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Logharmonic series: Difference between revisions

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__FORCETOC__
{{Mathematical interest}}


== Introduction ==
== Introduction ==
[[File:2-logharmonic vs harmonic.png|thumb|
[[File:2-logharmonic vs harmonic.png|thumb|
2-logharmonic series vs. harmonic series
2-logharmonic series vs. harmonic series
Line 11: Line 10:
<math>\qquad f(n) = log_b{n}
<math>\qquad f(n) = log_b{n}
</math>
</math>
Where <span><math>b > 1</math></span>.


At <span><math>f(1)</math></span>, any logharmonic series will be <span><math>0</math></span>, which is not useful as a frequency multiplier, since there is no such thing as 0 Hz. So, we ignore the first step of logharmonic series.
At <span><math>f(1)</math></span>, any logharmonic series will be <span><math>0</math></span>, which is not useful as a frequency multiplier, since there is no such thing as 0 Hz. So, we ignore the first step of logharmonic series.
Line 16: Line 17:
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 one.
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 one.


[[Logarithmic intonation]] is an analogue of [[just intonation]] which uses the base-e logharmonic series rather than the [[harmonic series]].
== Frequencies ==
{| class="wikitable"
{| class="wikitable"
|+
| colspan="6" |2-logharmonic series
| colspan="6" |harmonic series
|-
|-
|'''pitch #'''
| colspan="6" | 2-logharmonic series
|'''frequency multiplier (definition)'''
| colspan="6" | Harmonic series
|'''frequency multiplier (decimal)'''
|-
|'''pitch (¢)'''
! Pitch #
|'''pitch Δ (¢)'''
! Frequency multiplier (definition)
|'''octave reduced pitch (¢)'''
! Frequency multiplier (decimal)
|'''pitch #'''
! Pitch (¢)
|'''frequency multiplier (definition)'''
! Pitch Δ (¢)
|'''frequency multiplier (decimal)'''
! Octave reduced pitch (¢)
|'''pitch (¢)'''
! Pitch #
|'''pitch Δ (¢)'''
! Frequency multiplier (definition)
|'''octave reduced pitch (¢)'''
! Frequency multiplier (decimal)
! Pitch (¢)
! Pitch Δ (¢)
! Octave reduced pitch (¢)
|-
|-
|2
| 2
|log<sub>2</sub>2
| log<sub>2</sub>2
|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
|-
|-
|3
| 3
|log<sub>2</sub>3
| log<sub>2</sub>3
|1.584962501
| 1.584962501
|797.34
| 797.34
|797.34
| 797.34
|797.34
| 797.34
| colspan="6" |
| colspan="6" |  
|-
|-
|4
| 4
|log<sub>2</sub>4
| log<sub>2</sub>4
|2.00000000
| 2.00000000
|1200.00
| 1200.00
|402.66
| 402.66
|0.00
| 0.00
|2
| 2
|2
| 2
|2.000000
| 2.000000
|1200.00
| 1200.00
|1200.00
| 1200.00
|0.00
| 0.00
|-
|-
|5
| 5
|log<sub>2</sub>5
| log<sub>2</sub>5
|2.321928095
| 2.321928095
|1458.39
| 1458.39
|258.39
| 258.39
|258.39
| 258.39
| colspan="6" rowspan="3" |
| colspan="6" rowspan="3" |  
|-
|-
|6
| 6
|log<sub>2</sub>6
| log<sub>2</sub>6
|2.584962501
| 2.584962501
|1644.17
| 1644.17
|185.78
| 185.78
|444.17
| 444.17
|-
|-
|7
| 7
|log<sub>2</sub>7
| log<sub>2</sub>7
|2.807354922
| 2.807354922
|1787.05
| 1787.05
|142.88
| 142.88
|587.05
| 587.05
|-
|-
|8
| 8
|log<sub>2</sub>8
| log<sub>2</sub>8
|3.00000000
| 3.00000000
|1901.96
| 1901.96
|114.90
| 114.90
|701.96
| 701.96
|3
| 3
|3
| 3
|3.000000
| 3.000000
|1901.96
| 1901.96
|701.96
| 701.96
|701.96
| 701.96
|-
|-
|9
| 9
|log<sub>2</sub>9
| log<sub>2</sub>9
|3.169925001
| 3.169925001
|1997.34
| 1997.34
|95.38
| 95.38
|797.34
