Fifth-chroma temperaments - Revision history

FloraC: Text replacement - "Category:Temperament collections↵Category:Pages with mostly numerical content" to "Category:Temperament collections"

2025-11-27T12:26:54Z

Text replacement - "Category:Temperament collections↵Category:Pages with mostly numerical content" to "Category:Temperament collections"

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	[[Category:Temperament collections]]		[[Category:Temperament collections]]
	~~[[Category:Pages with mostly numerical content]]~~

	== Fifth-chroma temperaments ==		== Fifth-chroma temperaments ==

Lériendil: Text replacement - "Category:Temperament collections" to "Category:Temperament collections Category:Pages with mostly numerical content"

2025-06-24T00:41:10Z

Text replacement - "Category:Temperament collections" to "Category:Temperament collections Category:Pages with mostly numerical content"

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	[[Category:Temperament collections]]		[[Category:Temperament collections]]
			[[Category:Pages with mostly numerical content]]

	== Fifth-chroma temperaments ==		== Fifth-chroma temperaments ==

Godtone: fix link

2025-03-04T19:13:46Z

fix link

← Older revision		Revision as of 19:13, 4 March 2025
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	Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.		Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.
	==== Novamajor ====		==== Novamajor ====
	{{ See also \| ~~Augmented–chromatic~~ equivalence continuum }}		{{ See also \| User:Godtone/Augmented-chromatic equivalence continuum }}
	This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125\|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full no-31's 37-limit temperament (and possibly/likely higher) that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):		This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125\|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full no-31's 37-limit temperament (and possibly/likely higher) that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):
	* inframinor third: ~15/13		* inframinor third: ~15/13

Godtone: /* Fifth-chroma temperaments */ add seeds

2025-02-27T05:43:04Z

Fifth-chroma temperaments: add seeds

← Older revision		Revision as of 05:43, 27 February 2025
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	==== Primary ====		==== Primary ====
	* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]		* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]
			: * The "5-limit seed" of novamajor is by splitting 25/24 into five parts of ~sqrt(128/125) = (128/125)<sup>1/2</sup>.
	* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form		* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form
			: * The "5-limit seed" of absurdity is by tempering out (81/80)<sup>5</sup> / (25/24) (the absurdity comma) so that 25/24 is split into five flattened (underexaggerated) ~81/80's, which is equivalent to interpreting 1\7 as (10/9)/(81/80) because 2 / ((10/9)/(81/80) = 800/729)<sup>7</sup> is equal to the square of the absurdity comma.
	* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)		* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)
	==== Secondary ====		==== Secondary ====
	* 77 & 87: [[Restles]]		* 77 & 87: [[Restles]]
			: * The "2.3.13 seed" of restles is [[lesser tendoneutralic]], which observes (16/13)/(39/32) = 512/507 so that this interval is what divides 25/24 into five in 2.3.5.13 restles.
			* 84 & 87: [[Mutt]]
			: * The "5-limit seed" of mutt is by splitting 25/24 into 5 parts of ~cbrt(128/125) = (128/125)<sup>1/3</sup>.
	* 77 & 94: [[Tsaharuk]]		* 77 & 94: [[Tsaharuk]]
	* 80 & 87: "alt artoneutral": a version of [[Hemifamity temperaments#Artoneutral\|artoneutral]] that uses a different mapping for prime 7		* 80 & 87: "alt artoneutral": a version of [[Hemifamity temperaments#Artoneutral\|artoneutral]] that uses a different mapping for prime 7
			: * This has the same "5-limit seed" as artoneutral, but is actually more optimal for the 5-limit.
	* 80 & 94: unnamed? proposed name: "pseudohemipyth"		* 80 & 94: unnamed? proposed name: "pseudohemipyth"
	* 87 & 94: [[Artoneutral]]		* 87 & 94: [[Artoneutral]]
			: * The "5-limit seed" of artoneutral is by splitting 25/24 into 5 parts of ~sqrt(81/80) = (81/80)<sup>1/2</sup>.

	=== New temperament definitions ===		=== New temperament definitions ===

Godtone: /* Fifth-chroma temperaments */ typo

2025-02-27T05:09:59Z

Fifth-chroma temperaments: typo

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	=== Overview ===		=== Overview ===
	==== Primary ====		==== Primary ====
	* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]]		* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]
	* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form		* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form
	* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)		* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)

