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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.


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:
== ''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:


<math>\qquad f(n) = n^p
<math>\qquad f(n) = n^p
Line 8: Line 9:


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"
The harmonic series is technically a powharmonic series, but it is the trivial case, with the exponent equal to 1. <span><math>p</math><span> closer to 1 give 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.
|+
 
!pitch #
Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant ''c''. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS|AFS (arithmetic frequency sequence)]].
|'''frequency multiplier (definition)'''
 
|'''frequency multiplier (decimal)'''
Using a negative power for the exponent gives a similar, but inverted effect. <span><math>f(n) = n^{-1}</math></span> is simply the subharmonic series. Other negative powers give you the subharmonic equivalent of their (super) powharmonic counterpart. You could call these subpowharmonic series.
|'''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.


== log-base-b-of-a-powharmonic series ==
== log-base-''b''-of-''a''-Powharmonic series ==
[[File:Log-base-3-of-2-powharmonic series.png|thumb|
[[File:Log-base-3-of-2-powharmonic series.png|thumb|
log-base-3-of-2-powharmonic series
log-base-3-of-2-powharmonic series
]]
]]


=== description ===
=== Description ===
 
When we choose a <span><math>p</math></span> of the form <span><math>\log_{b}a</math></span>, the resulting scale will include every integer power of <span><math>a</math></span>, and the count of steps between each power of <span><math>a</math></span> will increase by a factor of <span><math>b</math></span>.
When we choose a <span><math>p</math></span> of the form <span><math>\log_{b}a</math></span>, the resulting scale will include every integer power of <span><math>a</math></span>, and the count of steps between each power of <span><math>a</math></span> will increase by a factor of <span><math>b</math></span>.


Extending the naming scheme ''p-powharmonic series'', we call this a ''log-base-b-of-a-powharmonic series''.
Extending the naming scheme '''''p''-powharmonic series''', we call this a '''log-base-''b''-of-''a''-powharmonic series'''.


=== pitches per period ===
=== Pitches per period ===
 
For example, the log-base-3-of-2-powharmonic series, where <span><math>p = log_{3}2</math></span>, will&mdash;like the harmonic series and by virtue of being "of 2"&mdash;include every octave (multiple of 2) of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:
For example, the log-base-3-of-2-powharmonic series, where <span><math>p = log_{3}2</math></span>, will — like the harmonic series — and by virtue of being "of 2" — include every octave (multiple of 2) of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:


<math>2, 4, 8, 16…
<math>2, 4, 8, 16…
</math>
</math>


they’ll — by virtue of being "base-3" — increase by a factor of 3:
they will&mdash;by virtue of being "base-3"&mdash;increase by a factor of 3:


<math>2, 6, 18, 54…
<math>2, 6, 18, 54…
</math>
</math>


=== equality explanation ===
=== Equality explanation ===
 
An equality involving exponents and logarithms helps us understand why:
An equality involving exponents and logarithms helps us understand why:


<math>\qquad x^{\log_{b}a} = a^{log_{b}x}
<math>\qquad n^{\log_{b}a} = a^{log_{b}n}
</math>
</math>


Breaking this down step by step:
Breaking this down step by step:


# <span><math>\log_{b}x</math></span> gives the power to which <span><math>b</math></span> must be raised to give <span><math>x</math></span>
# <span><math>\log_{b}n</math></span> gives the power to which <span><math>b</math></span> must be raised to give <span><math>n</math></span>
# whenever <span><math>x</math></span> is an integer power (squared, cubed, etc.) of <span><math>b</math></span>, <span><math>\log_{b}x</math></span> will be an integer
# whenever <span><math>n</math></span> is an integer power (squared, cubed, etc.) of <span><math>b</math></span>, <span><math>\log_{b}n</math></span> will be an integer
# whenever <span><math>\log_{b}x</math></span> is an integer, we raise <span><math>a</math></span> to an integer power
# whenever <span><math>\log_{b}n</math></span> is an integer, we raise <span><math>a</math></span> to an integer power
# <span><math>x</math></span>, being the pitch # or index, increments linearly by 1
# <span><math>n</math></span>, being the pitch # or index, increments linearly by 1
# it takes longer and longer each time for <span><math>x</math></span> to reach the next power of <span><math>b</math></span>
# it takes longer and longer each time for <span><math>n</math></span> to reach the next power of <span><math>b</math></span>


=== initial count ===
=== Initial count ===
The first period of the series, determined by <span><math>a</math></span>, will contain <span><math>b - 1</math></span> pitches. For example, the log-base-4-of-5-powharmonic series' first 5/1 interval will contain <span><math>4 - 1 = 3</math></span> pitches.


