Relative interval error - Revision history

FloraC: Note the distribution in direct approximation. Misc. cleanup

2025-08-15T09:47:14Z

Note the distribution in direct approximation. Misc. cleanup

← Older revision		Revision as of 09:47, 15 August 2025
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	The '''relative error''' of an [[interval]] in an [[edo]] is the interval's [[error]] in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.		The '''relative error''' of an [[interval]] in an [[edo]] is the interval's [[error]] in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.

	For example, in 24edo, 3/2 has an absolute error of about ~~−2¢~~, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is ~~50¢~~, and ~~−2~~ / 50 = ~~−0~~.04, therefore the relative error is about ~~−4~~% or ~~−4~~ relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of ~~−2~~%~~. (~~In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.)		For example, in 24edo, 3/2 has an absolute error of about −2{{c}}, meaning that the nearest interval in the edo is about 2{{c}} flat of 3/2. One edostep is 50{{c}}, and {{nowrap\| −2/50 {{=}} −0.04 }}, therefore the relative error is about −4% or −4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of −2%<ref group="note">In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.</ref>.

	== Computation ==		== Computation ==
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	The unit of relative error is ''relative cent'' or ''percent''.		The unit of relative error is ''relative cent'' or ''percent''.

	~~With direct approximation via the ratio's cents, the relative error ranges from −50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.~~

	=== In val mapping ===		=== In val mapping ===
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	== Linearity ==		== Linearity ==
	In val mapping, the relative error space {''Ɛ''<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and V = ''α''V<sub>1</sub> + ''β''V<sub>2</sub>, then		In val mapping, the relative error space {''Ɛ''<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and ''V'' = ''αV''<sub>1</sub> + ''βV''<sub>2</sub>, then

	<math>		<math>
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	</math>		</math>

	For example, the relative error map of 26edo using its 5-limit patent val is		In direct approximation, the relative error space is also linear, but modulo 100%. Furthermore, we can show the relative error of any individual interval is {{w\|equidistributed sequence\|equidistributed}} from −50% to +50%, according to the {{w\|equidistribution theorem}}.

			An application of these properties concerns the fact that we can add the relative error maps of two edos together to form the relative error map of their sum. For example, the relative error map of 26edo using its 5-limit patent val is

	<math>\mathcal {E}_\text {r} (26) = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>		<math>\mathcal {E}_\text {r} (26) = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>
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	<math>\mathcal {E}_\text {r} (27) = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>		<math>\mathcal {E}_\text {r} (27) = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>

	As 53 = 26 + 27, the relative error map of 53edo using its 5-limit patent val is		As {{nowrap\| 53 {{=}} 26 + 27 }}, the relative error map of 53edo using its 5-limit patent val is

	<math>\mathcal {E}_\text {r} (53) = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>		<math>\mathcal {E}_\text {r} (53) = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>

	We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.		We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.

	It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of −50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

	== See also ==		== See also ==
	* [[Relative cent]]		* [[Relative cent]]
	* [[Relative errors of small EDOs]]		* [[Relative errors of small EDOs]]

			== Notes ==
			<references group="note"/>

	[[Category:Terms]]		[[Category:Terms]]

ArrowHead294 at 14:51, 10 May 2024

2024-05-10T14:51:38Z

← Older revision		Revision as of 14:51, 10 May 2024
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	The '''relative error''' of an [[interval]] in an [[edo]] is the interval's [[error]] in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.		The '''relative error''' of an [[interval]] in an [[edo]] is the interval's [[error]] in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.

	For example, in 24edo, 3/2 has an absolute error of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12edo's absolute ~~and relative errors are always identical~~.)		For example, in 24edo, 3/2 has an absolute error of about −2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and −2 / 50 = −0.04, therefore the relative error is about −4% or −4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of −2%. (In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.)

	== Computation ==		== Computation ==
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	The unit of relative error is ''relative cent'' or ''percent''.		The unit of relative error is ''relative cent'' or ''percent''.

