
|ln(14)
|2.63905733
|1680.03
|49.31
|480.03
|H(14)
|3.251562327
|2041.36
|35.14
|841.36
|0.6125049969
|-
|15
|ln(15)
|2.708050201
|1724.71
|44.68
|524.71
|H(15)
|3.318228993
|2076.50
|32.31
|876.50
|0.6101787921
|-
|16
|ln(16)
|2.772588722
|1765.48
|40.77
|565.48
|H(16)
|3.380728993
|2108.80
|29.86
|908.80
|0.608140271
|-
|17
|ln(17)
|2.833213344
|1802.93
|37.45
|602.93
|H(17)
|3.439552523
|2138.67
|27.74
|938.67
|0.6063391786
|-
|18
|ln(18)
|2.890371758
|1837.51
|34.58
|637.51
|H(18)
|3.495108078
|2166.40
|25.88
|966.40
|0.6047363203
|-
|19
|ln(19)
|2.944438979
|1869.59
|32.09
|669.59
|H(19)
|3.547739657
|2192.28
|24.23
|992.28
|0.603300678
|-
|20
|ln(20)
|2.995732274
|1899.49
|29.90
|699.49
|H(20)
|3.597739657
|2216.51
|22.76
|1016.51
|0.6020073836
|-
|21
|ln(21)
|3.044522438
|1927.46
|27.97
|727.46
|H(21)
|3.645358705
|2239.27
|21.45
|1039.27
|0.600836267
|-
|22
|ln(22)
|3.091042453
|1953.71
|26.25
|753.71
|H(22)
|3.69081325
|2260.73
|20.27
|1060.73
|0.5997707969
|-
|23
|ln(23)
|3.135494216
|1978.43
|24.72
|778.43
|H(23)
|3.734291511
|2281.00
|19.21
|1081.00
|0.5987972952
|-
|24
|ln(24)
|3.17805383
|2001.77
|23.34
|801.77
|H(24)
|3.775958178
|2300.21
|18.24
|1100.21
|0.5979043474
|-
|25
|ln(25)
|3.218875825
|2023.87
|22.10
|823.87
|H(25)
|3.815958178
|2318.45
|17.36
|1118.45
|0.5970823529
|-
|26
|ln(26)
|3.258096538
|2044.84
|20.97
|844.84
|H(26)
|3.854419716
|2335.82
|16.56
|1135.82
|0.5963231782
|-
|27
|ln(27)
|3.295836866
|2064.77
|19.94
|864.77
|H(27)
|3.891456753
|2352.37
|15.82
|1152.37
|0.5956198872
|-
|28
|ln(28)
|3.33220451
|2083.77
|19.00
|883.77
|H(28)
|3.927171039
|2368.19
|15.13
|1168.19
|0.5949665288
|-
|29
|ln(29)
|3.36729583
|2101.91
|18.14
|901.91
|H(29)
|3.961653798
|2383.32
|14.51
|1183.32
|0.5943579676
|-
|30
|ln(30)
|3.401197382
|2119.25
|17.34
|919.25
|H(30)
|3.994987131
|2397.83
|13.92
|1197.83
|0.5937897493 ... -> ''γ ='' 0.5772156649
|}
<span><math>≈ 0.5772156649</math></span>
Retrieved from "https://en.xen.wiki/w/Logharmonic_series"