The first period of the series, determined by <span><math>a</math></span>, will contain <span><math>b - 1</math></span> pitches. For example, the log-base-4-of-5-powharmonic series' first 5/1 interval will contain <span><math>4 - 1 = 3</math></span> pitches.
=== Equivalences ===
The harmonic series features counts of pitches of increasing powers of 2 in each next octave, but it also contains counts of pitches of increasing powers of 3 in each next tritave, and counts of pitches in increasing powers of 5 in each next 5/1 interval, and so forth. This is because the harmonic series is equivalent to the log-base-2-of-2-powharmonic series, the log-base-3-of-3-powharmonic series, the log-base-5-of-5-powharmonic series, and so forth (the log-base-b-of-b-powharmonic series). This because any <span><math>\log_{b}b = 1</math></span>.


== edharmonic series ==
Any powharmonic series has infinite equivalent ways of being expressed. We can visualize the equivalences with the following coloration of powharmonic space:


=== prerequisite: ln-of-a-powharmonic series ===
[[File:Powharmonic space.png|378x378px]]


== ''a''-Edharmonic series ==
=== Prerequisite: ln-of-''a''-powharmonic series ===
[[File:Ln-of-2-powharmonic series.png|thumb|
[[File:Ln-of-2-powharmonic series.png|thumb|
ln-of-2-powharmonic series
ln-of-2-powharmonic series
Line 184: Line 70:
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>&mdash;in other words, to use a [[wikipedia:Natural_logarithm|natural logarithm]].


For example, the ''ln-of-2-powharmonic series'' fits <span><math>e</math></span> times as many many more pitches into each next octave as the previous octave. Because <span><math>e</math></span> is irrational, however, no integer multiples of the octave will ever be reached.
For example, the ''ln-of-2-powharmonic series'' fits <span><math>e</math></span> times as many many more pitches into each next octave as the previous octave. Because <span><math>e</math></span> is irrational, however, no integer multiples of the octave will ever be reached.


In fact, this series is equivalent to the example given in the introduction, because <span><math>ln(2) ≈ 0.69314718056</math></span>.
In fact, this series is equivalent to the example given in the introduction, because <span><math>ln(2) ≈ 0.69314718056</math></span>, and if any powharmonic series were to qualify to be referred to for short as "the" powharmonic series, this would be the one.
 
=== 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.


=== relation to ln-of-a-powharmonic series ===
=== Relation to ln-of-''a''-powharmonic series ===
 
The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches <span><math>2^γ ≈ 1.49196704047</math><span>, where <span><math>γ</math></span> is the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <span><math>a</math></span> is equivalent to a series of pitches <span><math>2^{H(n)}</math></span> where <span><math>H(n)</math></span> is the <span><math>n^{th}</math></span> [[wikipedia:Harmonic_number|harmonic number]]:
The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches <span><math>2^γ ≈ 1.49196704047</math><span>, where <span><math>γ</math></span> is the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <span><math>a</math></span> is equivalent to a series of pitches <span><math>2^{H(n)}</math></span> where <span><math>H(n)</math></span> is the <span><math>n^{th}</math></span> [[wikipedia:Harmonic_number|harmonic number]]:


Line 210: Line 94:
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.


