Powharmonic series: Difference between revisions

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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''.

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>

An equality involving exponents and logarithms helps us understand why:

# it takes longer and longer each time for <span><math>n</math></span> to reach the next power of <span><math>b</math></span>

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 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>.

== a-edharmonic series ==

[[File:Ln-of-2-powharmonic series.png|thumb|

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.

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.

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]]:

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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.  

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

Edharmonic series are to powharmonic series as the matharmonic series is to the [[Logharmonic series|logharmonic series]].

== See also ==

[[Logharmonic series|Logharmonic series]]