Godtone: include 46/37 as per its subgroup being at least no-31's 37-limit

2025-02-27T05:07:48Z

include 46/37 as per its subgroup being at least no-31's 37-limit

← Older revision		Revision as of 05:07, 27 February 2025
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	==== Novamajor ====		==== Novamajor ====
	{{ See also \| Augmented–chromatic equivalence continuum }}		{{ See also \| Augmented–chromatic equivalence continuum }}
	This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125\|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full 29-limit temperament that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):		This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125\|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full no-31's 37-limit temperament (and possibly/likely higher) that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):
	* inframinor third: ~15/13		* inframinor third: ~15/13
	* subminor third: ~7/6		* subminor third: ~7/6
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	* subneutral third: ~39/32~50/41~11/9		* subneutral third: ~39/32~50/41~11/9
	* superneutral third: ~27/22~16/13		* superneutral third: ~27/22~16/13
	* submajor third: ~36/29~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))		* submajor third: ~36/29~46/37~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))
	* major third: ~5/4		* major third: ~5/4
	* novamajor third: ~24/19~43/34~19/15		* novamajor third: ~24/19~43/34~19/15

Godtone: style

2025-02-27T05:03:07Z

style

Godtone: /* Fifthchroma */ add reminder

2025-02-27T05:01:46Z

Fifthchroma: add reminder

← Older revision		Revision as of 05:01, 27 February 2025
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	Another omission is 70 & 80 & 84e & 94 which reduces by tempering out [[540/539\|S12/S14]]; though the significance of this is less clear, it can be observed that it is similar to the other omission by having the latter two edos being an offset of 10 from the former two edos.		Another omission is 70 & 80 & 84e & 94 which reduces by tempering out [[540/539\|S12/S14]]; though the significance of this is less clear, it can be observed that it is similar to the other omission by having the latter two edos being an offset of 10 from the former two edos.

			As a reminder then, the remarkable feature is that ''all of these edo joins are equivalent''* and refer to the same 13-limit tempered structure: the one which splits 25/24 into five equal parts of ~144/143 so that 11/9 and 16/13 are made fifth-complements. This structure maps various high-limit things onto the 2.3.7.11 subgroup, and is home to a family of lower-rank temperaments that introduce various extra equivalences, discussed in [[#Fifth-chroma temperaments]].

			:: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)

	=== Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===		=== Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===

Godtone: /* New temperament definitions */ add novamajor

2025-02-27T04:55:56Z

New temperament definitions: add novamajor

← Older revision		Revision as of 04:55, 27 February 2025
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	{{ See also \| Absurdity }}		{{ See also \| Absurdity }}
	Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.		Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.
			==== Novamajor ====
			{{ See also \| Augmented–chromatic equivalence continuum }}
			This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125\|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full 29-limit temperament that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):
			* inframinor third: ~15/13
			* subminor third: ~7/6
			* neominor third: ~20/17 (note the absence of ~13/11, which here is ambiguous in mapping between novaminor third (77edo) and neominor third (80edo))
			* novaminor third: ~19/16 (note the absence of ~25/21, which here is ambiguous in mapping between minor third (77edo) and novaminor third (80edo))
			* minor third: ~6/5
			* supraminor third: ~35/29~29/24~23/19 (note the absence of ~17/14~40/33, which here is ambiguous in mapping between supraminor third (77edo) and subneutral third (80edo))
			* subneutral third: ~39/32~50/41~11/9
			* superneutral third: ~27/22~16/13
			* submajor third: ~36/29~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))
			* major third: ~5/4
			* novamajor third: ~24/19~43/34~19/15
			* neomajor third: ~14/11~37/29~23/18
			* supermajor third: ~9/7
			* ultramajor third: ~13/10
			Therefore, the reason for the name "novamajor" is that the structure that the aforementioned "guarantee of a certain kind of metaphorical colour palette with certain properties" via making those intervals equidistant relates it directly to [[User:Godtone#Intervals\|the interval colour names devised by User:Godtone]], with the most idiosyncratic term (that has seen least adoption by others) being "novamajor" and "novaminor", and as the novamajor third is the generator of this temperament, it seemed a fitting name. (The associated 5-limit temperament was discovered independently by User:Godtone with this high-limit application in mind, but isnes wasn't found at the time for various reasons including the ratio not being shown.)

Godtone: /* Fifth-chroma temperaments */ tweak rank 2 temperaments overview

2025-02-27T04:24:31Z

Fifth-chroma temperaments: tweak rank 2 temperaments overview

← Older revision		Revision as of 04:24, 27 February 2025
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	==== Primary ====		==== Primary ====
	* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]]		* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]]
	* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament		* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form
	* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)		* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)
	==== Secondary ====		==== Secondary ====
	* 77 & 87: [[Restles]]		* 77 & 87: [[Restles]]
	* 77 & 94: [[Tsaharuk]]		* 77 & 94: [[Tsaharuk]]
	* 80 & 87: a version of [[Hemifamity temperaments#Artoneutral\|artoneutral]] that ~~unmaps/~~uses a different mapping for prime 7		* 80 & 87: "alt artoneutral": a version of [[Hemifamity temperaments#Artoneutral\|artoneutral]] that uses a different mapping for prime 7
	* 80 & 94: unnamed? proposed name: "pseudohemipyth"		* 80 & 94: unnamed? proposed name: "pseudohemipyth"
	* 87 & 94: [[Artoneutral]]		* 87 & 94: [[Artoneutral]]
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	==== Absurdity ====		==== Absurdity ====
	{{ See also \| Absurdity }}		{{ See also \| Absurdity }}
	Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal		Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.