{| class="wikitable"
{| class="wikitable center-all"
|+
|+ style="font-size: 105%;" | Comparison of ln-of-2-Powharmonic with 2-Edharmonic series+
! rowspan="2" |pitch #
! rowspan="3" | Pitch #
! colspan="5" |ln-of-2-powharmonic series
! colspan="5" | ln-of-2-Powharmonic series
! colspan="5" |2-edharmonic series
! colspan="5" | 2-Edharmonic series
! rowspan="2" |ratio between frequency multipliers
! rowspan="3" | Ratio between frequency multipliers
|-
|-
|'''frequency multiplier (definition)'''
! colspan="2" | Frequency multiplier
|'''frequency multiplier (decimal)'''
! colspan="3" | Pitch
|'''pitch (¢)'''
! colspan="2" | Frequency multiplier
|'''pitch Δ (¢)'''
! colspan="3" | Pitch
|'''octave reduced pitch (¢)'''
|'''frequency multiplier (definition)'''
|'''frequency multiplier (decimal)'''
|'''pitch (¢)'''
|'''pitch Δ (¢)'''
|'''octave reduced pitch (¢)'''
|-
|-
|1
! Definition
|1<sup>ln(2)</sup> = 2<sup>ln(1)</sup>
! Decimal
|1
! Cents
|0.00
! Change (cents)
| -
! Octave-reduced (cents)
|0.00
! Definition
|2<sup>H(1)</sup> = 2<sup>1</sup>
! Decimal
|2
! Cents
|1200.00
! Change (cents)
| -
! Octave-reduced (cents)
|0.00
|2
|-
|-
|2
| 1
|2<sup>ln(2)</sup> = 2<sup>ln(2)</sup>
| 1<sup>ln(2)</sup> = 2<sup>ln(1)</sup>
|1.616806672
| 1
|831.78
| 0.00
|831.78
| &mdash;
|831.78
| 0.00
|2<sup>H(2)</sup> = 2<sup>3/2</sup>
| 2<sup>H(1)</sup> = 2<sup>1</sup>
|2.828427125
| 2
|1800.00
| 1200.00
|600.00
| &mdash;
|600.00
| 0.00
|1.749391052
| 2
|-
|-
|3
| 2
|3<sup>ln(2)</sup> = 2<sup>ln(3)</sup>
| 2<sup>ln(2)</sup> = 2<sup>ln(2)</sup>
|2.141486064
| 1.616806672
|1318.33
| 831.78
|486.56
| 831.78
|118.33
| 831.78
|2<sup>H(3)</sup> = 2<sup>11/6</sup>
| 2<sup>H(2)</sup> = 2<sup>3/2</sup>
|3.563594873
| 2.828427125
|2200.00
| 1800.00
|400.00
| 600.00
|1000.00
| 600.00
|1.664075677
| 1.749391052
|-
|-
|4
| 3
|4<sup>ln(2)</sup> = 2<sup>ln(4)</sup>
| 3<sup>ln(2)</sup> = 2<sup>ln(3)</sup>
|2.614063815
| 2.141486064
|1663.55
| 1318.33
|345.22
| 486.56
|463.55
| 118.33
|2<sup>H(4)</sup> = 2<sup>25/12</sup>
| 2<sup>H(3)</sup> = 2<sup>11/6</sup>
|4.237852377
| 3.563594873
|2500.00
| 2200.00
|300.00
| 400.00
|100.00
| 1000.00
|1.621174033
| 1.664075677
|-
|-
|5
| 4
|5<sup>ln(2)</sup> = 2<sup>ln(5)</sup>
| 4<sup>ln(2)</sup> = 2<sup>ln(4)</sup>
|3.05132936
| 2.614063815
|1931.33
| 1663.55
|267.77
| 345.22
|731.33
| 463.55
|2<sup>H(5)</sup> = 2<sup>137/60</sup>
| 2<sup>H(4)</sup> = 2<sup>25/12</sup>
|4.868014055
| 4.237852377
|2740.00
| 2500.00
|240.00
| 300.00
|340.00
| 100.00
|1.595374829
| 1.621174033
|-
|-
|6
| 5
|6<sup>ln(2)</sup> = 2<sup>ln(6)</sup>
| 5<sup>ln(2)</sup> = 2<sup>ln(5)</sup>
|3.462368957
| 3.05132936
|2150.11
| 1931.33
|218.79
| 267.77
|950.11
| 731.33
|2<sup>H(6)</sup> = 2<sup>49/20</sup>
| 2<sup>H(5)</sup> = 2<sup>137/60</sup>
|5.464161027
| 4.868014055
|2940.00
| 2740.00
|200.00
| 240.00
|540.00
| 340.00
|1.578156775
| 1.595374829
|-
|-
|7
| 6
|7<sup>ln(2)</sup> = 2<sup>ln(7)</sup>
| 6<sup>ln(2)</sup> = 2<sup>ln(6)</sup>
|3.852807616
| 3.462368957
|2335.09
| 2150.11
|184.98
| 218.79
|1135.09
| 950.11
|2<sup>H(7)</sup> = 2<sup>363/140</sup>
| 2<sup>H(6)</sup> = 2<sup>49/20</sup>
|6.032922891
| 5.464161027
|3111.43
| 2940.00
|171.43
| 200.00
|711.43
| 540.00
|1.56585106
| 1.578156775
