Powharmonic series - Revision history

Cmloegcmluin: /* = Equivalences */

2024-09-12T01:29:02Z

= Equivalences

← Older revision		Revision as of 01:29, 12 September 2024
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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>.

ArrowHead294 at 12:27, 15 July 2024

2024-07-15T12:27:24Z

← Older revision		Revision as of 12:27, 15 July 2024
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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

ArrowHead294 at 16:17, 3 June 2024

2024-06-03T16:17:31Z

← Older revision		Revision as of 16:17, 3 June 2024
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	== ''p''-Powharmonic series ==		== ''p''-Powharmonic series ==
	A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a ''p-powharmonic series'' is simply:		A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a '''''p''-powharmonic series''' is simply:

	<math>\qquad f(n) = n^p		<math>\qquad f(n) = n^p
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	The harmonic series is technically a powharmonic series, but it is the trivial case, with the exponent equal to 1. <span><math>p</math><span> closer to 1 give series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.		The harmonic series is technically a powharmonic series, but it is the trivial case, with the exponent equal to 1. <span><math>p</math><span> closer to 1 give series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.

	Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant c. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS\|AFS (arithmetic frequency sequence)]].		Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant ''c''. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS\|AFS (arithmetic frequency sequence)]].

	Using a negative power for the exponent gives a similar, but inverted effect. <span><math>f(n) = n^{-1}</math></span> is simply the subharmonic series. Other negative powers give you the subharmonic equivalent of their (super) powharmonic counterpart. You could call these subpowharmonic series.		Using a negative power for the exponent gives a similar, but inverted effect. <span><math>f(n) = n^{-1}</math></span> is simply the subharmonic series. Other negative powers give you the subharmonic equivalent of their (super) powharmonic counterpart. You could call these subpowharmonic series.

	== log-base-b-of-a-Powharmonic series ==		== log-base-''b''-of-''a''-Powharmonic series ==
	[[File:Log-base-3-of-2-powharmonic series.png\|thumb\|		[[File:Log-base-3-of-2-powharmonic series.png\|thumb\|
	log-base-3-of-2-powharmonic series		log-base-3-of-2-powharmonic series
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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 increase by a factor 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 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''.		Extending the naming scheme '''''p''-powharmonic series''', we call this a '''log-base-''b''-of-''a''-powharmonic series'''.

	=== Pitches per period ===		=== Pitches per period ===
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	== ''a''-Edharmonic series ==		== ''a''-Edharmonic series ==
	=== Prerequisite: ln-of-a-powharmonic series ===		=== Prerequisite: ln-of-''a''-powharmonic series ===
	[[File:Ln-of-2-powharmonic series.png\|thumb\|		[[File:Ln-of-2-powharmonic series.png\|thumb\|
	ln-of-2-powharmonic series		ln-of-2-powharmonic series
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	Irrational values can be used as <span><math>a</math></span> or <span><math>b</math></span>.		Irrational values can be used as <span><math>a</math></span> or <span><math>b</math></span>.

	In particular it may be of interest to use [[wikipedia:E_(mathematical_constant)\|<span><math>e</math></span>]] as <span><math>b</math></span> — in other words, to use a [[wikipedia:Natural_logarithm\|natural logarithm]].		In particular it may be of interest to use [[wikipedia:E_(mathematical_constant)\|<span><math>e</math></span>]] as <span><math>b</math></span>—in other words, to use a [[wikipedia:Natural_logarithm\|natural logarithm]].

	For example, the ''ln-of-2-powharmonic series'' fits <span><math>e</math></span> times as many many more pitches into each next octave as the previous octave. Because <span><math>e</math></span> is irrational, however, no integer multiples of the octave will ever be reached.		For example, the ''ln-of-2-powharmonic series'' fits <span><math>e</math></span> times as many many more pitches into each next octave as the previous octave. Because <span><math>e</math></span> is irrational, however, no integer multiples of the octave will ever be reached.
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	For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ln-of-2-powharmonic series. We could call this series the 2-edharmonic series.		For example, if we first move by a step of 1ed2 (1200¢), then by 2ed2 (600¢), then 3ed2 (400¢), etc. we will soon find that the deltas between steps of our series are very close to the deltas between steps of the ln-of-2-powharmonic series. We could call this series the 2-edharmonic series.

