Powharmonic series: Difference between revisions

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Edharmonic series are to powharmonic series as the matharmonic series is to the [[Logharmonic series|logharmonic series]].

=== Emulatory Series ===

=== Emulatory edharmonic Series ===

The 0th 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 <math>\frac11</math> to 0.

In accordance with this observation, it further seems reasonable that any a-edharmonic series could be prefixed with the frequency multiplier 1, rather than beginning straight away with the frequency multiplier <math>a</math>. 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. We therefore propose referring to ~~these variations~~ as "emulatory a-edharmonic series", because ~~they are emulating~~ the harmonic series.

In accordance with this observation, it further seems reasonable that any a-edharmonic series could be prefixed with the frequency multiplier 1, rather than beginning straight away with the frequency multiplier <math>a</math>.

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.

{| class="wikitable"

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! rowspan="2" |pitch #

! colspan="4" |harmonic series

! colspan="5" |edharmonic series

! colspan="5" |emulatory edharmonic series

|-

|'''frequency multiplier (decimal)'''

@@ Line 463: / Line 463: @@
 Edharmonic series are to powharmonic series as the matharmonic series is to the [[Logharmonic series|logharmonic series]].
-=== Emulatory Series ===
+=== Emulatory edharmonic Series ===
 The 0<sup>th</sup> 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 accordance with this observation, it further seems reasonable that any a-edharmonic series could be prefixed with the frequency multiplier 1, rather than beginning straight away with the frequency multiplier <span><math>a</math></span>. 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. We therefore propose referring to these variations as "emulatory a-edharmonic series", because they are emulating the harmonic series.
+In accordance with this observation, it further seems reasonable that any a-edharmonic series could be prefixed with the frequency multiplier 1, rather than beginning straight away with the frequency multiplier <span><math>a</math></span>.
+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.
 {| class="wikitable"
@@ Line 473: / Line 475: @@
 ! rowspan="2" |pitch #
 ! colspan="4" |harmonic series
-! colspan="5" |edharmonic series
+! colspan="5" |emulatory edharmonic series
 |-
 |'''frequency multiplier (decimal)'''