Tempering out - Revision history

FloraC: Interwiki

2026-03-24T11:10:04Z

Interwiki

← Older revision		Revision as of 11:10, 24 March 2026
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	\| en = Tempering out		\| en = Tempering out
	\| ja = テンパーアウト		\| ja = テンパーアウト
			\| zh = 调和
	}}		}}
	'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments a.k.a. [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref group="note">''Vanish'' is notably used throughout ''[[A Middle Path]]''. However, ''vanish'' is broader than ''temper out'', as any abstract interval can ''vanish''. For example, you could say "in [[5edo]], the small step of the [[5L 2s\|diatonic scale]] vanishes" without specifying a just interval at all. In this sense, ''being tempered out'' is a specific way of ''vanishing''. </ref>		'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments a.k.a. [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref group="note">''Vanish'' is notably used throughout ''[[A Middle Path]]''. However, ''vanish'' is broader than ''temper out'', as any abstract interval can ''vanish''. For example, you could say "in [[5edo]], the small step of the [[5L 2s\|diatonic scale]] vanishes" without specifying a just interval at all. In this sense, ''being tempered out'' is a specific way of ''vanishing''. </ref>

FloraC: More on the term vanish

2025-12-04T07:52:15Z

Dummy index: add interwiki

2025-08-23T12:11:18Z

add interwiki

← Older revision		Revision as of 12:11, 23 August 2025
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			{{interwiki
			\| en = Tempering out
			\| ja = テンパーアウト
			}}
	'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments aka [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>''Vanish'' is notably used throughout ''[[A Middle Path]]''.</ref>		'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments aka [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>''Vanish'' is notably used throughout ''[[A Middle Path]]''.</ref>

Aurame: Fix inverted superscripts in denominator of prime factors fraction of 81/80.

2025-03-13T03:48:20Z

Fix inverted superscripts in denominator of prime factors fraction of 81/80.

← Older revision		Revision as of 03:48, 13 March 2025
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	== Example ==		== Example ==
	The syntonic comma is 81/80. That is {{sfrac\|3<sup>4</sup>\|2 ~~× 5~~<sup>4</sup>}} or, in [[monzo]] form, {{monzo\| -4 4 -1}}.		The syntonic comma is 81/80. That is {{sfrac\|3<sup>4</sup>\|2<sup>4</sup> × 5}} or, in [[monzo]] form, {{monzo\| -4 4 -1}}.

	19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:		19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:

ArrowHead294 at 18:35, 16 January 2025

2025-01-16T18:35:49Z

← Older revision		Revision as of 18:35, 16 January 2025
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	== Tempering together ==		== Tempering together ==
	Two or more [[chord~~\|chords~~]] or intervals are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.		Two or more [[chord]]s or intervals are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.

	For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{nowrap\|{{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} {{=}} {{Frac\|6\|5}}}} and {{nowrap\|{{Frac\|81\|64}} {{=}} {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.		For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{nowrap\|{{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} {{=}} {{Frac\|6\|5}}}} and {{nowrap\|{{Frac\|81\|64}} {{=}} {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.

ArrowHead294 at 18:35, 16 January 2025

2025-01-16T18:35:36Z

← Older revision		Revision as of 18:35, 16 January 2025
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	== Overview ==		== Overview ==

	For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything.		For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything.

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	== Tempering together ==		== Tempering together ==

	Two or more [[chord\|chords]] or intervals are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.		Two or more [[chord\|chords]] or intervals are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.

	For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} = {{Frac\|6\|5}} and {{Frac\|81\|64}} = {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.		For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{nowrap\|{{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} {{=}} {{Frac\|6\|5}}}} and {{nowrap\|{{Frac\|81\|64}} {{=}} {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.

	== Example ==		== Example ==
			The syntonic comma is 81/80. That is {{sfrac\|3<sup>4</sup>\|2 × 5<sup>4</sup>}} or, in [[monzo]] form, {{monzo\| -4 4 -1}}.
	The syntonic comma is 81/80. That is ~~(3×3×3×3)~~ / ~~(5×2×2×2×2)~~ or, in [[monzo]] form, {{monzo\| -4 4 -1}}.

