48edo: Difference between revisions

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== Theory ==

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, ~~the equal temperament~~ tempers out [[20000/19683]], but [[34edo]] ~~does~~ a much better job for this temperament, known as [[tetracot]]. However in the 7-limit it can be used for [[Jubilismic_clan|doublewide temperament]], the ~~1/2~~ 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 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|~~optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 ~~cent~~ interval serving as both [[~~9/7|~~9/7]] and [[~~14/11|~~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]].

Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths.

=== Odd harmonics ===

{{Harmonics in equal|48|columns=14|start=15|collapsed=true|title=Approximation of odd harmonics in 48edo (continued)}}

=== As a tuning standard ===

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=== Subsets and supersets ===

48edo is the 10th [[highly composite edo]]. Since 48 factors into {{factorization|48}}, 48edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, and 24 }}.

~~== Notation ==~~

~~=== Ups and downs notation ===~~

~~48edo can be notated using [[ups and downs notation]]. This can be accomplished by combining [[Helmholtz-Ellis]] accidentals with semisharps and semiflats:~~

~~{{Sharpness-sharp4|48}}~~

~~===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]].~~

~~====Evo flavor====~~

~~<imagemap>~~

~~File:48-EDO_Evo_Sagittal.svg~~

~~desc none~~

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 140 106 [[736/729]]~~

~~rect 140 80 260 106 [[33/32]]~~

~~default [[File:48-EDO_Evo_Sagittal.svg]]~~

~~</imagemap>~~

~~====Revo flavor====~~

~~<imagemap>~~

~~File:48-EDO_Revo_Sagittal.svg~~

~~desc none~~

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 140 106 [[736/729]]~~

~~rect 140 80 260 106 [[33/32]]~~

~~default [[File:48-EDO_Revo_Sagittal.svg]]~~

~~</imagemap>~~

~~====Evo-SZ flavor====~~

~~<imagemap>~~

~~File:48-EDO_Evo-SZ_Sagittal.svg~~

~~desc none~~

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 300 0 519 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 140 106 [[736/729]]~~

~~rect 140 80 260 106 [[33/32]]~~

~~default [[File:48-EDO_Evo-SZ_Sagittal.svg]]~~

~~</imagemap>~~

== Intervals ==

{| class="wikitable center-1 right-2"

|-

! #

! Cents

! [[Ups and downs notation|Ups and ~~Downs Notation~~]]

! [[Ups and downs notation|Ups and downs notation]]

|-

| 0

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| {{UDnote|step=48}}

|}

== Notation ==

=== Ups and downs 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.

[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis 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]].

==== Evo flavor ====

File:48-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 140 106 [[736/729]]

rect 140 80 260 106 [[33/32]]

default [[File:48-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:48-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 140 106 [[736/729]]

rect 140 80 260 106 [[33/32]]

default [[File:48-EDO_Revo_Sagittal.svg]]

</imagemap>

==== Evo-SZ flavor ====

File:48-EDO_Evo-SZ_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 519 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 140 106 [[736/729]]

rect 140 80 260 106 [[33/32]]

default [[File:48-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>

== Regular temperament properties ==

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<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== ~~Zeta properties~~ ==

== Instruments ==

~~===Zeta peak index===~~

* [[Lumatone mapping for 48edo]]

~~{| class="wikitable"~~

* [[Skip fretting system 48 2 13]]

~~! colspan="3" |Tuning~~

~~! colspan="3" |Strength~~

~~! colspan="2" |Closest EDO~~

~~! colspan="2" |Integer limit~~

|-

~~!ZPI~~

~~!Steps per octave~~

~~!Step size (cents)~~

~~!Height~~

~~!Integral~~

~~!Gap~~

~~!EDO~~

~~!Octave (cents)~~

~~!Consistent~~

~~!Distinct~~

|-

|[[~~226zpi~~]]

~~|47.9883202080850~~

~~|25.0060847055410~~

~~|5.843508~~

~~|0.890480~~

|13~~.888482~~

~~|48edo~~

~~|1200.29206586597~~

|6

|}

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/w1m9PMj0Ato ''48edo layout''] (2025)

; [[E8 Heterotic]]

* [https://youtu.be/puADOWdfjSQ?si=FWEqYKwud-yoI6ae "Elements - Fire"] from ''Elements'' (2019–2020)

* [https://youtu.be/puADOWdfjSQ?si=FWEqYKwud-yoI6ae ''Elements - Fire''] from ''Elements'' (2019–2020)

; [[norokusi]]

* from ''Works for Strings Vol.1'' (2020)

** "Cruise for strings" – [https://norokusi.bandcamp.com/track/cruise-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/ayilWbYWr54?si=b-H-SR1inT00quT8 YouTube]

** ''Cruise for strings'' – [https://norokusi.bandcamp.com/track/cruise-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/ayilWbYWr54?si=b-H-SR1inT00quT8 YouTube]

** "Rigel for Strings" – [https://norokusi.bandcamp.com/track/rigel-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/XjjI6NVRVDs?si=-S_Vcx-8ypvxaODx YouTube]

** ''Rigel for Strings'' – [https://norokusi.bandcamp.com/track/rigel-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/XjjI6NVRVDs?si=-S_Vcx-8ypvxaODx YouTube]

* [https://youtu.be/bfPx121QqJ4?si=Yfspa9i9Rq2FKbZP ''Elysium Planitia for strings''] (2022)

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; [[Jon Lyle Smith]]

* [https://archive.org/download/Quincunx/Quincunx.mp3 ''Quincunx'']{{dead link}}

~~== Instruments ==~~

* [[Lumatone mapping for 48edo]]

* [[Skip fretting system 48 2 13]]

