Neutral temperaments - Revision history

Vector at 02:10, 2 June 2026

2026-06-02T02:10:59Z

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	== Rastmatic ==		== Rastmatic ==
	Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22.		[[File:Mohajira7.png\|thumb\|Mohajira heptatonic MOS]]
			Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. Mohajira extends rastmatic to 2.3.5.11.

	=== Etymology ===		=== Etymology ===
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	== Namo ==		== Namo ==
			[[File:Suhajira7.png\|thumb\|Suhajira heptatonic MOS]]
	''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.		''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.

	It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.		It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.

			Namo extends to 7 with the temperament Suhajira, which takes the fifth as an [[archy]] fifth.

	== Patent vals ==		== Patent vals ==

Inthar: /* History */

2026-05-18T20:56:31Z

History

← Older revision		Revision as of 20:56, 18 May 2026
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	=== History ===		=== History ===
	According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an ~~exotenperament~~ today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)		According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an exotemperament today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)

	== Namo ==		== Namo ==

Vector at 08:19, 18 May 2026

2026-05-18T08:19:33Z

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	=== Tuning considerations ===		=== Tuning considerations ===
	A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.		A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.

			=== History ===
			According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an exotenperament today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)

	== Namo ==		== Namo ==

Overthink: formatting

2026-04-08T20:11:52Z

formatting

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	== Dicot ==		== Dicot ==
	Dicot<~~sup~~>[[~~#Footnotes\|[a]~~]]</~~sup~~>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.		Dicot<ref group="note>The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].</ref>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.

	Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales\|dicot]] is named after it.		Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales\|dicot]] is named after it.
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	== Footnotes ==		== Footnotes ==
	~~[[#Dicot\|[a]]] The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].~~		<references group="note"/>

	{{Navbox regtemp}}		{{Navbox regtemp}}

Overthink: link footnote

2026-04-07T17:28:43Z

link footnote

← Older revision		Revision as of 17:28, 7 April 2026
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	== Dicot ==		== Dicot ==
	Dicot<sup>[a]</sup>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.		Dicot<sup>[[#Footnotes\|[a]]]</sup>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.

	Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales\|dicot]] is named after it.		Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales\|dicot]] is named after it.
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	== Footnotes ==		== Footnotes ==
	[a] The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].		[[#Dicot\|[a]]] The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].

	{{Navbox regtemp}}		{{Navbox regtemp}}

Vector at 08:47, 4 April 2026

2026-04-04T08:47:15Z

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	=== Etymology ===		=== Etymology ===
	Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.		Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.

	~~== Hemififths ==~~
	Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.

	== Dicot ==		== Dicot ==
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	\|685.7c		\|685.7c
	\|}		\|}

			== Other temperaments with neutral third generators ==

			=== Hemififths ===
			Hemififths, 41 & 58, is a temperament in the 2.3.5.7 subgroup which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament. Note that 7 & 10 in 2.3.5.7 is [[dichotic]].

	== Footnotes ==		== Footnotes ==

Inthar: /* Hemififths */

2026-04-04T08:10:45Z

Hemififths

← Older revision		Revision as of 08:10, 4 April 2026
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	== Hemififths ==		== Hemififths ==
	Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which tempers out 2401/2400 = S49 equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.		Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.

	== Dicot ==		== Dicot ==

Inthar: /* Hemififths */

2026-04-04T02:12:32Z

Hemififths

← Older revision		Revision as of 02:12, 4 April 2026
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	== Hemififths ==		== Hemififths ==
	Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which ~~equates~~ 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.		Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which tempers out 2401/2400 = S49 equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.

	== Dicot ==		== Dicot ==

Inthar: /* Rastmatic */

2026-04-03T22:28:01Z

Rastmatic

← Older revision		Revision as of 22:28, 3 April 2026
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	=== Etymology ===		=== Etymology ===
	Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.		Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.

			== Hemififths ==
			Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which equates 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.

	== Dicot ==		== Dicot ==

Inthar at 14:31, 22 March 2026

2026-03-22T14:31:43Z

← Older revision		Revision as of 14:31, 22 March 2026
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	'''Neutral temperaments''' are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the ''dicot'' temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (~~rastmatic~~) and 2.3.5 (Dicot) versions.		'''Neutral temperaments''' are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the ''dicot'' temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.

	10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot [[~~archy~~]] temperament.		10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot [[Archy]] temperament.

	== Notation ==		== Notation ==
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	== Rastmatic ==		== Rastmatic ==
	Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to ~~rastmatic~~ or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22.		Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22.

	=== Etymology ===		=== Etymology ===
	Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''~~rast,~~'' which utilizes a scale with several neutral intervals.		Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.

	== Dicot ==		== Dicot ==