	With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.		With direct approximation via the ratio's cents, the relative error ranges from −50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.

	=== In val mapping ===		=== In val mapping ===
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	=== Example ===		=== Example ===
	Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.		Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, −11.43% and −11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (−11.43%) − (−11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

	== Linearity ==		== Linearity ==
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	We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.		We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.

	It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.		It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of −50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

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

FloraC: Restylize and recategorize

2023-11-27T11:28:40Z

Restylize and recategorize

← Older revision		Revision as of 11:28, 27 November 2023
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	~~: ''This article is about~~ the error of intervals measured in relative cents~~. For~~ the relative error of temperaments~~, see [[~~Tenney-Euclidean temperament measures #TE simple badness~~]].''~~		{{About\|the error of intervals measured in relative cents\|the relative error of temperaments\|Tenney-Euclidean temperament measures #TE simple badness}}

	The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.		The '''relative error''' of an [[interval]] in an [[edo]] is the interval's [[error]] in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.

	For example, in 24edo, 3/2 has an ~~'''~~absolute error~~'''~~ of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12edo's absolute and relative errors are always identical.)		For example, in 24edo, 3/2 has an absolute error of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12edo's absolute and relative errors are always identical.)

	== Computation ==		== Computation ==
	=== In direct approximation ===		=== In direct approximation ===
	To find the relative error ''e'' of any [[JI]] [[ratio]] in direct approximation:		To find the relative error ''ε'' of any [[JI]] [[ratio]] in direct approximation:

	<math>e (n, r) = (\operatorname{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>		<math>\varepsilon (n, r) = (\operatorname{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>

	where ''n'' is the edo number and ''r'' is the ratio in question.		where ''n'' is the edo number and ''r'' is the ratio in question.
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	=== In val mapping ===		=== In val mapping ===
	Given ''n''-edo equipped with ''p''-limit val V = {{val\| ''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>π (''p'')</sub> }}, the relative error map E<sub>r</sub> of each prime harmonic is given by		Given ''n''-edo equipped with ''p''-limit val ''V'' = {{val\| ''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>π (''p'')</sub> }}, the relative error map ''Ɛ''<sub>r</sub> of each prime harmonic is given by

	<math>E_\text {r} = (V - nJ) \times 100\%</math>		<math>\mathcal {E}_\text {r} = (V - nJ) \times 100\%</math>

	where J = {{val\| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[~~JIP~~]].		where ''J'' = {{val\| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[just tuning map]].

	Thanks to the [[Monzos and interval space\|linearity of the interval space]], the relative error for any monzo m is given by		Thanks to the [[Monzos and interval space\|linearity of the interval space]], the relative error for any monzo '''m''' is given by

	<math>E_\text {r} \cdot \vec m</math>		<math>\mathcal {E}_\text {r} \cdot \vec m</math>

	=== Example ===		=== Example ===
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	== Linearity ==		== Linearity ==
	In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and V = ''α''V<sub>1</sub> + ''β''V<sub>2</sub>, then		In val mapping, the relative error space {''Ɛ''<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and V = ''α''V<sub>1</sub> + ''β''V<sub>2</sub>, then

	<math>		<math>
	\begin{align}		\begin{align}
	E_\text {r} &= (V - nJ) \times 100\% \\		\mathcal {E}_\text {r} &= (V - nJ) \times 100\% \\
	&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\		&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
	&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\		&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\
	&= \alpha E_\text {r1} + \beta E_\text {r2}		&= \alpha \mathcal {E}_\text {r1} + \beta \mathcal {E}_\text {r2}
	\end{align}		\end{align}
	</math>		</math>
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	For example, the relative error map of 26edo using its 5-limit patent val is		For example, the relative error map of 26edo using its 5-limit patent val is

	<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>		<math>\mathcal {E}_\text {r} (26) = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>

	That of 27edo using its 5-limit patent val is		That of 27edo using its 5-limit patent val is

	<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>		<math>\mathcal {E}_\text {r} (27) = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>