|-
|-
|8
| 7
|8<sup>ln(2)</sup> = 2<sup>ln(8)</sup>
| 7<sup>ln(2)</sup> = 2<sup>ln(7)</sup>
|4.226435818
| 3.852807616
|2495.33
| 2335.09
|160.24
| 184.98
|95.33
| 1135.09
|2<sup>H(8)</sup> = 2<sup>761/280</sup>
| 2<sup>H(7)</sup> = 2<sup>363/140</sup>
|6.578949063
| 6.032922891
|3261.43
| 3111.43
|150.00
| 171.43
|861.43
| 711.43
|1.556618708
| 1.56585106
|-
|-
|9
| 8
|9<sup>ln(2)</sup> = 2<sup>ln(9)</sup>
| 8<sup>ln(2)</sup> = 2<sup>ln(8)</sup>
|4.585962562
| 4.226435818
|2636.67
| 2495.33
|141.34
| 160.24
|236.67
| 95.33
|2<sup>H(9)</sup> = 2<sup>7129/2520</sup>
| 2<sup>H(8)</sup> = 2<sup>761/280</sup>
|7.105658007
| 6.578949063
|3394.76
| 3261.43
|133.33
| 150.00
|994.76
| 861.43
|1.549436549
| 1.556618708
|-
|-
|10
| 9
|10<sup>ln(2)</sup> = 2<sup>ln(10)</sup>
| 9<sup>ln(2)</sup> = 2<sup>ln(9)</sup>
|4.933409668
| 4.585962562
|2763.10
| 2636.67
|126.43
| 141.34
|363.10
| 236.67
|2<sup>H(10)</sup>
| 2<sup>H(9)</sup> = 2<sup>7129/2520</sup>
|7.615655686
| 7.105658007
|3514.76
| 3394.76
|120.00
| 133.33
|1114.76
| 994.76
|1.543690105
| 1.549436549
|-
|-
|11
| 10
|11<sup>ln(2)</sup> = 2<sup>ln(11)</sup>
| 10<sup>ln(2)</sup> = 2<sup>ln(10)</sup>
|5.270337212
| 4.933409668
|2877.47
| 2763.10
|114.37
| 126.43
|477.47
| 363.10
|2<sup>H(11)</sup>
| 2<sup>H(10)</sup>
|8.110986229
| 7.615655686
|3623.85
| 3514.76
|109.09
| 120.00
|23.85
| 1114.76
|1.538988096
| 1.543690105
|-
|-
|12
| 11
|12<sup>ln(2)</sup> = 2<sup>ln(12)</sup>
| 11<sup>ln(2)</sup> = 2<sup>ln(11)</sup>
|5.597981231
| 5.270337212
|2981.89
| 2877.47
|104.41
| 114.37
|581.89
| 477.47
|2<sup>H(12)</sup>
| 2<sup>H(11)</sup>
|8.593290568
| 8.110986229
|3723.85
| 3623.85
|100.00
| 109.09
|123.85
| 23.85
|1.535069557
| 1.538988096
|-
|-
|13
| 12
|13<sup>ln(2)</sup> = 2<sup>ln(13)</sup>
| 12<sup>ln(2)</sup> = 2<sup>ln(12)</sup>
|5.917342318
| 5.597981231
|3077.94
| 2981.89
|96.05
| 104.41
|677.94
| 581.89
|2<sup>H(13)</sup>
| 2<sup>H(12)</sup>
|9.063911377
| 8.593290568
|3816.16
| 3723.85
|92.31
| 100.00
|216.16
| 123.85
|1.531753765
| 1.535069557
|-
|-
|14
| 13
|14<sup>ln(2)</sup> = 2<sup>ln(14)</sup>
| 13<sup>ln(2)</sup> = 2<sup>ln(13)</sup>
|6.22924506
| 5.917342318
|3166.87
| 3077.94
|88.93
| 96.05
|766.87
| 677.94
|2<sup>H(14)</sup>
| 2<sup>H(13)</sup>
|9.523965051
| 9.063911377
|3901.87
| 3816.16
|85.71
| 92.31
|301.87
| 216.16
|1.528911603
| 1.531753765
|-
|-
|15
| 14
|15<sup>ln(2)</sup> = 2<sup>ln(15)</sup>
| 14<sup>ln(2)</sup> = 2<sup>ln(14)</sup>
|6.5343793
| 6.22924506
|3249.66
| 3166.87
|82.79
| 88.93
|849.66
| 766.87
|2<sup>H(15)</sup>
| 2<sup>H(14)</sup>
|9.974392624
| 9.523965051
|3981.87
| 3901.87
|80.00
| 85.71
|381.87
| 301.87
|1.526448369
| 1.528911603
|-
|-
|16
| 15
|16<sup>ln(2)</sup> = 2<sup>ln(16)</sup>
| 15<sup>ln(2)</sup> = 2<sup>ln(15)</sup>
|6.833329631
| 6.5343793
|3327.11
| 3249.66
|77.45
| 82.79
|927.11
| 849.66
|2<sup>H(16)</sup>
| 2<sup>H(15)</sup>
|10.41599671
| 9.974392624
|4056.87
| 3981.87
|75.00
| 80.00
|456.87
| 381.87
|1.524293028 ... -> 2γ = 1.49196704047
| 1.526448369
|-
| 16
| 16<sup>ln(2)</sup> = 2<sup>ln(16)</sup>
| 6.833329631
| 3327.11
| 77.45
| 927.11
| 2<sup>H(16)</sup>
| 10.41599671
| 4056.87
| 75.00
| 456.87
| 1.524293028 ... &rarr; 2<sup>&gamma;</sup> = 1.49196704047
|}
|}


