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Logharmonic series: Difference between revisions

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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.
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.
An analogous [https://en.xen.wiki/w/Powharmonic_series#Emulatory_edharmonic_series emulatory edharmonic series] exists.


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