31edo: Difference between revisions
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''Thirty-one tone equal temperament'', also called ''31-tET'', ''31-EDO'', ''31-et'', or ''tricesimoprimal meantone temperament'', is the scale derived by dividing the octave into 31 [[equal]]ly large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. | ''Thirty-one tone equal temperament'', also called ''31-tET'', ''31-EDO'', ''31-et'', or ''tricesimoprimal meantone temperament'', is the scale derived by dividing the octave into 31 [[equal]]ly large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. | ||
31edo is nearly optimal for the temperament for the 2.9.21.5 subgroup tempering out 1029/1024 (the difference between three 21/16's and one 9/8) and 81/80 which yields scales with an approximate 21/16 as a generator (also represented by 13edo and 18edo). | |||
31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]]. | 31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]]. | ||