← Older revision		Revision as of 05:03, 27 February 2025
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	* "superneutral", which principally represents ~27/22~16/13		* "superneutral", which principally represents ~27/22~16/13
	* "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37		* "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37
	:: <nowiki>*</nowiki> of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later		: <nowiki>*</nowiki> of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later
	The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.		The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.

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	That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]**, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].		That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]**, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].

	:: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].		: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].

	:: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.		: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.

	If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:		If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:
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	As a reminder then, the remarkable feature is that ''all of these edo joins are equivalent''* and refer to the same 13-limit tempered structure: the one which splits 25/24 into five equal parts of ~144/143 so that 11/9 and 16/13 are made fifth-complements. This structure maps various high-limit things onto the 2.3.7.11 subgroup, and is home to a family of lower-rank temperaments that introduce various extra equivalences, discussed in [[#Fifth-chroma temperaments]].		As a reminder then, the remarkable feature is that ''all of these edo joins are equivalent''* and refer to the same 13-limit tempered structure: the one which splits 25/24 into five equal parts of ~144/143 so that 11/9 and 16/13 are made fifth-complements. This structure maps various high-limit things onto the 2.3.7.11 subgroup, and is home to a family of lower-rank temperaments that introduce various extra equivalences, discussed in [[#Fifth-chroma temperaments]].

	:: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)		: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)

	=== Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===		=== Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===
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	The default badness (out-of-tune-ness) measure is a type of Mean-Square-Error applied to the set of intervals given, where the "error" in question is cent error weighted proportional to the square-root of the odd-limit of the interval, so that it can also be thought of as minimizing the square of the cent error with each squared error contribution of a given interval weighted proportional to its odd-limit. The mean-square-error is standard, but the odd-limit weighting needs explanation: a more in-depth discussion of its principles can be found in [[Talk:Marvel#Challenge on optimality of 53edo for FloraC\|the discussion page of marvel]] as the discussion also covers various tuning principles User:Godtone believes in, but the basic motivation is that the more complex an interval is, the more precise the tuning needs to be to approximate it; initially this seems extremely harsh, because one would expect that the precision required goes up with the square of the odd-limit, and though this is true for dyads, it needn't be true in practice when constructing harmonies, because the other notes that a tempered interval is placed with quickly eliminate possibilities for what the tempered interval might be approximating in favor of a comparatively small number of possible harmonic series chords, so that even weighting the unsquared error proportional to the odd-limit can give strangely biased results, but clearly more complex intervals need more tuning fidelity* even if we have a relatively forcing harmonic context, so the next best thing was the square-root which also can be interpreted as weighting the squared error proportional the odd-limit, which is how it was originally conceived.		The default badness (out-of-tune-ness) measure is a type of Mean-Square-Error applied to the set of intervals given, where the "error" in question is cent error weighted proportional to the square-root of the odd-limit of the interval, so that it can also be thought of as minimizing the square of the cent error with each squared error contribution of a given interval weighted proportional to its odd-limit. The mean-square-error is standard, but the odd-limit weighting needs explanation: a more in-depth discussion of its principles can be found in [[Talk:Marvel#Challenge on optimality of 53edo for FloraC\|the discussion page of marvel]] as the discussion also covers various tuning principles User:Godtone believes in, but the basic motivation is that the more complex an interval is, the more precise the tuning needs to be to approximate it; initially this seems extremely harsh, because one would expect that the precision required goes up with the square of the odd-limit, and though this is true for dyads, it needn't be true in practice when constructing harmonies, because the other notes that a tempered interval is placed with quickly eliminate possibilities for what the tempered interval might be approximating in favor of a comparatively small number of possible harmonic series chords, so that even weighting the unsquared error proportional to the odd-limit can give strangely biased results, but clearly more complex intervals need more tuning fidelity* even if we have a relatively forcing harmonic context, so the next best thing was the square-root which also can be interpreted as weighting the squared error proportional the odd-limit, which is how it was originally conceived.

	:: <nowiki>*</nowiki> This is not to touch on the debate of what intervals it is aesthetically more ''important'' to get right, where one might consider 3/2 as requiring more precise tuning simply because it's ''more obvious'' when mistuned due to its simplicity; the perspective of the default used by <code>optimal_edo_sequence</code> is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.		: <nowiki>*</nowiki> This is not to touch on the debate of what intervals it is aesthetically more ''important'' to get right, where one might consider 3/2 as requiring more precise tuning simply because it's ''more obvious'' when mistuned due to its simplicity; the perspective of the default used by <code>optimal_edo_sequence</code> is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.

	[[Category:Temperament collections]]		[[Category:Temperament collections]]