=== naming details ===
In yet other words, the definition of an ''a''-edharmonic series is:


<math> \qquad f(n) = a^{H(n)}
</math>
=== Naming details ===
We cross-pollinate the abbreviation for "[[wikipedia:Equal_temperament|equal division]]" with affiliation for the pronunciation of "[[wikipedia:Enharmonic|enharmonic]]" to get the name "edharmonic series".  
We cross-pollinate the abbreviation for "[[wikipedia:Equal_temperament|equal division]]" with affiliation for the pronunciation of "[[wikipedia:Enharmonic|enharmonic]]" to get the name "edharmonic series".  


Due to the dominance of octave in music, we can actually refer to the 2-edharmonic series simply as ''the edharmonic series'' for short.  
Due to the dominance of octave in music, we can actually refer to the 2-edharmonic series simply as ''the edharmonic series'' for short.  


=== other examples ===
=== Other examples ===
As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.
 
=== Analogy with matharmonic series ===
Edharmonic series are to powharmonic series as the matharmonic series is to the [[Logharmonic series|logharmonic series]].
 
=== Emulatory edharmonic series ===
The 0<sup>th</sup> harmonic number is not defined, however, if it were, it seems reasonable to assume it would be defined as 0; in other words, the first step of the harmonic series would be to add <span><math>\frac11</math></span> to 0.
 
In accordance with this observation, it further seems reasonable that any a-edharmonic series could be prefixed with the frequency multiplier 1, rather than beginning straight away with the frequency multiplier <span><math>a</math></span>.
 