	19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:		19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:

	=== 1. Counting steps of the val ===		=== 1. Counting steps of the val ===

	Because there are no primes larger than 5 in 81/80, we say it is a 5-limit comma. The 5-limit patent val for 19edo is {{val\| 19 30 44 }}. That means that you add 19 steps of 19edo to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1.		Because there are no primes larger than 5 in 81/80, we say it is a 5-limit comma. The 5-limit patent val for 19edo is {{val\| 19 30 44 }}. That means that you add 19 steps of 19edo to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1.

	Note that, because this is an edo, 19 steps gets you precisely to 2/1. We say that 30 steps of 19edo gets you to 3/1, but that is only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it is very useful error.		Note that, because this is an edo, 19 steps gets you precisely to 2/1. We say that 30 steps of 19edo gets you to 3/1, but that is only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it is very useful error.

	Getting to 81 is 3×3×3×3, or, with 19edo steps, 30 + 30 + 30 + 30 = 120 steps of 19edo.		Getting to 81 is 3×3×3×3, or, with 19edo steps, {{nowrap\|30 + 30 + 30 + 30 {{=}} 120}} steps of 19edo.

	Getting to 80 is 5×2×2×2×2, or, with 19edo steps, 44 + 19 + 19 + 19 + 19 = 120 steps of 19edo.		Getting to 80 is 5×2×2×2×2, or, with 19edo steps, {{nowrap\|44 + 19 + 19 + 19 + 19 {{=}} 120}} steps of 19edo.

	Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. 120 steps - 120 steps = 0 steps.		Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. {{nowrap\|120 steps − 120 steps {{=}} 0 steps}}.

	Applying the monzo to the val (also called getting the ''homomorphism'') is easier. Multiply the first number in the monzo (which represents the number of 2/1's in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: (-4 × 19) + (4 × 30) + (-1 × 44) = 0 steps.		Applying the monzo to the val (also called getting the ''homomorphism'') is easier. Multiply the first number in the monzo (which represents the number of 2/1's in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: {{nowrap\|(−4 × 19) + (4 × 30) + (−1 × 44) {{=}} 0 steps}}.

	Therefore, adding 81/80 to any interval in 19edo means adding 0 steps of 19edo to it. In other words, 81/80 is effectively zero: 81/80 is ''tempered out''.		Therefore, adding 81/80 to any interval in 19edo means adding 0 steps of 19edo to it. In other words, 81/80 is effectively zero: 81/80 is ''tempered out''.

	=== 2. Painstakingly doing the math ===		=== 2. Painstakingly doing the math ===
			We say that 30 steps of 19edo gets you to 3/1, but, as we say above, that is an error. One step of 19edo is the 19th root of 2, or 2<sup>1/19</sup>, or approximately 1.037155. (That is 63.157895 cents.) If you multiply that by itself 19 times, you get exactly 2. But if you multiply that by itself 30 times, you do not get 3: You get 2.987518. Similarly, multiplying it by 44 steps gets you 4.97877 instead of 5.
	We say that 30 steps of 19edo gets you to 3/1, but, as we say above, that is an error. One step of 19edo is the 19th root of 2, or 2<sup>1/19</sup>, or approximately 1.~~03715504445~~. (That is 63.~~15789474~~ cents.) If you multiply that by itself 19 times, you get exactly 2. But if you multiply that by itself 30 times, you do not get 3: You get 2.~~98751792330896~~. Similarly, multiplying it by 44 steps gets you 4.~~97877035785607~~ instead of 5.

	If we plug in these values into 81/80, we see that 81/80 is tempered out:		If we plug in these values into 81/80, we see that 81/80 is tempered out:
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	81/80 = 3333 / 52222 = (3^4) / (5)(2^4). // Substitute our values and you get		81/80 = 3333 / 52222 = (3^4) / (5)(2^4). // Substitute our values and you get

	(2.~~98751792330896~~ ^ 4) / (4.~~97877035785607~~)*(2^4)		(2.987518 ^ 4) / (4.97877)*(2^4)
	= 79.~~66032573~~ / (4.~~97877035785607~~ * 16)		= 79.660326 / (4.97877 * 16)
	= 79.~~66032573~~ / 79.~~66032573~~		= 79.660326 / 79.660326
	= 1/1.		= 1/1
	</pre>		</pre>

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	== Notes ==		== Notes ==
	<references/>		<references />

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Lériendil: /* Tempering together */

2024-08-31T06:49:11Z

Tempering together

← Older revision		Revision as of 06:49, 31 August 2024
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	== Tempering together ==		== Tempering together ==

	Two or more [[chord\|chords]] are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.		Two or more [[chord\|chords]] or intervals are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.

