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 related to the next integer power~~ of <math>b</math>.

=== description ===

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 increase by a factor 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 — 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:

=== pitches per period ===

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 (multiple of 2) of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:

<math>2, 4, 8, 16…

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<math>2, 6, 18, 54…

</math>

=== equality explanation ===

An equality involving exponents and logarithms helps us understand why:

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# <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>

=== initial count ===

The first period of the series, determined by <math>a</math>, will contain <math>b - 1</math> pitches. For example, the log-base-4-of-5-powharmonic series' first 5/1 interval will contain <math>4 - 1 = 3</math> pitches.

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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 related to the next integer power of <span><math>b</math></span>.
+=== 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>.
 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 of the fundamental. However, instead of the counts of pitches per octave increasing by a factor of 2:
+=== pitches per period ===
+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>2, 4, 8, 16…
@@ Line 150: / Line 154: @@
 <math>2, 6, 18, 54…
 </math>
+=== equality explanation ===
 An equality involving exponents and logarithms helps us understand why:
@@ Line 163: / Line 169: @@
 # <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>
+=== 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.