| 797.34
| colspan="6" rowspan="7" |
| colspan="6" rowspan="7" |  
|-
|-
|1
| 10
|log<sub>2</sub>10
| log<sub>2</sub>10
|3.321928095
| 3.321928095
|2078.43
| 2078.43
|81.09
| 81.09
|878.43
| 878.43
|-
|-
|11
| 11
|log<sub>2</sub>11
| log<sub>2</sub>11
|3.459431619
| 3.459431619
|2148.64
| 2148.64
|70.22
| 70.22
|948.64
| 948.64
|-
|-
|12
| 12
|log<sub>2</sub>12
| log<sub>2</sub>12
|3.584962501
| 3.584962501
|2210.35
| 2210.35
|61.71
| 61.71
|1010.35
| 1010.35
|-
|-
|13
| 13
|log<sub>2</sub>13
| log<sub>2</sub>13
|3.700439718
| 3.700439718
|2265.24
| 2265.24
|54.89
| 54.89
|1065.24
| 1065.24
|-
|-
|14
| 14
|log<sub>2</sub>14
| log<sub>2</sub>14
|3.807354922
| 3.807354922
|2314.55
| 2314.55
|49.31
| 49.31
|1114.55
| 1114.55
|-
|-
|15
| 15
|log<sub>2</sub>15
| log<sub>2</sub>15
|3.906890596
| 3.906890596
|2359.23
| 2359.23
|44.68
| 44.68
|1159.23
| 1159.23
|-
|-
|16
| 16
|log<sub>2</sub>16
| log<sub>2</sub>16
|4.00000000
| 4.00000000
|2400.00
| 2400.00
|40.77
| 40.77
|0.00
| 0.00
|4
| 4
|4
| 4
|4.000000
| 4.000000
|2400.00
| 2400.00
|498.04
| 498.04
|0.00
| 0.00
|-
|-
|17
| 17
|log<sub>2</sub>17
| log<sub>2</sub>17
|4.087462841
| 4.087462841
|2437.45
| 2437.45
|37.45
| 37.45
|37.45
| 37.45
| colspan="6" rowspan="15" |
| colspan="6" rowspan="15" |  
|-
|-
|18
| 18
|log<sub>2</sub>18
| log<sub>2</sub>18
|4.169925001
| 4.169925001
|2472.03
| 2472.03
|34.58
| 34.58
|72.03
| 72.03
|-
|-
|19
| 19
|log<sub>2</sub>19
| log<sub>2</sub>19
|4.247927513
| 4.247927513
|2504.11
| 2504.11
|32.09
| 32.09
|104.11
| 104.11
|-
|-
|20
| 20
|log<sub>2</sub>20
| log<sub>2</sub>20
|4.321928095
| 4.321928095
|2534.01
| 2534.01
|29.90
| 29.90
|134.01
| 134.01
|-
|-
|21
| 21
|log<sub>2</sub>21
| log<sub>2</sub>21
|4.392317423
| 4.392317423
|2561.98
| 2561.98
|27.97
| 27.97
|161.98
| 161.98
|-
|-
|22
| 22
|log<sub>2</sub>22
| log<sub>2</sub>22
|4.459431619
| 4.459431619
|2588.23
| 2588.23
|26.25
| 26.25
|188.23
| 188.23
|-
|-
|23
| 23
|log<sub>2</sub>23
| log<sub>2</sub>23
|4.523561956
| 4.523561956
|2612.95
| 2612.95
|24.72
| 24.72
|212.95
| 212.95
|-
|-
|24
| 24
|log<sub>2</sub>24
| log<sub>2</sub>24
|4.584962501
| 4.584962501
|2636.29
| 2636.29
|23.34
| 23.34
|236.29
| 236.29
|-
|-
|25
| 25
|log<sub>2</sub>25
| log<sub>2</sub>25
|4.64385619
| 4.64385619
|2658.39
| 2658.39
|22.10
| 22.10
|258.39
| 258.39
|-
|-
|26
| 26
|log<sub>2</sub>26
| log<sub>2</sub>26
|4.700439718
| 4.700439718
|2679.35
| 2679.35
|20.97
| 20.97
|279.35
| 279.35
|-
|-
|27
| 27
|log<sub>2</sub>27
| log<sub>2</sub>27
|4.754887502
| 4.754887502
|2699.29
| 2699.29
|19.94
| 19.94
|299.29
| 299.29
|-
|-
|28
| 28
|log<sub>2</sub>28
| log<sub>2</sub>28
|4.807354922
| 4.807354922
|2718.29
| 2718.29
|19.00
| 19.00
|318.29
| 318.29
|-
|-
|29
| 29
|log<sub>2</sub>29
| log<sub>2</sub>29
|4.857980995
| 4.857980995
|2736.43
| 2736.43
|18.14
| 18.14
|336.43
| 336.43
|-
|-
|30
| 30
|log<sub>2</sub>30
| log<sub>2</sub>30
|4.906890596
| 4.906890596
|2753.77
| 2753.77
|17.34
| 17.34
|353.77
| 353.77
|-
|-
|31
| 31
|log<sub>2</sub>31
| log<sub>2</sub>31
|4.95419631
| 4.95419631
|2770.38
| 2770.38
|16.61
| 16.61
|370.38
| 370.38
|-
|-
|32
| 32
|log<sub>2</sub>32
| log<sub>2</sub>32
|5.00000000
| 5.00000000
|2786.31
| 2786.31
|15.93
| 15.93
|386.31
| 386.31
|5
| 5
|5
| 5
|5.000000
| 5.000000
|2786.31
| 2786.31
|386.31
| 386.31
|386.31
| 386.31
|}
|}


