205edo - Revision history

Overthink: /* Odd harmonics */ add another table

2026-04-10T21:14:27Z

Odd harmonics: add another table

← Older revision		Revision as of 21:14, 10 April 2026
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	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|205}}		{{Harmonics in equal\|205\|columns=11}}
			{{Harmonics in equal\|205\|columns=11\|start=12\|collapsed=1\|title = Approximation of odd harmonics in 205edo (continued)}}

	=== Structural properties ===		=== Structural properties ===

Overthink: /* Theory */ expand

2026-04-08T22:30:56Z

Theory: expand

← Older revision		Revision as of 22:30, 8 April 2026
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	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|205}}		{{Harmonics in equal\|205}}

			=== Structural properties ===
			205edo contains a very accurate approximation of the [[2.3.5.11 subgroup]], inheriting the perfect fifth from 41edo. The patent val mappings of primes 7 and 13 can then be found by mapping [[7/5]] to the Pythagorean diminished fifth, and [[13/11]] at the Pythagorean minor third, thus tempering out [[5120/5103]] and [[352/351]], as well as [[847/845]] and [[2080/2079]]. In fact, it is the last edo tempering out 5120/5103 to map both [[7/5]] and [[1024/729]] consistently. It also supports the [[counterpyth]] mapping of prime 19.

			Its step size represents several important intervals, such as the septimal kleisma [[225/224]], and the keenanisma [[385/384]]. Notably, the mappings of primes 5, 7, 11, 13, and 19 all differ from their nearest 41edo step by 1 step of 205edo, so 205edo can be considered as 41edo with fine-tuning, similarly to how [[217edo]] can be considered as 31edo with fine-tuning. The intervals [[11/10]], [[12/11]], [[13/12]], [[14/13]], and [[15/14]] are mapped equidistant, corresponding to [[121/120]], [[144/143]], [[169/168]], and [[196/195]] all being mapped to 2 steps. The mappings of 17 and 19 are accurate, with 15/14, [[16/15]], [[17/16]], [[18/17]], [[19/18]], and [[20/19]] all spaced apart from each other by one step. Overall, despite the sharpness of its 7 and 13, 205edo does fairly well in a range of prime limits.

	=== Temperament generators and Tonal Plexus ===		=== Temperament generators and Tonal Plexus ===

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:46Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
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	\| [[Countercomp]]		\| [[Countercomp]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==

Hotcrystal0: this is better

2026-01-21T18:34:01Z

this is better

← Older revision		Revision as of 18:34, 21 January 2026
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	205edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.		205edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.

	The 119\205 meantone fifth is extremely close to the [[~~quarter~~-comma meantone]] fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino\|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.		The 119\205 meantone fifth is extremely close to the [[1/4-comma meantone]] fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino\|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

	=== Subsets and supersets ===		=== Subsets and supersets ===

Hotcrystal0: linking 1/4-comma meantone

2026-01-21T18:32:33Z

linking 1/4-comma meantone

← Older revision		Revision as of 18:32, 21 January 2026
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	205edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.		205edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.

	The 119\205 meantone fifth is extremely close to the ~~1/4~~-comma fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino\|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.		The 119\205 meantone fifth is extremely close to the [[quarter-comma meantone]] fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino\|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

	=== Subsets and supersets ===		=== Subsets and supersets ===

Lucius Chiaraviglio: /* Theory */ Fix grammar

2026-01-18T09:43:47Z

Theory: Fix grammar

← Older revision		Revision as of 09:43, 18 January 2026
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	Using its alternative mapping {{val\| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val\| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val\| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]] and [[364/363]] to the comma list.		Using its alternative mapping {{val\| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val\| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val\| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]] and [[364/363]] to the comma list.

	Anyway, ~~assume~~ the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.		Anyway, assuming the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.

	=== Odd harmonics ===		=== Odd harmonics ===

Lucius Chiaraviglio: Add interval table

2025-11-07T10:00:07Z

Add interval table

← Older revision		Revision as of 10:00, 7 November 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	205 factors into primes as [[5edo\|5]] × [[41edo\|41]], a fact some advocates of the division make use of; it is also [[2460edo\|2460/12]], so that a single step is precisely 12 [[mina]]s.		205 factors into primes as [[5edo\|5]] × [[41edo\|41]], a fact some advocates of the division make use of; it is also [[2460edo\|2460/12]], so that a single step is precisely 12 [[mina]]s.

			== Intervals ==
			{{Interval table}}

	== Notation ==		== Notation ==

FloraC: No need to remind readers of what a regular temperament is everywhere

2025-04-11T11:18:37Z

No need to remind readers of what a regular temperament is everywhere

← Older revision		Revision as of 11:18, 11 April 2025
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	== Theory ==		== Theory ==
	~~''It is recommended to read the page [[regular temperament]] first to understand this section.''~~

	Since {{nowrap\|205 {{=}} 5 × 41}}, 205edo shares its [[3/2\|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.		Since {{nowrap\|205 {{=}} 5 × 41}}, 205edo shares its [[3/2\|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.

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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	~~''It is recommended to read the page [[regular temperament]] first to understand this section.''~~

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator

BudjarnLambeth: /* Rank-2 temperaments */

2025-04-11T07:45:28Z

Rank-2 temperaments

← Older revision		Revision as of 07:45, 11 April 2025
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
			''It is recommended to read the page [[regular temperament]] first to understand this section.''

	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator

BudjarnLambeth: /* Theory */ ''It is recommended to read the page regular temperament first to understand this section.''

2025-04-11T07:45:14Z

Theory: ''It is recommended to read the page regular temperament first to understand this section.''

← Older revision		Revision as of 07:45, 11 April 2025
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	== Theory ==		== Theory ==
			''It is recommended to read the page [[regular temperament]] first to understand this section.''

	Since {{nowrap\|205 {{=}} 5 × 41}}, 205edo shares its [[3/2\|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.		Since {{nowrap\|205 {{=}} 5 × 41}}, 205edo shares its [[3/2\|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.