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>.

~~By extension of~~ 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''.

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 — include every octave of the fundamental. However, instead of the octaves containing counts of pitches in increasing powers of 2:

<math>2, 4, 8, 16…

</math>

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

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

<math>3, 9, 27, 81…

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# whenever <math>x</math> is an integer power (squared, cubed, etc.) of <math>b</math>, <math>\log_{b}x</math> will be an integer

# whenever <math>\log_{b}x</math> is an integer, we raise <math>a</math> to an integer power

# <math>x</math> increments linearly by 1

# <math>x</math>, being the pitch # or index, increments linearly by 1

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

@@ Line 139: / Line 139: @@
 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>.
-By extension of 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''.
-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 — include every octave of the fundamental. However, instead of the octaves containing counts of pitches in increasing powers of 2:
 <math>2, 4, 8, 16…
 </math>
-they’ll contain counts of pitches in increasing powers of 3
+they’ll contain counts of pitches in increasing powers of 3:
 <math>3, 9, 27, 81…
@@ Line 161: / Line 161: @@
 # 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>\log_{b}x</math></span> is an integer, we raise <span><math>a</math></span> to an integer power
-# <span><math>x</math></span> increments linearly by 1
+# <span><math>x</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>