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Powharmonic series: Difference between revisions

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Irrational values can be used as <span><math>a</math></span> or <span><math>b</math></span>.  
Irrational values can be used as <span><math>a</math></span> or <span><math>b</math></span>.


In particular it may be of interest to use [[wikipedia:E_(mathematical_constant)|<span><math>e</math></span>]] as <span><math>b</math></span> — in other words, to use a [[wikipedia:Natural_logarithm|natural logarithm]].
In particular it may be of interest to use [[wikipedia:E_(mathematical_constant)|<span><math>e</math></span>]] as <span><math>b</math></span> — in other words, to use a [[wikipedia:Natural_logarithm|natural logarithm]].
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=== description ===
=== description ===


Perhaps even more interestingly, a ln-of-a-powharmonic series can be approximated by moving by steps of increasing equal divisions of <span><math>a</math></span>.  
Perhaps even more interestingly, a ln-of-a-powharmonic series can be approximated by moving by steps of increasing equal divisions of <span><math>a</math></span>.


For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ln-of-2-powharmonic series. We could call this series the 2-edharmonic series.
For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ln-of-2-powharmonic series. We could call this series the 2-edharmonic series.
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In other words, if we have gone by a step of 1ed2, we are at <span><math>2^1</math></span>. If we then go by a step of 2ed2, we have gone by <span><math>2^1 · 2^{\frac12} = 2^{\frac32}</math></span>. And a further step of 3ed2 gets us to <span><math>2^1 · 2^{\frac12} · 2^{\frac13} = 2^{\frac{11}{6}}</math></span>, etc.
In other words, if we have gone by a step of 1ed2, we are at <span><math>2^1</math></span>. If we then go by a step of 2ed2, we have gone by <span><math>2^1 · 2^{\frac12} = 2^{\frac32}</math></span>. And a further step of 3ed2 gets us to <span><math>2^1 · 2^{\frac12} · 2^{\frac13} = 2^{\frac{11}{6}}</math></span>, etc.


(insert chart with edharmonic series, and maybe a few columns comparing it with ln-of-2 powharmonic series)
{| class="wikitable"
|+
! rowspan="2" |pitch #
! colspan="5" |ln-of-2-powharmonic series
! colspan="5" |2-edharmonic series
! rowspan="2" |ratio 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
|1<sup>ln(2)</sup> = 2<sup>ln(1)</sup>
|1
|0.00
| -
|0.00
|2<sup>H(1)</sup>
|2
|1200.00
| -
|0.00
|2
|-
|2
|2<sup>ln(2)</sup> = 2<sup>ln(2)</sup>
|1.616806672
|831.78
|831.78
|831.78
|2<sup>H(2)</sup>
|2.828427125
|1800.00
|600.00
|600.00
|1.749391052
|-
|3
|3<sup>ln(2)</sup> = 2<sup>ln(3)</sup>
|2.141486064
|1318.33
|486.56
|118.33
|2<sup>H(3)</sup>
|3.563594873
|2200.00
|400.00
|1000.00
|1.664075677
|-
|4
|4<sup>ln(2)</sup> = 2<sup>ln(4)</sup>
|2.614063815
|1663.55
|345.22
|463.55
|2<sup>ln(4)</sup>
|4.237852377
|2500.00
|300.00
|100.00
|1.621174033
|-
|5
|5<sup>ln(2)</sup> = 2<sup>ln(5)</sup>
|3.05132936
|1931.33
|267.77
|731.33
|2<sup>ln(5)</sup>
|4.868014055
|2740.00
|240.00
|340.00
|1.595374829
|-
|6
|6<sup>ln(2)</sup> = 2<sup>ln(6)</sup>
|3.462368957
|2150.11
|218.79
|950.11
|2<sup>ln(6)</sup>
|5.464161027
|2940.00
|200.00
|540.00
|1.578156775
|-
|7
|7<sup>ln(2)</sup> = 2<sup>ln(7)</sup>
|3.852807616
|2335.09
|184.98
|1135.09
|2<sup>ln(7)</sup>
|6.032922891
|3111.43
|171.43
|711.43
|1.56585106
|-
|8
|8<sup>ln(2)</sup> = 2<sup>ln(8)</sup>
|4.226435818
|2495.33
|160.24
|95.33
|2<sup>ln(8)</sup>
|6.578949063
|3261.43
|150.00
|861.43
|1.556618708
|-
|9
|9<sup>ln(2)</sup> = 2<sup>ln(9)</sup>
|4.585962562
|2636.67
|141.34
|236.67
|2<sup>ln(9)</sup>
|7.105658007
|3394.76
|133.33
|994.76
|1.549436549
|-
|10
|10<sup>ln(2)</sup> = 2<sup>ln(10)</sup>
|4.933409668
|2763.10
|126.43
|363.10
|2<sup>ln(10)</sup>
|7.615655686
|3514.76
|120.00
|1114.76
|1.543690105
|-
|11
|11<sup>ln(2)</sup> = 2<sup>ln(11)</sup>
|5.270337212
|2877.47
|114.37
|477.47
|2<sup>ln(11)</sup>
|8.110986229
|3623.85
|109.09
|23.85
|1.538988096
|-
|12
|12<sup>ln(2)</sup> = 2<sup>ln(12)</sup>
|5.597981231
|2981.89
|104.41
|581.89
|2<sup>ln(12)</sup>
|8.593290568
|3723.85
|100.00
|123.85
|1.535069557
|-
|13
|13<sup>ln(2)</sup> = 2<sup>ln(13)</sup>
|5.917342318
|3077.94
|96.05
|677.94
|2<sup>ln(13)</sup>
|9.063911377
|3816.16
|92.31
|216.16
|1.531753765
|-
|14
|14<sup>ln(2)</sup> = 2<sup>ln(14)</sup>
|6.22924506
|3166.87
|88.93
|766.87
|2<sup>ln(14)</sup>
|9.523965051
|3901.87
|85.71
|301.87
|1.528911603
|-
|15
|15<sup>ln(2)</sup> = 2<sup>ln(15)</sup>
|6.5343793
|3249.66
|82.79
|849.66
|2<sup>ln(15)</sup>
|9.974392624
|3981.87
|80.00
|381.87
|1.526448369
|-
|16
|16<sup>ln(2)</sup> = 2<sup>ln(16)</sup>
|6.833329631
|3327.11
|77.45
|927.11
|2<sup>ln(16)</sup>
|10.41599671
|4056.87
|75.00
|456.87
|1.524293028
|}


=== naming details ===
=== naming details ===
Retrieved from "https://en.xen.wiki/w/Powharmonic_series"