View source for 48edo

{{Infobox ET}}
{{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, 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]]. 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. Since 16edo supports [[mavila]] temperament, and 48edo has two viable major and minor thirds, 48edo could be used for an unexplored system of "adaptive mavila".

=== Odd harmonics ===
{{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 ===
A step of 48edo is known as a '''doamu''' (second MIDI-resolution unit, 2mu, 2<sup>2</sup> = 4 equal divisions of the 12edo semitone). The internal data structure of the 2mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or &minus;) showing the direction of the pitch-bend up or down, and three other bits which are not used. 

=== 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 }}.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! &#35;
! Cents
! [[Ups and downs notation|Ups and downs notation]]
|-
| 0
| 0.0
| {{UDnote|step=0}}
|-
| 1
| 25.0
| {{UDnote|step=1}}
|-
| 2
| 50.0
| {{UDnote|step=2}}
|-
| 3
| 75.0
| {{UDnote|step=3}}
|-
| 4
| 100.0
| {{UDnote|step=4}}
|-
| 5
| 125.0
| {{UDnote|step=5}}
|-
| 6
| 150.0
| {{UDnote|step=6}}
|-
| 7
| 175.0
| {{UDnote|step=7}}
|-
| 8
| 200.0
| {{UDnote|step=8}}
|-
| 9
| 225.0
| {{UDnote|step=9}}
|-
| 10
| 250.0
| {{UDnote|step=10}}
|-
| 11
| 275.0
| {{UDnote|step=11}}
|-
| 12
| 300.0
| {{UDnote|step=12}}
|-
| 13
| 325.0
| {{UDnote|step=13}}
|-
| 14
| 350.0
| {{UDnote|step=14}}
|-
| 15
| 375.0
| {{UDnote|step=15}}
|-
| 16
| 400.0
| {{UDnote|step=16}}
|-
| 17
| 425.0
| {{UDnote|step=17}}
|-
| 18
| 450.0
| {{UDnote|step=18}}
|-
| 19
| 475.0
| {{UDnote|step=19}}
|-
| 20
| 500.0
| {{UDnote|step=20}}
|-
| 21
| 525.0
| {{UDnote|step=21}}
|-
| 22
| 550.0
| {{UDnote|step=22}}
|-
| 23
| 575.0
| {{UDnote|step=23}}
|-
| 24
| 600.0
| {{UDnote|step=24}}
|-
| 25
| 625.0
| {{UDnote|step=25}}
|-
| 26
| 650.0
| {{UDnote|step=26}}
|-
| 27
| 675.0
| {{UDnote|step=27}}
|-
| 28
| 700.0
| {{UDnote|step=28}}
|-
| 29
| 725.0
| {{UDnote|step=29}}
|-
| 30
| 750.0
| {{UDnote|step=30}}
|-
| 31
| 775.0
| {{UDnote|step=31}}
|-
| 32
| 800.0
| {{UDnote|step=32}}
|-
| 33
| 825.0
| {{UDnote|step=33}}
|-
| 34
| 850.0
| {{UDnote|step=34}}
|-
| 35
| 875.0
| {{UDnote|step=35}}
|-
| 36
| 900.0
| {{UDnote|step=36}}
|-
| 37
| 925.0
| {{UDnote|step=37}}
|-
| 38
| 950.0
| {{UDnote|step=38}}
|-
| 39
| 975.0
| {{UDnote|step=39}}
|-
| 40
| 1000.0
| {{UDnote|step=40}}
|-
| 41
| 1025.0
| {{UDnote|step=41}}
|-
| 42
| 1050.0
| {{UDnote|step=42}}
|-
| 43
| 1075.0
| {{UDnote|step=43}}
|-
| 44
| 1100.0
| {{UDnote|step=44}}
|-
| 45
| 1125.0
| {{UDnote|step=45}}
|-
| 46
| 1150.0
| {{UDnote|step=46}}
|-
| 47
| 1175.0
| {{UDnote|step=47}}
|-
| 48
| 1200.0
| {{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.
{{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 ===
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>
== 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 ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 5\48
| 125.0
| 16/15
| [[Negri]]
|-
| 1
| 7\48
| 175.0
| 10/9
| [[Tetracot]]
|-
| 1
| 13\48
| 325.0
| 77/64
| [[Orgone]]
|-
| 1
| 17\48
| 425.0
| 9/7
| [[Squares]]
|-
| 1
| 19\48
| 475.0
| 21/16
| [[Buzzard]]
|-
| 2
| 13\48
| 325.0
| 6/5
| [[Doublewide]]
|}
<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 ==
Most of 48edo's simple [[prime]]s have low error, but its 5 is substantially flat, so 48edo can benefit from slight [[octave stretching]]. Some stretched-octave 48edo tunings (least to most stretched) include [[zpi|266zpi]], [[ed12|172ed12]], [[ed6|124ed6]], [[76edt]] or [[28edf]].

== Instruments ==
* [[Lumatone mapping for 48edo]]
* [[Skip fretting system 48 2 13]]

== Music ==
=== 48 equal divisions of the octave (48edo proper) ===
; [[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)

; [[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]
** ''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)

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=HHozjGjWI-Q ''Octatonic Groove''] (2020) &ndash; 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]]
* ''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)]

; [[Carlo Serafini]]
* ''Neutral Steel'' (2009) &ndash; [http://www.seraph.it/blog_files/dae37b1f7663cf6fb349aebd57f16446-18.html blog] | [http://www.seraph.it/dep/det/Neutral%20Steel.mp3 play]
* ''Two At Once'' (2009) &ndash; [http://www.seraph.it/blog_files/a0aafbec9519cfe9600e7a82118da2ee-26.html blog] | [http://www.seraph.it/dep/det/twoatonce.mp3 play]
* ''Tim's Flutes'' (2009) &ndash; [http://www.seraph.it/blog_files/a8435fb03236157e8a60b047e9892594-27.html blog] | [http://www.seraph.it/dep/det/Tim%27sFlutes.mp3 play]
* ''Two At Once 2'' (2018) &ndash; [http://www.seraph.it/blog_files/7412388219d6c3414606cc8429542cd1-254.html blog] | [http://www.seraph.it/dep/det/TwoAtOnce2.mp3 play]
* ''The Dolomites'' (2018) &ndash; [http://www.seraph.it/blog_files/dfe021db3e7b290d12277273ab68a722-258.html blog] | [https://www.youtube.com/watch?v=74iVIc0sbhk&feature=youtu.be YouTube]

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

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=nYrLvB_T2g4 ''October...walking alone''] (2025)

=== Well-tempered derivatives of 48edo ===
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=sstmTuK3Vyo ''passing the edge of the universe, my thoughts were of you (for microtonal orchestra)''] (2026 &mdash; 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 ==
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[24edo]] (1mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)

== External links ==
* [http://tonalsoft.com/enc/number/2mu.aspx 2mu / doamu (also 48-ed2 / 48-edo / 48-ET / 48-tone equal-temperament)] on [[Tonalsoft Encyclopedia]]

[[Category:Doublewide]]
[[Category:Jubilee]]
[[Category:Listen]]
[[Category:Squares]]
[[Category:Subgroup temperaments]]