	For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} = {{Frac\|6\|5}} and {{Frac\|81\|64}} = {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.		For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} = {{Frac\|6\|5}} and {{Frac\|81\|64}} = {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.

Bcmills: add related term “tempering together”, which was already widely used but not specifically defined throughout the wiki

2024-08-31T03:08:02Z

add related term “tempering together”, which was already widely used but not specifically defined throughout the wiki

← Older revision		Revision as of 03:08, 31 August 2024
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	In both cases, that implies that we are introducing some error into our tunings: where we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. There are also options to temper the octave, such as [[TOP tuning]].		In both cases, that implies that we are introducing some error into our tunings: where we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. There are also options to temper the octave, such as [[TOP tuning]].

			== Tempering together ==

			Two or more [[chord\|chords]] are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical.

			For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac\|32\|27}}{{dot}}{{Frac\|81\|80}} = {{Frac\|6\|5}} and {{Frac\|81\|64}} = {{Frac\|5\|4}}{{dot}}{{Frac\|81\|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4.

	== Example ==		== Example ==

	The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo\| -4 4 -1 }}.		The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo\| -4 4 -1}}.

	19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:		19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:

FloraC: Recategorize

2024-01-09T10:14:39Z

Recategorize

← Older revision		Revision as of 10:14, 9 January 2024
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	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:~~Commas~~]]		[[Category:Comma]]
	[[Category:Method]]		[[Category:Method]]
	[[Category:Terms]]		[[Category:Terms]]

FloraC: Style

2022-12-28T04:40:29Z

Style

← Older revision		Revision as of 04:40, 28 December 2022
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	'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments aka [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>~~https://dkeenan~~.~~com/Music/MiddlePath.pdf~~</ref>		'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments aka [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>''Vanish'' is notably used throughout ''[[A Middle Path]]''.</ref>

	== Overview ==		== Overview ==

	For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio ~~doesn't~~ change anything.		For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything.

	~~For a tone measured in cents to "disappear", it must become 0 cents, so that adding it doesn't change anything.~~		In both cases, that implies that we are introducing some error into our tunings: where we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. There are also options to temper the octave, such as [[TOP tuning]].

	In both cases, that implies that we~~'re~~ introducing some error into our tunings: ~~Where~~ we would use 3, for instance, we use a number slightly larger or smaller than 3. You can introduce error into any prime, and when tempering out a single comma you can choose to leave any given prime pure. In practice, many people leave 2 pure to achieve pure octaves. There are also options to temper the octave, such as [[TOP tuning]].

	== Example ==		== Example ==

	The syntonic comma is 81/80. That's 3×3×3×3 / 5×2×2×2×2 or, in [[monzo]] form, {{monzo\| -4 4 -1 }}.		The syntonic comma is 81/80. That is (3×3×3×3) / (5×2×2×2×2) or, in [[monzo]] form, {{monzo\| -4 4 -1 }}.

	~~19 EDO~~ tempers out 81/80. (Technically, we should say that ~~19 EDO~~ tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:		19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways:

	=== 1. Counting steps of the val ===		=== 1. Counting steps of the val ===

	Because there are no primes larger than 5 in 81/80, we say it's a 5-limit comma. The 5-limit patent val for ~~19 EDO~~ is {{val\| 19 30 44 }}. That means that you add 19 steps of ~~19 EDO~~ to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1.		Because there are no primes larger than 5 in 81/80, we say it is a 5-limit comma. The 5-limit patent val for 19edo is {{val\| 19 30 44 }}. That means that you add 19 steps of 19edo to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1.

	Note that, because this is an ~~EDO~~, 19 steps gets you precisely to 2/1. We say that 30 steps of ~~19 EDO~~ gets you to 3/1, but that's only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it's very useful error.		Note that, because this is an edo, 19 steps gets you precisely to 2/1. We say that 30 steps of 19edo gets you to 3/1, but that is only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it is very useful error.

