48edo - Revision history

Lucius Chiaraviglio: /* Theory */ Fix link to Doublewide

2026-06-18T06:18:20Z

Theory: Fix link to Doublewide

← Older revision		Revision as of 06:18, 18 June 2026
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	Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the [[5-limit]], but [[tempering out\|tempers out]] [[2401/2400]] in the [[7-limit]], making it a tuning for [[squares]] temperament. In the [[11-limit]] we can add [[99/98]] and [[121/120]] to the list, and in the [[13-limit]], [[66/65]]. While [[31edo]] can also do 13-limit squares, 48 might be preferred for some purposes.		Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the [[5-limit]], but [[tempering out\|tempers out]] [[2401/2400]] in the [[7-limit]], making it a tuning for [[squares]] temperament. In the [[11-limit]] we can add [[99/98]] and [[121/120]] to the list, and in the [[13-limit]], [[66/65]]. While [[31edo]] can also do 13-limit squares, 48 might be preferred for some purposes.

	Using its best major third, 48edo tempers out [[20000/19683]], but [[27edo]] and [[34edo]] do a much better job for this temperament, known as [[tetracot]], since (for example) 48edo's [[7L 6s]] has a rather awkward 6:1 step ratio, while 27edo and 34edo have 3:1 and 4:1 step ratios for the same scale. However in the 7-limit it can be used for [[Jubilismic_clan\|doublewide temperament]], the half-octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the [[optimal patent val]]. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament [[Jubilismic_family\|jubilee]], which tempers out 50/49 and 99/98.		Using its best major third, 48edo tempers out [[20000/19683]], but [[27edo]] and [[34edo]] do a much better job for this temperament, known as [[tetracot]], since (for example) 48edo's [[7L 6s]] has a rather awkward 6:1 step ratio, while 27edo and 34edo have 3:1 and 4:1 step ratios for the same scale. However in the 7-limit it can be used for [[Jubilismic_clan#Doublewide\|doublewide temperament]], the half-octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the [[optimal patent val]]. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament [[Jubilismic_family\|jubilee]], which tempers out 50/49 and 99/98.

	If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the [[optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425{{c}} interval serving as both [[9/7]] and [[14/11]].		If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the [[optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425{{c}} interval serving as both [[9/7]] and [[14/11]].

FloraC: /* Notation */ improve description

2026-05-12T14:16:35Z

Notation: improve description

← Older revision		Revision as of 14:16, 12 May 2026
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	== Notation ==		== Notation ==
	=== Stein–Zimmermann–Gould notation ===		=== Stein–Zimmermann–Gould notation ===
	[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:		[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
	{{Sharpness-sharp4-szg\|48}}		{{Sharpness-sharp4-szg\|48}}

FloraC: /* Notation */ SZG notation

2026-05-12T13:27:04Z

Notation: SZG notation

← Older revision		Revision as of 13:27, 12 May 2026
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	== Notation ==		== Notation ==
	=== ~~Ups~~ and downs notation ===		=== Stein–Zimmermann–Gould notation ===
	48edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.		[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
			{{Sharpness-sharp4-szg\|48}}

			=== Kite's ups and downs notation ===
			48edo can also be notated with [[Kite's ups and downs notation\|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
	{{Ups and downs sharpness}}		{{Ups and downs sharpness}}

	~~[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:~~
	~~{{Sharpness-sharp4\|48}}~~

	=== Sagittal notation ===		=== Sagittal notation ===
	This notation is a superset of the notations for ~~EDOs~~ [[24edo#Sagittal notation\|24]], [[12edo#Sagittal notation\|12]], [[8edo#Sagittal notation\|8]], and [[6edo#Sagittal notation\|6]].		This notation is a superset of the notations for edos [[24edo #Sagittal notation\|24]], [[12edo #Sagittal notation\|12]], [[8edo #Sagittal notation\|8]], and [[6edo #Sagittal notation\|6]].

	==== Evo flavor ====		==== Evo flavor ====
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	default [[File:48-EDO_Evo-SZ_Sagittal.svg]]		default [[File:48-EDO_Evo-SZ_Sagittal.svg]]
	</imagemap>		</imagemap>

	== Approximation to JI ==		== Approximation to JI ==
	=== Interval mappings ===		=== Interval mappings ===

Yourmusic Productions: /* Music */ add music

2026-05-09T10:56:40Z

Music: add music

← Older revision		Revision as of 10:56, 9 May 2026
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	; [[Ray Perlner]]		; [[Ray Perlner]]
	* [https://www.youtube.com/watch?v=HHozjGjWI-Q ''Octatonic Groove''] (2020) – jubilismic[8] in 48edo tuning		* [https://www.youtube.com/watch?v=HHozjGjWI-Q ''Octatonic Groove''] (2020) – jubilismic[8] in 48edo tuning

