48edo: Difference between revisions

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The ''48 equal division'' divides the [[Octave|octave]] into 48 1/8th tones of precisely 25 [[cent|cent]]s each. 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|5-limit]], but [[tempering_out|tempers out]] [[2401/2400|2401/2400]] in the [[7-limit|7-limit]], making it a tuning for [[Meantone_family|squares temperament]]. In the [[11-limit|11-limit]] we can add [[99/98|99/98]] and [[121/120|121/120]] to the list, and in the [[13-limit|13-limit]], [[66/65|66/65]]. While [[31edo|31edo]] can also do 13-limit squares, 48 might be preferred for some purposes.

Using its best major third, 48 tempers out 20000/19683, but [[34edo|34edo]] does a much better job for this temperament, known as [[Tetracot_family|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|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.

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.

* [http://archive.org/download/Quincunx/Quincunx.mp3 Quincunx] by Jon Lyle Smith