	=== Relation to ln-of-a-powharmonic series ===		=== Relation to ln-of-''a''-powharmonic series ===
	The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches <span><math>2^γ ≈ 1.49196704047</math><span>, where <span><math>γ</math></span> is the [[wikipedia:Euler–Mascheroni_constant\|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)\|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <span><math>a</math></span> is equivalent to a series of pitches <span><math>2^{H(n)}</math></span> where <span><math>H(n)</math></span> is the <span><math>n^{th}</math></span> [[wikipedia:Harmonic_number\|harmonic number]]:		The ratio between pitches of the ln-of-2-powharmonic series and the 2-edharmonic series approaches <span><math>2^γ ≈ 1.49196704047</math><span>, where <span><math>γ</math></span> is the [[wikipedia:Euler–Mascheroni_constant\|Euler-Mascheroni constant]], <span><math>≈ 0.5772156649</math></span>, which represents the difference between the natural logarithm and the [[wikipedia:Harmonic_series_(mathematics)\|mathematical harmonic series]] (as opposed to the musical harmonic series). This is because moving by steps of increasing equal divisions of <span><math>a</math></span> is equivalent to a series of pitches <span><math>2^{H(n)}</math></span> where <span><math>H(n)</math></span> is the <span><math>n^{th}</math></span> [[wikipedia:Harmonic_number\|harmonic number]]:

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	\|}		\|}

	In yet other words, the definition of an a-edharmonic series is:		In yet other words, the definition of an ''a''-edharmonic series is:

	<math> \qquad f(n) = a^{H(n)}		<math> \qquad f(n) = a^{H(n)}

ArrowHead294 at 13:28, 3 June 2024

2024-06-03T13:28:18Z

← Older revision		Revision as of 13:28, 3 June 2024
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	=== Naming details ===		=== Naming details ===

	We cross-pollinate the abbreviation for "[[wikipedia:Equal_temperament\|equal division]]" with affiliation for the pronunciation of "[[wikipedia:Enharmonic\|enharmonic]]" to get the name "edharmonic series".		We cross-pollinate the abbreviation for "[[wikipedia:Equal_temperament\|equal division]]" with affiliation for the pronunciation of "[[wikipedia:Enharmonic\|enharmonic]]" to get the name "edharmonic series".

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	=== Other examples ===		=== Other examples ===

	As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.		As another example, the 3-edharmonic series would be moving first by a tritave (1ed3), then by 2ed3, 3ed3, 4ed3, etc.

	=== Analogy with matharmonic series ===		=== Analogy with matharmonic series ===

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

	=== Emulatory edharmonic 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.		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.

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	== See also ==		== See also ==

	[[Harmonotonic tunings]]: powharmonic series are non-[[Arithmetic tunings\|arithmetic]] harmonotonic tunings.		[[Harmonotonic tunings]]: powharmonic series are non-[[Arithmetic tunings\|arithmetic]] harmonotonic tunings.

ArrowHead294 at 13:27, 3 June 2024

2024-06-03T13:27:53Z

Show changes

ArrowHead294 at 13:15, 3 June 2024

2024-06-03T13:15:39Z

@@ Line 1: / Line 1: @@
 == Introduction ==
 A powerharmonic series, like the [[wikipedia:Harmonic_series_(music)|harmonic series]], is an infinitely ascending set of pitches from which scales can be drawn.
-== p-powharmonic series ==
+== ''p''-Powharmonic series ==
 A powharmonic series can be built on any number <span><math>p</math></span>, whether it is rational or irrational, positive or negative. The formula for a ''p-powharmonic series'' is simply:
@@ Line 11: / Line 9: @@
 For example, the 0.69314718056-powharmonic series looks like this:
+{{powharmonic series|12|expo=0.69314718056}}
 The harmonic series is technically a powharmonic series: the 1-powharmonic series. <span><math>p</math><span> closer to 1 give powharmonic series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.