	As 53 = 26 + 27, the relative error map of 53edo using its 5-limit patent val is		As 53 = 26 + 27, the relative error map of 53edo using its 5-limit patent val is

	<math>E_\text {r, 53} = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>		<math>\mathcal {E}_\text {r} (53) = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>

	We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.		We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.
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	[[Category:Error]]		[[Category:Error]]
	[[Category:Approximation]]		[[Category:Approximation]]
	[[Category:Relative ~~measure~~]]		[[Category:Relative measures]]

FloraC: Styling improvements, basically

2023-05-13T08:20:56Z

Styling improvements, basically

← Older revision		Revision as of 08:20, 13 May 2023
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	''This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see [[Tenney-Euclidean temperament measures #TE simple badness]].''		: ''This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see [[Tenney-Euclidean temperament measures #TE simple badness]].''

	The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.		The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.
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	== Computation ==		== Computation ==
	=== In direct approximation ===		=== In direct approximation ===
	To find the relative error of any [[JI]] ratio in direct approximation:		To find the relative error ''e'' of any [[JI]] [[ratio]] in direct approximation:

	<math>e (n, r) = (\~~text~~{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>		<math>e (n, r) = (\operatorname{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>

	where ''n'' is the edo number and ''r'' is the ~~targeted [[frequency~~ ratio]].		where ''n'' is the edo number and ''r'' is the ratio in question.

	The unit of relative error is ''relative cent'' or ''percent''.		The unit of relative error is ''relative cent'' or ''percent''.

	With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a val mapping via [[patent val]] or other vals, it can be ~~farther~~.		With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.

	=== In val mapping ===		=== In val mapping ===
	Given ''n''-edo equipped with ''p''-limit val A = {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> … ''a''<sub>π (''p'')</sub> }}, the relative error map E<sub>r</sub> of each prime harmonic is given by		Given ''n''-edo equipped with ''p''-limit val V = {{val\| ''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>π (''p'')</sub> }}, the relative error map E<sub>r</sub> of each prime harmonic is given by

	<math>E_\text {r} = (A - nJ) \times 100\%</math>		<math>E_\text {r} = (V - nJ) \times 100\%</math>

	where J = {{val\| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[JIP]].		where J = {{val\| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[JIP]].

	Thanks to the [[Monzos and interval space\|linearity of the interval space]], the relative error for any monzo b is given by		Thanks to the [[Monzos and interval space\|linearity of the interval space]], the relative error for any monzo m is given by

	<math>E_\text {r} \vec b</math>		<math>E_\text {r} \cdot \vec m</math>

	=== Example ===		=== Example ===
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	== Linearity ==		== Linearity ==
	In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and A = ''α''A<sub>1</sub> + ''β''A<sub>2</sub>, then		In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and V = ''α''V<sub>1</sub> + ''β''V<sub>2</sub>, then

	<math>		<math>
	E_\text {r} = (A - nJ) \times 100\% \\		\begin{align}
	= ((\alpha ~~A_1~~ + \beta ~~A_2~~) - (\alpha n_1 + \beta n_2)J) \times 100\% \\		E_\text {r} &= (V - nJ) \times 100\% \\
	= \alpha (~~A_1~~ - n_1 J) \times 100\% + (\beta (~~A_2~~ - n_2 J) \times 100\% \\		&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
	= \alpha E_\text {r1} + \beta E_\text {r2}		&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\
			&= \alpha E_\text {r1} + \beta E_\text {r2}
			\end{align}
	</math>		</math>

	~~Here is an~~ example~~. The~~ relative ~~errors~~ of 26edo in its 5-limit patent val is		For example, the relative error map of 26edo using its 5-limit patent val is

	<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>		<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>

	That of 27edo in its 5-limit patent val is		That of 27edo using its 5-limit patent val is

	<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>		<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>

	As 53 = 26 + 27, the relative ~~errors~~ of 53edo in its 5-limit patent val is		As 53 = 26 + 27, the relative error map of 53edo using its 5-limit patent val is

	<math>E_\text {r, 53} = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>		<math>E_\text {r, 53} = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>
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	We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.		We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.

