Powharmonic series: Difference between revisions

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When we choose a <math>p</math> of the form <math>\log_{b}a</math>, the resulting scale will include every integer power of <math>a</math>, and the count of steps between each power of <math>a</math> will be ~~equal~~ to the next integer power of <math>b</math>.

When we choose a <math>p</math> of the form <math>\log_{b}a</math>, the resulting scale will include every integer power of <math>a</math>, and the count of steps between each power of <math>a</math> will be related to the next integer power of <math>b</math>.

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 <math>p = log_{3}2</math>, will — like the harmonic series — include every octave of the fundamental. However, instead of the ~~octaves containing~~ counts of pitches in increasing ~~powers~~ of 2:

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

<math>2, 4, 8, 16…

</math>

they’ll ~~contain counts~~ of ~~pitches in increasing powers~~ of 3:

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

<math>3, 9, 27, ~~81…~~

<math>2, 6, 18, 54…

</math>

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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 be equal to the next integer power 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 be related to the next integer power 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 — include every octave of the fundamental. However, instead of the octaves containing counts of pitches in increasing powers 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 of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:
 <math>2, 4, 8, 16…
 </math>
-they’ll contain counts of pitches in increasing powers of 3:
+they’ll — by virtue of being "base-3" — increase by a factor of 3:
-<math>3, 9, 27, 81…
+<math>2, 6, 18, 54…
 </math>