Powharmonic series: Difference between revisions

== 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.

<math>\qquad f(n) = n^p

For example, the 0.69314718056-powharmonic series looks like this:

[[File:Log-base-3-of-2-powharmonic series.png|thumb|

log-base-3-of-2-powharmonic series

=== 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>.

=== Pitches per period ===

<math>2, 4, 8, 16…

</math>

<math>2, 6, 18, 54…

=== Equality explanation ===

An equality involving exponents and logarithms helps us understand why:

=== 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.

=== 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>.

[[File:Powharmonic space.png|378x378px]]

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

ln-of-2-powharmonic series

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

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.

=== 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.

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]]:

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.

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<math> \qquad f(n) = a^{H(n)}

=== 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".  

=== 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^th 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 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.

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== See also ==

http://anaphoria.com/harm-subharm.pdf