	It is somewhat applicable to direct approximation~~, too~~, but if the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.		It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

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

Fredg999 category edits: Removing from Category:Measure using Cat-a-lot

2023-02-26T06:34:33Z

Removing from Category:Measure using Cat-a-lot

← Older revision		Revision as of 06:34, 26 February 2023
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	[[Category:Error]]		[[Category:Error]]
	[[Category:Approximation]]		[[Category:Approximation]]
	~~[[Category:Measure]]~~
	[[Category:Relative measure]]		[[Category:Relative measure]]

Fredg999 category edits: Copying from Category:Terms to Category:Error using Cat-a-lot

2023-02-26T06:27:19Z

Copying from Category:Terms to Category:Error using Cat-a-lot

← Older revision		Revision as of 06:27, 26 February 2023
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	[[Category:Terms]]		[[Category:Terms]]
			[[Category:Error]]
	[[Category:Approximation]]		[[Category:Approximation]]
	[[Category:Measure]]		[[Category:Measure]]
	[[Category:Relative measure]]		[[Category:Relative measure]]

FloraC: More rework

2022-01-29T16:44:02Z

More rework

← Older revision		Revision as of 16:44, 29 January 2022
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	<math>E_\text {r} = (A - nJ) \times 100\%</math>		<math>E_\text {r} = (A - nJ) \times 100\%</math>

	where J is the [[JIP]].		where J = {{val\| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[JIP]].

	~~The~~ relative error for any monzo b is given by		Thanks to the [[Monzos and interval space\|linearity of the interval space]], the relative error for any monzo b is given by

	<math>E_\text {r} \vec b</math>		<math>E_\text {r} \vec b</math>

			=== Example ===
			Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

	== Linearity ==		== Linearity ==
	~~=== Linearity of the relative error space ===~~
	In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and A = ''α''A<sub>1</sub> + ''β''A<sub>2</sub>, then		In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and A = ''α''A<sub>1</sub> + ''β''A<sub>2</sub>, then

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	It is somewhat applicable to direct approximation, too, but if the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.		It is somewhat applicable to direct approximation, too, but if the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

	~~=== Linearity of the interval space ===~~
	~~{{See also\| Monzos and interval space }}~~

	~~Another linearity is actually about the interval space {b}. This enables us to find the relative error of any ratio in a given tuning of an equal temperament.~~

	Let us try finding the relative error of 6/5 in 19edo's patent val. We first find the errors of 2/1, 3/1 and 5/1 in 19edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

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

FloraC: /* Linearity */ correction

2022-01-29T16:26:04Z

Linearity: correction

← Older revision		Revision as of 16:26, 29 January 2022
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	</math>		</math>

	Here is an example. The relative errors of 26edo in its 7-limit patent val is		Here is an example. The relative errors of 26edo in its 5-limit patent val is

	<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>		<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>
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	Another linearity is actually about the interval space {b}. This enables us to find the relative error of any ratio in a given tuning of an equal temperament.		Another linearity is actually about the interval space {b}. This enables us to find the relative error of any ratio in a given tuning of an equal temperament.

	Let us try finding the relative error of 6/5 in 19edo's patent val. We first find the errors of 2/1, 3/1 and 5/1 in 19edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.		Let us try finding the relative error of 6/5 in 19edo's patent val. We first find the errors of 2/1, 3/1 and 5/1 in 19edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

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

FloraC: Rework

2022-01-29T16:21:43Z

Rework

← Older revision		Revision as of 16:21, 29 January 2022
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	== Computation ==		== Computation ==
	To find the relative error of any [[JI]] ~~interval~~ in direct approximation:		=== In direct approximation ===
			To find the relative error of any [[JI]] ratio in direct approximation:

	<math>e (n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>		<math>e (n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>
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	The unit of relative error is ''relative cent'' or ''percent''.		The unit of relative error is ''relative cent'' or ''percent''.