In the case of the (2-)edharmonic series, doing so brings it closer in similarity to the (musical) harmonic series; the first step is exactly an octave, the second step a fifth (701.96¢ vs 600.00¢), the third step a fourth (498.04¢ vs 400.00¢), the fourth step a third, (386.31¢ vs 300¢), etc. 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 center-all"
|+ style="font-size: 105%;" | Comparison of harmonic series and emulatory edharmonic series
! rowspan="3" | Pitch #
! colspan="4" | Harmonic series
! colspan="5" | Emulatory edharmonic series
|-
! Frequency multiplier
! colspan="3" | Pitch
! colspan="2" | Frequency multiplier
! colspan="3" | Pitch
|-
! Decimal
! Cents
! Change (cents)
! Octave-reduced (cents)
! Definition
! Decimal
! Cents
! Change (cents)
! Octave-reduced (cents)
|-
| 1
| 1.000000
| 0
| &mdash;
| 0
| 2<sup>H(0)</sup> = 2<sup>0</sup>
| 1.000000000
| 0
| &mdash;
| 0
|-
| 2
| 2.000000
| 1200
| 1200
| 0
| 2<sup>H(1)</sup> = 2<sup>1</sup>
| 2.000000000
| 1200.00
| 1200.00
| 0.00
|-
| 3
| 3.000000
| 1901.955001
| 701.955001
| 701.955001
| 2<sup>H(2)</sup> = 2<sup>3/2</sup>
| 2.828427125
| 1800.00
| 600.00
| 600.00
|-
| 4
| 4.000000
| 2400
| 498.044999
| 0
| 2<sup>H(3)</sup> = 2<sup>11/6</sup>
| 3.563594873
| 2200.00
| 400.00
| 1000.00
|-
| 5
| 5.000000
| 2786.313714
| 386.313714
| 386.313714
| 2<sup>H(4)</sup> = 2<sup>25/12</sup>
| 4.237852377
| 2500.00
| 300.00
| 100.00
|-
| 6
| 6.000000
| 3101.955001
| 315.6412870
| 701.955001
| 2<sup>H(5)</sup> = 2<sup>137/60</sup>
| 4.868014055
| 2740.00
| 240.00
| 340.00
|-
| 7
| 7.000000
| 3368.825906
| 266.8709056
| 968.825906
| 2<sup>H(6)</sup> = 2<sup>49/20</sup>
| 5.464161027
| 2940.00
| 200.00
| 540.00
|-
| 8
| 8.000000
| 3600
| 231.1740935
| 0
| 2<sup>H(7)</sup> = 2<sup>363/140</sup>
| 6.032922891
| 3111.43
| 171.43
| 711.43
|-
| 9
| 9.000000
| 3803.910002
| 203.9100017
| 203.910002
| 2<sup>H(8)</sup> = 2<sup>761/280</sup>
| 6.578949063
| 3261.43
| 150.00
| 861.43
|-
| 10
| 10.000000
| 3986.313714
| 182.4037121
| 386.313714
| 2<sup>H(9)</sup> = 2<sup>7129/2520</sup>
| 7.105658007
| 3394.76
| 133.33
| 994.76
|-
| 11
| 11.000000
| 4151.317942
| 165.0042285
| 551.317942
| 2<sup>H(10)</sup>
| 7.615655686
| 3514.76
| 120.00
| 1114.76
|-
| 12
| 12.000000
| 4301.955001
| 150.6370585
| 701.955001
| 2<sup>H(11)</sup>
| 8.110986229
| 3623.85
| 109.09
| 23.85
|-
| 13
| 13.000000
| 4440.527662
| 138.5726609
| 840.527662
| 2<sup>H(12)</sup>
| 8.593290568
| 3723.85
| 100.00
| 123.85
|-
| 14
| 14.000000
| 4568.825906
| 128.2982447
| 968.825906
| 2<sup>H(13)</sup>
| 9.063911377
| 3816.16
| 92.31
| 216.16
|-
| 15
| 15.000000
| 4688.268715
| 119.4428083
| 1088.268715
| 2<sup>H(14)</sup>
| 9.523965051
| 3901.87
| 85.71
| 301.87
|-
| 16
| 16.000000
| 4800
| 111.7312853
| 0
| 2<sup>H(15)</sup>
| 9.974392624
| 3981.87
| 80.00
| 381.87
|}


As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.
An analogous [https://en.xen.wiki/w/Logharmonic_series#Emulatory_matharmonic_series emulatory matharmonic series] exists.


== equivalent powharmonic series ==
== See also ==
[[Harmonotonic tunings]]: powharmonic series are non-[[Arithmetic tunings|arithmetic]] harmonotonic tunings.


The harmonic series features counts of pitches of increasing powers of 2 in each next octave, but it also contains counts of pitches of increasing powers of 3 in each next tritave, and counts of pitches in increasing powers of 5 in each next 5/1 interval, and so forth. This is because the harmonic series is equivalent to the log-base-2-of-2-powharmonic series, the log-base-3-of-3-powharmonic series, the log-base-5-of-5-powharmonic series, and so forth (the log-base-b-of-b-powharmonic series). This because any <span><math>\log_{b}b = 1</math></span>.
[[Logharmonic series]]: another type of non-arithmetic harmonotonic tuning.


Include Jacob chart and point about all harmonic series being the same or per octave per tritave
[[Xenharmonic_series]]


== see also ==
http://anaphoria.com/harm-subharm.pdf


logharmonic series
[[Category:Otonality and utonality]]
[[Category:Otonality]]
[[Category:Harmonic]]
[[Category:Harmonic series‏‎]]
[[Category:Utonality]]
[[Category:Subharmonic]]
[[Category:Subharmonic series‏‎]]
[[Category:Equal-step tuning‏‎]]
[[Category:Equal divisions of the octave‏‎]]
[[Category:Xenharmonic series]]
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