Powharmonic series: Difference between revisions
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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 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 == | === 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>. | 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>. | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+ Comparison of ln-of-2-Powharmonic with 2-Edharmonic series+ | |+ style="font-size: 105%;" | Comparison of ln-of-2-Powharmonic with 2-Edharmonic series+ | ||
! rowspan="3" | Pitch # | ! rowspan="3" | Pitch # | ||
! colspan="5" | ln-of-2-Powharmonic series | ! colspan="5" | ln-of-2-Powharmonic series | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+ Comparison of harmonic series and emulatory edharmonic series | |+ style="font-size: 105%;" | Comparison of harmonic series and emulatory edharmonic series | ||
! rowspan="3" | Pitch # | ! rowspan="3" | Pitch # | ||
! colspan="4" | Harmonic series | ! colspan="4" | Harmonic series | ||