	With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a val mapping via [[patent val]] or other vals, it can be farther ~~from zero. To obtain the relative error in patent val mapping, first find the relative errors of each prime, and then find the dot product of this vector with the ratio's monzo~~.		With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a val mapping via [[patent val]] or other vals, it can be farther.

	== ~~Additivity~~ ==		=== In val mapping ===
	In val ~~mapping~~, ~~there are two additivities~~ of ~~relative errors.~~		Given ''n''-edo equipped with ''p''-limit val A = {{val\| ''a''<sub>1</sub> ''a''<sub>2</sub> … ''a''<sub>π (''p'')</sub> }}, the relative error map E<sub>r</sub> of each prime harmonic is given by

	~~First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''<sub>3</sub> = ''r''<sub>1~~<~~/sub~~>''r~~''<sub>2</sub> for ''n'', then ''e'' (''n'', ''r''<sub>3</sub>)~~ = ~~''e''~~ (~~''n'', ''r''<sub>1</sub>~~) ~~+ ''e'' (''n'', ''r''<sub>2~~</~~sub~~>).		<math>E_\text {r} = (A - nJ) \times 100\%</math>

	~~Second, for the same ratio, an edo which~~ is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then ''e'' (''n''<sub>3</sub>, ''r'') = ''e'' (''n''<sub>1</sub>, ''r'') + ''e'' (''n''<sub>2</sub>'','' ''r'').		where J is the [[JIP]].

	~~In either case, if the~~ error ~~exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there~~ is ~~a discrepancy in val mapping and direct approximation, so is the error. To find the error in direct approximation, modulo the previous result~~ by ~~100%.~~		The relative error for any monzo b is given by

	~~An example of the first additivity is shown as follows. The errors of 2~~/1, 3/1 and 5/1 in 19edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.		<math>E_\text {r} \vec b</math>

	Here is an example ~~for the second additivity~~. The errors of ~~3/1 for~~ 26edo ~~and 27edo are~~ -20.90% ~~and~~ +20.60%, ~~respectively~~, and ~~their~~ sum -0.30% is the error of 3/1 ~~for 53edo~~.		== Linearity ==
			=== Linearity of the relative error space ===
			In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and A = ''α''A<sub>1</sub> + ''β''A<sub>2</sub>, then

			<math>
			E_\text {r} = (A - nJ) \times 100\% \\
			= ((\alpha A_1 + \beta A_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
			= \alpha (A_1 - n_1 J) \times 100\% + (\beta (A_2 - n_2 J) \times 100\% \\
			= \alpha E_\text {r1} + \beta E_\text {r2}
			</math>

			Here is an example. The relative errors of 26edo in its 7-limit patent val is

			<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>

			That of 27edo in its 5-limit patent val is

			<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>

			As 53 = 26 + 27, the relative errors of 53edo in its 5-limit patent val is

			<math>E_\text {r, 53} = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>

			We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.

			It is somewhat applicable to direct approximation, too, but if the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

			=== Linearity of the interval space ===
			{{See also\| Monzos and interval space }}

			Another linearity is actually about the interval space {b}. This enables us to find the relative error of any ratio in a given tuning of an equal temperament.

			Let us try finding the relative error of 6/5 in 19edo's patent val. We first find the errors of 2/1, 3/1 and 5/1 in 19edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

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

FloraC: Using "val mapping" and "direct approximation" and other style improvements

2021-08-30T15:30:08Z

Using "val mapping" and "direct approximation" and other style improvements

← Older revision		Revision as of 15:30, 30 August 2021
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	''This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see [[Tenney-Euclidean temperament measures #TE simple badness]].''		''This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see [[Tenney-Euclidean temperament measures #TE simple badness]].''

	The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[~~Relative~~ cent~~\|relative cents~~]].		The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[relative cent]]s.

