Powharmonic series: Difference between revisions

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== edharmonic series ==

Perhaps even more interestingly, ln-of-a powharmonic series can be approximated ~~by series constructed~~ by moving by steps of increasing equal divisions of a. For example, if we first move by a step of 1ed2, then by 2ed2, then 3ed2, 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-a series. The ~~difference~~ between the ~~frequency multipliers on our fundamental will approach~~ the Euler-~~Masceroni~~ constant, which represents the difference between the natural logarithm and the mathematical harmonic series. This is because moving by steps of increasing equal divisions of a is equivalent to a series of pitches 2^H(n) where H(n) is the ~~nth~~ harmonic number.

=== description ===

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

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 ===

The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], 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 <math>a</math> is equivalent to a series of pitches <math>2^{H(n)}</math> where <math>H(n)</math> is the <math>n^{th}</math> [[wikipedia:Harmonic_number|harmonic number]]:

<math>

\qquad H(1) = 1 \\

\qquad H(2) = \frac{3}{2} = 1 + \frac{1}{2} \\

\qquad H(3) = \frac{11}{6} = 1 + \frac{1}{2} + \frac{1}{3} \\

\qquad H(4) = \frac{25}{12} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\

\qquad …

</math>

In other words, if we have gone by a step of 1ed2, we are at <math>2^1</math>. If we then go by a step of 2ed2, we have gone by <math>2^1 · 2^{\frac12} = 2^{\frac32}</math>. And a further step of 3ed2 gets us to <math>2^1 · 2^{\frac12} · 2^{\frac13} = 2^{\frac{11}{6}}</math>, etc.

(insert chart with edharmonic series, and maybe a few columns comparing it with ln-of-2 powharmonic series)

We can refer to the 2-edharmonic series as the edharmonic series for short. ~~The~~ 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.

=== 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".

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 ===

As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.

== equivalent powharmonic series ==

@@ Line 182: / Line 182: @@
 == edharmonic series ==
-Perhaps even more interestingly, ln-of-a powharmonic series can be approximated by series constructed by moving by steps of increasing equal divisions of a. For example, if we first move by a step of 1ed2, then by 2ed2, then 3ed2, 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-a series. The difference between the frequency multipliers on our fundamental will approach the Euler-Masceroni constant, which represents the difference between the natural logarithm and the mathematical harmonic series. This is because moving by steps of increasing equal divisions of a is equivalent to a series of pitches 2^H(n) where H(n) is the nth harmonic number.
+=== 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>.
+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 ===
+The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches the [[wikipedia:Euler–Mascheroni_constant|Euler-Mascheroni constant]], 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]]:
+<math>
+\qquad H(1) = 1 \\
+\qquad H(2) = \frac{3}{2} = 1 + \frac{1}{2} \\
+\qquad H(3) = \frac{11}{6} = 1 + \frac{1}{2} + \frac{1}{3} \\
+\qquad H(4) = \frac{25}{12} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\
+\qquad …
+</math>
+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)
-We can refer to the 2-edharmonic series as the edharmonic series for short. The 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.
+=== 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".
+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 ===
+As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.
 == equivalent powharmonic series ==