Line 291: Line 295:


== Matharmonic series ==
== Matharmonic series ==
The logharmonic series can be approximated by pitches taken from the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series):
The logharmonic series can be approximated by pitches taken from the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series):


Line 307: Line 310:


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


The matharmonic series is to the logharmonic series as an edharmonic series is to a ln-of-a-[[Powharmonic series|powharmonic series]].
The matharmonic series is to the logharmonic series as an edharmonic series is to a ln-of-a-[[Powharmonic series|powharmonic series]].
== Emulatory matharmonic series ==
The first two steps of the matharmonic series are 1 and 3/2, which have the same ratio as the second and third steps of the harmonic series, 2 and 3. To make them align the matharmonic series may be rebased onto 2, starting it a step late, inserting a 1 before it starts. This brings it closer in similarity to the harmonic series; this similarity could be useful when using the entire series as a scale rather than drawing scales from it. We therefore propose referring to this variation as the "emulatory edharmonic series", because it emulates the harmonic series.
{| class="wikitable"
|-
| rowspan="2" ! Pitch #
| colspan="4" rowspan="1" ! Harmonic series
| colspan="5" ! Emulatory matharmonic series
|-
! Frequency multiplier (decimal)
! Pitch (¢)
! Pitch Δ (¢)
! Octave reduced pitch (¢)
! Frequency multiplier (definition)
! Frequency multiplier (decimal)
! Pitch (¢)
! Pitch Δ (¢)
! Octave reduced pitch (¢)
|-
! 1
| 1.000000
| 0.00
|  -
| 0.000000
| 1
| 1
| 0.00
| 0.00
| 1200.00
|-
! 2
| 2.000000
| 1200.00
| 1200.00
| 0.000000
| 2⋅H(1)
| 2
| 1200.00
| 0.00
| 701.96
|-
! 3
| 3.000000
| 1901.96
| 701.96
| 701.955001
| 2⋅H(2)
| 3
| 1901.96
| 701.96
| 347.41
|-
! 4
| 4.000000
| 2400.00
| 498.04
| 0.000000
| 2⋅H(3)
| 3.666666667
| 2249.36
| 1049.36
| 221.31
|-
! 5
| 5.000000
| 2786.31
| 386.31
| 386.313714
| 2⋅H(4)
| 4.166666667
| 2470.67
| 70.67
| 158.70
|-
! 6
| 6.000000
| 3101.96
| 315.64
| 701.955001
| 2⋅H(5)
| 4.566666667
| 2629.37
| 229.37
| 121.97
|-
! 7
| 7.000000
| 3368.83
| 266.87
| 968.825906
| 2⋅H(6)
| 4.9
| 2751.34
| 351.34
| 98.11
|-
! 8
| 8.000000
| 3600.00
| 231.17
| 0.000000
| 2⋅H(7)
| 5.185714286
| 2849.45
| 449.45
| 81.51
|-
! 9
| 9.000000
| 3803.91
| 203.91
| 203.910002
| 2⋅H(8)
| 5.435714286
| 2930.96
| 530.96
| 69.37
|-
! 10
| 10.000000
| 3986.31
| 182.40
| 386.313714
| 2⋅H(9)
| 5.657936508
| 3000.33
| 600.33
| 60.14
|-
! 11
| 11.000000
| 4151.32
| 165.00
| 551.317942
| 2⋅H(10)
| 5.857936508
| 3060.47
| 660.47
| 52.92
|-
! 12
| 12.000000
| 4301.96
| 150.64
| 701.955001
| 2⋅H(11)
| 6.03975469
| 3113.39
| 713.39
| 47.13
|-
! 13
| 13.000000
| 4440.53
| 138.57
| 840.527662
| 2⋅H(12)
| 6.206421356
| 3160.51
| 760.51
| 42.39
|-
! 14
| 14.000000
| 4568.83
| 128.30
| 968.825906
| 2⋅H(13)
| 6.36026751
| 3202.90
| 802.90
| 38.45
|-
! 15
| 15.000000
| 4688.27
| 119.44
| 1088.268715
| 2⋅H(14)
| 6.503124653
| 3241.36
| 841.36
