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48edo: Difference between revisions

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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]].
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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 combined with quartertone accidentals and 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 ==
=== 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 ==
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; [[Ray Perlner]]
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=HHozjGjWI-Q ''Octatonic Groove''] (2020) &ndash; jubilismic[8] in 48edo tuning
* [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]]
; [[Carlo Serafini]]
Retrieved from "https://en.xen.wiki/w/48edo"