Powharmonic series: Difference between revisions

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== Introduction ==
== Introduction ==
A powerharmonic series, like the [[wikipedia:Harmonic_series_(music)|harmonic series]], is an infinitely ascending set of pitches from which scales can be drawn.
A powerharmonic series, like the [[wikipedia:Harmonic_series_(music)|harmonic series]], is an infinitely ascending set of pitches from which scales can be drawn.


== p-powharmonic series ==
== ''p''-Powharmonic series ==
 
A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a ''p-powharmonic series'' is simply:
A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a ''p-powharmonic series'' is simply:


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For example, the 0.69314718056-powharmonic series looks like this:
For example, the 0.69314718056-powharmonic series looks like this:
 
{{powharmonic series|12|expo=0.69314718056}}
{| class="wikitable"
|+
!pitch #
|'''frequency multiplier (definition)'''
|'''frequency multiplier (decimal)'''
|'''pitch (¢)'''
|'''pitch Δ (¢)'''
|'''octave reduced pitch (¢)'''
|-
|1
|1<sup>0.69314718056</sup>
|1
|0.00
| -
|0.00
|-
|2
|2<sup>0.69314718056</sup>
|1.616806672
|831.78
|831.78
|831.78
|-
|3
|3<sup>0.69314718056</sup>
|2.141486064
|1318.33
|486.56
|118.33
|-
|4
|4<sup>0.69314718056</sup>
|2.614063815
|1663.55
|345.22
|463.55
|-
|5
|5<sup>0.69314718056</sup>
|3.05132936
|1931.33
|267.77
|731.33
|-
|6
|6<sup>0.69314718056</sup>
|3.462368957
|2150.11
|218.79
|950.11
|-
|7
|7<sup>0.69314718056</sup>
|3.852807616
|2335.09
|184.98
|1135.09
|-
|8
|8<sup>0.69314718056</sup>
|4.226435818
|2495.33
|160.24
|95.33
|-
|9
|9<sup>0.69314718056</sup>
|4.585962562
|2636.67
|141.34
|236.67
|-
|10
|10<sup>0.69314718056</sup>
|4.933409668
|2763.10
|126.43
|363.10
|-
|11
|11<sup>0.69314718056</sup>
|5.270337212
|2877.47
|114.37
|477.47
|-
|12
|12<sup>0.69314718056</sup>
|5.597981231
|2981.89
|104.41
|581.89
|-
|13
|13<sup>0.69314718056</sup>
|5.917342318
|3077.94
|96.05
|677.94
|-
|14
|14<sup>0.69314718056</sup>
|6.22924506
|3166.87
|88.93
|766.87
|-
|15
|15<sup>0.69314718056</sup>
|6.5343793
|3249.66
|82.79
|849.66
|-
|16
|16<sup>0.69314718056</sup>
|6.833329631
|3327.11
|77.45
|927.11
|}


The harmonic series is technically a powharmonic series: the 1-powharmonic series. <span><math>p</math><span> closer to 1 give powharmonic series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.
The harmonic series is technically a powharmonic series: the 1-powharmonic series. <span><math>p</math><span> closer to 1 give powharmonic series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.