	For example, in ~~24-edo~~, 3/2 has an '''absolute error''' of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, ~~12-edo~~ has the same absolute error, but a smaller relative error of -2%. (In fact, ~~12-edo~~'s absolute and relative errors are always identical.)		For example, in 24edo, 3/2 has an '''absolute error''' of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12edo's absolute and relative errors are always identical.)

	== ~~Formula~~ ==		== Computation ==
	~~The formula for closest mapping~~ of any [[JI]] interval is		To find the relative error of any [[JI]] interval in direct approximation:

	<math>e(n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>		<math>e (n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>

	where ''n'' is the edo number and ''r'' is the targeted [[frequency ratio]].		where ''n'' is the edo number and ''r'' is the targeted [[frequency ratio]].
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	The unit of relative error is ''relative cent'' or ''percent''.		The unit of relative error is ''relative cent'' or ''percent''.

	With a direct ~~mapping~~ via the ratio's cents, the relative error ranges from -50% to +50%. With ~~an indirect~~ mapping via [[patent val]] or other ~~val~~, it can be farther from zero. To obtain the relative error in patent val mapping, first find the relative errors of each prime, and then find the dot product of this vector with the ratio's monzo.		With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a val mapping via [[patent val]] or other vals, it can be farther from zero. To obtain the relative error in patent val mapping, first find the relative errors of each prime, and then find the dot product of this vector with the ratio's monzo.

	== Additivity ==		== Additivity ==
	In ~~indirect~~ mapping, there are two additivities of relative errors.		In val mapping, there are two additivities of relative errors.

	First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''<sub>3</sub> = ''r''<sub>1</sub>''r''<sub>2</sub> for ''n'', then ''e'' (''n'', ''r''<sub>3</sub>) = ''e'' (''n'', ''r''<sub>1</sub>) + ''e'' (''n'', ''r''<sub>2</sub>).		First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''<sub>3</sub> = ''r''<sub>1</sub>''r''<sub>2</sub> for ''n'', then ''e'' (''n'', ''r''<sub>3</sub>) = ''e'' (''n'', ''r''<sub>1</sub>) + ''e'' (''n'', ''r''<sub>2</sub>).
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	Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then ''e'' (''n''<sub>3</sub>, ''r'') = ''e'' (''n''<sub>1</sub>, ''r'') + ''e'' (''n''<sub>2</sub>'','' ''r'').		Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then ''e'' (''n''<sub>3</sub>, ''r'') = ''e'' (''n''<sub>1</sub>, ''r'') + ''e'' (''n''<sub>2</sub>'','' ''r'').

	In either case, if the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in val mapping and direct ~~mapping~~, so is the error. To find the error in ~~closest mapping~~, modulo the previous result by 100%.		In either case, if the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in val mapping and direct approximation, so is the error. To find the error in direct approximation, modulo the previous result by 100%.

	An example of the first additivity is shown as follows. The errors of 2/1, 3/1 and 5/1 in ~~19-edo~~ are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows ~~19-edo~~ has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.		An example of the first additivity is shown as follows. The errors of 2/1, 3/1 and 5/1 in 19edo are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

	Here is an example for the second additivity. The errors of 3/1 for ~~26-edo~~ and ~~27-edo~~ are -20.90% and +20.60%, respectively, and their sum -0.30% is the error of 3/1 for ~~53-edo~~.		Here is an example for the second additivity. The errors of 3/1 for 26edo and 27edo are -20.90% and +20.60%, respectively, and their sum -0.30% is the error of 3/1 for 53edo.

	== See also ==		== See also ==
	* [[Relative cent]]		* [[Relative cent]]
	* [[Relative errors of small ~~edos\|Relative Errors of Small Edos~~]]		* [[Relative errors of small EDOs]]

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:~~approximation~~]]		[[Category:Approximation]]
	[[Category:~~measure~~]]		[[Category:Measure]]
	[[Category:~~relative~~ measure]]		[[Category:Relative measure]]