512/507 - Revision history

Hotcrystal0: fifth complement, not octave complement

2026-06-01T21:24:41Z

fifth complement, not octave complement

← Older revision		Revision as of 21:24, 1 June 2026
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. It can thus be seen as the 2.3.13 analogue of 2.3.11 [[Neutral (temperament)\|neutral]], which tempers out [[243/242]] and equates [[11/9]] with its ~~octave~~ complement, but instead equating 16/13 with 39/32, and preferring sharper fifths over flatter ones.		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. It can thus be seen as the 2.3.13 analogue of 2.3.11 [[Neutral (temperament)\|neutral]], which tempers out [[243/242]] and equates [[11/9]] with its fifth complement, but instead equating 16/13 with 39/32, and preferring sharper fifths over flatter ones.

	However, it has significantly more damage than neutral, similarly to how [[archy]] has more damage than [[meantone]]. For example, the neutral third in 2.3.13 CWE tuning is 7.7{{C}} flat of 16/13, but also 4.4{{C}} sharp of 11/9. This makes it ambiguous, but since 11/9 is simpler and closer, the neutral third is likely to be heard as 11/9 rather than 16/13, which shows the higher damage of harmoneutral. Using a tuning sharper than "optimal", such as [[17edo]] or [[61edo\|78edo]], makes the tridecimal nature of the intervals more apparent, but only to an extent. Compare that to in neutral, where the neutral third in 2.3.11 CWE tuning is 3.1{{C}} sharp of 11/9 and 8.9{{C}} flat of 16/13, so it is much more clearly 11/9. Harmoneutral also has much greater errors, with a (2.3.13) 13-odd-limit [[minimax]] error of 6.80{{C}} compared to neutral's (2.3.11) 11-odd-limit minimax error of 2.86{{C}}.		However, it has significantly more damage than neutral, similarly to how [[archy]] has more damage than [[meantone]]. For example, the neutral third in 2.3.13 CWE tuning is 7.7{{C}} flat of 16/13, but also 4.4{{C}} sharp of 11/9. This makes it ambiguous, but since 11/9 is simpler and closer, the neutral third is likely to be heard as 11/9 rather than 16/13, which shows the higher damage of harmoneutral. Using a tuning sharper than "optimal", such as [[17edo]] or [[61edo\|78edo]], makes the tridecimal nature of the intervals more apparent, but only to an extent. Compare that to in neutral, where the neutral third in 2.3.11 CWE tuning is 3.1{{C}} sharp of 11/9 and 8.9{{C}} flat of 16/13, so it is much more clearly 11/9. Harmoneutral also has much greater errors, with a (2.3.13) 13-odd-limit [[minimax]] error of 6.80{{C}} compared to neutral's (2.3.11) 11-odd-limit minimax error of 2.86{{C}}.

Overthink: Expanded temperaments section

2025-10-14T16:56:35Z

Expanded temperaments section

← Older revision		Revision as of 16:56, 14 October 2025
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. Tempering it out along with 4096/3993 and 361/352 in 2.3.11.13.19 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. It can thus be seen as the 2.3.13 analogue of 2.3.11 [[Neutral (temperament)\|neutral]], which tempers out [[243/242]] and equates [[11/9]] with its octave complement, but instead equating 16/13 with 39/32, and preferring sharper fifths over flatter ones.

			However, it has significantly more damage than neutral, similarly to how [[archy]] has more damage than [[meantone]]. For example, the neutral third in 2.3.13 CWE tuning is 7.7{{C}} flat of 16/13, but also 4.4{{C}} sharp of 11/9. This makes it ambiguous, but since 11/9 is simpler and closer, the neutral third is likely to be heard as 11/9 rather than 16/13, which shows the higher damage of harmoneutral. Using a tuning sharper than "optimal", such as [[17edo]] or [[61edo\|78edo]], makes the tridecimal nature of the intervals more apparent, but only to an extent. Compare that to in neutral, where the neutral third in 2.3.11 CWE tuning is 3.1{{C}} sharp of 11/9 and 8.9{{C}} flat of 16/13, so it is much more clearly 11/9. Harmoneutral also has much greater errors, with a (2.3.13) 13-odd-limit [[minimax]] error of 6.80{{C}} compared to neutral's (2.3.11) 11-odd-limit minimax error of 2.86{{C}}.

			Another possible option is to temper out both 512/507 and 243/242, and therefore [[144/143]], leading to [[namo]]. This temperament is generated by a neutral third like harmoneutral and neutral, except all of the neutral intervals are now ambiguously 11-limit or 13-limit.

			Tempering it out along with 4096/3993 and 361/352 in 2.3.11.13.19 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.

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

VectorGraphics at 00:36, 30 August 2025

2025-08-30T00:36:59Z

← Older revision		Revision as of 00:36, 30 August 2025
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. Tempering it out along with 4096/3993 and 361/352 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. Tempering it out along with 4096/3993 and 361/352 in 2.3.11.13.19 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.

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

VectorGraphics at 00:36, 30 August 2025

2025-08-30T00:36:46Z

← Older revision		Revision as of 00:36, 30 August 2025
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement.		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. Tempering it out along with 4096/3993 and 361/352 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.

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

VectorGraphics at 00:35, 30 August 2025

2025-08-30T00:35:25Z

← Older revision		Revision as of 00:35, 30 August 2025
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. ~~A possible weak extension to the 2.3.11.13.19 subgroup, ''deltahexacot'', also tempers out 4096/3993 and 361/352.~~		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement.

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

VectorGraphics at 00:34, 30 August 2025

2025-08-30T00:34:54Z

← Older revision		Revision as of 00:34, 30 August 2025
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement.		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. A possible weak extension to the 2.3.11.13.19 subgroup, ''deltahexacot'', also tempers out 4096/3993 and 361/352.

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

VectorGraphics at 18:33, 1 June 2025

2025-06-01T18:33:39Z

← Older revision		Revision as of 18:33, 1 June 2025
Line 8:		Line 8:

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the ~~harmonic~~ neutral third together with its fifth complement.		[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement.

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

FloraC: Cleanup. Note proposal maker

2025-05-31T13:11:59Z

Cleanup. Note proposal maker

← Older revision		Revision as of 13:11, 31 May 2025
Line 5:		Line 5:
	}}		}}

	'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also 2.3.13 subgroup) comma with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].		'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also [[2.3.13 subgroup\|2.3.13-subgroup]]) [[comma]] with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].

	Tempering out this ~~interval~~ equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and ~~can~~ be called '''harmoneutral''' temperament, as it tempers the harmonic neutral third together with its fifth complement.		== Temperaments ==
			[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics\|Vector]] in 2025, as it tempers the harmonic neutral third together with its fifth complement.

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

VectorGraphics at 09:06, 31 May 2025

2025-05-31T09:06:28Z

← Older revision		Revision as of 09:06, 31 May 2025
Line 6:		Line 6:

	'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also 2.3.13 subgroup) comma with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].		'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also 2.3.13 subgroup) comma with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].

			Tempering out this interval equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and can be called '''harmoneutral''' temperament, as it tempers the harmonic neutral third together with its fifth complement.

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

BudjarnLambeth at 23:12, 15 November 2024

2024-11-15T23:12:46Z

← Older revision		Revision as of 23:12, 15 November 2024
Line 9:		Line 9:
	== See also ==		== See also ==
	* [[Small comma]]		* [[Small comma]]

			[[Category:Commas named for the intervals they stack]]