BudjarnLambeth: Added to category "xenharmonic series"

2023-04-26T05:16:58Z

Added to category "xenharmonic series"

← Older revision		Revision as of 05:16, 26 April 2023
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	[[Category:Equal-step tuning‏‎]]		[[Category:Equal-step tuning‏‎]]
	[[Category:Equal divisions of the octave‏‎]]		[[Category:Equal divisions of the octave‏‎]]
			[[Category:Xenharmonic series]]

Fredg999: Categories

2021-08-30T04:01:53Z

Cmloegcmluin: /* p-powharmonic series */

2021-03-25T23:08:45Z

p-powharmonic series

← Older revision		Revision as of 23:08, 25 March 2021
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	Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant c. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS\|AFS (arithmetic frequency sequence)]].		Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant c. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS\|AFS (arithmetic frequency sequence)]].

			Using a negative power for the exponent gives a similar, but inverted effect. <span><math>f(n) = n^{-1}</math></span> is simply the subharmonic series. Other negative powers give you the subharmonic equivalent of their (super) powharmonic counterpart. You could call these subpowharmonic series.

	== log-base-b-of-a-powharmonic series ==		== log-base-b-of-a-powharmonic series ==

Cmloegcmluin: /* p-powharmonic series */

2021-03-24T19:10:31Z

p-powharmonic series

← Older revision		Revision as of 19:10, 24 March 2021
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	The harmonic series is technically a powharmonic series: the 1-powharmonic series. <span><math>p</math><span> closer to 1 give powharmonic series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.		The harmonic series is technically a powharmonic series: the 1-powharmonic series. <span><math>p</math><span> closer to 1 give powharmonic series closer to the harmonic series, in case a series is desired which is close enough to the harmonic series to evoke it but has some finely alternately tuned characteristics.

	Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant c. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢.		Multiplying the exponent of a powharmonic series by some constant c is equivalent to multiplying each of its pitches' cents by that constant c. For example, the 1.5-powharmonic series would be like stretching each octave of the harmonic series from 1200¢ to 1800¢. If you were to instead manipulate a harmonic series by adding or subtracting frequency, rather than exponentiating it, you instead get an [[AFS\|AFS (arithmetic frequency sequence)]].

	== log-base-b-of-a-powharmonic series ==		== log-base-b-of-a-powharmonic series ==

← Older revision		Revision as of 12:27, 15 July 2024
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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

← Older revision		Revision as of 04:01, 30 August 2021
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	http://anaphoria.com/harm-subharm.pdf		http://anaphoria.com/harm-subharm.pdf

	[[Category:~~Overtone]]~~		[[Category:Otonality and utonality]]
	~~[[Category:Overtone‏‎ series~~]]
	[[Category:Otonality]]		[[Category:Otonality]]
	[[Category:Harmonic]]		[[Category:Harmonic]]
	[[Category:Harmonic series‏‎]]		[[Category:Harmonic series‏‎]]

	~~[[Category:Undertone]]~~
	~~[[Category:Undertone series]]~~
	[[Category:Utonality]]		[[Category:Utonality]]
	[[Category:Subharmonic]]		[[Category:Subharmonic]]
	[[Category:Subharmonic series‏‎]]		[[Category:Subharmonic series‏‎]]

	~~[[Category:Otonality and utonality]]~~

	[[Category:Equal-step tuning‏‎]]		[[Category:Equal-step tuning‏‎]]
	[[Category:Equal divisions of the octave‏‎ ]]		[[Category:Equal divisions of the octave‏‎]]