== See also ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|48}}
+{{ED intro}}
 == Theory ==
 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, the equal temperament tempers out [[20000/19683]], but [[34edo]] does a much better job for this temperament, known as [[tetracot]]. However in the 7-limit it can be used for [[Jubilismic_clan|doublewide temperament]], the 1/2 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&nbsp;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.
-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|optimal patent val]]. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 cent interval serving as both [[9/7|9/7]] and [[14/11|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]].
 Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths.
 === Odd harmonics ===
-{{Harmonics in equal|48}}
+{{Harmonics in equal|48|columns=14}}
+{{Harmonics in equal|48|columns=14|start=15|collapsed=true|title=Approximation of odd harmonics in 48edo (continued)}}
 === As a tuning standard ===
@@ Line 19: / Line 20: @@
 === Subsets and supersets ===
 edo is the 10th [[highly composite edo]]. Since 48 factors into {{factorization|48}}, 48edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, and 24 }}.
-== Notation ==
-=== Ups and downs notation ===
-edo can be notated using [[ups and downs notation]]. This can be accomplished by combining [[Helmholtz-Ellis]] accidentals with semisharps and semiflats:
-{{Sharpness-sharp4|48}}
-===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]].
-====Evo flavor====
-<imagemap>
-File:48-EDO_Evo_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 140 106 [[736/729]]
-rect 140 80 260 106 [[33/32]]
-default [[File:48-EDO_Evo_Sagittal.svg]]
-</imagemap>
-====Revo flavor====
-<imagemap>
-File:48-EDO_Revo_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 140 106 [[736/729]]
-rect 140 80 260 106 [[33/32]]
-default [[File:48-EDO_Revo_Sagittal.svg]]
-</imagemap>
-====Evo-SZ flavor====
-<imagemap>
-File:48-EDO_Evo-SZ_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 519 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 140 106 [[736/729]]
-rect 140 80 260 106 [[33/32]]
-default [[File:48-EDO_Evo-SZ_Sagittal.svg]]
-</imagemap>
 == Intervals ==
 {| class="wikitable center-1 right-2"
 |-
-! #
+! &#35;
 ! Cents
-! [[Ups and downs notation|Ups and Downs Notation]]
+! [[Ups and downs notation|Ups and downs notation]]
 |-
 | 0
@@ Line 266: / Line 224: @@
 | {{UDnote|step=48}}
 |}
+== Notation ==
+=== Ups and downs notation ===
+edo 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.
+{{Sharpness-sharp4a}}
+[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+{{Sharpness-sharp4}}
+=== 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]].
+==== Evo flavor ====
+<imagemap>
+File:48-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[736/729]]
+rect 140 80 260 106 [[33/32]]
+default [[File:48-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:48-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[736/729]]
+rect 140 80 260 106 [[33/32]]
+default [[File:48-EDO_Revo_Sagittal.svg]]
+</imagemap>
+==== Evo-SZ flavor ====
+<imagemap>
+File:48-EDO_Evo-SZ_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 519 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 140 106 [[736/729]]
+rect 140 80 260 106 [[33/32]]
+default [[File:48-EDO_Evo-SZ_Sagittal.svg]]
+</imagemap>
 == Regular temperament properties ==
@@ Line 316: / Line 316: @@
 <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-== Zeta properties ==
+== Instruments ==
-===Zeta peak index===
+* [[Lumatone mapping for 48edo]]
-{| class="wikitable"
+* [[Skip fretting system 48 2 13]]
-! colspan="3" |Tuning
-! colspan="3" |Strength
-! colspan="2" |Closest EDO
-! colspan="2" |Integer limit
-|-
-!ZPI
-!Steps per octave
-!Step size (cents)
-!Height
-!Integral
-!Gap
-!EDO
-!Octave (cents)
-!Consistent
-!Distinct
-|-
-|[[226zpi]]
-|47.9883202080850
-|25.0060847055410
-|5.843508
-|0.890480
-|13.888482
-|48edo
-|1200.29206586597
-|6
-|6
-|}
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/w1m9PMj0Ato ''48edo layout''] (2025)
 ; [[E8 Heterotic]]
-* [https://youtu.be/puADOWdfjSQ?si=FWEqYKwud-yoI6ae "Elements - Fire"] from ''Elements'' (2019–2020)
+* [https://youtu.be/puADOWdfjSQ?si=FWEqYKwud-yoI6ae ''Elements - Fire''] from ''Elements'' (2019–2020)
 ; [[norokusi]]
 * from ''Works for Strings Vol.1'' (2020)
-** "Cruise for strings" &ndash; [https://norokusi.bandcamp.com/track/cruise-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/ayilWbYWr54?si=b-H-SR1inT00quT8 YouTube]
+** ''Cruise for strings'' &ndash; [https://norokusi.bandcamp.com/track/cruise-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/ayilWbYWr54?si=b-H-SR1inT00quT8 YouTube]
-** "Rigel for Strings" &ndash; [https://norokusi.bandcamp.com/track/rigel-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/XjjI6NVRVDs?si=-S_Vcx-8ypvxaODx YouTube]
+** ''Rigel for Strings'' &ndash; [https://norokusi.bandcamp.com/track/rigel-for-strings-48-tone-microtonal-music-48edo-48tet Bandcamp] | [https://youtu.be/XjjI6NVRVDs?si=-S_Vcx-8ypvxaODx YouTube]
 * [https://youtu.be/bfPx121QqJ4?si=Yfspa9i9Rq2FKbZP ''Elysium Planitia for strings''] (2022)
@@ Line 368: / Line 345: @@
 ; [[Jon Lyle Smith]]
 * [https://archive.org/download/Quincunx/Quincunx.mp3 ''Quincunx'']{{dead link}}
-== Instruments ==
-* [[Lumatone mapping for 48edo]]
-* [[Skip fretting system 48 2 13]]
 == See also ==