128/125 - Revision history

Yourmusic Productions: further extension.

2026-08-31T11:35:10Z

further extension.

← Older revision		Revision as of 11:35, 31 August 2026
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	=== As an interval ===		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. The simplest way to accomplish this is tempering out [[126/125]] and [[4125/4096]] to equate it with [[64/63]] and [[33/32]]. Alternately, tempering out [[5632/5625]] equates it with [[45/44]] in a more accurate but complex way.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. The simplest way to accomplish this is tempering out [[126/125]] and [[4125/4096]] to equate it with [[64/63]] and [[33/32]]. Alternately, tempering out [[5632/5625]] equates it with [[45/44]] in a more accurate but complex way. Extending to the [[13-limit]] happens even more naturally by tempering out [[625/624]] to equate it with [[40/39]].

	== Trivia ==		== Trivia ==

Yourmusic Productions: Another possible comma basis.

2026-08-31T11:17:22Z

Another possible comma basis.

← Older revision		Revision as of 11:17, 31 August 2026
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	=== As an interval ===		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. The simplest way to accomplish this is tempering out [[126/125]] and [[4125/4096]] to equate it with [[64/63]] and [[33/32]].		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. The simplest way to accomplish this is tempering out [[126/125]] and [[4125/4096]] to equate it with [[64/63]] and [[33/32]]. Alternately, tempering out [[5632/5625]] equates it with [[45/44]] in a more accurate but complex way.

	== Trivia ==		== Trivia ==

Yourmusic Productions: commas needed to use the diesis to approximate higher limit intervals.

2026-08-31T10:52:54Z

commas needed to use the diesis to approximate higher limit intervals.

← Older revision		Revision as of 10:52, 31 August 2026
Line 19:		Line 19:

	=== As an interval ===		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony~~{{clarify}}~~ into 5-limit scales.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. The simplest way to accomplish this is tempering out [[126/125]] and [[4125/4096]] to equate it with [[64/63]] and [[33/32]].

	== Trivia ==		== Trivia ==

Eliora: /* Temperaments */

2026-08-23T19:30:49Z

Temperaments

← Older revision		Revision as of 19:30, 23 August 2026
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	=== As an interval ===		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony{{clarify}} into 5-limit scales.

	== Trivia ==		== Trivia ==

FloraC: /* Temperaments */ link

2025-09-11T04:24:40Z

Temperaments: link

← Older revision		Revision as of 04:24, 11 September 2025
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	== Temperaments ==		== Temperaments ==
	=== As a comma ===		=== As a comma ===
	[[Tempering out]] this comma leads to [[augmented]] temperament. See [[~~augmented~~ family]] for the family where it is tempered out.		[[Tempering out]] this comma leads to [[augmented (temperament)\|augmented]] temperament. See [[Augmented family]] for the family where it is tempered out.

	=== As an interval ===		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.

	== Trivia ==		== Trivia ==

BudjarnLambeth at 07:02, 3 November 2024

2024-11-03T07:02:05Z

← Older revision		Revision as of 07:02, 3 November 2024
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	[[Category:Augmented]]		[[Category:Augmented]]
	[[Category:Sonifications]]		[[Category:Sonifications]]
			[[Category:Commas named after their interval size]]

FloraC: Revert to original section order (I don't see anything wrong with this)

2024-01-04T07:39:09Z

Revert to original section order (I don't see anything wrong with this)

← Older revision		Revision as of 07:39, 4 January 2024
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	The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)<sup>3</sup>.		The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)<sup>3</sup>.

			== Approximations ==
			This interval is fairly accurately represented by a single step in [[28edo\|28-]], [[31edo\|31-]] or [[34edo]], and by two steps of [[53edo\|53-]], [[59edo\|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason.

	== Temperaments ==		== Temperaments ==
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	=== As an interval ===		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.

	~~== Approximations ==~~
	It is fairly accurately represented by a single step in [[28edo\|28-]], [[31edo\|31-]] or [[34edo]], and by two steps of [[53edo\|53-]], [[59edo\|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason.

	== Trivia ==		== Trivia ==

FloraC: Add back original section titles; adopt template: Interwiki

2024-01-04T07:26:37Z

Add back original section titles; adopt template: Interwiki

← Older revision		Revision as of 07:26, 4 January 2024
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			{{Interwiki
			\| en = 128/125
			\| de = 128/125
			}}
	{{Infobox Interval		{{Infobox Interval
	\| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma		\| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
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	}}		}}

	The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)<sup>3</sup>.		The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)<sup>3</sup>.

	== As a comma ==		== Temperaments ==
	Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out.		=== As a comma ===
			[[Tempering out]] this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out.

	== As an interval ==		=== As an interval ===
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.

Line 16:		Line 21:
	It is fairly accurately represented by a single step in [[28edo\|28-]], [[31edo\|31-]] or [[34edo]], and by two steps of [[53edo\|53-]], [[59edo\|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason.		It is fairly accurately represented by a single step in [[28edo\|28-]], [[31edo\|31-]] or [[34edo]], and by two steps of [[53edo\|53-]], [[59edo\|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason.

	== ~~Relation to binary~~ ==		== Trivia ==
	This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000.		This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000.

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	[[Category:Augmented]]		[[Category:Augmented]]
	[[Category:Sonifications]]		[[Category:Sonifications]]

	~~<!-- interwiki: links are being placed in the sidebar -->~~
	~~[[https://de.xen.wiki/w/128/125]]~~
	~~[[Category:Augmented family]]~~

BudjarnLambeth: Add to category augmented family

2024-01-04T02:55:33Z

Add to category augmented family

← Older revision		Revision as of 02:55, 4 January 2024
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	<!-- interwiki: links are being placed in the sidebar -->		<!-- interwiki: links are being placed in the sidebar -->
	[[de:128/125]]		[[https://de.xen.wiki/w/128/125]]
			[[Category:Augmented family]]

BudjarnLambeth: Improved layout

2024-01-04T02:53:50Z

Improved layout

← Older revision		Revision as of 02:53, 4 January 2024
Line 7:		Line 7:
	The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)<sup>3</sup>.		The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)<sup>3</sup>.

	== ~~Approximation~~ ==		== As a comma ==
			Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out.

			== As an interval ==
	If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.		If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.

			== Approximations ==
	It is fairly accurately represented by a single step in [[28edo\|28-]], [[31edo\|31-]] or [[34edo]], and by two steps of [[53edo\|53-]], [[59edo\|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason.		It is fairly accurately represented by a single step in [[28edo\|28-]], [[31edo\|31-]] or [[34edo]], and by two steps of [[53edo\|53-]], [[59edo\|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason.

	== ~~Temperaments ==~~		== Relation to binary ==
	~~Tempering out this comma leads~~ to ~~[[augmented]] temperament. See [[augmented family]] for the family where it is tempered out.~~

	~~== Trivia~~ ==
	This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000.		This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000.

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	<!-- interwiki: links are being placed in the sidebar -->		<!-- interwiki: links are being placed in the sidebar -->
	[[de:128/125]]		[[de:128/125]]

	~~[[Category:Todo:improve layout]]~~