			; [[Nils Potet]]
			* [https://www.youtube.com/watch?v=bYbVXX067A4 ''Ricercar, Premier Moment''] (2011)
			* [https://www.youtube.com/watch?v=5Pq0j4HUv6w ''Ricercar, Deuxième Moment''] (2011)
			* [https://www.youtube.com/watch?v=mz5ufHKekjc ''Perdu dans le Landmannalaugar''] (2012)

	; [[Romeolz]]		; [[Romeolz]]

Fitzgerald Lee: /* Notation */Added Interval mappings for 48p and 48c

2026-04-25T10:53:57Z

Notation: Added Interval mappings for 48p and 48c

← Older revision		Revision as of 10:53, 25 April 2026
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	default [[File:48-EDO_Evo-SZ_Sagittal.svg]]		default [[File:48-EDO_Evo-SZ_Sagittal.svg]]
	</imagemap>		</imagemap>
			== Approximation to JI ==
			=== Interval mappings ===
			{{Q-odd-limit intervals\|48}}
			{{Q-odd-limit intervals\|48.1\|apx=val\|header=none\|tag=none\|title=15-odd-limit intervals by 48c val mapping}}

	== Regular temperament properties ==		== Regular temperament properties ==

BudjarnLambeth: /* Music */

2026-04-12T09:03:18Z

Music

← Older revision		Revision as of 09:03, 12 April 2026
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	; [[Romeolz]]		; [[Romeolz]]
	* ''Boundary Breaker''		* ''Boundary Breaker'' (2025)
	** 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audio and graphic score (BeepBox)]
	** [https://www.reddit.com/r/Beepbox/comments/1mqa0j3/my_first_proper_track_vanilla_48edo_microtonal/ description (Reddit)]		** [https://www.reddit.com/r/Beepbox/comments/1mqa0j3/my_first_proper_track_vanilla_48edo_microtonal/ description (Reddit)]

BudjarnLambeth: /* Music */

2026-04-12T09:02:56Z

Music

← Older revision		Revision as of 09:02, 12 April 2026
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	* ''Boundary Breaker''		* ''Boundary Breaker''
	** 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audio and graphic score (BeepBox)]		** 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audio and graphic score (BeepBox)]
	** [https://www.reddit.com/r/Beepbox/comments/1mqa0j3/my_first_proper_track_vanilla_48edo_microtonal/ description (Reddit)		** [https://www.reddit.com/r/Beepbox/comments/1mqa0j3/my_first_proper_track_vanilla_48edo_microtonal/ description (Reddit)]

	; [[Carlo Serafini]]		; [[Carlo Serafini]]

BudjarnLambeth: /* 48 equal divisions of the octave (48edo proper) */ add 1 (long url)

2026-04-12T09:02:44Z

48 equal divisions of the octave (48edo proper): add 1 (long url)

← Older revision		Revision as of 09:02, 12 April 2026
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	; [[Ray Perlner]]		; [[Ray Perlner]]
	* [https://www.youtube.com/watch?v=HHozjGjWI-Q ''Octatonic Groove''] (2020) – jubilismic[8] in 48edo tuning		* [https://www.youtube.com/watch?v=HHozjGjWI-Q ''Octatonic Groove''] (2020) – jubilismic[8] in 48edo tuning

			; [[Romeolz]]
			* ''Boundary Breaker''
			** 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audio and graphic score (BeepBox)]
			** [https://www.reddit.com/r/Beepbox/comments/1mqa0j3/my_first_proper_track_vanilla_48edo_microtonal/ description (Reddit)

	; [[Carlo Serafini]]		; [[Carlo Serafini]]

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"

2026-03-13T13:32:36Z

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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	\| [[Doublewide]]		\| [[Doublewide]]
	\|}		\|}
	<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

	== Octave stretch or compression ==		== Octave stretch or compression ==

Lucius Chiaraviglio: /* Well-tempered derivatives of 48edo */ Randy Wells' 'passing the edge of the universe, my thoughts were of you (for microtonal orchestra)'': added tuning information

2026-01-06T11:15:30Z

Well-tempered derivatives of 48edo: Randy Wells' 'passing the edge of the universe, my thoughts were of you (for microtonal orchestra)'': added tuning information

← Older revision		Revision as of 11:15, 6 January 2026
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	=== Well-tempered derivatives of 48edo ===		=== Well-tempered derivatives of 48edo ===
	; [[Randy Wells]]		; [[Randy Wells]]
	* [https://www.youtube.com/watch?v=sstmTuK3Vyo ''passing the edge of the universe, my thoughts were of you (for microtonal orchestra)''] (2026)		* [https://www.youtube.com/watch?v=sstmTuK3Vyo ''passing the edge of the universe, my thoughts were of you (for microtonal orchestra)''] (2026 — video description specifies quarter-tones, but in comments, intervals of ±20{{c}} and ±33{{c}} are listed, and while these can be fudged together for consideration as a 48-well-tempered tuning system, an even higher number of divisions of the octave may be needed)

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