	Getting to 81 is 3×3×3×3, or, with ~~19 EDO~~ steps, 30 + 30 + 30 + 30 = 120 steps of ~~19 EDO~~.		Getting to 81 is 3×3×3×3, or, with 19edo steps, 30 + 30 + 30 + 30 = 120 steps of 19edo.

	Getting to 80 is 5×2×2×2×2, or, with ~~19 EDO~~ steps, 44 + 19 + 19 + 19 + 19 = 120 steps of ~~19 EDO~~.		Getting to 80 is 5×2×2×2×2, or, with 19edo steps, 44 + 19 + 19 + 19 + 19 = 120 steps of 19edo.

	Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. 120 steps - 120 steps = 0 steps.		Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. 120 steps - 120 steps = 0 steps.

	Applying the monzo to the val (also called getting the ''homomorphism'') is easier. Multiply the first number in the monzo (which represents the number of 2/1s in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: (-4 × 19) + (4 × 30) + (-1 × 44) = 0 steps.		Applying the monzo to the val (also called getting the ''homomorphism'') is easier. Multiply the first number in the monzo (which represents the number of 2/1's in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: (-4 × 19) + (4 × 30) + (-1 × 44) = 0 steps.

	Therefore, adding 81/80 to any interval in ~~19 EDO~~ means adding 0 steps of ~~19 EDO~~ to it. In other words, 81/80 is effectively zero: 81/80 is ''tempered out''.		Therefore, adding 81/80 to any interval in 19edo means adding 0 steps of 19edo to it. In other words, 81/80 is effectively zero: 81/80 is ''tempered out''.

	=== 2. Painstakingly doing the math ===		=== 2. Painstakingly doing the math ===

	We say that 30 steps of ~~19 EDO~~ gets you to 3/1, but, as we say above, that's an error. One step of ~~19 EDO~~ is the 19th root of 2, or 2<sup>1/19</sup>, or approximately 1.03715504445. (That's 63.15789474 cents.) If you multiply that by itself 19 times, you get exactly 2. But if you multiply that by itself 30 times, you ~~don't~~ get 3: You get 2.98751792330896. Similarly, multiplying it by 44 steps gets you 4.97877035785607 instead of 5.		We say that 30 steps of 19edo gets you to 3/1, but, as we say above, that is an error. One step of 19edo is the 19th root of 2, or 2<sup>1/19</sup>, or approximately 1.03715504445. (That is 63.15789474 cents.) If you multiply that by itself 19 times, you get exactly 2. But if you multiply that by itself 30 times, you do not get 3: You get 2.98751792330896. Similarly, multiplying it by 44 steps gets you 4.97877035785607 instead of 5.

	If we plug in these values into 81/80, we see that 81/80 is tempered out:		If we plug in these values into 81/80, we see that 81/80 is tempered out:
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	== See also ==		== See also ==

	* [[Fudging]] – or ''virtual tempering''		* [[Fudging]] – or ''virtual tempering''

	== Notes ==		== Notes ==
			<references/>

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

← Older revision		Revision as of 07:52, 4 December 2025
Line 3:		Line 3:
	\| ja = テンパーアウト		\| ja = テンパーアウト
	}}		}}
	'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments ~~aka~~ [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref>''Vanish'' is notably used throughout ''[[A Middle Path]]''.</ref>		'''Tempering out''' is what a [[regular temperament]] (including rank-1 temperaments a.k.a. [[equal temperament]]s) does to a small interval like a [[comma]]: it makes it disappear, or as some authors put it, ''vanish''.<ref group="note">''Vanish'' is notably used throughout ''[[A Middle Path]]''. However, ''vanish'' is broader than ''temper out'', as any abstract interval can ''vanish''. For example, you could say "in [[5edo]], the small step of the [[5L 2s\|diatonic scale]] vanishes" without specifying a just interval at all. In this sense, ''being tempered out'' is a specific way of ''vanishing''. </ref>

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

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Tempering out - Revision history

FloraC: Interwiki

FloraC: More on the term *vanish*

Dummy index: add interwiki

Aurame: Fix inverted superscripts in denominator of prime factors fraction of 81/80.

ArrowHead294 at 18:35, 16 January 2025

ArrowHead294 at 18:35, 16 January 2025

Lériendil: /* Tempering together */

Bcmills: add related term “tempering together”, which was already widely used but not specifically defined throughout the wiki

FloraC: Recategorize

FloraC: Style

FloraC: More on the term vanish