| 35.14
|-
! 16
| 16.000000
| 4800.00
| 111.73
| 0.000000
| 2⋅H(15)
| 6.636457986
| 3276.50
| 876.50
| 34.44
|}
An analogous [https://en.xen.wiki/w/Powharmonic_series#Emulatory_edharmonic_series emulatory edharmonic series] exists.
== Sublogharmonic series ==
Just like the subharmonic series can be found by using one over the frequency of any step of the harmonic series, an equivalent sublogharmonic series may be found by using one over the frequency of any step of a logharmonic series.
== Sound of the logharmonic series as a timbre ==
The usual harmonic series can be thought of as being derived from a periodic timbre, such as a sawtooth wave. In general, we can derive the sawtooth wave as
<math>\sum_{n=1}^\infty \frac{\sin(nt)}{n}</math>
We can change the "rolloff" on the harmonics to get different derived waveforms which instead have the partials rolling off at a rate of <math>n^s</math>:
<math>\sum_{n=1}^\infty \frac{\sin(nt)}{n^s}</math>
As <math>s</math> increases, so does the rolloff (and hence the brightness of the timbre), so that as we get <math>s \to \infty</math>, our sawtooth wave goes to a sine wave.
If we instead replace our <math>\sin(nt)</math> term with <math>\sin(\log(n) t)</math>, we instead get the Riemann Zeta function, treated as an audio sound wave:
<math>\sum_{n=1}^\infty \frac{\sin(\log(n) t)}{n^s} = \mathbf{Im}[-\zeta(s+it)]</math>
This is easy to see using the Dirichlet series representation:
<math>\zeta(s+it) = \sum_n \frac{1}{n^{s+it}} = \sum_n \frac{n^{-it}}{n^s} = \sum_n \frac{\exp(-it \log(n))}{n^s}</math>
And since we have <math>-\sin(t \log(n)) = \exp(-it \log(n))</math>, this completes our proof.
Note that the above only converges if we have <math>s > 1</math>, e.g. if the rolloff is greater than that of a sawtooth. However, we can use the analytic continuation of the Riemann Zeta function to procure a function even for <math>s < 1</math>. Here is an example of the critical line of the Riemann zeta function (e.g. <math>s = 0.5</math> played as a sound wave):
<youtube>https://www.youtube.com/watch?v=-iZhaAlmoFE</youtube>
(Mike Battaglia's note: I believe this is technically the real part of the critical line, which sounds basically the same as the imaginary part, only with the phases shifted from sine waves to cosine waves.)
== Listening ==
[https://www.youtube.com/watch?v=VoPv-YziWoc The Apples in Stereo - C.P.U.]


== See also ==
== See also ==
* [[Harmonotonic tunings]]: logharmonic series are non-[[Arithmetic tunings|arithmetic]] harmonotonic tunings.
* [[Powharmonic series]]: another type of non-arithmetic harmonotonic tuning.
* [[Xenharmonic series]]
* [[Logarithmic intonation]]
* {{W|Robert Schneider#Non-Pythagorean scale|Robert Schneider's non-Pythagorean scale on Wikipedia}}


[[Powharmonic series|Powharmonic series]]
[[Category:Otonality and utonality]]
[[Category:Otonality]]
[[Category:Harmonic]]
[[Category:Harmonic series]]
[[Category:Utonality]]
[[Category:Subharmonic]]
[[Category:Subharmonic series]]
[[Category:Xenharmonic series]]
Retrieved from "https://en.xen.wiki/w/Logharmonic_series"